Highly Excited Strings I: Generating Function

This is the first of a series of detailed papers on string amplitudes with highly excited strings (HES). In the present paper we construct a generating function for string amplitudes with generic HES vertex operators using a fixed-loop momentum formalism. We generalise the proof of the chiral splitting theorem of D'Hoker and Phong to string amplitudes with arbitrary HES vertex operators (with generic KK and winding charges, polarisation tensors and oscillators) in general toroidal compactifications $\mathcal{E}=\mathbb{R}^{D-1,1}\times \mathbb{T}^{D_{\rm cr}-D}$ (with generic constant K\"ahler and complex structure target space moduli, background Kaluza-Klein (KK) gauge fields and torsion). We adopt a novel approach that does not rely on a"reverse engineering"method to make explicit the loop momenta, thus avoiding a certain ambiguity pointed out in a recent paper by Sen, while also keeping the genus of the worldsheet generic. This approach will also be useful in discussions of quantum gravity and in particular in relation to black holes in string theory, non-locality and breakdown of local effective field theory, as well as in discussions of cosmic superstrings and their phenomenological relevance. We also discuss the manifestation of wave/particle (or rather wave/string) duality in string theory.

Contrary to field theoretic intuition, it has also recently been shown [35] that in semirealistic heterotic string compactifications with spontaneously broken supersymmetry and exponentially small values for the cosmological constant, the global structure (or "shape") of the effective potential (around certain self-dual points in moduli space, corresponding to extrema of the effective potential) is strongly influenced by contributions from massive string modes (as well as non-level matched string states), thus further highlighting the importance of the HES contributions even in low energy effective field theories.
In this context HES are referred to as cosmic superstrings, and if produced they can have a wide variety of signatures, most notably gravitational wave signatures [49,50], including gravitational wave bursts produced from cusps and kinks [51][52][53][54][55][56], covering a wide frequency range that can be probed by pulsar timing arrays, ground-based interferometers (such as LIGO) and the much anticipated eLISA [49,50], to mention a handful. In addition, even though the Planck satellite has placed strong constraints from the temperature data of the cosmic microwave background (CMB), the CMB still offers observational prospects via polarisation and non-Gaussianity [48], see also [40,42] and references therein for a more complete list of observational signals. One major uncertainty [42] is the eventual destination of the energy of a string network, gravitational and possibly massive radiation (and the associated backreaction) which is believed to play a major role in determining the average size of the produced string loops. Furthermore, incorporating backreaction in theoretical predictions for gravitational wave bursts from individual loops is still an unresolved issue that is suspected may play a major role in their observational prospects.
Partly motivated by the above developments, let us now zoom in further on HES in the context of string perturbation theory in particular. From this viewpoint, the first step will be to set-up an efficient construction that will directly yield string amplitudes in the presence of HES vertex operators, a complete set of which (in a coherent state basis) was first constructed in [57,58]. In [56] we used these vertex operators to compute decay rates and power associated to massless radiation, while also making contact with low energy effective theory (in a certain IR limit). The tools we have constructed are powerful enough to capture a wide range of phenomena, including, e.g., radiative backreaction corrections to the classical gravitational wave results [51][52][53][54][55][56], cross sections and decay rates associated to HES, including loop corrections, etc. The current document is the first of a series of technical papers on string amplitudes with HES [59][60][61][62][63].
A fundamental tool that we make use of is the chiral splitting theorem of D'Hoker and Phong [64], whereby string amplitudes at fixed-loop momenta chirally factorise. 1 We will make use of and generalise this framework in a number of ways, but to motivate further our approach in this document let us begin with some introductory technical comments.
In a series of recent papers [65][66][67][68][69][70][71][72] Sen and collaborators have revisited various aspects of superstring theory (unitarity of string amplitudes, mass and wavefunction renormalisation [67,71,72], perturbation theory around dynamically shifted string vacua [70], offshell string amplitudes [73], Wick rotations and analytic continuations [65][66][67], one-particle ir-reducible (1PI) quantum effective actions, etc.). In a very careful and complete study [65] Pius and Sen derived Cutkosky rules for superstring field theory amplitudes to all orders in perturbation theory by providing a prescription for taking integration contours of loop energies in the complex plane (which would naively otherwise yield divergent results for the corresponding S-matrix elements). In [66] Sen then showed that this prescription of introducing loop momenta and appropriately deforming the integration contours is equivalent to that of Berera [74] and Witten [75], (see also [76][77][78][79][80] and in particular [81] for related discussions on analyticity of string amplitudes), in the worldsheet approach to superstring theory to all orders in perturbation theory. In the approach of Witten [75] one is to deform the integration cycles over moduli space of punctured Riemann surfaces into a complexified moduli space, and this establishes consistency of the former fixed-loop momenta approach with S-matrix unitarity. Sen then recently also discussed [67] an application of the fixedloop momenta approach, building on earlier work [71,72] and in particular [65], namely mass renormalisation of unstable massive string states (where a naive computation yields divergent results for the two-point one-loop amplitude), explaining how to obtain finite results that are consistent with unitarity. The basic reason for the aforementioned divergences are ultimately due to the fact that the analogue of the 'i ' prescription of quantum field theory is somewhat subtle in string theory [75] because string amplitudes are most naturally defined in Euclidean space where they are real [81]. Therefore, e.g., potential imaginary parts (that are required by unitarity) in Lorentzian signature string amplitudes show up as divergences in the corresponding Euclidean space amplitudes. Motivated by string field theory, Sen [66] and Pius and Sen [65] have provided a well-defined prescription for dealing with such analytic continuations by reformulating string amplitudes in terms of fixed-loop momenta and deforming their integration cycles into the complex plane following a specific prescription (whereby loop energy contours are pinned down at ±i∞ following a non-trivial but well-defined path in between, leading to Lorentzian signature amplitudes).
Drawing from analogies with string field theory [65], the introduction of fixed-loop momenta is central to Sen's analytic continuation approach to string amplitudes (which are traditionally given as integrals over moduli space, in the "Schwinger parametrisation" with implicitly integrated loop momenta). Fixed-loop momenta amplitudes have a long history in string theory that dates back to the old dual models, see e.g. [82] and references therein, but it was not until Dijkgraaf, E. Verlinde and H. Verlinde [83] (building on [84]) that fixed-loop momenta appeared in the path integral formulation of string theory, where various interesting properties were also noted, one such property being that (taking into account also the Belavin-Knizhnik theorem [85]) bosonic amplitudes with tachyonic external vertex operators in simple toroidal compactifications chirally factorise. This observation was later explored in a much more complete manner and in the full superstring context by D'Hoker and Phong [64] (although even here the explicit results were derived for massless external states, and also non-compact flat spacetime). This study led to a chiral splitting theorem: when string amplitudes are written in terms of integrals over loop momenta 2 and fermion zero modes (when present), the corresponding integrands chirally factorise (in terms of their supermoduli, abelian differentials, worldsheet coordinates of vertex operators, polarisation tensors and momenta).
One thing we would like to highlight is that in the approach of D'Hoker and Phong [64] and Sen [67], the fixed-loop momenta amplitudes are constructed by a "reverse engineering" method, whereby string amplitudes are computed using the conventional approach [86], and it is only at a later stage of the computation (after integrating out the path integral fields and hence obtaining the "Schwinger form" of amplitudes) that it is noted that explicit string amplitudes under consideration can be written as integrals over loop momenta.
Unfortunately, such an approach is almost hopeless when considering amplitudes with arbitrarily massive HES vertex operators, because the fixed-loop momenta amplitudes will typically have quite a complicated form that one is to somehow guess, hence the name 'reverse engineering' mentioned above. (A systematic approach to "guessing" the correct loop momentum integrand given the Schwinger parametrisation was given by Sen at one-string loop in [67], but this is somewhat tedious and messy and requires a case-by-case study.) D'Hoker and Phong [64] Given that these effective rules were derived explicitly by making use of the 'reverse engineering' method, it is important to show that the prescription for the effective rules does not depend on it. 5 Of course, one expects the chiral splitting theorem to also hold for generic vertex operators and in generic toroidally compactified string theories, but as the derivation of D'Hoker and Phong [64] was carried out explicitly in non-compact Minkowski (or Euclidean) spacetime (and in [83,90] in the context on Z 2 orbifold compactifications) and for massless vertex operator insertions it would be desirable to discuss amplitudes with generic vertex operators more explicitly. 6 Furthermore, it would be desirable to adopt an approach that does not rely on the reverse engineering method outlined above, and hence show in particular that the potential ambiguities discussed by Sen [67] are absent for generic vertex operators when using the effective chiral splitting rules of D'Hoker and Phong. The ambiguity we are referring to is the following. In [67] Sen has shown that the same Schwinger parameter representation of a given string amplitude (which arises directly from the usual path integral formulation of string theory) can be represented in more than one way from a fixed-loop momenta representation, see the discussion associated to equation (2.17) in [67] and also the footnote below (3.20) there, leading one to question whether this reverse engineering method that has been adopted to-date could potentially be ambiguous for generic vertex operators. Sen then went on to argue that this ambiguity will actually not be visible in the result for the full amplitude after having integrated out the loop momenta using the prescription for avoiding singularities [65] in the loop momenta integrations.
Although this is a very important step forward (given in particular the subtleties of the loop momentum contour integrations in Lorentzian-signature target spacetimes), there are situations where it is not desirable to integrate out the loop momenta completely, and identify the loop momentum integrand with a physical observable. This at first sight may seem unphysical, not least because the integrand of the loop momentum integrals are not modular invariant (defining loop momenta requires specifying a homology basis [89], which transforms non-trivially under the mapping class group), but nevertheless there do exist physical observables that are sensitive to this integrand. Put simply, strings in loops can go onshell and can therefore also appear in the detector of a given observer. Their momenta can thus be measured, and so it does make sense to consider the integrands of loop momentum integrands of string amplitudes as being physical. One can also make this argument by appealing to the optical theorem and unitarity. A good example is the following.
Consider the power emitted from a massive string per unit solid angle. Here we imagine a generic highly excited string that is unstable and emitting radiation while it decays, usually anisotropically, with massless radiation often being the dominant decay channel.
We place a detector far from the interaction region and measure the power absorbed by the detector as a function of energy or frequency of the radiation, and so extract a spectrum which will contain information about the radiating string state (allowing one in principle to reconstruct the quantum numbers of the decaying string). The orientation of the emitting string can also be determined in this manner, due to the anisotropy of the decay; this is particularly relevant for gravitational wave emission from strings with cusps where the associated burst of radiation is highly anisotropic, and this ultimately provides one of the strongest signals in cosmic string phenomenology; this has a long history, see [51][52][53][54][55] for an effective theory computation and [56] for a corresponding analytical string theory computation, the two being in precise agreement when backreaction is neglected and one is confined to low energies. 7 The observable in this thought experiment can, e.g., be extracted from the imaginary part of the two-point (say one-loop at weak coupling) amplitude at fixed-loop momenta, Im M T 2 (O zz , O zz ; P), with two vertex operator insertions (related by Euclidean conjugation, more about which will be discussed in a sequel [59]). In particular, for D non-compact dimensions, the power associated to decay products of momentum P µ , centred around some arbitrary spatial directionP, can be extracted from [56] where it is seen that the loop momenta, P µ , actually get identified with the momentum of the decay products, some of which end up in the detector, as alluded to above. Here L is the length [59] and L 2πα the corresponding mass of the emitting string whose vertex operator, O zz , is normalised by the leading singularity in the OPE: There are a number of other approaches in the literature to decay rate computations of highly excited strings, but these typically rely on numerical approximations or saddle-point evaluations of integrated-loop momentum two-point one-loop amplitudes (but also tree-level amplitudes), in order to obtain order of magnitude estimates, and they also consider leading Regge trajectory states only, see [78,79,91] and more recently [80,92,93] and [94][95][96][97], and also [98] for a review on string decay. 8 For this computation choosing the correct contour for the P 0 integral is crucial and has been discussed very carefully and clearly for generic loop amplitudes in [65,67]. 9 Newton's constant, G D , is related to the gravitational coupling, κ D , via κ 2 D = 8πG D . In D = 4, κ −1 4 = 2.4 × 10 18 GeV is the reduced Planck mass. Note that g D = κ D /(2π).
An 'overline' represents taking the Euclidean adjoint, see [86,99] and in particular [59] for a refined discussion of this notion (in the presence of compact dimensions where there are some additional phases that are absent in [86,99]), and we have decomposed the loop momentum integral as follows, d D P µ = dP 0 d|P| |P| D−2 dΩ S D−2 . Extracting the imaginary part will give rise (according to the Cutkosky rules [65]) to two delta functions, one of which places the emitted radiation onshell, and the second delta function quantises the spectrum of decay products (in the case of massless radiation), leading to integer-valued energies, P 0 = ω n , of the form: and n is summed over (subject to energy conservation), as expected, e.g., for gravitational waves from strings. This procedure was first carried out by the present authors in [56], where a brief summary of our results can be found, as well as the effective low energy theory that reproduces them. Clearly, if we want to extract information about the energydependence of the emitted radiation we do not want to perform the sum over n. If we were to integrate out the loop momentum completely the ω n dependence of the power would be lost. We compute observables in explicitly in follow-up articles [61,62], but the purpose of the discussion in this paragraph is to show that it is sometimes desirable to not integrate out the loop momenta completely, and that this is of interest even in calculations of physical observables. Therefore, the resolution of the aforementioned ambiguity of Sen [67] is not totally satisfactory (because it relies on the assumption that it is of interest to integrate out loop momenta completely). Summarising, it would be desirable to adopt an approach that does not rely on the reverse engineering derivation of D'Hoker and Phong [64] and that of Sen [67] and show explicitly how to resolve this potential ambiguity observed by Sen in his study of two-point amplitudes of massive strings.
In this article we resolve this particular ambiguity completely 10 , for completely generic vertex operator insertions (with arbitrary winding and KK charges), and for generic toroidally compactified backgrounds (with generic constant Kähler and complex structure target space moduli, background KK gauge fields and torsion), and to any finite order in the string loop expansion. Therefore, our derivation applies to all closed string amplitudes in target spacetimes R D−1,1 ×T Dcr−D . We focus on the bosonic string for simplicity (or the bosonic sector of the superstring, the conclusions being independent of the chiral splitting statements given that up to fermion zero modes the fermionic contribution is already chirally factorised, as are the ghost contributions). 10 There remain field redefinition ambiguities familiar already from the string field theory context, see Sec. 4 in [67], and the authors are greatful to Ashoke Sen for extensive discussions on this.
Specifically, our approach will be to drop the 'reverse engineering' approach of D'Hoker and Phong [64] and Sen [67] altogether, and to rather construct the fixed-loop momentum representations directly and explicitly, starting from a generic worldsheet path integral, leaving no room for ambiguities. This will be achieved by inserting momentum-conserving delta functions into the worldsheet path integral that explicitly determine the loop momentum contribution associated to A I -cycle strings in the loops, as shown in the second equality in (3.14) below. These loop momenta constitute an independent set and their presence are also a fundamental ingredient in obtaining a handle on the energies and momenta that contribute to loop corrections, thus for example bridging the gap between Wilson's approach to quantum field theory [100,101], see e.g. [102], and an analogous approach in string theory (although we will not explore this connection further here), and Wilson's approach adapted to string field theory has recently been discussed by Sen in [68]. Unless one wants to work directly with string field theory or in the old operator approach, instead of adopting the first-quantised covariant path integral formalism that we consider here, string amplitudes with fixed-loop momenta is the closest one can get to the corresponding field theory amplitudes without considering explicit pant decompositions and degenerations of the worldsheet.
Fixed-loop momenta amplitudes are, arguably, one of the most natural approaches to (at least closed) perturbative string amplitude computations. First and foremost, it is considerably easier to write down amplitudes at fixed-loop momenta than it is to adopt the traditional approach [86] and write down the corresponding integrated loop momenta amplitudes directly -a fact that is certainly not widely appreciated in the literature. This is largely due to the chiral splitting theorem of D'Hoker and Phong [64]; recall that correlation functions for the worldsheet fields are carried out using the chiral propagators exhibited above (analogous relations for the fermionic sector or in terms of superfields can be found in [64]), where also zero mode subtractions are absent (it is useful to compare with the non-chiral genus-g propagator (3.54)). Secondly, it is (apparently [56,61,62]) considerably easier (from a technical point of view) to analytically compute explicit amplitudes in the chiral fixed-loop momenta formalism than it is to extract the corresponding integrated loop momenta expressions, as we briefly summarised in [56]. The physical reason is that in the integrated loop momenta approach one is automatically resumming all momentum contributions inside loops (so that it is difficult to take a low energy limit as loop energies are already integrated out), and as a result one ends up having to resort to saddle-point approximations or numerical methods in order to extract some physics or even order of magnitude estimates [78][79][80][91][92][93][94][95][96][97], whereas in the fixed-loop momenta approach (after adopting a coherent vertex operator basis for external states [56][57][58], see in particular [59] for a recent analysis), things tend to resum into, e.g., Bessel functions, exponentials and related special functions [56]. As the latter have been studied by mathematicians for centuries, with various of their properties examined in detail (such as asymptotic expansions, series and integral representations, etc.), this provides a useful and novel working handle on generic string amplitudes.
Amplitudes with HES should be expected to reproduce various classical or effective field theory results in certain limits (with non-local stringy sources), and adopting the correct toolset is absolutely fundamental to exposing this simple structure, while also providing an explicit approach to computing various stringy and quantum corrections which may or may not be large compared to the effective description -the effective approach and its link to string amplitudes with HES vertex operators will be presented in [63], where again the basic connection to the effective theory was presented in [56], building on an earlier conjecture by Dabholkar, Gibbons, Harvey and Ruiz [103,104], see also [105] for a very insightful complementary decription. The emphasis on the naturalness of adopting a coherent vertex operator basis in particular when discussing amplitudes with HES will be explained in sequels in much greater detail [59][60][61][62]. The present contribution will not rely on any particular vertex operator basis, and also (with a bit a care and tweaking [73], see also [99,106]) will apply to offshell as well as onshell string amplitudes.
Let us also re-emphasise that our initial focus will be on bosonic string theory in this series of papers, because as string amplitudes with coherent vertex operators is a novel and unexplored area of research it will be easiest to first focus on the bosonic string and understand that case well before moving on to the much more interesting but also more involved superstring framework. The bosonic string already contains most of the nontrivial features associated to HES and will provide the basic physics, with the additional complications associated to [64,73,89,[107][108][109][110][111][112] supermoduli space, gauge fixing, picture changing, etc., of the superstring providing a sharpening of the bosonic string results (by eliminating a tachyon, introducing supersymmetry to stabilise the vacuum and eliminate massless tadpoles, etc.), but it will not change much of the essential physics picture.
Finally, for the more philosophically-minded readers, we will also discuss how wave/particle duality (or rather wave/string duality) manifests itself in string theory. We will discuss a simple example (the standard one-loop vacuum amplitude) and then generalise the argument to all string amplitudes at any string loop order. The resulting picture is rather simple, but we are not aware of it having been discussed before in the literature.
The result is that fixed-loop momenta amplitudes can be thought of as corresponding to a wave picture, whereas the corresponding integrated-loop momenta amplitudes provide the corresponding string picture. There are four natural representations for string amplitudes in toroidally compactified spacetimes, corresponding to the fact that loop momenta in the compact or non-compact dimensions can be either integrated or fixed, so there are four possibilities. The intuitive statement, as we shall explain, is roughly that summing over all trajectories of a loop of string in (say) a compact target space, including the number of times a closed string can traverse the compact space is equivalent to summing over all frequencies of a standing wave in a "box", thus making wave/string duality manifest. This therefore provides a physical interpretation of the standard Poisson resummation in string partition functions (for the compact dimensions), and there is an analogous statement in the non-compact dimensions.
In Sec. 2 we provide a brief overview of our results (skipping almost all of the subtle and technical points). In Sec. 3 we derive the generating function for generic string amplitudes in generic toroidal compactifications associated to arbitrary vertex operator insertions and at arbitrary string loop order; this is where the majority of the work lies. The result here is extremely simple. In Sec. 4 we discuss the string theory manifestation of wave/particle duality of quantum mechanics, which is closely related to the presence or absence of fixedloop momenta (in both compact and non-compact target spacetime dimensions). Sec. 5 is a corollary of the preceding sections and completes the derivation of the D'Hoker and Phong chiral splitting theorem for generic HES vertex operator insertions (including KK and winding charges and general polarisation tensors and oscillators spanning all spacetime dimensions) and generic constant target space Kähler and complex structure moduli, KK gauge fields, as well as spacetime torsion.

Overview
In this section we provide a brief overview of the main results of the current document.
The main objective underlying this series of papers is to provide a working handle on string amplitudes with HES vertex operator insertions. The first step in this direction, which is presented here, is thus to construct a generating function, A(j), for generic string amplitudes in generic toroidal compactifications: where D denotes the number of non-compact dimensions (e.g., D = 4) and D cr the critical number of dimensions (e.g., D cr = 26 or 10, in the bosonic string or superstring respectively). We will consider the exact (in the fundamental string length, s := √ α ) string backgrounds where the spacetime metric, G M N , antisymmetric tensor, B M N , dilaton, Φ, and tachyon, U , are general (bare) constants 11 , subject only to the requirement that string perturbation theory is applicable, see (3.16). The first two of these contain [113] the Kähler, complex structure moduli and background Kaluza-Klein (KK) gauge fields associated to the compactification (2.2), as well as torsion, B µν , all of which will be allowed to be turned on. In order to make contact with the NS sector of low energy supergravity, see e.g. [114], it will sometimes be convenient to consider the parametrisation, 12 µ are a subset of the aforementioned Kaluza-Klein gauge fields, the remaining ones being B µa . We always raise and lower indices with G M N , the inverse being defined by Using the fixed-loop momenta approach of D'Hoker and Phong [64], the first goal will be to show that generic correlation functions associated to asymptotic vertex operators with generic instanton contributions, KK and winding charges, and generic polarisation tensors can all be extracted from the following genus-g contribution to the generating function in the aforementioned background: 13 where, in a canonical intersection basis [89] for the 2g homology cycles of the compact genus-g Riemann surface, {A 1 , B 1 , . . . , A g , B g }, the sum/integral appearing in the second equality is over loop momenta, Q I M ,Q I M , associated to A I -cycle (with I = 1, . . . , g) strings that span the full target spacetime E, see (3.76) and (3.48). I gh is the usual b, c ghost action (3.72), the µ j are Beltrami differentials and specify a gauge slice in the space of worldsheet metrics [89], whereas, I(x|j), encodes the standard matter contribution with a source, Constants meaning that they are independent of worldsheet and target space embedding coordinates. 12 Detailed definitions appear in the main section and Appendices. 13 The first equality here is in Euclidean worldsheet and target space signature whereas the second equality is in Lorentzian target spacetime signature and Euclidean worldsheet signature (which is always possible at generic points in the moduli space of the Riemann surface under consideration [66,75]).
where j M (z,z) is a generic source term, such that functional derivatives of A(j) with respect to it generate all (matter) correlation functions of interest (see below). In going from the first to the second equality in (2.4) we have inserted loop-momentum conserving delta functions, see (3.18), expanded the embedding coordinate into a zero mode, instantons, and quantum fluctuations, as discussed below (3.26), before finally integrating out x 0 ,x, and performing a Poisson resummation in the instanton sector. We also keep the constant tachyon background implicit throughout (this will play an explicit role in tadpole cancellation [60]). The effective coupling appearing in (2.4) at fixed-loop momenta (in Lorentzian signature) is given by: whereas the delta function constraint in (2.4) enforces overall charge neutrality, see (3.77). The quantity Z g is determined entirely from the ghost contributions, see (3.71) and (3.73).
For example, at g = 1, Z 1 = η(τ ) −24 , where η(τ ) is the Dedekind eta function and τ the complex structure modulus of the torus [86]. The prime form [89] is denoted by E(z, w), and is the unique holomorphic function defined on a Riemann surface that has precisely one (simple) zero, which is at z = w.
where M I a , N a I ∈ Z are summed over. 14 When considering string amplitudes associated to HES vertex operator insertions it is extremely useful to have the result for a generic correlation function. Denoting expectation values by: with A(j) defined in (2.4), we will show (using point-splitting) that generic correlation functions chirally factorise: 15 The {D j ,D j } are arbitrary worldsheet derivative operators, which, together with the j LM , j RM (anti-)chiral sources are (with an appropriate point-splitting procedure) read off from the specific vertex operator insertions of interest (in their chiral representation [59]).
We want to emphasise that the left-hand side of (2.9) contains insertions of the full path integral field, x M = x M 0 + x M cl +x M , and its derivatives, including zero modes, instantons and quantum fluctuations (with Green function (3.54)), whereas the chiral fields, x M ± , of the right-hand side are defined by their correlation functions, according to the rule that Wick contractions are carried out using (anti-)chiral propagators, , and do not contain zero modes or instantons. The latter have already been taken into account in writing down (2.9).
Clearly, using the chiral representation on the right-hand side vastly simplifies computa-tions. The result for the chiral half on the right-hand side of (2.9) is given by, where the argument in the exponential equals 2 s 4 d 2 z d 2 z j L + H · j L + H ln E(z, z ), and similarly for the anti-chiral half, (2.11) S I is the symmetric group of degree I [115], the group of all permutations of I elements, and the equivalence relation '∼' is such that π i ∼ π j with π i , π j ∈ S I when they define the same element in (2.10), and similarly for (2.11). In the case of coherent vertex operator insertions, as we will see in [59][60][61], the sum over permutations can be carried out explicitly, and the various quantities appearing can be rewritten in terms of exponentials and special functions, thus vastly simplifying amplitude computations compared to the traditional approach in the literature that adopts a momentum eigenstate basis for vertex operators.
One can think of the fixed-loop momenta representation of the generating function (2.4) as defining a Hamiltonian formulation of string theory, because the zero mode momenta in all spacetime dimensions are manifest. Integrating out the loop momenta leads to a Lagrangian formulation, which is the usual starting point for string amplitude computations in the path integral formalism. In addition to these two there are also two natural hybrid formulations (also called Routhian formulations by analogy to classical mechanics) whereby the loop momenta are manifest in the compact dimensions but integrated out in the non-compact dimensions and vice versa. All these cases are discussed explicitly in Sec. 4, where it is also argued that (by direct analogy to point-particle quantum mechanics) the Hamiltonian formulation may be regarded as a 'wave formulation of string theory', whereas the Lagrangian formulation may correspondingly be thought of as a string formulation. The equivalence of all four formulations can thus be regarded as a stringy manifestation of 'wave/particle duality' of quantum mechanics, and so by analogy we refer to it as 'wave/string duality'. For instance, we will argue that (2.9) may be regarded as a string theory statement of wave/string duality, where the left-hand side is in a string picture whereas the right-hand side is the corresponding wave picture. As one should expect (from our experience with point-particle quantum mechanics, such as the double-slit experiment), certain questions are more easily addressed in a wave rather than a string picture and vice versa. We provide flesh to this claim by explicit decay rate computations (in both pictures) whose details will be presented elsewhere [61].

Generating Function
The starting point is to obtain a simple expression for the generating function of interest, A(j), that is crucial in the discussion of string amplitudes, cross sections and decay rates, associated to generic HES vertex operator insertions. It is defined by (in Euclidean target space and worldsheet signature): (3.12) and we reserve the notation A(j) for the corresponding Lorentzian signature quantity, see below. The (complex) number of moduli and conformal Killing vectors (CKV) are: The b, c are the Grassmann-odd ghosts, (c,c) = (c z (dz) −1 , cz(dz) −1 ) and (b,b) = (b zz (dz) 2 , bzz(dz) 2 ), whereas the Beltrami differentials, (µ j ,μ j ) = (µ z z (dz) −1 dz, µz z dz(dz) −1 ) j , provide a parametrisation of the space of metrics on the Riemann surface, Σ g , and define a gauge slice. 16 There are as many insertions of, | µ j , b | 2 , as there are moduli (equivalently b zero modes), and the pairing, µ j , b , is defined with respect to the natural inner product of the space and is independent of a metric, µ, b = Σ d 2 z µ z z b zz , see (A.121). Similarly, in our approach it will be convenient to have as many insertions of cc as there are conformal Killing vectors (CKV) (equivalently c zero modes) on the Riemann surface, i.e. the minimal number of allowed ccghost zero insertions. More general ghost insertions are also of interest [73,106,107,116,117], and it is straightforward to extend the results of this paper to include also these cases (although strict chiral splitting may be lost in these more general situations). In turn, every x(z,z) represents an embedding of the worldsheet into spacetime, In general, the (worldsheet) matter and ghost contributions factorise, so let us focus initially on the matter contribution, with, The constant tachyon background term, 1 2πα d 2 zg zz U , will be kept implicit throughout, but it will play a role in tadpole cancellations as we will see in the context of coherent vertex operator 2-point amplitudes in [60]. Notation-wise, it will be convenient to define I m := I(x|0), so that the full (source-free) action reads I = I m + I gh . We now define the various quantities appearing in (3.14) and (3.15).
The quantity j M is a (possibly physical, either real or complex, possibly local) source, and as we also discuss below functional derivatives with respect to it (upon adopting an appropriate point-splitting procedure) generate the correlation functions and amplitudes of interest. The one condition it must satisfy will be: d 2 zj M = 0, which is usually associated to charge and momentum conservation.
We consider the exact (in α ) string background where the spacetime metric, G M N , antisymmetric tensor, B M N , dilaton, Φ, and tachyon, U , are generic 17 constants, The first two of these parametrise [113] the Kähler and complex structure moduli of the target space torus, T Dcr−D (contained in G ab and B ab ), as well as KK gauge fields (contained in G µa and B µa ) and torsion (contained in B µν ). We work in Euclidean signature (to make sense of the path integral over x 0 ) and eventually analytically continue back to Lorentzian signature. 18 Modulo this comment, index contractions will henceforth be carried out using the spacetime metric, , so that we raise and lower indices with the full metric G M N . We will state explicitly when we rotate to Lorentzian signature.
The coefficient of the constant dilaton, Φ, in the action is a topological invariant, equal to the Euler character χ(Σ g ) = 2 − 2g of the Riemann surface; see (A.125) and note that the Ricci tensor R zz is related to the Ricci scalar R (2) in (A.114). It is convenient to also define the string coupling in the standard manner: and so there is an overall factor g In the second equality in (3.14) we have inserted the unit operator: whereP µ I is the standard momentum operator andŴ µ I the winding operator. For a generic homology cycle C of the compact genus-g Riemann surface these are defined by: The operatorŴ M C [86] measures the winding of a string whose spacelike (worldsheet) dimension traverses a generic cycle C of the worldsheet. The eigenvalues W M C will be non-vanishing when the spacetime embedding of this string (associated to the homology cycle C of interest) wraps topologically non-trivial cycles of the spacetime torus, T Dcr−D . Let us also define the chiral and anti-chiral halves,Q M I ,Q M I , respectively, such that: and so,Q Choosing a canonical intersection basis for the 2g homology cycles of the compact genus-g Riemann surface [89], see Appendix A.1, the operators appearing in the 2Dg delta functions 18 Wick-rotating to Lorentzian signature can be achieved by replacing G M N are used to raise and lower indices before and after this replacement respectively. Note however that one has to be extremely careful when trying to interpret the energy integrals of the loop momenta and this has been analysed in detail by Pius and Sen [65]; see also Witten [75] for an alternative approach.
in (3.14) or (3.18) correspond to the specific choice of contours C = A I , with I = 1, . . . , g.
For simplicity we writeP I ≡P A I , andŴ I ≡Ŵ A I . In the corresponding eigenvalues we omit the 'ˆ'.
We want an expression for the amplitude at fixed-loop momenta, but in the presence of a B M N field, P M is not the physical momentum (P M is not the charge associated to spacetime translations). In particular, the Noether current [86] associated to rigid spacetime translations, x M → x M + a M of the theory (3.15), reads (with a stringy normalisation [86]), and so the associated conserved charge flowing through an arbitrary closed contour, C, of the Riemann surface instead reads, When B M N = 0 the quantityP M C indeed measures spacetime momentum, but in the presence of a B M N field the notion of momentum is modified,P M C being replaced byΠ C,M , the two being related as in (3.22). This is much like the momentum of a particle of mass m, namely mṙ, is replaced by mṙ + eA in the presence of a U(1) charge, e, (corresponding to W) and associated vector potential A (corresponding to B M N ). These statements hold for a generic closed contour C, and holomorphicity allows one to continuously deform this across the various homology cycles of the Riemann surface, or it may be taken to encircle one or more punctures at which vertex operators are inserted. Momentum and winding conservation is of course closely related to this notion of holomorphicity [86]. As mentioned above, we identify the contour, C, with the A I -cycles in the above delta functions.
We are aiming for an expression for the generating function, A(j), at physical fixed-loop momenta, and on account of the above discussion we should think of Π Iµ as the physical momentum (i.e. the momentum dumped into a detector) and so insert one more delta function constraint into the amplitude (3.14): before finally integrating out P µ I and W µ I , after having evaluated the path integral over embeddings in (3.14) at fixed loop momenta. One thing to note is that when B µa = G µa = 0, then Π µ = G µν P ν (there will also be an independent delta function constraint δ Dg (W µ I ) as we discuss momentarily), so when this is the case it is natural to define P µ := Π µ . In the current document however all components of background fields will be kept generic.
Of course, strings cannot wrap around a non-compact dimension, and so W µ I should vanish identically. An important consistency check therefore will be to show that in fact: On the other hand, winding in the compact dimensions will generically be non-trivial (see below), and so Π Iµ will also receive contributions from W a I when B µa = 0, as can be seen from (3.22).
Before embarking on the evaluation of the path integral it will be important to make two final remarks. Even though the quantity Π M is the physical momentum, the quantities that appear most naturally in loop amplitudes will actually be Q M ,Q M , and these are also the quantities that enter the mass formulas and vertex operators. It will be useful for later reference to have at hand an expression for the latter in terms ofΠ M andŴ M , and these follow from the above expressions by trivial rearrangement. That chiral and anti-chiral vertex operator momenta are actually constructed out of eigenvalues ofQ M , where the (anti-)chiral fields x + (z), (x − (z)) are related to the full path integral field x(z,z) by a very subtle and indirect (yet remarkable) relation that we derive below, see (5.100), and as is well-known it is not correct to identify x(z,z) with x + (z) + x − (z) in general, although this may sometimes be justified. 19 We then choose the contour C to encircle the vertex operator (with any other features or insertions outside the contour), use holomorphicity to shrink the contour, in which case only the leading singular piece contributes, and similarly for the anti-chiral sector of vertex operators.
Because it isΠ M that generates spacetime translations, the usual argument concerning single-valuedness of the wavefunction [86] implies that eigenvalues ofΠ M must be discrete.
To make this statement sharp, note that we absorb all Kähler and complex structure moduli 19 Suffice it to say here that (as mentioned in the Overview section) x(z,z) contains zero modes, instanton contributions and quantum fluctuations, x(z,z) = x 0 + x cl (z) + x cl (z) +x(z,z), whereas the chiral fields x + (z), x − (z) are defined (for any genus g = 0, 1, . . . ) by their correlation functions, into the background fields, G M N , B M N . This allows us to compactify the x a on a (D cr −D)dimensional hypercube, such that for any a spanning T Dcr−D , we make the identification: x a ∼ x a + 2π s , with s = √ α the string length. (In this approach the actual compactification radius is determined by the moduli in G ab , B ab , and it is not s as one might naively conclude.) Then, under a lattice translation x a → x a + 2π s , the equation e i(2π s)Πa = 1 (for every a spanning T Dcr−D ) should hold as an operator statement in the string Hilbert space, so that its eigenvalues must be quantised in units of 1/ s : The position of the indices is important; recall that we generically raise and lower spacetime indices with G M N , and we do not assume G µa = 0 (or B µa = 0) in this document.
We are now ready to evaluate the matter generating function (3.14) in where we denote quantum fluctuations byx M and have also extracted out a constant zero mode x M 0 . Before inserting this into the action (3.15), and then into the path integral (3.14), let us determine the classical instanton solution. There are various subtleties (as well as new features) that are not discussed in the standard literature, so we will be fairly explicit.
Note primarily that x cl encodes the information that closed cycles on the worldsheet may wrap around non-trivial cycles of the torus T Dcr−D . As discussed above, all Kähler and complex structure moduli will be absorbed into G M N , B M N , and so we are free to normalise the x a cl such that x a cl ∼ x a cl + 2π s , for all a spanning T Dcr−D . The quantity x M cl by definition satisfies the classical equations of motion 20 of (3.15), 27) and is transverse to the constant zero mode, x M 0 . First consider the case j M = 0, the case of interest always being d 2 zj M = 0, that ensures overall charge neutrality (this is enforced upon us by the zero mode integrals 21 ). The solution that describes the soliton contribution of interest can be expanded in a complete basis, ω I ,ω I , as follows: where ℘,℘ ∈ Σ g denote an arbitrary reference point on which amplitudes do not depend (see below), the ω I = ω I (z)dz, (with I = 1, . . . , g and an implicit sum over repeated indices) denote a basis for the g abelian holomorphic differentials associated to a compact genus-g Riemann surface, normalised by their A I -cycles, A I ω J = δ IJ , and similarly for ω I =ω I (z)dz, namely A Iω J = δ IJ . The existence of the ω I ,ω I is guaranteed by the Atiyah-Singer-Riemann-Roch index theorem, see (A.124) and the discussion following (A.126).
Working with a canonical intersection basis (A.126) we denote the corresponding period 1 for a more explicit overview of conventions). In order to arrive at (3.28) and (3.29), note that in toroidal compactifications as we go around an A I -or B I -cycle of the worldsheet, the spacetime embedding should return to itself up to an integer multiple of 2π s , where we write d = dz∂ z + dz∂z for total differentials in the (z,z) coordinate system, with z * = z. We solve these constraints by expanding in a complete basis, ∂x a cl = I γ a I ω I , ∂x a cl = Iγ a Iω I , and then the γ a I ,γ a I are determined immediately from (3.30), (A.127) and (A.128), leading to (3.29). As a consistency check, notice that under A I -cycle translations 22 where we have used 21 It is sometimes of interest to relax momentum conservation either at an intermediate stage in a calculation (see [118] and recently emphasised in [119], as a means of regularisation while preserving onshell conditions), or relax momentum conservation all together (which is of interest for perturbation theory on non-trivial backgrounds), so we try to state explicitly throughout where momentum conservation is assumed so as to allow for appropriate generalisations. 22 We are being a little bit sloppy here. The coordinate z should really be thought of as the image z(℘) of a point ℘ ∈ Σ g under the Jacobi (or Abel) map, I : ℘ → z(℘) = ℘ ℘0 ω 1 , . . . , ℘ ℘0 ω g , with ℘ 0 some (universal) reference point on which physical observables do not depend. In particular, by transport z around a cycle A I we mean z(℘) → z(℘ + A I ), and similarly for the B-cycles. The vector z is an element of the complex torus J(Σ g ) ≡ C h /(Z h + ΩZ h ).
(A.127), and similarly for translations around B-cycles, under z → z + B I on account of (A.128) we have x a cl → x a cl +2πM a I s . Given the identification x a ∼ x a +2π s , the embedding of the worldsheet into spacetime is single-valued under z → z + A I and z → z + B I . Now let us turn on a general source term, j M (z,z), subject to d 2 zj M (z,z) = 0, and consider the set of solutions to (3.27). Making use of the defining equation for the Green function transverse to zero modes, see Appendices A.2 and A.1, (we have factored out the zero mode x M 0 as displayed in (3.26)), the soliton solution of interest that solves the full equation of motion (3.27) can now be seen to take the form: with γ M I ,γ M I as displayed above. 23 This satisfies all the monodromy requirements, given that (in addition to the above observations concerning the j = 0 piece) the Green function is by construction periodic under translations z → z + A I and z → z + B I (see Appendix For a given source term, j M (z,z), the set of topologically distinct classical solutions is still classified by the set of integers in γ M I ,γ M I , (i.e. the topological winding numbers associated to A I and B I cycles wrapping T Dcr−D ) as in (3.28). Secondly, the theory is Gaussian and so on account of the decomposition (3.26) we are free to absorb the j-dependent terms in (3.31) into a redefinition of the fluctuations,x M (z,z), without affecting the background around which we are expanding. (We will give this last comment more flesh at the end of this section, where we will derive the effect of this shift in the final answer for the generating function.) 24 This amounts to the simultaneous shifts x cl → y cl andx → y, with: 23 In principle we could add to x M cl (z,z) arbitrary well-behaved functions f (a I z ω I )+f (ā I zω I ) subject to (3.30), but the aforementioned choice (which will be referred to as the 'basic' one below) will be sufficient for our purposes.
where note that y cl + y = x cl +x. In particular, y M cl (z,z) is identified with (3.28) and y M (z,z) is the new quantum field. Given that such a shift (3.32b), being field-independent, will certainly leave the path integral measure invariant, the quantity (3.28), equivalently y M cl , can be taken to be the complete set of the basic classical solutions. We next substitute: into the full action (3.15), without dropping any boundary terms, so that on account of (3.32) and (3.31) we can recast the result into the form: where we have defined (note that dz ∧ dz = 0): Let us now return to the full path integral over matter fields (3.14). It is necessary to integrate out the zero modes, x M 0 , first, because this leads to constraints that will be enforced when integrating out y M and y M cl . 25 We expand x M 0 + y M (z,z) in a complete orthonormal basis {φ α } as follows, and ω 2 0 := 0 defining the constant zero mode, x 0 = A 0 φ 0 . The natural measure is then, Dx = "d Dcr x 0 Dy y cl " = α∈Z d Dcr A α √ DetG M N y cl , and we factor out the zero modes, d Dcr remaining fluctuation contribution and a sum over topologically distinct classical instanton contributions, y cl . The zero mode integral then factorises into a piece associated to R D and one associated to T Dcr−D :

36)
25 Not integrating out the zero modes, x M 0 , at this stage of the calculation is certainly interesting, as it is relevant for string scattering in curved backgrounds (in a background field expansion sense) for strings whose spatial extent is smaller than any background curvature scale, and we hope to return to this point in the future. and for convenience we have also included the dilaton contribution in the definition of Ψ 0 Eucl . The d-dimensional Kronecker delta is denoted by δ d (·),0 , and δ d (·) is a d-dimensional Dirac delta function (whose argument has indices "downstairs" 26 ), that arise from T Dcr−D and R D respectively. The identifications x a ∼ x a + 2π s lead to the factor (2π s ) Dcr−D . Rotating back to Lorentzian signature target spacetime amounts to replacing (G M N ) Eucl by , so that the right-hand side of (3.36) in Lorentzian target space signature reads: the branch of the square root being convention-dependent (our choice is in agreement with Polchinski [86]). Note that the source (Dirac and Kronecker) delta functions enforce charge neutrality for the asymptotic states, Having determined the zero mode contribution, let us turn our attention to the ydependent pieces, starting from the y-dependent integrals in (3.35). It is sometimes convenient to write the holomorphic one forms, ω I ,ω I , in terms of the Abel map g I ,ḡ I : On the cut Riemann surface [88], see We might therefore be tempted to drop these integrals, given that the integration domain is a compact Riemann surface, but the integrand has non-trivial monodromies around A I and B I cycles, and there is also the possibility of dy contributing poles that may lead to a non-vanishing result, see e.g. [88] (p. 150) and also Appendix A of [120]. Given that y is a quantum field, what we need to check is whether F I ,F I contribute to correlation functions. That is, if we can show that (when d 2 zj M (z,z) = 0): 26 When G µa = 0, we can raise indices in the Dirac delta function using the following rule: . We do not assume G µa = 0 in this section however. Figure 1: Pictorial representation of a genus-2 Riemann surface, Σ 2 , (on the left) and the corresponding cut surface,Σ 2 , (on the right), obtained from the former by smoothly (isotopically inΣ 2 ) dragging all cycles associated to the canonical intersection basis of the homology group so that they meet at a point (an 8-point vertex, see the first image), and subsequently "deleting" these homology lines from the surface. This leads to the cut Riemann surface,Σ 2 , which has a boundary, ∂Σ 2 , on which functions are single-valued. The point p indicates a point at which the integrand (3.41) is singular and the disc D of infinitesimal radius is defined to be centred at p (denoted by local coordinates w,w in the text) with boundary ∂D. A similar picture is to be understood for any genus g ≥ 1 surface where the resulting cut surface is a 4g polygon.
and similar expressions withF I replacing F I , then (given the theory is free) the following equality holds (correlators being with respect to the y path integral), for any set of constants 27 C IN ,C IN , and we can effectively set F I =F I = 0 from the outset.
That (3.39) holds is indeed the case, but because the reasoning is somewhat subtle we will be explicit. Considering first F IFJ and F I F J , we make use of the explicit expression for the propagator, y M (z,z)y N (w,w) = G M N G(z, w), see (3.54), the Riemann bilinear identity, Σg ω I ∧ω J = −2i(ImΩ) IJ , and the monodromy properties of the prime form [64,87,88], see also Appendix A.1: It is straightforward to then show that the following contractions of F I ,F I vanish identically: The remaining correlators we need to compute in order to establish (3.40) are j M y MF N J and j M y M F N I . Given that j M = 0, all we need to check is that the correlator 27 The case of interest above being: y(w,w)F I is w,w-independent (and similarly for the anti-chiral sector) -that is, the w orw derivative of this correlator must vanish. Therefore, carrying out the contraction using the full propagator, it suffices to check that the integrals: vanish, and similar expressions withḡ I replacing g I , and also ∂ w replacing ∂w; four integrals in total, but two are related by complex conjugation. Here dG ≡ dG(z, w), d = dz∂ z +dz∂z, and G(z, w) is the full propagator on the genus-g surface. For these boundary integrals one may consider the polygon representation of the cut surface,Σ g , see Fig. 1, cut out small discs (D) of infinitesimal radius | | centred around the pole that comes from dG(z, w) for z → w, and write the integral over the full cut surface as an integral overΣ g − D plus an integral over D. Both of these can be written as boundary integrals using Stoke's theorem and a careful consideration of each of the terms that arise (along the lines of Mumford [88] or Lugo and Russo [120], although in the present context the integrand contains both meromorphic and anti-meromorphic quantities), after various cancellations, leads to the vanishing of the correlator in question, Therefore, repeating the argument forF N I as well the naive assumption that we can set F I =F I = 0 is, effectively, correct. Note that onΣ g − D, ∂ z ∂w ln |E(z, w)| 2 = 0, whereas on D, ∂ z ∂w ln |E(z, w)| 2 = −2πδ 2 (z − w). We also use the fact that g I is on A −1 J the same as g I on A J plus B J ω I = Ω IJ , and that g I is on B −1 J the same as g I on B J plus A −1 x = x 0 + y cl + y factorises the action into three distinct pieces, and substituting these into (3.14), leads to: where Ψ cl Eucl and Ψ q Eucl are to be identified with the second and third parentheses respectively in the second or third lines, and we have defined: Im 43) with the understanding that, according to the above discussion, the Φ I a ,Φ I a in (3.35) reduce to (generalising the definition to generic spacetime components for later convenience): We next evaluate each of the two remaining factors, Ψ cl Eucl and Ψ q Eucl , below in the generic case where all components of the source j M are potentially non-vanishing (or possibly physical). This latter point will ensure that we can extract amplitudes from A x (j) by functional differentiation with respect to the source j associated to S-matrix elements whose asymptotic states have generic Neveu-Schwarz (NS) charges: we allow for vertex operators whose polarisation tensors and momenta potentially span the full space R D−1,1 × T Dcr−D or any subspace of interest, with generic KK and winding charges.
Let us now consider the classical instanton contributions, with γ a I ,γ a I given explicitly in (3.29). To make the loop momenta in the compact dimensions manifest it is desirable to perform a Poisson resummation on the integers {M a I } appearing in γ a I ,γ a I , the particular identity of interest being (when the matrix A ab IJ is not necessarily diagonal in 'ab' or 'IJ'), with 'Det' with respect to the 'ab' indices and 'det' with respect to 'IJ'. (This can be thought of as a first step towards a Hamiltonian formulation of string theory as opposed to a Lagrangian formulation, because this Poissson resummation makes the compact loop momenta manifest, see below.) The invertible (g × g, real and symmetric) matrices of interest are A IJ ab = G ab (ImΩ) −1 IJ , and the complex vectors B I a = (2ImΩ) −1 which follows from the defining relations G M N G N L ≡ δ M L and (G ab ) −1 G bc ≡ δ a c . After a certain amount of algebra we learn that: (3.47) The (non-chirally split) exponent in the last factor in (3.47) is closely related to the zero mode of the multi-loop propagator, see below. 28 We have defined the quantities: (ii) integrate out y M in resulting expression; (iii) evaluate resulting A I -cycle contour integrals, see (3.19); (iv) integrate out the λ, in resulting expression. 28 We often use the convention that under complex conjugation one is to take (in addition to z →z, ω I →ω I , Ω IJ →Ω IJ ) Q →Q and j a → j a (independently of whether j M or Q is real or complex), as this allows us to rewrite the two exponents in the second line of (3.47) as exp(. . . ) 2 .
Step (i), introducing an integral representation for the delta functions, leads to: where we made use of (3.19) in order to define the quantity B M = (B µ , B a ) with: There are two slightly subtle points that go into deriving the equality (3.50). The first concerns an apparent interchange of the orders of integration: for some generic homology cycle, C, we have been somewhat cavalier in going from the first to the second equality in: without discussing the issue of absolute convergence. However, the key word in the above statement is the word 'apparent', because in evaluating these integrals (after integrating out x) it will always be understood that we first carry out the area integrals and subsequently the contour integrals. The second subtlety is potentially more serious, namely a real interchange in the order of integration: Dy d Dg λ(. . . ) = d Dg λ Dy(. . . ). This interchange is potentially subtle (given that, e.g., we have not addressed the issue of absolute convergence of the y integral), but will nevertheless proceed in this manner and rest assured on the fact that our result for A x (j) will be consistent (in certain limiting cases) with the result of D'Hoker and Phong [64] who proceed without introducing such delta function insertions, and so this procedure is not expected to introduce any spurious terms.
Step ( Note that the determinant, Det −1/2 G M N , crucially, has cancelled out of (3.53). Here gg αβ ∂ β is the standard Laplacian (which in the z,z coordinates reduces to −2g zz ∂ z ∂z), the prime on the determinant indicates that ∆ (0) acts in the space orthogonal to zero modes. The (multi-loop) propagator, G(z, w) := α =0 2πα ω 2 α φ α (z,z)φ α (w,w), and sat- √ g , with the completeness relation 1 √ g δ 2 (z − w) = α∈Z φ α (z,z)φ α (w,w); see also Appendix A.2. For example, for compact genus-g Riemann surfaces (the case of interest in the current document) we can take [84], The zero mode piece appearing here is the main obstruction to chiral splitting in amplitudes.
Fixing the loop momenta [64] removes these zero mode contributions from correlation functions, thus significantly facilitating amplitude computations as we shall see, especially in the context of highly excited strings.
Step (iii), the evaluation of the A I -cycle contour integrals is carried out by noting that in the exponent in (3.53) there arise the terms: and from the definition of B µ therefore, we must interpret the quantities A I dz∂ z G(z, z ), and A I dz∂zG(z, z ), and various related combinations -these are the contour integrals referred to in item (iii) above. According to (A.143), the prime form is periodic around the A I -cycles, and therefore the sole contribution will come from zero modes, see (A.146). (If we had instead fixed the B-cycle momenta, or a linear combination of A-and B-cycle momenta, the non-zero mode components would also contribute, and we would have to add additional pieces to the propagator. 29 ) Analytically continuing z,z to independent variables, a short calculation 30 using the defining relations for the holomorphic differentials, A I ω J = δ IJ and with ℘ ∈ Σ g a reference point on which amplitudes do not depend. This step is somewhat naive (due to the continuation of z,z to independent variables), but it gives the correct 29 DPS thanks Eric D'Hoker for a discussion on this point. 30 Here we write Im answer. One of the relevant integrals for the second (for M = µ) and third (for M = a) terms in the above exponential is then, (3.56) where the dots denote the contribution coming from the terms − 1 4 (ImΩ) II in (3.55), which do not contribute because d 2 zj M (z,z) = 0, see (3.36). Notice we are not assuming the cross terms, G µa , vanish. Furthermore, the difference in sign in the chiral and antichiral halves in the second terms of the right-hand sides on (3.55) is crucial: it implies that theλ-dependent terms in (3.51) precisely cancel out (on account of the constraint Finally, there is an integral that is quadratic in B µ (z,z). This is equivalent to (3.56) (up to a factor of two) but with B M (z,z) replacing j M (z,z), Substituting this into (3.53), which is in turn substituted back into (3.50), we obtain the following expression for the delta function expectation values in the presence of a source: (3.59) Step (iv) of the computation is to carry out the remaining integrations over the λ Iµ , λ Iµ in (3.59). 31 After some trivial rearrangement, where we note that theλ integrals lead to the delta function constraint δ Dg W µ I , which on account of (3.20) allows us to identify P µ I with Q µ I orQ µ I , given that in the absence of winding all these are equivalent.
The second and third lines of the RHS in (3.61) are chirally split, whereas the last line is not. (The non-chirally split terms in the first line will cancel when ghost contributions are included.) The term in the last line will ultimately cancel a similar quantity that arose from the instanton contribution, Ψ cl Eucl , see the last line in (3.47), but to make this cancellation manifest let us consider the quantity: (3.61). When M and/or N span R D this quantity vanishes, given that by definition (G µν ) −1 G νρ ≡ δ ρ µ . Therefore, only if both M and N span T Dcr−D will the non-chirally split term in the last line of (3.61) contribute. That is, (3.62) But according to (3.46) the quantity in the brackets, G ab − G aµ (G µν ) −1 G νb , is precisely (G ab ) −1 , and so the non-chirally split factor (3.62) is also equal to: This exponent is (up to a crucial minus sign) identical to that in the last line of (3.47), implying that in the product Ψ cl Eucl Ψ q Eucl the non-chirally split exponential will cancel out 31 Defining A µν IJ := G µν 2πα ImΩ −1 , the following integral is required: The full result for the quantum fluctuations reads, on account of (3.61) and (3.49) and the above discussion, Let us now gather all the results for the various terms appearing in (3.42), starting from the chirally split exponentials in Ψ q Eucl in (3.63) and Ψ cl Eucl in (3.47). It is straightforward to show (using G M N G N L = δ M L , (G ab ) −1 G bc ≡ δ a c and (G µν ) −1 G νσ ≡ δ σ µ , always raising and lowering indices with G M N , and taking into account the delta function constraint, δ Dg (W µ I ), in Ψ q Eucl which enforces P µ = Q µ =Q µ ) that: A Eucl (3.65) where we used (3.44) and took into account that δ D ( s ∫ j µ ) 1 DetG µν = δ D ( s ∫ j µ ) (given that although we raise indices with the full metric, G M N , we also have j a = 0 implying that effectively G µM j M = G µν j ν ). The (dimensionless) sum/integral over (Q,Q) should be understood as an integral over non-compact momenta, P µ I (with Q µ I =Q µ I ), and a sum over Clearly, from (3.65) we see that the natural expansion parameter at fixed-loop momenta is: The G ab and g s dependence of g eff is as expected, since this combination has precisely the form required in order for g eff to be invariant under T-duality, more about which later. The G µν dependence is novel and deserves further elaboration; we elaborate on this below. Note that g eff is also precisely the dimensionless version of the coupling g D that appears in vertex operators, with g D = g eff A Eucl and we have defined: To complete the story we now include the ghost contribution, A gh , to extract the full generating function, A Eucl (j) = A gh A Eucl x (j). At this point the ghost insertions can be completely general, but we restrict here to the minimal number of ghost insertions that lead to a non-vanishing result, with,

72)
A gh has been well-studied for arbitrary genus [84,89] and we have nothing new to add here, so we will be brief. Suffice it to say that the operator-product expansions (OPE's) (5.107) imply A gh has various obvious zeros and poles due to the explicit b, c insertions. In addition, viewed as a function of, say, w 1 , it has g additional zeros that are determined uniquely by the Jacobi inversion theorem [84], while the related Riemann vanishing theorem [89] further ensures that A gh can be expressed entirely in terms of Riemann theta functions and related 32 The result (3.68) is consistent with the chiral splitting theorem of D'Hoker and Phong [64] and reproduces the tachyon n-point amplitude of [83] in D cr = 26 when: j(z,z) = i k L,i zi ℘ duδ 2 (u − z)∂ z + k R,i zī ℘ dūδ 2 (u − z)∂z , for vertex operators with total momentum k i = 1 2 (k L + k R ) i and winding w i = 1 2 (k L − k R ) i , and (℘,℘) a universal generic point on the worldsheet on which amplitudes do not depend due to momentum conservation, i k i = 0 that in turn arises from the delta-function constraint. When the external states have zero winding, w i = 0, with w = 1 2 (k L − k R ), the source reduces to j(z,z) = i δ 2 (z i − z)k i . quantities, allowing it to be evaluated explicitly [84,85,89]; see also [121] for a pedagogical account. What we will make use of here is the following generic result (when g > 0): 33 (3.73) As above, a prime on determinants always signifies that it is to be computed in the space transverse to zero modes of the associated operator. That A gh chirally factorises up to the term in the parenthesis (and a Liouville factor that we are suppressing) is well understood and holds for arbitrary genus g. The quantity Z g is in turn a certain combination of modular functions. For example, at genus g = 1, it may be written in terms of the Dedekind eta function, with τ = τ 1 + iτ 2 the complex structure modulus of the torus [86]. The quantity in the parenthesis can also be expressed in terms of modular functions [84,89], but in fact the above form will be more useful in what follows, because it precisely cancels a similar factor from the matter sector in the critical dimension at fixed internal loop momenta, as we will elaborate on explicitly next.
Let us now collect all the pieces and write down an expression for the full (g > 0) generating function (3.12) for closed string scattering in target spacetimes R D ×T Dcr−D with generic Kähler and complex structure moduli, background KK gauge fields and torsion, see In the critical dimension of bosonic string theory, D cr = 26, the non-chirally split terms cancel out completely, and (unless we include a fermionic sector to extend this result to the superstring) this is precisely where this computation is valid. In non-critical bosonic string theory, where D cr = 26, there is an additional Liouville factor that contributes to restore Weyl invariance. In what follows we focus on D cr = 26, but we emphasise that the 33 We are neglecting the contribution of the Liouville factor that will always cancel in the final answer for the full amplitude in the critical dimension.
above expression for A Eucl (j) holds true also for the superstring when D cr = 10 and the sources j M are shifted by worldsheet fermions [64]. The superstring will be discussed in detail elsewhere.
Wick-rotating A Eucl (j) to Lorentzian target space signature 34 , and denoting the resulting object by A(j), we can then very concisely write the full result for the generating function as follows: The (dimensionless) sum/integral over (Q,Q) should be understood as an integral over the non-compact momenta, P µ I (with Q µ I =Q µ I ), and a sum over the compact momenta, Q Ia ,Q Ia , which lie on the genus-g torus lattice, Γ g  with the propertyδ(j s ) = −D sδ (j). It is customary for S-matrix calculations, see (5.113), to work in terms of the dimensionful delta functionδ(j) = (2π) Dδ (j), i.e., It should also be understood that all implicit spacetime index contractions in (3.75) are carried out with the full metric G M N , which now has Lorentzian signature, and that the 34 As discussed above, Wick rotating back to Lorentzian target spacetime signature can be achieved by replacing G Eucl µν → G Mink µν and DetG µν Eucl → i −DetG µν Mink (the branch of the square root being convention-dependent). This is equivalent to starting from a Euclidean signature generating function and then interpreting all spacetime contractions as being with respect to a Lorentzian signature metric, G Mink µν , while rotating the coupling (g 2 eff ) Eucl → −i(g 2 eff ) Mink , leaving other quantities unchanged, with (g eff ) Mink positive definite as defined in (3.79). This approach leads to an overall factor of i 1−g , with the i displayed explicitly in (3.75) and the i −g absorbed into (3.76). Upon rotating to Lorentzian signature, the contours of energy loop integrals are to be interpreted as in [65][66][67].
T-duality invariant g eff is defined in terms of this: The above holds for arbitrary constant backgrounds, G M N , B M N , but a standard example is the following (torsion-free) background: where now g eff = g s (DetG ab ) − 1 4 = g s a ( s /R a ) 1 2 , and T-dualising along one compact dimension of radius R 1 [86]: The full generating function is explicitly dimensionless, as it should be. Vertex operators should of course also then be dimensionless in order to lead to a dimensionless S matrix whose modulus-square yields a probability. This is indeed the case when kinematic factors for each of the vertex operators are included, 1/ 2k 0 V D−1 , (with V D−1 = (2π) D−1 δ D−1 (0) the formal volume of non-compact space that always cancels out of observables (precisely as in standard field theory [123]) and k 0 the expectation value of the energy of the vertex operator) whose mass dimension precisely cancels that of the string coupling g D ≡ g eff We would like to end this subsection by briefly returning to the discussion associated to the shift of quantum fluctuations (3.32) that subtracts the source dependent piece from the classical solitons (3.31). We stated there that (being a field redefinition) such a shift will not affect amplitudes or the generating function. The manner in which this invariance manifests itself is quite interesting, so we will discuss it briefly. Suppose that instead of computing quantum fluctuations around the soliton solution y cl we computed quantum fluctuations around the original soliton solution, By explicit computation one can show that the effect of the new source-dependent shift in the classical soliton sector, i d 2 wG M N j N (w,w)G(z, w), is to undo the chiral splitting in the sense that the exponential, in (3.65) or (3.75) would get replaced by: where the term in the parenthesis in the latter expression is precisely the propagator for the full non-chirally split quantum fluctuations (3.54

Wave/String Duality
In this subsection we discuss the sense in which wave/particle duality of point-particle quantum mechanics arises in string theory. The analogous relation in string theory will be referred to as wave/string duality, because in string perturbation theory the notion of particle is replaced by the notion of string. In passing we will also elaborate on some aspects of target space effective actions. We will now argue that the displayed equalities (4.82) may be regarded as a stringy manifestation of wave/string duality, generalising the well-known wave/particle duality of quantum mechanics; a principle that applies to all scattering amplitudes in string theory, to all orders in perturbation theory. In this language the correspondences are: 'F' = wave picture, and 'I' = string picture.
In the above discussion A(j) corresponds to a (wave, wave) formulation of the generating function, A(j) = A(j) (F,F) , but considering also the other three pictures provides some additional insight.
The (string, string) generating function, A(j) (I,I) , is the formulation that naturally arises out of the Lagrangian formulation of string theory, which is the usual starting point for string calculations in the path integral language [86]. Here one (generically) sums over all string trajectories for some fixed set of boundary conditions and asymptotic states, so it is natural to associate this with a string picture (analogous to a particle picture in the Feynman formulation of quantum mechanics where one sums over all trajectories of one or more particles given a set of boundary conditions). A good example that provides some further insight arises from considering the one-loop partition function in the Lagrangian formulation. Let us in particular focus on the compact dimensions, there being analogous statements in the non-compact dimensions. This contains the instanton action associated to classical trajectories (3.28). Setting N a I = 0, one can compute the associated momentum of an A I -cycle string using (3.19), and hence notice that the winding number M a I that one is summing over, see (3.29) and (3.30), may be interpreted as the number of times an A I -cycle closed string traverses a compact dimension of size 2πR in worldsheet time interval Im Ω 11 (more precisely in the analytically continued worldsheet real time interval −iIm Ω 11 , recall the worldsheet theory is in Euclidean signature).
Let us now think about the corresponding interpretation in the (wave, wave) picture where the relevant quantity is A(j) (F,F) . This is closely related to a Hamiltonian formulation of string theory, given that all (independent) loop momenta in this formulation are explicit. Considering again the one-loop partition function referred to above, A(j) ( The remaining two (hybrid) cases, A(j) (F,I) and A(j) (I,F) may be thought of as Routhians (analogous to the Routhians of classical mechanics) given that they correspond to a Hamiltonian formulation in the non-compact and compact dimensions respectively, with a Lagrangian formulation in the remaining dimensions.
In examining further these four pictures let us primarily zoom in on the G µν dependence in the effective coupling (3.67). The quantity Det G µν is present in g eff because the associated generating function (3.65), being in the (F,F) picture, has fixed non-compact loop momenta. To see that this enters in precisely the expected manner it proves useful to consider the usual dimensional reduction on R D × T Dcr−D of the NS-NS sector of low energy supergravity, see e.g. [114,124]. The relevant metric decomposition that leads to a natural expression for the dimensionally reduced target space effective action is: where the A a µ are Kaluza-Klein gauge fields. It is useful to compare with (3.16). The usefulness of this parametrisation is that: We are free to define what we mean by g µν , and it is consistent [114] to simply define g µν := (g µν ) −1 . (Note however that generically G ab = (G ab ) −1 , because G ab is the ab component of G M N that is completely fixed by the defining relation G M N G N L = δ M L .) We can then rewrite the following term in (3.74) in terms of g µν , where following standard practice we have defined a dimensionally-reduced dilaton: Fourier transforming the depicted delta function in (4.84) and the integrand of the g A Icycle loop momentum integrals leads to factors: which is clearly a collection of natural position space measures with a dilaton dependence that is precisely that expected from low energy supergravity [114], whose tree-level in g s contribution contains the universal factor: For the reader that is trying to make contact with the quantum effective action of quantum field theory [125] note that the momenta of strings propagating through the various isotopically (in Σ g ) distinct cycles of the underlying genus-g Riemann surface with expects from a perturbative expansion of the field theory path integral [125]). Hence there will be a universal factor in this particular degeneration:

87)
g + 1 integrals of which are manifest from the derived explicit factor (4.86) above, while the remaining g + n − 3 integrals are not manifest in the above decomposition because we have only fixed the independent (in particular A I -cycle) loop momenta (and implicitly the vertex operator momenta). This is a well-known peculiarity of string theory [81], in that it is not so natural to exhibit all intermediate propagators in a string theory amplitude until we reach a field theory limit. The remaining internal momenta are nevertheless all fixed by momentum conservation and so can be made explicit by introducing momentum conserving delta functions for a given pant decomposition. From standard field theory and Feynman diagram topology considerations Fourier transforming the resulting momentum integrands must lead precisely to an overall factor (4.87), thus making the quantum effective action and corresponding field theory limit manifest. Clearly, the g = 0 terms all have one overall factor d D A much more complete discussion on some of these aspects can be found in [73].
From these considerations it is clear that the Det G µν dependence in (3.74) is completely natural and necessary in order to make contact with quantum field theory considerations. Given that the Det G µν dependence is (according to the above discussion) associated to the explicit presence of internal non-compact loop momenta, P µ I , we can remove it by integrating them out. Returning to the original parametrisation of the metric (3.16), on account of DetG µν (recall that j a = 0 and that in Euclidean space G µν is positive definite), and a slight variation of the Gaussian integral (3.60) we obtain the (string, wave) (or (I, F)) representation, 36 A Eucl where G(z, z ) denotes the full Green function (3.54). We have presented the result for the matter contribution for clarity, the full generating function, A Eucl (j) (I,F) , being obtained by multiplying the right-hand side of (4.89) by the ghost contribution (3.73). (In reconstructing the worldsheet Green function we have made use of the constraint d 2 zj M = 0.) It is seen 36 The following relations (and analogous expressions obtained by interchanges (µ, ν) ↔ (a, b)) are useful: that the effective coupling in this representation is g s DetG ab − 1 4 , as one expects from Kaluza-Klein reduction of low energy supergravity [114,124] on R D × T Dcr−D .
The corresponding generating function in the (wave, string) (or (F, I)) picture is similarly obtained from (3.36), (3.63) and (3.45), where we have defined the compactification volume: whereas for the (string, string) picture generating function (i.e. (I, I)) we may consider (4.89), and perform a (or more precisely undo the) Poisson resummation in the compact dimensions. This will also remove the G ab determinant from the effective coupling, . Making further use of the relations for determinants (4.88) in the footnote and taking (3.47) into account we obtain the (string, string) (or (I, I)) picture representation, It is important to mention that when g = 1 the above expressions for the generating function assume there is at least one vertex operator insertion. Let us briefly discuss the vacuum amplitude which (although standard) is the one example where this is not the case.
In the absence of vertex operators and for g > 1 integrating over moduli gives the vacuum amplitude, usually denoted by Z g , 92) because in this case there are no CKV's; see (5.109) for the definition of dM g . The genus g cosmological constant, Λ g D , is defined by: (4.93) The first equality is in the (I, I) picture when the instanton contribution, Ψ cl | j=0 , is identified with (3.45) evaluated at Φ I a =Φ I a = 0, whereas the second is in the (I, F) picture. For g = 1 however, where Ω IJ → τ = τ 1 + iτ 2 , we should include a further factor of 2τ 2 in the denominator in the absence of vertex operators to obtain the vacuum amplitude [86], with the genus-1 dimensionally reduced cosmological constant, For all amplitudes with at least one vertex operator insertion there is no additional factor of 2τ 2 in the denominator. The moduli space measure dM 1 = 1 2 d 2 τ (the additional factor of 1/2 here being due to the remaining Z 2 isometry, see Appendix B). We have taken into account that T 2 d 2 z = 2τ 2 and made use of the presence of one CKV in order to write all vertex operators in the integrated picture, and then set the number of vertex operators to zero, i.e. n γ=1 d 2 z γ V zγzγ → 1. Furthermore, we have defined d D x 0 := (2π) D δ D (0), with 37 δ D (0) = δ D (j µ )| jµ=0 , which has dimensions of L D . The above expression for the vacuum amplitude is in precise agreement with standard conventions [86,126] and serves as a non-trivial check of the normalisation of A(j) at g = 1.
We next discuss correlation functions for generic vertex operator insertions.

Correlation Functions
Given the result for the generating function (3.75), whose defining equation is (3.12), let us now start to think about generic correlation functions, setting the stage in particular for correlation functions of highly excited strings. We define: where note that this implicitly includes the minimum number of ghost insertions required to make amplitudes not vanish trivially, see (3.12). To compute correlation functions of generic operators we take functional derivatives of A(j) with respect to j M (z,z), and subsequently set the source equal to the value of interest, see the footnote on p. 36. For instance, given a set of operators {D i } (that commute with the path integral) we can extract correlation functions from A(j) as follows: The operators {D i } may denote a set of worldsheet derivatives, e.g., {∂ z , ∂ 2 z , ∂ w , ∂z, ∂w, . . . }, or (e.g. in the case of coherent vertex operators [59]) they may be more complicated but linear operators [60]. In order for this procedure to be useful in the case of composite operators (where multiple D i x's may be inserted at the same location on the worldsheet), we use the notion of point splitting, see e.g. [91]; that is, we write a normal-ordered operator : , calculate the correlators as specified in (5.97), subtract the terms singular in (z 2 − z 1 ), and take the limit z 1 → z, z 2 → z. We refer to the latter step as point merging.
Carrying out the functional derivatives as specified in (5.97) on account of (3.75), (3.69) 37 That it is natural to identify the integral over the zero modes, x µ 0 , with (2π) D δ D (j µ )| jµ=0 (with indices downstairs) follows from the integral representation of the delta function. and (3.70) leads to: 38 which on account of (3.70) may be viewed as a functional generalisation of, It is also possible to show that for fixed k the number of terms that appear in the sum over permutations in (5.98) before point merging is indeed: as one would expect from the finite dimensional formula. The notation J /2 in the sum over k indicates that the maximum value of k is the integer that saturates the inequality k ≤ J /2. S J is the symmetric group of degree J [115], the group of all permutations of J elements, and the equivalence relation '∼' is such that π i ∼ π j with π i , π j ∈ S J when they define the same element in (5.98).
The point merging procedure will give rise to contact terms, i.e. terms that only contribute when two or more vertex operators are coincident, e.g. from contractions of the form ∂ z ∂w ln |E(z, w)| 2 = −2πδ 2 (z − w). Following a standard argument, in view of the (assumed 39 ) analyticity of string amplitudes in external momenta [81], and the fact that 38 Notation-wise, (DD ln |E| 2 ) π(2l−1)π(2l) ≡ D π(2l−1) D π(2l) ln |E(z π(2l−1) , z π(2l) )| 2 and j M π(q) (D ln |E| 2 ) π(q) ≡ d 2 zj M π(q) (z,z)D π(q) ln |E(z π(q) , z)| 2 . 39 It is not obvious whether analyticity in external momenta is present for generic amplitudes [81], and one needs to check this on a case by case basis. In fact, the tachyon and massless tadpoles often cause trouble [81] in searching for absolute convergence in bosonic string amplitudes, and we one has to adopt a certain prescription in order to extract physical observables. In addition, certain degeneration limits of the worldsheet moduli can lead to trouble when one or more internal lines are forced to be onshell by momentum conservation (such as the separating degeneration of a two-loop two-point amplitude with the amplitude always contains a factor of the form it follows that such terms will not contribute even after the vertex operator positions have been integrated over, and will thus be set to zero [64]. 40 A very important implication of this latter observation is that of all the permutations that we are to sum over in (5.98), the only ones that will give a non-zero contribution will be those that respect chiral splitting. That is, we can partition the full set of operators and then (denoting the worldsheet coordinates where the chiral and anti-chiral operators are inserted by {z j ,z j } and {w j ,w j } respectively) a careful consideration of the single sum over k in (5.98) shows that it factorises into two independent sums. In turn, these two independent sums can be extracted from two completely independent correlation functions one vertex operator on either component, or tadpole degeneration limits). These cases require particular care [73,99,106,116], such as an offshell description [73], vertex operators in a larger Hilbert space than the conformally invariant one [73,99,106,116], and one must introduce a local coordinate dependence [73,99] (i.e. abandon conformal invariance) that cancels [73] (see also [71,72]) out of observables. The matter contribution to the generating function introduced here is still applicable in such cases, but one typically needs to consider more general ghost insertions than the minimal number. In this document we assume a region does exist in the complex momentum plane (or complex 'Mandelstam variables', or appropriate generalisations thereof for n-point amplitudes) of absolute convergence, and physical amplitudes are then obtained by analytic continuation from this region. In sequels [60,61] tachyon divergences will be carefully identified and some (in particular tadpoles) will be absorbed by background shifts and others dropped by brute force. 40 We mention the argument for completeness. Notice that the exponent of |E(z i , z j )| can always be made positive by analytic continuation and that when two vertex insertion points come close together, E(z i , z j ) z i − z j . Therefore, given that (symbolically) d 2 z|w − z| ki·kj δ 2 (w − z) = 0 when Re k i · k j > 0 it follows from a famous theorem of complex analysis that the entire expression will vanish for all k j . In amplitudes involving coherent vertex operators one also encounters exponentials of contact terms, and so one also needs to consider multiple delta functions. Similar reasoning to the above leads also to the vanishing of multiple delta functions, e.g.
To see this write this expression as lim →0 d 2 z|w − z + | ki·kj δ 2 (w − z)δ 2 (w − z + ). Performing the z integration leads to lim →0 | | ki·kj δ 2 ( ), which vanishes for the following two reasons: the integral d 2 | | ki·kj δ 2 ( ) = 0 and the corresponding integrand is non-negative -therefore, the integrand must vanish. Extending this reasoning to three or more delta function insertions implies that (unless the momenta under consideration are constrained to vanish identically by momentum conservation, such as in the case of tadpoles) contact terms do not contribute to the amplitudes and will be dropped. as follows: where j = j L = j R was assumed in the above derivation -we will discuss the extension to j L = j R momentarily. We want to emphasise that on the left-hand side the x N (z,z) = x N 0 + y N cl (z,z) + y N (z,z) appearing contain the zero modes, instanton contributions and quantum fluctuations. Recall the analysis following (3.33). On the right-hand side however, a chiral and anti-chiral field appears, x N + (z) and x N − (w) respectively, which does not contain any zero mode or instanton contributions. These zero mode and instanton contributions are rather contained inδ(j s ) and H,H respectively. The relevant correlators on the righthand side of (5.100) are defined with respect to the "chiral propagators" [64], which unlike the full propagator G(z, w) have [87,88] non-trivial monodromies around B I cycles but not around A I cycles, see (A.143), The chiral correlator in (5.100) reads explicitly: where the argument in the exponential equals 2 s 4 d 2 z d 2 z j L + H · j L + H ln E(z, z ), and similarly for the anti-chiral half, As mentioned above, in the derivation of (5.100) we assumed j = j L = j R , but in fact using the (anti-)chiral representation of amplitudes enables one to consider more general insertions for which asymptotic vertex operators can have non-trivial winding. That is, using the chirally split generating function it is almost obvious how to insert vertex operators of the form: with k M =k M , simply by taking j L on the right-hand side of (5.100) to be independent of j R . The corresponding insertion on the left-hand side of (5.100) however is not so obvious, given that here vertex operators associated to (5.104) should be functionals of the full path integral field, x M (z,z). When k M =k M , it is clear that to every vertex operator insertion (5.104) on the right-hand side is associated a vertex operator insertion, 105) on the left-hand side with total momentum 1 2 (k M +k M ). In order to extend insertions on the left-hand side to vertex operators with non-trivial winding where k M =k M we need to integrate over all x M (z,z) with source j M (z,z), and constrain the integration to fields with non-trivial winding. This may be achieved [83] by a j L -, j R -dependent shift in the classical instanton solutions x M cl (z,z) of (3.28). Therefore, with this shift vertex operators of the form (5.105) remain valid insertions even in the presence of non-trivial winding. We will not work out the details of this procedure here as there exists a simpler approach. In particular, we will instead enforce chiral splitting of the source, the prescription being the following. 41 Suppose we consider an amplitude with n vertex operator insertions, each of which (in the chiral representation (5.104)) carries an exponential of the form: e ik γ ·x + (zγ ) e ik γ ·x − (zγ ) , 41 The authors would like to thank Joe Polchinski for suggesting this alternative procedure.
with γ = 1, . . . , n, in addition to some polynomial of derivatives of x + (z) and x − (z). (More generally, every vertex operator will be a superposition of such momentum eigenstates, as is the case for coherent vertex operators for instance.) The statement is that insertions with exponentials of the form: n γ=1 e ik γ ·x + (zγ ) e ik γ ·x − (zγ ) , on the right-hand side of (5.100) (with k γ M =k γ M generically) correspond to evaluating the source on the left-hand side of (5.100) at: so that with this choice of source there exists the correspondence: even though on the left-hand side the embedding field, x(z,z), contains (potentially) also instanton or soliton contributions whereas the right-hand side does not. So the prescription is to consider j M (z,z) on the left-hand side as an operator and act with the derivatives, ∂ z , ∂z, of (5.106) before carrying out the line integrals. Using the representation for the source (5.106) makes is obvious that we can simply substitute (5.106) into the right-hand side of the j = j L = j R expression (5.100), and then the operator nature of the decomposition (5.106) will ensure that only j L appears in chiral terms and j R in anti-chiral terms, and so we can legitimately extend the result (5.100) to the case where j = j L = j R . This is the desired result.
Having understood how to insert vertex operators with non-trivial winding using either the chiral fixed-loop momenta or the non-chiral integrated-loop momenta representation, a crucial remark is that making use of 'chiral vertex operators' (5.104) that are constructed out of x ± and correspondingly the chiral fixed-loop momenta representation of amplitudes (i.e. working in terms of the right-hand side of (5.100)) vastly simplifies computations while preserving complete generality.
That the fixed-loop momenta generating function chirally factorises in the critical dimension is in line with the Belavin-Knizhnik theorem [64,85] combined with the chiral splitting theorem [64], although the existing proof of chiral splitting had been established explicitly only for generic genus-g massless and exponential external physical vertex operators. Here we have extended this result to all correlation functions of operators inserted on generic compact Riemann surfaces. Notice also that this statement is independent of whether the vertex operator insertions are onshell, and given that correlation functions of generic ghost insertions factorise in the same way as above, where Z g (Z g ) may be replaced by more general superpositions of (anti-)chiral ghost correlators, we have shown that generic offshell amplitudes [73] also respect chiral splitting.
It is worth re-emphasising that (5.100) is truly a remarkable statement, and it is due to this relation that it is justified to use vertex operators that are constructed out of the chiral fields x + (z), x − (z) (and also c z (z),cz(z) and b zz (z),bzz(z)). To compute any string amplitude, for the matter sector we can use either vertex operators constructed out of the full path integral fields, x(z,z), or the (anti-)chiral fields, x + (z), x − (z), and this choice depends on whether we want to extract correlation functions using the left-hand side of (5.100) or the right-hand side respectively. However, the natural representation for vertex operators that arises from the operator-state correspondence is in terms of the (anti-)chiral fields. Notice that we have not appealed to any onshell condition in order to split the field in the path integral x(z,z) into chiral and anti-chiral pieces, x + (z), x − (z). The best way to think of the latter is as fields that arise effectively after properly taking into account all zero mode contributions (and instantons if they are present) associated to the full field x(z,z). 42 The above analysis makes it completely manifest when this is justified and why: fixing the loop momenta (in both compact and non-compact dimensions) is the key to realising these statements. Another point to emphasise is that when vertex operators have winding charges, KK charges and/or polarisations in compact directions we need not expand the fields x + (z), x − (z) that vertex operators are constructed out of around zero mode or classical instanton contributions, and in addition the simple correlators, are exact in the limit z → z and should be used to carry out the operator product expansions that map states to vertex operators -this will be discussed in more detail in [59]. This appears to be somewhat miraculous, but it is nevertheless true (for arbitrary-genus string amplitudes).
These observations are of course direct generalisations of the classic result of D'Hoker and Phong [64], the differences being that here: (a) we consider generic correlation functions (rather than massless asymptotic states) associated to arbitrarily excited string vertex operators (potentially with winding and KK charges and general polarisation tensors and oscillators); (c) we consider generic target spaces R D−1,1 × T Dcr−D (rather than R Dcr−1,1 ), implying that there are also instanton contributions (worldsheets that wrap T Dcr−D ) that are absent in D = D cr and that were hence not made manifest in [64]. The latter were discussed in [83], building on earlier results [84], but the focus there was entirely on exponential insertions, and also there target space moduli were fixed and background gauge fields were absent.
(d) we derived these results directly without using the "reverse engineering" approach, as discussed in the introduction, thus eliminating the potential ambiguity of the type discussed by Sen [67].
Finally, for completeness let us discuss how to extract connected S-matrix elements. Given a set of n external states described by general old covariant quantisation (OCQ) (possibly coherent) vertex operators V zγzγ (with γ = 1, 2, . . . , n), connected (dimensionless) S-matrix elements are extracted from (for n ≥ 2): Here it is implied that vertex operators are inserted at (z γ ,z γ ) in Σ g (or the covering space, Σ g , thereof, see e.g. Fig. 1 on p. 26), and normalised by the leading singularity of the OPE, where an overline denotes taking the Euclidean adjoint [59]. It is conventional to extract out the kinematic factors and define V zz = 1 √ 2k 0 V D−1 O zz , so that invariant amplitudes, M fi (1, . . . , n), defined below, are most naturally written in terms of O zz 's (recall the discussion on p. 7). The quantity δ C f i represents the interaction-free contribution to the connected S-matrix elements, and given that S C f i only contains connected contributions δ C f i should be non-vanishing only for n = 2 asymptotic states, because for n > 2 the interaction-free terms cannot be connected. We have defined the measure: N g being the order of the unfixed global worldsheet diffeomorphisms [127], e.g. at g = 1 this is N 1 = 2, corresponding to the fact that our gauge choice, ds 2 = |dz| 2 , leaves z → −z of SL(2, Z) unfixed (the space of global diffeomorphisms being SL(2, Z)/Z 2 ), see Appendix B for further details where also our genus-one conventions are presented. 43 The full S-matrix elements, S fi , are in turn extracted from products of these and sums where the sum is (according to the cluster decomposition principle) over all distinct par- . . } of f| and over distinct partitions {|i 1 , |i 2 , . . . } of |i , with the "incoming" states, |i γ associated to vertex operators V zγzγ , and the "outgoing" states f γ | associated to Euclidean adjoints, V zγzγ . Our conventions are such that |S fi | 2 is interpreted as a transition probability associated to going from |i to f|, whereas S-matrix unitarity corresponds to the statements: The precise interpretation of the sum over states, h , and also of the delta function, δ fi , requires specifying a basis (and for coherent vertex operators in particular an overcomplete basis) and will be discussed elsewhere [59].
Note that (even when both 'f' and 'i' represent multi-string states) there are generically [128] also vacuum-to-vacuum contributions in this partitioning [129,130], denote these by S C 00 , as well as explicit tadpole contributions, S C 0i (and/or S C f 0 ) if 'i' (and/or 'f ') are single string states, in addition to implicit ones (that arise in various regions of the boundary of moduli space where internal lines are forced to lie on the mass shell) that may already be present in S C f i . Summing over distinct partitions in (5.110) shows that the former exponentiate, so there is [130] an overall factor e S C 00 in S fi , and this is analogous to the exponentiation of the D-instanton amplitude in [129,130], ultimately suggesting a breakdown of the worldsheet in that context.
For instance, generic n = 2-point S-matrix elements are of the form, For n = 2 (and n = 3) there is therefore (up to the overall universal factor e S C 0,0 ) no distinction between the two sets of S-matrix elements, S ii and S C ii , in the absence of tadpoles, S C i0 = 0, as only connected diagrams exist, but for n > 3 there is a distinction. The tadpole contributions, S C 0i , and the vacuum-to-vacuum contribution, e S C 00 , are pathological in the bosonic string (due to the presence of a tachyon in the spectrum and also massless tadpoles) and these will be absent in the superstring (when the vacuum of interest is stable under quantum corrections).
Finally, let us also note that for momentum eigenstates and when G µν = η µν it is conventional to extract out the kinematic factors and a momentum conserving delta function and define an invariant amplitude, M fi (1, . . . , n), as follows, The argument of the delta function, e.g.δ(j) := (2π) D δ D (∫ j µ )δ Dcr−D (∫ ja s),0 , see (3.78), enforces momentum conservation, as well as conservation of any other charges (such as KK and winding charges) that may be present in the external states, but note that for coherent vertex operators there will be a sum over such delta function contributions. Factorisation, normalisation and unitarity of string amplitudes (with coherent vertex operator insertions) and related concepts will be discussed in [60] where we focus on n = 2.
The n external states are assumed to have well-defined energy expectation values 45 denoted by k 0 γ , for γ = 1, . . . , n, and V D−1 := (2π) D−1 δ D−1 (0) denotes the formal (infinite) spatial volume of R D−1 . Another point to emphasise is that (as mentioned above) the formal volume V D−1 will always cancel out and does not appear in the observables of interest (cross sections, decay rates, etc.), just as in field theory [123]. Finally, generically there will be additional delta-function (or Kronecker-delta) constraints (implicit in M fi (1, . . . , n)) in addition to that appearing explicitly in (5.113), associated to the fact that the full invariant amplitude also contains disconnected pieces, i.e. if n > 3, depending on context, as exhibited in (5.110).

Discussion
We have constructed a generating function (and associated correlation functions) for string amplitudes in generic constant string backgrounds, G M N , B M N , Φ and U , on R D−1,1 × T Dcr−D , so that also all Kähler and complex structure moduli (of the target space torus, T Dcr−D ) contained in G ab , B ab , background KK gauge fields, A a µ and B µa , spacetime torsion, B µν and also spacetime metric, G µν , are allowed to be turned on. In the process, we have derived the chiral splitting theorem of D'Hoker and Phong [64] for string amplitudes, which we have generalised to the aforementioned background and with arbitrarily excited string vertex operator insertions (with generic KK and winding charges, as well as polarisation tensors associated to generic oscillators and spacetime indices 46 ).
Our approach differs from that of D'Hoker and Phong [64] (and also Sen [67]), in that we did not make use of the "reverse engineering" approach (where the target spacetime embedding fields, x M (z,z), are first integrated out and only at a later stage of the computation is it noted that the result can be written as an integral whose integration variables get interpreted as A I -cycle loop momenta). As pointed out in a recent paper by Sen [67], such a reverse engineering approach could potentially lead to ambiguities (because the same integrated-loop momenta amplitude can be written in more than one way as an integral over loop momenta [67]). Sen went on to explain that these ambiguities will not be visible in the final amplitudes after integrating out the loop momenta (while adopting an appropriate analytic continuation for the loop momentum integral contours [65]), and that these ambiguities are therefore immaterial. However, as discussed in the Introduction above, it is sometimes desirable to not integrate out the loop momenta, and that this is also of interest for the computation of some physical observables, such as the spectrum of massless radiation associated to a decaying string. Therefore, the reverse engineering method could potentially lead to ambiguous results for observables. In our approach we have resolved this potential ambiguity, in that we introduced loop momenta associated to A I -cycle strings from the outset (by explicit momentum conserving delta function insertions into the original path integral where there is no room for this ambiguity), and have thus shown that the result of D'Hoker and Phong (that one can replace the target space fields, x M (z,z), by a set of effective chiral fields, x M + (z), x M − (z) for the left-and right-moving degrees of freedom, appropriately modified so as to apply to generic backgrounds, R D−1,1 × T Dcr−D , and vertex operators) is fully justified and leads to the correct un-ambiguous result for the fixed-loop momentum amplitudes. 47 Let us now zoom in on the statement (5.100). Here it is crucial to note that the left-hand side denotes the usual path integral over matter, x M (z,z), and ghost fields, b, c, whereas on the right-hand side the matter and ghost fields have been integrated out, and the result has been written in terms of Wick contractions of effective (anti-)chiral fields, x M + (z), (x − (z)), whose correlation functions are determined from the chiral propagators (5.101), with the results given in (5.102) and (5.103). What we want to emphasise here is that on the lefthand side of (5.100) the target space embedding field appearing in vertex operators and the worldsheet action contains (generically) zero modes, instanton (or soliton) contributions, as well as quantum fluctuations, whereas the chiral fields on the right hand side are defined by their correlation functions, so that x M ± do not contain information about zero modes or instanton contributions. The latter have nevertheless been fully taken into account and appear in the overall delta function and loop momenta respectively. Therefore, using the chiral representation of amplitudes significantly simplifies amplitude computations.
Finally, we have also discussed how wave/particle (or rather wave/string) duality is manifested in string theory, and we have shown that the fixed-loop momenta representation can be thought of as the 'wave picture', the integrated loop momenta expression yielding the 'string picture'. There are also hybrid formulations (or Routhians) whereby the compact and non-compact dimensions are in the wave or string picture, leading to four natural possibilities in total. In a forthcoming article [61] we will show that adopting a wave picture leads to significant simplifications and explicit analytic results (a string picture being much less tractable analytically).
The objective here has been to provide a working and efficient handle on computing string amplitudes involving HES vertex operators. In [59] we construct chiral HES coherent vertex operators (which is a very natural basis for excited strings) and discuss the notion of Euclidean adjoint vertex operators (which refines the rule of thumb of Polchinski [86,99], a refinement that is necessary in order for all vertex operators to have positive norm 48 ). These vertex operators are then [60] used to derive a generic expression for two-point amplitudes (where we keep the genus of the worldsheet generic in order to study generic properties), 47 Note that there are still expected to be field-redefinition ambiguities that one expects from insight from string field theory, see Sec. 4 in [67]. The authors thank Ashoke Sen for an extensive discussion of this point. 48 The rule of thumb [86,99] that to obtain the Euclidean adjoint of a vertex operator one is to conjugate all explicit factors of 'i = √ −1' is not sufficient when vertex operators have winding N −N ∈ 2Z + 1, in that there are some additional phases (here N,N are level numbers). whose imaginary part at one loop [61] yields decay rates and power emitted into massless and massive radiation (including radiative backreaction and in particular α corrections), the real part giving mass shifts (relevant for black hole physics [32]). In [62] we discuss decay rates associated to gravitational radiation in particular and in [63] we make the connection to low energy effective field theory.
where the components of Riemann curvature tensor in terms of the Christofel symbol read, and in the above coordinate system the only non-vanishing Christofel symbols are Γ z zz = ∂ z ln g zz and Γzzz = ∂z ln g zz , R z zzz = −∂zΓ z zz = −∂z∂ z ln g zz , (A. 115) so that: A tensor V of conformal weight (h,h) is of the form: so that K (h,h) is the space of tensors of weight (h,h) and spin h −h = 1 2 Z. The components of V are sometimes referred to as conformal primary operators. Examples used in the main text are: Define K (n,0) ≡ K n (and K (0,n) ≡K n ). Using the metric g zz to raise and lower indices there is an isomorphism (n − m, 0) ∼ (n, m) ∼ (0, m − n), and one may therefore express all tensors in terms of holomorphic indices, e.g. we write, g zz Vz = V z , with g zz g zz = 1.
Covariant derivatives satisfy 49 ∇ (n) It is straightforward to show, using the explicit expression for the Christoffel symbols above (A.115), that (A.118) is equivalent to: In addition, there is the Cauchy-Riemann operator ∂z; formally ∇ n z : K n → K n,1 , According to the above identification we could also have written the Cauchy-Riemann operator as ∇ z (n) : K n → K n−1 , We now move on to discuss certain global topological aspects of Riemann surfaces. A key relation is the Atiyah-Singer-Riemann-Roch index theorem: dim C ker ∇ (n) z − dim C ker ∇ z (n+1) = 1 2 (2n + 1)χ(Σ g ), (A.124) and this relates the number of zero modes of tensors in K n , tensors in K n+1 , (for n ∈ 1 2 Z) and the Euler characteristic, χ(Σ g ), of the Riemann surface. For compact Riemann surfaces the latter reads: χ(Σ g ) = 1 2π Σg d 2 zR zz = 2 − 2g. (A.125) Following D'Hoker and Phong [89] (see also [133]), we parametrise the genus-g compact The space H g = {Ω ∈ C g | Ω IJ = Ω JI , ImΩ > 0} is the Siegel upper half space. Fixing the loop momenta in amplitudes breaks manifest modular invariance, but of course integrating out the loop momenta restores it. In order to keep track of this, let us briefly mention how modular transformations act on the various ingredients that appear in amplitudes [84]. The first of these follows from requiring that A I ω J = δ IJ remains invariant, whereas the second follows from the first and the definition B I ω J ≡ Ω IJ , but see also [133,134]. Note that Ω IJ is also an element of H g when Ω IJ is.
Period matrices related as in (A.132) refer to the same Riemann surface, but in fact restricting to the quotient H g /Sp(2g, Z) is still a redundant description of the moduli space, F g , which is contained in H g /Sp(2g, Z) in a rather complicated manner for generic genus g surfaces, see e.g. [135,136] for a detailed discussion and [137] for a broader overview, and also [89,133] for discussions with a physics-motivated approach. A detailed discussion of the moduli space would take us far afield, but it is useful to always keep in mind the physical picture whereby different points in F g correspond to distinct deformations of the Riemann surface (i.e. that cannot be undone by using a symmetry transformation, namely global and local diffeomorphisms and Weyl transformations of Σ g ), whereas the boundary of moduli space (upon compactification, F g →F g ) can be identified with the set of degenerations whereby one or more isotopically distinct cycles in Σ g (with cycles encircling vertex operator insertions considered non-trivial) are shrunk to points.
Given any base point ℘ 0 we may associate to every point ℘ on Σ a complex g-component vector z by the Jacobi map (referred to also as the Abel map): This vector is unique up to periods (A.127), (A.128). We associate to Ω a lattice L Ω ⊂ C g , such that L Ω ≡ Z g + ΩZ g . The vector z is an element of the complex torus J(Σ), also known as the Jacobian variety of Σ, In terms of the Riemann theta function, ϑ[ a b ] (z, Ω) = exp 2πi and correspond to a spin bundle associated to [ a which, up to terms of the form f (z,z) + g(w,w) which do not contribute to amplitudes (in spacetimes for which charge and momentum is conserved), is determined uniquely by the requirement that it be single-valued around A I and B I cycles, and that it have the correct singular behaviour as z → w, G(z, w) − ln |z − w| 2 + . . . . The prime form is quasi-periodic on Σ g , see (A.143).
B The Torus, T 2
(We are working with the critical string where the non-chiral Liouville action [84] is absent.) This is the standard result for the ghost contribution at genus g = 1, but in the main text we are rather interested in the quantity Z 1 , defined in (3.73), and so from the above, on account of det ImΩ IJ g=1 = τ 2 and Σ 1 d 2 z √ g = τ 2 , it follows immediately that we can also write (B.152) as follows,

B.4 Jacobi Theta Functions
The Jacobi theta function is defined as [88]: A useful quantity that appears in the definition of the prime form, E(z, z ), is ϑ 1 (0|τ ) ≡ ∂ z ϑ 1 (z|τ )| z=0 , an explicit expression for which follows directly from (B.158c): The quantity ϑ 1 (z|τ ) is odd under parity, ϑ 1 (z|τ ) = −ϑ 1 (−z|τ ), and hence ϑ 1 (0|τ ) = 0. In fact, the zeros of ϑ 1 (z|τ ) are located at: It is clear that E(z) has a simple zero at z = 0. In fact, for generic v it follows immediately from (B.162a) that: 2πiE(z) z→0 2πiz + 1 24 − n>0 v n 1 − 2v n + v 2n (2πiz) 3 + O(z 5 ), and, in fact, the prime form is the unique holomorphic object on a Riemann surface that has a simple zero at z = 0 and is non-vanishing elsewhere (modulo lattice periodicities, see below). E(z) therefore generalises the notion of distance on topologically non-trivial Riemann surfaces. In addition, the prime form has the following monodromies, where in the second line we have exhibited another combination that appears in string amplitudes. It is convenient to consider these expressions as a series expansion in v, which is useful in discussing the τ 2 → ∞ boundary of moduli space (with σ 1 , σ 2 generic). Defining S(u) ≡ u 1/2 − u −1/2 , C(u) ≡ u 1/2 + u −1/2 ,