Correlators of arbitrary untwisted operators and excited twist operators for N branes at angles

We compute the generic correlator with L untwisted operators and N (excited) twist fields for branes at angles on T^2 and show that it is given by a generalization of the Wick theorem. We give also the recipe to compute efficiently the generic OPE between an untwisted operator and an excited twisted state.


Introduction and conclusions
Since their introduction, D-branes have been very important in the formal development of string theory as well as in attempts to apply string theory to particle phenomenology and cosmology. However, the requirement of chirality in any physically realistic model leads to a somewhat restricted number of possible D-brane set-ups. An important class is intersecting brane models where chiral fermions can arise at the intersection of two branes at angles. An important issue for these models is the computation of Yukawa couplings and flavour changing neutral currents.
Besides the previous computations many other computations often involve correlators of twist fields and excited twist fields. It is therefore important and interesting in its own to be able to compute these correlators also because it is annoying to be able to compute, at least theoretically, all possible correlators involving all kinds of excited spin fields while not being able to do so with twist fields. As known in the literature [1] and explicitly shown in [2] for the case of magnetized branes these computations boil down to the knowledge of the Green function in presence of twist fields and of the correlators of the plain twist fields. In many previous papers correlators with excited twisted fields have been computed on a case by case basis without a clear global picture, see for example ([3], [4], [5]) In this technical paper we have analyzed the N excited twist fields amplitudes with L boundary vertices at tree level for open strings localized at D-branes intersections on R 2 (or T 2 ) using the classical path integral approach ( [1], [6]) which is more efficient than the also classical sewing approach ( [7], [8]). This approach has been explored in many papers in the branes at angles setup as well as the T dual magnetic branes setup see for example ([9], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22]). We will nevertheless follow a slightly different approach, the so called Reggeon vertex [23], which allows to compute the generating function of all correlators, in particular we will use the formulation put forward in [24]. This paper is organized as follows. In section 2 we review the geometrical framework of branes at angles and fix our conventions. In the same section we discuss carefully how to make use of the doubling trick in presence of multiple cuts and the existence of local and global constraints. In section 3 we compute the OPE of chiral and boundary vertex operators with an arbitrary excited twist operator by relying on the operator to state correspondence. We propose also a better notation for excited twist operators than that usually used which requires a new symbol for any excited twist operator. In this same section and for use in the fourth we establish also which chiral operators are best suited to obtain excited twist operators in the easiest way. Finally in section 4 we compute the generating function of correlators of N excited twist operators with L boundary operators. We do this in steps by first computing the interaction of boundary and chiral vertices with twist field operators and then computing the desired correlators by letting appropriate combinations of chiral vertex operators collide with twist fields. Our main result is the generating function given in eq. (108) which shows that all correlators can be computed once the N plain twist operators correlator together with the Green function in presence of these N twists are known. This expression remains nevertheless quite formal since it requires the precise knowledge of the Green function 1 and its regularized versions. Therefore somewhat explicit expressions of these quantities are given in appendix E (see also appendixes C and D) and completely explicit expressions for all involved quantities in the N = 3 case are given in appendix F. From these expressions it is clear that the computation of amplitudes, i.e. moduli integrated correlators, with (untwisted) states carrying momenta are very unwieldy because Green functions can at best be expressed as sum of product of type D Lauricella functions. This should however not be a complete surprise since in [25] it was shown that twist fields correlators in orbifold setup are connected to loop amplitudes which, up to now, have not been expressed in term of simpler functions.

Review of branes at angles
The Euclidean action for a string configuration is given by where u ∈ H, the upper half plane, d 2 u = e 2τ E dτ E dσ = du dū 2i and I = 1, 2 or z,z so that X = X z = 1 √ 2 (X 1 + iX 2 ),X = Xz = X * . The complex string coordinate is a map from the upper half plane to a closed polygon Σ in C, i.e. X : H → Σ ⊂ C. For example in fig. 1 we have pictured the interaction of N = 4 branes at angles D t with t = 1, . . . N . The interaction between brane D t and D t+1 is in f t ∈ C. We use the rule that index t is defined modulo N . As shown in [2] given the number of twist fields N there are N − 2 different where 0 < t < 1 (we define also¯ t = 1 − t for simplifying the expressions) are the twists defined as in eq. (8) and correspond to the angles measured from brane D t to brane D t+1 when they are labeled clockwise as in figure  3. It is also possible to have the very same geometrical configuration where branes are simply labeled counterclockwise as shown in figure 4 for which we have M ccw = N − M when we still measure angles from brane D t to brane D t+1 . To understand the meaning we notice that if we relabel the branes as in the clockwise case, i.e. we consider the branes labeled as in figure 3 but with the angles measured as in figure 4 and we use the conventions we use for the local description, described in the next section, the interactions on worldsheet would take place in the reversed world sheet time order or in the same order but on the other boundary as shown in figure 5. Since the physics must be connected and actually the two correlators are connected by complex conjugation and exchange ↔ 1− we have chosen to measure them clockwise. Figure 3: A polygon Σ with an reflex angle and branes labeled clockwise with N = 4 and M = 1. Figure 4: A polygon Σ with an reflex angle and branes labeled counterclockwise with N = 4 and M ccw = 3.

The local description
Locally at the interaction point f t ∈ ∂Σ ⊂ C the boundary conditions for the brane D t are given by where g t ∈ R is the distance from the line parallel to the brane going through the origin. Similarly the boundary conditions for the brane D t+1 is given by The interaction point is then When we write the Minkowskian string expansion as X(σ, τ ) = X L (τ + σ) + X R (τ − σ) the previous boundary conditions imply (and not become since they are not completely equivalent because of zero modes) X L loc (ξ) = e i2παt X R loc (ξ), X L loc (ξ + π) = e i2πα t+1 X R loc (ξ − π) (6) or in a more useful way in order to explicitly compute the mode expansion where we have defined so that 0 < t < 1 and there is no ambiguity in the phase e i2π t entering the boundary conditions. The quantity π t is the angle between the two branes D t and D t+1 measured counterclockwise as shown in fig. 6. A consequence of this definition is that becomes 1 − when we flip the order of two branes. For example the angles in fig. 4 become those in fig. 3 when we reverse the order we count the branes, i.e. when we follow the boundary clockwise instead of counterclockwise.
We introduce as usual the Euclidean fields X loc (u loc ,ū loc ),X loc (u loc ,ū loc ) by a worldsheet Wick rotation in such a way they are defined on the upper half plane by u loc = e τ E +iσ ∈ H. The previous choice of having brane D t at σ = 0 (3) and brane D t+1 at σ = π (4) implies that in the local description where the interaction point is at x = 0 D t is mapped into x > 0 and D t+1 into x < 0. The boundary conditions (6) can then immediately be written as and similarly relations forX which can be obtained by complex conjugation. When we add to the previous conditions the further constraints we obtain a system of conditions which are equivalent to the original ones (3,4). In order to express the boundary conditions (7) in the Euclidean formulation it is better to introduce the local fields defined on the whole complex plane by the doubling trick as ∂X loc (z loc ) = ∂ u X loc (u loc ) z loc = u loc with Im z loc > 0 or z loc ∈ R + e i2παt∂ūX loc (ū loc ) z loc =ū loc with Im z loc < 0 or z loc ∈ R + ∂X loc (z loc ) = ∂ uXloc (u loc ) z loc = u loc with Im z loc > 0 or z loc ∈ R + e −i2παt∂ū X loc (ū loc ) z loc =ū loc with Im z loc < 0 or z loc ∈ R + (11) In this way we can write eq.s (7) as Notice that the two Minkowskian boundary conditions in eq.s (7) can be mapped one into the other by complex conjugation while the corresponding ones in the Euclidean version given in eq.s (12) are independent and each is mapped into itself by complex conjugation therefore the Euclidean classical solutions for X loc andX loc are independent. The quantization of the string with given boundary conditions yields with¯ t = 1 − t and we can interpret f t as the classical solution since it is the only solution to the equations of motion with finite euclidean action. We find also the non trivial commutation relations (n, m ≥ 0) [α (t)n+ t , α † (t)m+ t ] = (n + t )δ m,n , [ᾱ (t)n+¯ t ,ᾱ † (t)m+¯ t ] = (n +¯ t )δ m,n (15) and the vacuum is defined in the usual way by At first sight it may seem that the vacuum encodes the t information only but as we show in eq. (52) it contains also information about f t and α t and α t+1 which can be extracted from the proper OPEs.

Global description
In the local description, where the interaction point is at x loc = 0, D t is mapped into x loc > 0 and D t+1 into x loc < 0 this means that in the global description the world sheet interaction points are mapped on the boundary of the upper half plane so that x t+1 < x t . The global equivalent of the local boundary conditions given in eq.s (9) which is useful for the path integral formulation where we cannot use the equations of motion is To the previous constraints one must also add the global equivalent to eq.
in order to get a system of boundary conditions equivalent to the original ones. When using an operatorial approach the global equivalent of eq.s (9) become If we introduce the global fields defined on the whole complex plane by the doubling trick as the local boundary conditions (7) can be written in the global formulation as Finally it is worth noticing the behavior of the previously introduced fields under complex conjugation when z is restricted to where ∂X (z →z) means that the holomorphic ∂X (z) is evaluated atz. The previous expressions also show that it is not necessary to introduce the antiholomorphic fields∂X (z) and∂X (z) which it is possible to construct applying the doubling trick on∂ūX(ū) and∂ūX(ū) respectively.

Twisted Fock Space and OPEs
Given the vacuum |T 2 defined as usual in eq. (16) and the expansions (13) we can immediately write a normalized basis element of the Fock space as We want now to explore the state to operator correspondence. To the vacuum |T we associate the twist field σ ,f (x) which depends both on the twist and on the position f ∈ C so that with normalization T |T = 1 (25) For the other states in Fock space it is however better to introduce the non normalized states √ 2α e −iπαt to which we make correspond the (generically non primary) chiral operators This notation can be partially misleading since, for example, it is not true that (see eq. (37) for the right OPE) 3 , 2 In this section we have dropped the dependence on t as much as possible to simplify the notation e.g. t → .
3 Notice that on shell ∂ 2 u X(u,ū) = 1 2 ∂ 2 u X L (u) and therefore the operator (27) is just a way to write the operator which corresponds to the state (26) under the operator to state correspondence. The advantage of this notation is that it clearly shows which state corresponds to which operator and that it is consistent with the usual untwisted state to operator correspondence for which We have in particular, as we soon show, that so that the excited twist operators (∂ u Xσ ,f )(x) and (∂ uX σ ,f )(x) are what usually written as τ (x) andτ (x). This observation also hints to another reason why to use the notation (27) which is to avoid the proliferation of new symbols, one for each excited twist operator.
Finally notice that almost all boundary operators can be recovered from chiral ones, for example when x t < x < x t−1 and with the help of the boundary conditions (17) we get where both ∂ u X(u,ū)| u=x and ∂ uX (u,ū)| u=x are chiral operators computed on the boundary. A notable exception is however e ikX(x,x)+ikX(x,x) which is intrinsically non chiral since it cannot be expressed off shell using chiral operators. In particular from the previous eq. ( 31) it follows that boundary operators have more complex OPEs, such as

Chiral OPEs from local Fock space
Let us now see how to compute OPEs of a chiral operator with an excited twist field using the local operatorial formalism. We will make several examples to make clear the way to proceed. We start considering the simplest example, i.e. the OPE ∂ u X(u,ū)σ ,f (x).
In order to find this OPE we compute 4 lim x→0 ∂ u X(u,ū)σ ,f (x)|0 SL(2) = ∂ u loc X loc (u loc )|T (33) where ∂ u X loc (u loc ) is the operator in the twisted Fock space which corresponds to the abstract operator ∂ u X(u,ū) and we can identify u loc = u − x. Now using the explicit expansion (13) we get from which we deduce not only the leading order of the OPE (30) but also the higher order terms As a second example we consider the OPE ∂ 2 u X(u,ū)σ ,f (x). Proceeding as before and using the fact that the local Fock space operator which corresponds to ∂ 2 u X(u,ū) is ∂ 2 u loc X loc (u loc ) we can find so that we can deduce the OPE which shows clearly what stated before about the wrongness of eq. (28). Obviously eq. (37) is compatible with eq. (36) since the former can be obtained from the latter by taking the derivative ∂ u .
In the previous examples the local Fock space operator has the same functional form of the abstract one but as discussed in [26] for the T dual configuration of branes with magnetic field this is not always the case. The correct mapping, which is rederived in section 3.3, is given by where the chiral generating function S c (c,c) is given by c n ∂ n u loc X loc (u loc ,ū loc ) + c n ∂ n u locX loc (u loc ,ū loc ) : where : · · · : is the normal ordering and is the regularized Green function with the twist σ ,f at x loc = 0 and the antitwist σ¯ ,f in x loc = ∞ and G zz U (t) (u loc ,ū loc , v loc ,v loc ) is the Green function for the untwisted string with boundary conditions corresponding to D t (see appendix A for more details) 5 . It is worth discussing how ∆ c has to be interpreted either as a kind of generating function or as a difference of Green functions, one of which associated to a couple of twist fields. This point of view is important in order to avoid confusions which could arise when considering the role of ∆ c in correlators with many twist fields. The answer is given in the derivation of eq. (80) which shows that ∆ c is a difference of Green functions.
As an example of the consequences of the previous expression (39) we can compute the local operator which corresponds to the energy-momentum tensor T (u) = − 2 α ∂ u X∂ uX . We find the local operator Then we can compute the OPE T (u)σ ,f (x) from 5 For the chiral correlators there is no difference in using the untwisted Green function for D t or D t+1 since the G zz U (t) is the only piece of the untwisted Green function G IJ U which contributes and it is insensitive to the angle at which the brane is rotated, this is not anymore true for the boundary correlators as we discuss later. from which we read both the conformal dimension of σ ,f and that k k¯ ∂ x σ ,f = ∂ u X∂ uX σ ,f (x). It is noteworthy that the double pole comes from the extra piece ∂∂∆ c in eq. (41) which would not be present in a naive state to operator correspondence.
As a last example to make clear the algorithm we consider the more at the leading order. First we compute the operatorial realization of (∂ 2 u X ∂ u X ∂ uX )(u) to be then we associate the state k¯ ᾱ † +2 |T to the excited twist ∂ 2 uX σ ,f (x) then an easy computation gives where the leading order contribution comes from the terms linear in X loc .

Boundary OPEs from local Fock space
In the computation of the interaction of twisted states with untwisted ones quite often we are not interested in chiral operators but in boundary operators such as e ikX(x,x)+ikX(x,x) . For this case we must extend the analysis given in the previous section. The correct mapping is then given by where the generating function S(c,c) is given by where : · · · : is the normal ordering and we defined the Green function 6 with the twist σ ,f at x loc = 0 and the anti-twist σ¯ ,f in x loc = ∞ and G IJ U (t) (u loc ,ū loc , v loc ,v loc ) is the Green function of the untwisted string with both ends on D t and the need to distinguish x > 0 from x < 0 is due to the different boundary conditions of the twisted string in this ranges As shown in eq. (126) in appendix A all ∆ IJ bou are equal to a real symmetric function ∆ bou (x 1 , x 2 ) up to phases which combine to allow the previous generating function to be written as When x loc < 0 we get a very similar expression but with the substitution c Dt → c D t+1 since in this case the vertex is on the brane D t+1 . Given the previous results we can now compute the operatorial realization of the the vertex e ik·X(x,x) to be, similarly to the results ( [26], [27]) being the digamma function. Notice that we have not required the momentum k to be tangent to the brane since the normal part gives simply a phase due to the boundary conditions nevertheless, as commented before, the explicit expression for ∆ IJ implies that only the momentum parallel to the D t brane k Dt = e −iπα t k+e iπα tk 6 We normalize the Green function such that ∂∂G(u loc ,ū loc , v loc ,v loc ) = −α 2 δ 2 (u loc − v loc ).
Using the previous operatorial realization we can then obtain the OPEs and similarly for the x loc < 0 case which corresponds to The previous OPE justify writing not only σ t (x t ) but σ t,ft (x t ) since the f t andf t can be computed using the phases of the leading order terms with different momenta. Moreover the phases e −iπαt and e −iπα t+1 can be extracted from the projections k Dt , k D t+1 of momentum k.
In a similar way we can compute the operator associated to ∂ It is worth stressing that the term proportional to is fundamental in getting the right interaction among a gluon and twisted matter ( [28], [26], [27]) when the amplitude is computed in operatorial formalism.

Short derivation of the generating function S
We will now quickly review the motivations to write down the generating vertex (47) and why it works.
Our aim is to get some hints on how to regularize the contact divergences that appear in the path integral computation of amplitudes in presence of twist fields. We consider the boundary case since all the others can be treated in an analogous manner. In the untwisted case the operatorial generating function is simply the Sciuto-Della Selva-Saito vertex [23] Dropping the loc indication we can rewrite it as where we have introduced the current J I (x a ) which must be set to J I (x a ) = ∞ n=0 c In ∂ n x δ(x a − x) when we want to reproduce the original vertex but can also be taken more general as we do in the following.
We can now compute the OPE of two such generating functions (vertices) which is valid for generic currents as long as they have compact support and the points in the support of J 1 have bigger absolute values than those in the support of J 2 so that the operatorial product is radial ordered. Now the non can be roughly understood as the generating function for the OPE coefficients and we want to check that it is reproduced when using the generating function (47) which is defined in the twisted Fock space. In particular we consider the generating function with a more general current given by It is then immediate to compute the product where we have added and subtracted to the exponent in order to complete the prefactor in eq. (57) in the case where J = J 1 + J 2 and used the symmetry ∆ IJ bou (x a , x b ) = ∆ JI bou (x b , x a ). Having verified that the generating function (47) gives vertices with the same OPEs as the untwisted ones we can exam the reason which leads to it ( [25], [29], [26]). In operatorial formalism the normal ordered vertex (55) for the untwisted case can be obtained from a regularized non normal ordered generating function by a multiplicative renormalization as where the regularized generating function is defined by a point splitting as without normal ordering and the multiplicative renormalization is given by the inverse of the factor we get by normal ordering the regularized generating function Now the generating function for the twisted string is defined in an analogous way by regularizing the non normal ordered generating function and then renormalizing in a minimal way using the same renormalization factor as in the untwisted case (61), explicitly

Getting excited twists
We are interested in excited twist states hence we would now write a kind of SDS vertex which generates these states (26). The main observation is then that therefore a normal ordered products of these operators gives directly an excited twist state, e.g.
The generating function for products of these operators is obviously Comparing with eq. (39) we realize that there is not the exponent quadratic in d, this means that the abstract operator corresponding to e.g. eq. (65) is in fact computing its OPE with the twist field σ ,f (x) as in section 3.1 we get Notice that this OPE explains why we can use u − x as argument of ∆ zz and not another function which behaves as We conclude therefore that the generating function for the abstract operators which give excited twists is given by The path integral approach The classic method [1] to compute twists correlators is by the path integral is the space of string configurations satisfying the boundary conditions (19) and (18). Since the integral is quadratic we can then efficiently separate the classical fields from the quantum fluctuations as where X cl satisfies the previous boundary conditions (17,18) while X q satisfies the same boundary conditions but with all f t = 0. After this splitting we obtain where the factor N (x t , t ) is the quantum contribution. In particular when all f t are equal, i.e. f t = f the classical action S E,cl (x t , t , f t = f ) is zero since the branes are the boundary of a zero area polygon therefore we can identify which is also true for the quantum fluctuation for which f = 0. Our aim is now to compute correlators with both (excited) twist field operators and untwisted operators V ξ i (x i ) (i = 1 . . . L) associated with the untwisted state ξ i of which the ones in eq. (30) are a particular case. We start with correlators with plain twist field operators which can be computed as To do so we notice that it is by far easier not to compute the previous correlator but to compute the generating function of all the correlators, i.e. the Reggeon vertex, in the form of the previous path integral (72) plus linear sources since all untwisted operators can be obtained by taking derivatives with respect to the coefficients c (i)nI as explained in the previous section. This starting point is very similar to ([30], [24]) where it was recognized that the generator for all closed (super)string amplitudes is a quadratic path integrals. The idea in the previous papers is that the appropriate boundary condition for R and/or NS sector can be obtained simply by inserting linear sources with the desired boundary conditions. Because of this assumption the quantum fluctuations are the same for all the amplitudes: from the purely NS to the mixed ones. It was later realized that this prescription misses a proper treatment of quantum fluctuations [31] and that when this part is considered the amplitudes factorize correctly [8].

Boundary correlators with non excited twists on R 2
Our strategy is therefore to compute the path integral (77) by properly defining it in order to regularize the divergences which arise as usual because of current self interactions. Taking inspiration from what is done for deriving the generating function we first regularize the δ functions in the currents, a step which corresponds to the point splitting in operatorial formalism and then subtract the self interaction of the untwisted string. We are therefore led to consider the path integral where the regularized currents are defined as is the untwisted Green function with boundary conditions which depends on the brane on which the point x i is. This dependence is the reason why we have written U (t i ).
It is then immediate to compute the previous path integral by using the splitting of X into quantum and classical part (73) and get where we have introduced the boundary Green function in presence of N twist fields σ t,ft=0 (x t ) (with f t = 0 since these terms come from the quantum fluctuations) in the sector M = N t=1 t and the regularized Green function where c n Dt i are defined as in eq. (50) and t i is chosen by the brane on which

Some consequences
There are now two immediate consequences. The first and more trivial is that all correlators with just one derivative vertex have contribution only from the classical solution X cl [5], i.e The second and more interesting is that all correlators can be essentially computed with Wick theorem plus classical contributions plus self interactions which are absent in Wick theorem. This implies that a string in presence of not excited twist fields (defects) is free and can be quantized almost in the usual manner.

Some examples
As a first example we want to compute the correlator of a tachyon with N twist fields which shows that untwisted matter sees a kind of form factor of the interacting twisted matter in accord with the result from OPE and what discussed in [26] for the case of a stringy instanton. Again a priori we can take k I not parallel to the brane D t but the explicit form of the Green function implies a projection in the direction parallel to D t . It is interesting to compare the previous result with what we can get using the OPE (52) in the previous correlator. Using the OPE at the leading order we get Now using the obvious behavior and the result shown in app. D in the exact expression we find the expected consistency with the OPE result. Secondly let us consider the correlator of a gauge boson with twisted matter which exhibits the same structure as the vertex (53) and the OPE it can be computed using it. Next we consider the interaction of two tachyons with the twisted matter in order to show the Wick-like expression in a simple case where G (N,M ),bou (x 1 , x 2 ) is the common factor of all the components of the boundary Green function G IJ (N,M ),bou (x 1 , x 2 ) and is given in eq. (146) whose explicit expression (145) implies that only the momenta parallel to the brane on which the vertex lies contributes.
Finally, we consider a more lengthy example

Boundary correlators with non excited twists on T 2
The wrapping contributions have been studied for the pure twist field correlators in [32] for the N=3 case and in [14] for the case M = N − 2 and there is not any difference among the different M values therefore the results obtained there are valid. Let us anyhow quickly review them. Given a minimal Npolygon in T 2 with vertices {f t }, i.e. with all vertices in the fundamental cell, we can consider non minimal polygons which wrap the T 2 . These can be easierly described as polygons which have vertices {f t } in the covering R 2 where T 2 ≡ R 2 /Λ with the lattice defined as Λ = {n 1 e 1 + n 2 e 2 |n 1 , n 2 ∈ Z}. These configurations give an additive contribution to the classical path integral as In order to determine the possible vertices {f t } without redundancy it is necessary to keep a vertex fixed and then expand the polygon. For definiteness we keep fixed the vertexf 1 = f 1 which lies at the intersection between D N and D 1 . We then move the next vertex f 2 along the D 1 brane. Explicitly we writẽ f 2 =f 1 + (f 2 − f 1 ) + n 1 t 1 = f 2 + n 1 t 1 with n 1 ∈ Z and t 1 the shortest tangent vector to D 1 which is in Λ. We can now continue for all the other vertices for which we havef For consistency we need requiringf N +1 ≡f 1 = f 1 , therefore the possible wrapped polygons are obtained from the solution of the Diophantine equation which cannot be solved in general terms but only on a case by case basis as discussed in [14].

Chiral correlators with non excited twists.
As a warming up for the computation of excited twist fields which we perform in the next section we consider the interaction of chiral vertices with plain twists.
We can now follow the same strategy we used in section 4.1 and compute the path integral with the insertion of an arbitrary number L c of currents which act as generating functions for the chiral vertex operators. As done for the boundary correlators in section 4.1 we first regularize the currents and then subtract the self interaction of the untwisted string. We are therefore led to consider the path integral where in the second line we have written the regularization factor analogous to the one used in section 4.1 which regularizes the currents in the third line.
In the previous expression the regularized currents are defined as with δ 2 (u − u c ; η c ) a regularization of the δ 2 () such that lim ηc→0 δ 2 (u; η c ) = δ 2 (u). Notice that we need using a δ 2 () in the previous expression since we use "directional" derivatives along u. To subtract the self interaction of the untwisted string we used G IJ U (tc) (u a ,ū a ; u b ,ū b ), the untwisted Green function computed for an arbitrary t c since we chose to consider currents with at least one derivative ∂ u so that only G zz U give with a non vanishing contribution which is also independent on the brane D t .
Finally performing the path integral we get where we have written the dependence on the complex conjugate variables such asū even if the derivatives are independent in order to be consistent with the notation used in the boundary case. The regularized chiral Green function is defined as expected as and we notice that the subtraction term is different from zero only when IJ = zz or IJ =zz because of the derivatives.

Correlators of excited twists on R 2
Finally we can compute the correlators of excited twist fields by letting the appropriate chiral currents collide with the twist fields. We follow the same strategy we used in section 4.1 and in the previous section 4.3 and compute the path integral with the insertion of one generating function (70) for each twist field. As done for the boundary correlators in section 4.1 we first regularize the δ 2 () functions in the currents and then subtract the self interaction of the untwisted string. We are therefore led to consider the path integral where in the second line we have written the regularization factor analogous to the one used in section 4.1 which regularizes the third line and finally in the last line we have the necessary subtraction term which is in eq. (70). In the previous expression the regularized currents are defined as with δ 2 (u − u t ; η t ) a regularization of the δ 2 () as in previous section. For writing a more compact expression we have also used tI defined as Finally performing the path integral we get where we have defined the regularized Green function at the twist fields t to be and we used G IJ to write the last line and again we have written the dependence onū andv even if the derivatives are independent on them for having a consistent notation. Actually because of the chiral derivatives the previous expression simplifies in two cases to It is interesting to notice that regularized Green functions are got by subtracting the divergent part with the proper monodromy at the point of regularization which at the points where a twist field is located means G N =2 while in all other points means G U . In particular both M ) are analytic functions at u = x t whose explicit expression is given in appendix E.
More explicitly the previous generating function can be written as

Some examples
Let us consider the simplest non trivial correlator with one excited twist field where the limit is not strictly necessary since the expression (u t −x t )¯ t ∂ u X cl (u t ) is regular in x t . The explicit expression can be easily computed and it is given in appendix F for some cases. Nevertheless correlators with only one excited twist can be less trivial as Finally we can also consider This correlator is the correlator of eq. (4.83) of [5], i.e. τ α ( when we set N = 4, n = m = 1 and 1 = α, 2 = 1 − α, 3 = β and 4 = 1 − β.

Correlators of boundary operators and excited twists on R 2
Finally we can assemble the results from previous section to write the generating function for correlators of boundary operators and excited twists on where the last line is the interaction between the twist fields and the boundary operators. The generating function for correlators on T 2 can be formally easily obtained as done in section 4.2 by summing over all possible wrapping contributions as in eq. 92.

Correlators of bulk operators and excited twists on R 2
We can now make an educated guess of the generating function of the correlators of bulk operators and excited twists. As long as the bulk vertex operators do not involve momenta there is no doubt on the result since the bulk field can be written as the product of a chiral vertex times an antichiral vertex therefore the generating function is nothing else but the product of the generating function for the chiral part times the generating function for the antichiral part times the obvious interaction of the chiral current with the antichiral. What requires an educated guess is when the bulk vertex operators involve momenta since in this case we know that the local description requires to separate the right moving from the left moving part and normal order them separately, i.e. to the abstract vertex e ik I X I (u,ū) corresponds the local version : e ik I X I , [26] but see also [34]) Hence we guess that in the presence of momenta the twisted Green function must be split in its chiral-chiral, chiral-antichiral and so on pieces, i.e. G = G (LL) + G (LR) + G (RL) + G (RR) . This is obviously consistent with the case where derivatives are present since applying a derivative like ∂ u of the Green function is actually projecting it on the chiral piece. We therefore guess that the generating function of the correlators of L c bulk operators and N excited twists read up to phases due to cocycles Using the previous generating function it would be interesting deriving the boundary state with N twist fields. This could be done as in [35] and would be an interesting generalization of the boundary state with open string interactions derived in [36]. This boundary state could be used as in [37] to derive useful information about the long distance spacetime geometry generated by branes at angles.

Rewriting the Reggeon vertex using auxiliary Fock spaces
In the previous section we have given the explicit form of the generating function for correlators of boundary operators and excited twists. Traditionally and for sewing a different expression is used where to any operator insertion, i.e. external leg is associated an auxiliary Fock space. If we associate to any twist operator an auxiliary Fock space with vacuum |T t and we identify 8 and we do the same for any boundary vertex operator to which we associate an auxiliary Fock space with vacuum |0 a , p (j) = 0 and we identify (n ≥ 0) n!c (j)n ↔ −2 α k t j α (j,aux)n , n!c (j)n ↔ −2 α k¯ t jᾱ (j,aux)n (111) 8 The normalization is chosen in the usual way such that applying the map d to auxiliary operators on eq. ( we can write the generating function as a usual Reggeon vertex as where we have used the doubled fields χ I (z) defined as in eq. 11 for the twisted case and in the usual way for the untwisted one. Notice that the terms t =t and i = j must be regularized by properly computing the two contour integrals as discussed in [24] Acknowledgments The author thanks R. Richter for useful comments on a preliminary draft. This work is supported in part by the Compagnia di San Paolo contract "Modern Application in String Theory" (MAST) TO-Call3-2012-0088 A Details and useful formula for the untwisted string and N = 2 case We start considering the untwisted string associated to the D t brane, i.e. the string with both ends on D t . This has boundary conditions in the upper half plane H and has the expansion We have the non trivial commutators where x1, p1 are the zero mode position and momentum of string X1 = 1 √ 2 (e −iπαt X +e +iπαtX ) with N N boundary condition. The vacuum is defined in the usual way by even if care must be taken in order to deal with the DD zero modes ***[].
Then we can compute the untwisted Green functions where k t = −i 1 2 √ 2α e iπαt and k¯ t = −i 1 2 √ 2α e −iπαt as in the main text. Notice that G zz U (t) does not feel that the brane is rotated while both G zz and Gzz U (t) do because of the phase in k t 2 and k¯ t 2 . In a similar way we can compute the N = 2 twisted Green functions where we have used as follows from the general expression for the hypergeometric function 2 F 1 (a, b; c; x) = ∞ n=0 which follow from the hypergeometric transformation properties in particular 2 F 1 (1, ; 1 + ; x) = ¯ x 2 F 1 (1,¯ ; 1 +¯ ; 1/x).
For future use f.x. in eq. (154) we notice that In order to write ∆ IJ , the regularized Green function, in a more compact and transparent way we introduce the quantity which can be expanded around x = 1 as where ψ(x) = d log Γ(x)/dx is the digamma function. All of this expansion but the constant term can be easily obtained by computing D (x; ) and the integrating on x. To get the constant term is necessary to use the We can now write the boundary ∆ IJ bou defined as where we have defined the common factor (127) We have to consider the cases x 1 , x 2 > 0 and x 1 , x 2 < 0 because, for example, v loc u loc = x 2 x 1 when x 1 , x 2 > 0 and v loc u loc = |x 2 | |x 1 | e i2π when x 1 , x 2 < 0 and this gives raise to different phases. This is issue is not present for ∆ zz bou because v loc u loc = x 2 x 1 independently on x 1 , x 2 > 0 or x 1 , x 2 < 0. Notice that these phases are fundamental for projecting an arbitrary momentum (k,k) in the direction parallel to the D t brane as shown explicitly in section 4.1.
In a similar way we define the regularized Green function with the twist σ ,f at x loc = 0 and the anti-twist σ¯ ,f in x loc = ∞ used in chiral operators correlators. In particular because of the fact that there are at least ∂ u ∂ v the only piece which contributes is which is again independent on the phase of k t .

B Classical solutions
In this appendix we would like to summarize the results of the previous work [22] (see also [15] and [14]). Defined the anharmonic ratio for a complex variable z ∈ C to be so that ω 1 = 1, ω 2 = 0 and ω N = −∞, a basis of the derivatives of zero modes of the two dimensional laplacian satisfying the boundary conditions (17) is so that we can write the classical solution which satisfies also the global constraints (18). The real coefficients e −iπα 1 a n (ω t ) and e −iπα 1 b r (ω t ) are fixed by the constraints The "reality" of the coefficients can be easily seen once we introduce the following functions which are real on the real axis and their integrals (which can be expressed using the type D Lauricella functions) so that we can write the constraints as where the quantity between square brackets on the right hand side is real. Finally the classical action can be written as

C Green function
In ( [2]) following previous works on the subjects we defined the derivatives of the Green function on the whole complex plane C using the doubling trick as with expansions where we have used the anharmonic ratio as defined in eq. (129) and we have extended the range of definition from N − M − 2 to N − M for X (n) and from M − 2 to M forX (s) in order to write in a more compact way g (N,M ) and have therefore used hatted indexes. The previous quantities are subject to the constraints anŝ(ω t =1,2,N )ωn z ωŝ w = 0 (139) because g must have only a double pole with coefficient 1 and due to the boundary conditions. This set of equations is an overdetermined but consistent one as discussed in [22]. Using the previous quantities we wrote that the Green function in presence of N twists is given by and and where the normalization is needed in order to match the singularity of the untwisted Green function (117) 9 . The arbitrariness of the lower integration limit is due to the constraints (140) which allow to change x t 1 → x t 3 for whichever t 3 ans similarly for x t 2 . We would now justify this result since it is important in computing correlators involving momenta. The reason why we fixed the lower integration limit to one of the twist location is because we want as follows from the boundary condition (18) in the case of the quantum fluctuation where f t → 0.

D Boundary Green functions and their regularized version ∆ (N,M ),bou
For the computation of the boundary correlators it is interesting and useful to notice that all the components of the Green function are proportional, analogously to eq.s (126) we have 9 This is obvious for G zz (N,M ) but for G zz (N,M ) there would seem to be a mismatch of phases since ∂ u ∂vG zz U (t) = e i2παt −2 α 1 (u−v) 2 has a phase which depends on the brane while naively ∂ u ∂vG zz The issue is solved by noticing that the singularity is only there when u,v → x ∈ R and therefore g (N,M ) (x + iδ 1 , x − iδ 2 ) (δ 1,2 > 0) has the singularity with the required phase, explicitly for where the point x 1 is on the brane D t 1 , i.e. x t 1 < x 1 < x t 1 −1 and similarly for x 2 and we have defined the common real symmetric function G (N,M ),bou (x 1 , x 2 ) = G (N,M ),bou (x 2 , x 1 ) 10 to be where ∂ ω X (n) (ω) and ∂ ω X (s) (ω) are the functions (133) which are real when ω ∈ R and we have introduced the integers which enter the game because of the way is defined in eq. (8).
Another interesting point is to study the behavior of ∆ (N,M ),bou (x 1 , x 2 ) with x 1 , x 2 ∈ (x t , x t−1 ), i.e. they are on D t when x 1 , x 2 → x t . This is an important check since we should recover both the singularity and the form factor R 2 ( t ) of the function ∆ bou given in eq. (127). In the limit x 1 , x 2 → x t only the last two lines of eq. (146) contributes, if we change variables as ω z = ω t + (ω x 1 − ω t )y 1 and ω w = ω t + (ω x 1 − ω t )y 2 in the third line and in a similar way in the forth one we get where we used eq.s (139) which imply N −M =¯ t u =t,N (ω t − ω u ) and the analogous 10 The easiest way to verify that it is symmetric is to notice that G zz (N,M ),bou (x 1 , x 2 ) is symmetric.
In the previous equation we have introduced alsoŷ = ω 2 −ωt are not the location of the twist fields but the points where the Green function is evaluated. This expression has to be compared with the analogous for N = 2 which can be written as then we can write In the limit x 1 , x 2 → x t we notice thatŷ = x 1 −xt

E Functions entering excited twist fields correlators
If we look at eq. (104) and the more general eq. (108) we see immediately that the key quantities are We would now like to give a more explicit expression for these quantities while a completely explicit expression is given in the next section for few cases. Actually the previous expressions except the cases with only one derivative can be written as and and The remaining cases which involve only one derivative and are needed for computing the interaction among untwisted vertices with momenta and excited twists require a little more work. The easiest cases can be written as then the IJ = zz,zz cases are We see therefore that all the previous cases boil down to computing the blocks Then we can write the Green function and and −2 α Gzz (3,1) (u,ū; v,v; {xt, t }) = e −i2πα 1ω 2 u (ω u − 1) 1 · ωv 0;ω∈H which clearly shows the logarithmic singularity as u → v. The double integral of g (3,1) can be expressed as the product of two single integrals by using an integration by part and then rewriting the new resulting integral. If we would not use this procedure we would obtained an integral of Lauricella function which is by far more complex. The idea is simple and amounts to write the new integral as (ω w − ω t ) − t P olynomial 1 (ω z , ω w ) where the key point is that P olynomial 2 (ω w ) depends only on ω w . The The boundary Green function reads where the integer N t is defined in eq. (147) and the sign entering the definition of the Green boundary function is chosen consistently with eq.s 145. The regularized version of the boundary Green function at x 1 (remember that in this case both x 1 and x 2 are on the same brane) is which is nothing else but the unregularized boundary Green function to which we have subtracted the logarithm.
We also have the basic blocks for the twisted computations which correspond to eq.s 158 (u − x 1 )¯ 1 ∂ u X (n) (ω u ) = e iπ 1 x N −n 12 with x tt = x t − xt andn = 0, 1, 2. Similarly for ∂ u X (r) (ω u ) which can be obtained fro the previous ones with ↔¯ andn ↔r. It is also a good check that in the limit u, v → x t the Green function gives the expected singularities. This can be obtained with a simple change of variable or, essentially in the same way, rewriting the Green function using the Lauricella functions, for example where for Re c > Re a > 0 (175) to get in the limit u, v → x 2 , ω v /ω u constant the desired behavior of the Green function for N = 2 given in eq.s (118) upon the use of ωv ωu → v−x 2 u−x 2 . For the limit u, v → x 1,3 we can proceed in the same way but we have to use the relation which connects the hypergeometric computed at x to that computed in 1/x or to use the symmetries (120).