D-Brane Scattering of N=2 Strings

The amplitudes for emission and scattering of N=2 strings off D-branes are calculated. We consider in detail the amplitudesandfor the different types of D-branes. For some D-branes we find massive poles in the scattering spectrum that are absent in the ordinary N=2 spectrum.


Introduction
It has become obvious in the last couple of years that D-branes are of utmost importance for our understanding of N = 1 string theory. In his pathbreaking paper [1], Polchinski showed that p-dimensional extended objects -the D-p-branes -are the the long sought carriers of the Ramond-Ramond charges in N = 1 string theory. In the context of scattering amplitude calculations the most important property of D-branes is that their quantum fluctuations are described by open strings moving on the brane and therefore are under good control at weak coupling (for reviews on D-branes see e.g. [2,3] and literature cited within). This allows to calculate amplitudes for emission and scattering of closed fundamental strings from D-branes by "pre-revolutionary" methods that have been invented more than a decade ago and are well understood [4]. For the N = 1 string these computations have been performed e.g. in refs. [5,6,7,8,9] and considerably contributed to the understanding of D-brane physics. It is the purpose of this note to perform a similar analysis for N = 2 strings.
That this analysis has not been undertaken so far for the N=2 string finds its reason in the lack of Ramond-Ramond fields in the string spectrum. The N = 2 superconformal algebra with c = 6 serves as constraint algebra and is powerful enough to remove all string excitations from the spectrum leaving the center of mass motion (which is not tachyonic but massless in this case) as the only physical degree of freedom [10]. This has immediate consequences for possible interactions. All n-point functions vanish [11], the only non-vanishing tree-level amplitude is the 3-point function. The corresponding field theory is self-dual gravity for closed strings and self-dual Yang-Mills theory for the open string sector. The critical dimension of the N=2 string is four but with "unphysical" signature (2,2), making it possible to identify the four real with two complex dimensions.
Although lacking the necessary Ramond-Ramond fields, it is still possible to formally define Dbranes in N=2 string theory by imposing Dirichlet boundary conditions in certain target space directions. The obvious question is then whether the closed N=2 strings feel the presence of the branes. This note gives an answer to this question by performing a scattering analysis similar to the one undertaken in [7] for N = 1 strings.
Moreover, we introduce the matrices J k + ·p − = k·p + ik·J·p.
(2) (J acts as a self-dual complex structure. It is J 02 = J 13 = 1, J 13 = J 02 = −1, all other elements = 0), and This matrix D is related to the flat target space metric η by a change of sign in the directions transverse to the D-brane. Example: Let x 2 be the only direction transverse to the D-brane. Then D = diag(− + ++).
Emission and scattering off D-branes is conveniently calculated by evaluating correlators between vertex operators on the upper half plane. Open strings are represented by holomorphic vertex operators restricted to live on the real axis whereas closed string vertex operators factorize into holomorphic and antiholomorphic parts, Here z lies inside the upper half plane. The sum of each picture number has to add up to −2 inside a non-vanishing scalar product. We will use vertex operators in the (−1, −1), (−1, 0) and (0, −1) picture [12]:

The general calculations
The separate propagators for holomorphic and antiholomorhic fields are standard. However, due to the presence of a world sheet boundary, there are also non-vanishing correlation functions between holomorphic and antiholomorphic fields [7,13,14]: The example of the fermionic fields shows how this translates in the {±}-basis: with the definition G ± = D ± J · D · J. The D-brane respects the complex structure in target space The new feature here (as compared e.g. to the mixed amplitudes) is that for Dirichlet boundary conditions in general one gets poles in the operator product expansion between holomorphic and antiholomorphic fields both having a + or − index.

A cc
It was shown in ref. [7] that the scattering amplitude of two N = 1 closed strings off a D-brane can be obtained from the N = 1 open string 4-point function by simply interchanging certain momenta. Thus the amplitude takes the form of an Euler-Beta-function of the Mandelstam variables that can be expanded as an infinite series of closed string poles in the t-channel or of open string poles in the s-channel and leads to the soft high energy behavior of the amplitude [5]. This result is intuitively clear since the interaction of closed strings with a D-brane is mediated by exchange of closed strings travelling between the passing closed string and the D-brane, or -via world sheet duality -by open strings moving along the brane. This argument should also be true in N = 2 string theory. But it is difficult to imagine what a dual amplitude could look like in a theory with only a single degree of freedom. We therefore expect the scattering amplitude of a closed N = 2 string off a D-brane to vanish 1 .
The scattering amplitude of two closed strings off a D-brane for the N = 2 string is given by the integral of the correlation function of two closed string vertex operators with the correct quantum numbers over the upper half-plane H + : Momentum conservation holds only in directions parallel to the brane: where p 1 and p 2 denote the momenta of the incoming and outgoing strings, respectively.
In analogy to 4-particle scattering the amplitude can be parametrized by the following Mandelstam variables: Obviously s is the momentum transfer along the brane and t is the amount of momentum absorbed by the brane. As usual s + t + u = 0.
SL(2, R) invariance of the correlation functions on the upper half plane allows us to fix three of the four variables of the vertex operators. For A cc we choose z = iy (y ∈ R + ) and w = i. The correct integration measure is 1 0 dy(1 − y 2 ). The resulting expression can be transformed into well known integral-representations of Euler-Beta-functions using the "miracle"-substitution The final result is with

A ooc
The amplitude for two open strings on the brane joining into an outgoing closed string is x and y are integrated along the real axis in such a way that x is always left of y. The momenta k i of the open strings have to be parallel to the brane which implies k i = D·k i . Momentum conservation in this case reads There is only a single kinematical variable s = k 1 ·k 2 = 1 4 p·D·p = − 1 2 p·k 1 = − 1 2 p·k 2 . To evaluate the amplitude A ooc we set z = i and x = −y (x, y ∈ R) and use 2s = −u = −t. The relevant integrals can all be evaluated with the formulae This expression is completely analogous to that of the N = 1 theory.

Evaluating the general results for each D-brane-type
In this section the above amplitudes are explicitly analyzed for each type of D-brane. Due to the special signature (2, 2) of our space-time we denote the D-branes by p + q, where p and q are the number of spatial and time directions, respectively, in which the D-brane lives.

A cc
In this case the brane fills all of space-time and we have ordinary interaction between open and closed strings that has been considered in [15,16]. A cc has the interpretation of the lowest order quantum correction to closed string propagation. Momentum conservation holds in all directions for closed string scattering off the 2 + 2 brane. We cannot use our result, though, since by fixing three real parameters before integrating we did not divide out the volume of the whole symmetry group, which is, as we are dealing with a closed string topology SL(2, C) rather than SL(2, R). Naive application of our result (12) would lead to A cc = 0, while the real amplitude is known to be constant.

A ooc
For the process of joining of two open strings into a closed string momentum conservation implies that p·k 1 = p·k 2 = k 1 ·k 2 = 0. Since s = 1 4 p 2 = 0 the amplitude (13) reduces to coinciding with the well-known result [15].

A cc
The 1 + 2 brane divides space-time into two halves and is analogous to the 8-brane in N = 1 string theory. There is only one transverse dimension which we choose to be the third. Momentum conservation together with the mass-shell condition fixes the momenta in the closed string scattering process almost entirely.
There are two cases; either the uninteresting case of no scattering at all, i.e. p 1 = −p 2 , or the case The Mandelstam variables are s = −t = 1 2 (p 3 1 ) 2 and u = 0. What one finds from eq. (12) is that the first two terms vanish because the denominator diverges at u = 0. The third term reduces to Now using Γ(s)Γ(1 − s) = π sin(πs) we see that this expression has infinitely many simple poles

A ooc
Again we demand Dirichlet boundary conditions in the 3-direction. G − = diag(0, −2, 0, −2). The kinematics read k 3 1 = k 3 2 = 0. We have to distinguish between two cases a) k 3 3 = 0 Here t = 0, thus we end up with a finite amplitude: This amplitude has a tachyonic pole.

A cc
For this kind of brane the matrix D satisfies the relation D = J·D·J which implies G − = C = 0 and G + = 2D. The closed string scattering amplitude becomes To further analyze the kinematical prefactor one recalls that in 2 + 2 dimensions four momentum vectors with 4 1 k i = k 2 i = 0 satisfy the relation [11] (k It is this equation that is responsible for the vanishing of the 4-point function in open and closed N = 2 string theory. Setting and using the fact that (Dp 1 ) − ·(Dp 2 ) + = p + 1 ·p − 2 for this particular form of D, one finds that What remains is In general this kinematical prefactor does not vanish and again the result has massive poles.

A ooc
Since G − = 0 we have A ooc = 0.

A cc
Two time dimensions with Dirichlet boundary conditions imply that D = −J·D·J and therefore G + = A = 0 and G − = 2D = 2diag(+ + ++). The scattering amplitude is It is remarkable that the prefactor is related to eq. (19) by making the replacements and using (Dp 1 ) + ·(Dp 2 ) − = p + 1 ·p − 2 . We therefore see that in this case where the brane does not break the complex structure in target space the scattering amplitude vanishes,

A ooc
Open strings on this kind of brane are non-dynamical since the metric on the brane is euclidean such that the masslessness of the open strings implies the vanishing of their momentum. The 0 + 2 brane should therefore be thought of as a completely rigid object. As one easily sees from inspection of the kinematical prefactor in eq.(13) the amplitude for closed string emission vanishes: A ooc = 0.

A cc
In this case all three terms in (12) contribute to the scattering amplitude A cc which can be rewritten as We checked that the kinematical prefactor does not vanish for the values of s and t where the Beta-function has its poles.

A ooc
Again this amplitude vanishes trivially.

A cc
Dirichlet boundary conditions in all directions imply for the scattering process that there is no relation between the momenta of the incoming and outgoing closed strings. Since D = −η the Mandelstam variables and kinematical factors become The scattering amplitude in this case is leading to a simple 1 t pole at t = 0. This pole can be obtained by either taking the s → 0 limit in eq. (25) or by recalculating the amplitude with s = 0 from the very beginning.
The single simple pole at t = 0 clearly is due to closed string exchange between the passing closed string and the D-instanton. The kinematical prefactor (p + 1 ·p − 2 ) 2 is precisely the 3-point function of self dual gravity, as described by the Plebanski equation. From a field theory point of view the process should be considered as the scattering of gravitons off a pointlike (in space and time(s)) source, which can be identified with the D-instanton.

A ooc
Needless to say that the amplitude for emission of a closed string off the D-instanton vanishes.

A oooc
For completeness we add our results we obtained calculating the amplitude for emission of a closed string from a D-brane on which three open strings interact.
We find that A oooc = 0 for all branes but the 1 + 2-brane. In that case we are left with an integral of the type: . So far we have not been able yet to solve this integral.

Results
We find that if the D-brane breaks the complex structure in target space additional correlation functions appear in the calculation which are absent for the usual Neumann boundary conditions. The result for the amplitude A cc is nevertheless an Euler Beta-function multiplied by a kinematical prefactor. A closer look at this kinematical factor shows that the amplitude vanishes only for the 2 + 2 and the 0 + 2 brane and has a single simple pole at t = 0 for the D-instanton which is due to closed string exchange. The scattering amplitudes of branes that break the complex structure in target space, i.e. the 1 + 2, the 1 + 1 and 0 + 1 brane all have poles that do not correspond to states in the spectrum of the N = 2 string.
How do we interpret these results? In this paper we have considered the N = 2 string in its gauge fixed NSR formulation. Massive poles in the scattering spectrum seem to be inconsistent with this type of string. The inconsistency can be traced back to the fact the presence of these branes conflicts with N = 2 world sheet supersymmetry. This is due to the fact that a fermion ψ obeying Dirichlet boundary conditions cannot be in the same multiplet as a fermion obeying Neumann boundary conditions. This breaking of N = 2 world sheet supersymmetry hence seems to leave no space for these type of branes in the gauge-fixed NSR formulation of the N = 2 string.
This result is also consistent with T-duality. Recall that Neumann and Dirichlet boundary conditions are interchanged upon performing a T-duality transformation in a toroidally compactified space-time. However, for N=2 string propagation only Ricci-flat Kähler manifolds with (2, 2) signature are allowed. This leaves only the possibility to compactify one or both complex directions. Compactification of one or three real coordinates breaks the complex structure and yields an illegal background. Fortunately the three relevant branes, namely the 2 + 2-and 0 + 2-brane and the D-instanton, form a closed set under the action of T-duality in the allowed backgrounds.
Apart from the 2+ 2-brane the other relevant branes are non-dynamical since open strings attached to them have vanishing momentum 2 . This means that dynamical D-branes do not exist in the NSR formulation of N = 2 string theory in accordance with the absence of the corresponding differential forms and solutions of the classical equations of motion. We want to mention, though, that our formulation is not the only one that is able to describe the N = 2 string. As was shown by Berkovitz and Vafa [17] and Siegel [18] there exists as well a more general formulation of the N = 2 string in terms of the so-called topological N = 4 string. This formulation admits more degrees of freedom and hence there might be a way how the forbidden branes can be consistently incorporated in N = 2 string theory. But so far no attempt in this direction has been made, leaving room for further work and speculations.