Two-loop Yang-Mills diagrams from superstring amplitudes

Starting from the superstring amplitude describing interactions among D-branes with a constant world-volume field strength, we present a detailed analysis of how the open string degeneration limits reproduce the corresponding field theory Feynman diagrams. A key ingredient in the string construction is represented by the twisted (Prym) super differentials, as their periods encode the information about the background field. We provide an efficient method to calculate perturbatively the determinant of the twisted period matrix in terms of sets of super-moduli appropriate to the degeneration limits. Using this result we show that there is a precise one-to-one correspondence between the degeneration of different factors in the superstring amplitudes and one-particle irreducible Feynman diagrams capturing the gauge theory effective action at the two-loop level.


Introduction
The study of scattering amplitudes has played a central rôle in the development of string theory since its very beginning. In the seventies and the eighties it was instrumental in showing that superstring theories provide perturbative gravitational models that, at loop level, are free of ultraviolet divergences and anomalies. The analysis of string amplitudes was also crucial in the discovery of D-branes and in the development of the web of dualities among different superstring theories. It is, then, not surprising that this field continues to be under intense study. Recently, there has been renewed interest in several aspects of string perturbation theory in the RNS formalism, with particular focus on contributions beyond one loop: for example, higher-loop diagrams with Ramond external states were discussed in Refs. [1,2]; further, Refs. [3,4,5] focused on the off-shell extension of amplitudes, studying various situations where this is necessary; finally, Ref. [6,7] derived an explicit result for the D 6 R 4 term in the type-IIB effective action, checking the predictions following from S-duality and supersymmetry. Another interesting approach to the pointlike limit of closed string amplitudes as a 'tropical' limit was discussed in [8]. For recent reviews on multiloop string amplitudes, with a more complete list of references, we refer the reader to [9,10].
Two themes in particular have been at the center of much important progress in our understanding of string interactions: the study of the mathematical properties of the world-sheet formulation of string amplitudes, and their relation to the effective actions describing the light degrees of freedom present in the theory. In this paper, we touch on both these aspects by studying in detail the open string degeneration limits of two-loop amplitudes described by a world-sheet with three borders and no handles. In particular, we expand upon the results of [11]: starting from the Neveu-Schwarz (NS) sector of the open superstring partition function in the background of a constant magnetic field strength, we derive the Euler-Heisenberg effective action for a gauge theory coupled to scalar fields in the 'Coulomb phase'. The idea of using string theory to investigate effective actions in constant electromagnetic fields has a long history, and was studied at one loop in [12,13,14], with some results for the bosonic theory at two loops given in [15]. In our analysis we find exact agreement between calculations in field theory and string theory, in the infinite-tension limit, for the two-loop correction to the effective action. Furthermore, we find that the correspondence holds not just for the whole amplitude, but we can precisely identify the string origin of all individual one-particle irreducible (1PI) Feynman diagrams contributing to the effective action. In order to do so, on the string theory side we need to use appropriate world-sheet super-moduli, respecting the symmetry of the Feynman graphs, while on the field theory side we need to use a version of the non-linear gauge condition introduced by Gervais and Neveu in [16], modified by dimensional reduction to involve the scalars also, and given here in Eq. (5.10).
On the formal side, it is advantageous to use the formalism of super Riemann surfaces [17,18,19,20,21,9], in which the complex structure is generalized to a superconformal structure, with local super-conformal coordinates (z|θ). We follow this approach by constructing the two-loop amplitude in the Schottky parametrization, since there is a close relationship between Schottky super-moduli, in particular the 'multipli-ers', and the sewing parameters of plumbing fixtures. This in turn relates the bosonic world-sheet moduli to the Schwinger parameters associated to the propagators in Feynman graphs, which provides the ideal framework for studying the connection between string integrands and field theory Feynman diagrams. In the bosonic case, it is possible to describe genus h Riemann surfaces as quotients of the Riemann sphere (with a discrete set of points removed) by a discrete (Schottky) group, freely generated by h Möbius transformations. Heuristically, quotienting the Riemann sphere by a Möbius transformation has the effect of cutting out a pair of circles and gluing them to each other along their boundaries. Schottky groups arose naturally in the early treatment of multi-loop string amplitudes [22,23,24,25,26,27] and remained useful [28,18,29,30,31,32] even after alternative methods of analysis were found. In the supersymmetric case, higher genus super Riemann surfaces are similarly generated by quotienting the super manifold CP 1|1 (with a discrete set of points removed) by a discrete group, generated by h 'super-projective' OSp(1|2) transformations.
As is well known, the presence of a constant background field strength in the spacetime description of the amplitudes translates on the world-sheet side into the presence of non-trivial monodromies along either the a or the b cycles of the Riemann surface. It is thus not surprising that the amplitudes we are interested in involve super 1|1-forms (sections of the Berezinian bundle) with twisted periodicities, also known as Prym differentials. The bosonic counterparts of these objects was discussed, in the Schottky parametrization, in [33], and their periods along the untwisted cycles appear in any string amplitude where the fields have non-trivial monodromies [34,35,15,36]. We extend these past results in two directions: first we generalise the twisted period matrix to the supersymmetric case; then we must calculate the supersymmetric version of the twisted determinant to sufficiently high order in the complete degeneration limit, so as to obtain the gauge theory Feynman graphs with multiple gluon propagators. In order to do this, we introduce an alternative formulation of the twisted super-determinant in terms of an integral along a Pochhammer contour, and we show that this simplifies drastically its perturbative evaluation in the Schottky parametrization.
The main result of this paper is to show how the two-loop 1PI Feynman diagrams listed in Fig. 1 arise from the degeneration limits of the superstring result. The graphical notation for the field propagators is explained in detail in Appendix C; here we note in particular that we are using two different types of edges to denote gluons, depending on whether they are polarized parallel or perpendicular to the plane of the background field. We note also that some of the graphs (those in Figs. 1i-1l) include vertices with an odd number of scalars: these vertices arise because of the non-vanishing scalar vacuum expectation values (to which these graphs are proportional); these diagrams appear automatically in the string calculation, and they appear on the field-theory side as a result of having imposed the gauge condition of Gervais and Neveu [16] before dimensional reduction. Our investigation is thus also a contribution to a long-standing program aimed to use string theory to gain insights into field-theory amplitudes, which was started in Ref. [37] in the language of dual models, and generalized to the superstring framework in [38]. The practical usefulness of string theory as an organizing principle for tree-level gauge-theory amplitudes was first noticed and applied in [39,40]. At genus one, several results are available in the literature: they include the derivation of the leading contribution to the Callan-Symanzik β-function of pure Yang-Mills theory in [41], as well as a general analysis of one-loop scattering amplitudes in [42,43,44,45,46]. This was later used to calculate the one-loop five-gluon amplitude in QCD for the first time in [47]. String theory also inspired many developments in the world-line approach to perturbative quantum field theory (QFT), starting with the work of Strassler in [48], with subsequent progress in [49,50], summarized in [51], and more recently, for example, in [52,53]. Bosonic strings were also used to compute Yang-Mills renormalization constants at one loop in [54], and one-loop off-shell gluon Green's functions in [55]. At the two-loop level much less is known: explicit QFT amplitudes with only scalar fields were obtained from bosonic strings in [56] and [57,58]. Two-loop amplitudes with gluons, however, have proved difficult to study with this technology [59,60,61]. Our analysis here marks significant progress in this direction, showing that the prescriptions discussed in [11] are indeed sufficient to derive from string theory all the bosonic two-loop 1PI gauge-theory diagrams listed in Fig. 1. The structure of the paper is as follows. In Section 2 we describe the D-brane setup in which our calculations are carried out. In Section 3 we recall the integration measure for the NS sector of open superstrings in the super Schottky parametrization and explain how to modify it in order to accommodate our background. In Section 4 we expand the measure in powers the Schottky multipliers, and then we identify the appropriate parametrizations to describe the two degenerations of the Riemann surface which are relevant for our purposes: the symmetric degeneration leading to the diagrams with the topology of Figs. 1a-1l, and the incomplete degeneration, leading to diagrams with only two field-theory propagators and a four-point vertex, depicted in Figs. 1m-1r. An analysis of the various factors contributing to the string amplitude, arising from different worldsheet conformal field theories, then enables to unambiguously identify each diagram in the field-theory limit. In Section 5 we obtain and discuss the Lagrangian for the world-volume QFT in the appropriate non-linear gauge, and we use it to compute example Feynman diagrams. Finally, in Section 6.1 we compare our string-theory and QFT calculations, and in Section 6.2 we discuss the differences between the present calculation and the analogous calculation using the bosonic string. In Appendix A we discuss super-projective transformations and the super Schottky group, in Appendix B we give the calculation of the twisted (Prym) super period matrix, and in Appendix C we list the values of all of the Feynman graphs in Fig. 1 with our choice of background fields.

The string theory setup
We consider a stack of N parallel d-dimensional D-branes embedded in a D-dimensional Minkowski space-time, where, as usual, D = 10 for type II theories and D = 26 for bosonic string theory. When d < D − 2, and provided the string coupling g s is small, so that g s N 1, this configuration can be described in terms of open strings moving in flat space and being supported by the D-branes. We will work generically in the 'Coulomb phase' where the D-branes are spatially separated from each other in the directions perpendicular to their world-volumes. Furthermore, on each of the D-branes we switch on a uniform U(1) background field in the {x 1 , x 2 } plane, with a field strength tensor given by where B A is a constant 'magnetic' field on the A-th brane (thus A = 1, . . . , N ). The positions of the D-branes in the transverse directions will be labelled by Y A I , with I = d, d + 1, . . . , D − 1. Such a D-brane configuration is depicted from various viewpoints in Figs. 2 and 3. A string stretched between branes A and B will have squared length and will receive a classical contribution m AB to its mass from the elastic potential energy associated with the stretching of the string, given by where T is the string tension and α the related Regge slope. These strings will also be charged under the magnetic fields B A and B B , with the sign of the charge depending on . . . x their orientation. Open strings that start and end on the same D-brane are uncharged and their mass is independent of Y A I . For generic values of Y A I , this configuration breaks the symmetry of the world-volume theory from U(N ) to U(1) N .
The theory describing open strings supported by this D-brane configuration is free [12,13]. The constant background magnetic fields on the D-brane world-volumes manifest themselves in the world-sheet picture by altering the boundary conditions of string coordinates in the magnetized plane. On the double cover of the surface, this gives twisted boundary conditions, or, in other words, non-trivial monodromies, to the zero modes in the two magnetized space directions. To describe this setup, we will use the conventions of Section 3 of Ref. [11], which we summarize below.
To begin with, let us briefly consider the spectrum of low-lying string excitations. In the bosonic case, the world-sheet theory, in a covariant approach, comprises D embedding coordinates X µ and the ghost system (b, c). The holomorphic components of these fields admit the mode expansions In the presence of constant abelian background fields, the theory remains free, but string coordinates in directions parallel to the magnetized plane acquire twisted boundary conditions and must be treated separately. Considering strings ending on branes A and B, it is convenient to introduce the combinations Z ± AB = (X 1 AB ± iX 2 AB )/ √ 2. These combinations diagonalize the boundary conditions and yield the mode expansions After canonical quantization, the modes introduced above satisfy standard commutation relations, except for magnetized directions, where one finds As usual in covariant quantization, not all states in the Fock space obtained by acting with the creation modes on the SL(2, R)-invariant vacuum |0 are physical: we need to select only the states belonging to the cohomology of the world-sheet BRST charge In the bosonic theory, the lowest-lying physical state is a tachyon |k ≡ c 1 |k, 0 , with mass-shell condition k 2 = −m 2 = 1/α . The next mass level comprises (D + 2) massless states, which will be the focus of our analysis in the field theory limit: one finds two unphysical states, two null states, and (D − 2) physical polarization states appropriate for massless gauge bosons. A crucial ingredient of our analysis is the mapping between these string states and the space-time states in the limiting quantum field theory: as noticed for instance in Chapter 4 of [62], the action of the worldsheet BRST charge (2.8) on the (D + 2) massless states mirrors the linearized action of the space-time BRST charge for the U(N ) gauge symmetry: in particular, the states created by world-sheet ghost oscillators, c −1 |k and b −1 |k , behave as the spacetime ghosts C and C. Acting with the α M −1 oscillators, on the other hand, generates d states along the D-brane, and n s = D − d states associated to the n s directions transverse to the D-brane, representing respectively the d polarisations of the gauge vectors (including two unphysical ones), and n s adjoint scalars. To be precise, the world-sheet BRST charge Q W B acts as while the linearised space-time BRST transformation δ B acts as where a is an adjoint index, Q a µ and Q a I stand for a gluon mode and a scalar, depending on whether X M is parallel or perpendicular to the D-brane, and k M = {k µ , 0}.
This simple relation between world-sheet and space-time states is preserved in perturbation theory, when the string coupling is switched on and non-linear terms in the BRST operators must be taken into account. This is expected, since, in a perturbative analysis, fields propagating between interaction vertices are free. In practice, we will test this statement by calculating a string diagram with the world-sheet topology of a degenerating double-annulus, and identifying the contributions coming from the various massless states listed above, as they propagate through the diagram. We will show that each contribution matches the gauge theory result, where the corresponding space-time fields propagate in the matching edge of the relevant Feynman diagram, provided that the gauge used in field theory is the nonlinear Gervais-Neveu gauge, introduced in [16]. In this way, we can identify individual Feynman diagrams in the target field theory directly at the level of the string amplitude, picking a specific boundary of the string moduli space, and identifying the string states as they propagate along the degenerating surface.
A similar analysis holds also in the superstring case. In the RNS formalism one needs to introduce the extra world-sheet fields ψ µ , β and γ, that are the partners under worldsheet supersymmetry of the ∂X µ , b and c fields mentioned above. The monodromies for these new fields will be the same as those of their partners, except for a possible extra sign, which is allowed for fields of half-integer weight, and distinguishes the Ramond from the Neveu-Schwarz sectors. In this paper we will focus on the Neveu-Schwarz contributions: the analysis of the states at the first mass level, above the tachyonic ground state |k , parallels that of the bosonic case. The only difference is that the relevant modes are ψ −1/2 , β −1/2 and γ −1/2 : in the superstring partition function, the low energy limit will be performed by focusing on the contributions of states with half-integer weight.

The superstring partition function for the NS sector
From the world-sheet point of view, the interaction among D-branes is described by the string vacuum amplitude (the partition function) with boundaries, as depicted in Fig. 2.
The case of two magnetized D-branes, corresponding to a one loop-amplitude, has been well studied [12,13,63,14]. Here we will focus on planar world-sheets, and most of what we will say in this section applies to surfaces with (h + 1) borders, corresponding to hloop open superstring diagrams, but restricted to the NS sector, where the super-Schottky formalism described in Ref. [30] can be used. In particular, as discussed in Section 2, we consider parallel magnetized D-branes that can be separated in the directions transverse to their world-volumes. As a consequence, and as depicted in Fig. 4 for a (two-loop) surface with three boundaries, the partition function depends on two set of variables: the relative distances among D-branes, and the magnetic field gradients between pairs of D-branes.
To be precise, let us label the (h + 1) world-sheet borders with i = 0, 1, . . . , h. Then we can label the D-brane to which the i-th border is attached with the integer A i , with A i ∈ {1, . . . , N }, and A i ≤ A i+1 . To get the full amplitude, we will have to sum over the A i 's. Having fixed A 0 , . . . , A h , we can take the A 0 -th brane as a reference and define The variables i thus form an h-dimensional vector, which we will denote by ; similarly, the variables d i I , which have dimension of length, form n s h-dimensional vectors, which we will label d I . The classical mass of the string stretching between the A 0 -th brane and the A i -th brane is then given by (3.2) Notice finally that, for h = 2, as depicted in Fig. 4, we make a slight variation in this notation by flipping the sign of the second component of the two-dimensional vectors and d I , which will be useful to take full advantage of the extra symmetry at two loops. The string partition function in this setup can be written as follows. For our purposes, it is useful to keep separate the contributions of the different conformal field theory sectors, which leads to the expression is a field-dependent normalization factor, to be discussed in Section 4.4, and we denoted the contributions of the world-sheet ghost systems b, c and β, γ by F gh , that of the string fields X I , ψ I perpendicular to the D-branes by F ( d ) scal , while the contribution of the fields along the D-branes has been separated into sectors parallel (F ( ) ) and perpendicular (F ⊥ ) to the magnetized directions. Finally, µ denotes collectively the supermoduli: here we use the super-Schottky formalism, reviewed in Appendix A, where the supermoduli are the sewing parameters e iπς i k i 1/2 (with ς i ∈ {0, 1}) and the fixed points (u i |θ i ), (v i |φ i ) of h super-projective transformations i = 1, . . . , h. Note that we explicitly associate with each Schottky multiplier k i the phase ς i associated with the NS spin structure around the b i homology cycle. In this parametrization the measure dµ h reads [30] where we denote superconformal coordinates in boldface, and the notation v i −u i indicates the supersymmetric difference The square parenthesis in Eq. (3.4) takes into account the super-projective invariance of the integrand, which allows us to fix three bosonic and two fermionic variables. Θ v 1 u 1 v 2 is the fermionic super-projective invariant which can be constructed with three fixed points, defined in Refs. [64,20], and given explicitly in Eq. (A.12). If we specialize Eq. (3.4) to h = 2 we find Let us now examine in turn the various factors in the integrand of Eq. (3.3). The ghost contribution is independent of both the magnetic fields and the D-brane separations, so we can use the result of Ref. [30], which reads 1 + e iπς 1 k (3.7) In Eq. (3.7), the notation α means that the product is over all primary classes of the super Schottky group: a primary class is an equivalence class of primitive super Schottky group elements, i.e. those elements which cannot be written as powers of another element; two primitive elements are in the same primary class if one is related to the other by a cyclic permutation of its factors, or by inversion. The vector N α has h integer-valued components, and is defined as follows: the i-th entry counts how many times the generator S i enters in the element of the super Schottky group T α : more precisely, we define N i α = 0 for T α = 1 and N i α = N i β ± 1 for T α = S ±1 i T β . Finally, also ς is a vector with h components, with the i-th component denoting the spin structure along the b i cycle, as noted above.
In fact, we need to be more precise about the notation in Eq. (3.7), because the half-integer powers of k α could indicate either of the two branches of the function. The notation is to be understood in the following way: when the spin structure is ς = 0, we define the eigenvalue of the Schottky group element T α with the smallest absolute value to be −k 1/2 α , see Eq (A.16). In particular, we take k 1/2 i to be positive 5 for i = 1, . . . , h. This corresponds to the fact that spinors are anti-periodic around a homology cycle with zero spin structure (see, for example, Ref. [65]). Furthermore, we expect the partition function to be symmetric under the exchange of the homology cycles b 1 , b 2 and b −1 1 · b 2 (depicted in Fig. 5), and one can verify that k 1/2 (S −1 1 S 2 ) is always positive whenever k 1/2 1 and k 1/2 2 have the same sign. Our convention puts all three multipliers on the same footing. Note that k 1/2 α is not in general positive when T α is not a generator: for example, the eigenvalues of T α = S 1 S 2 are positive when the spin structure is zero, so that k 1/2 (S 1 S 2 ), as computed in Eq. (A.25), is negative.
The scalar contribution to Eq. (3.3) depends on the separation between the D-branes in the transverse directions, as shown in Fig. 3. We can write F ( d ) scal as a product over the super Schottky group, capturing the non-zero mode contribution, times a new factor Y(µ, d ), as (3.8) The explicit form of Y can be found by repeating the calculation performed in Ref. [66] for the bosonic theory, and replacing the period matrix τ with the super-period matrix τ discussed in Appendix A.2.2. We find It is instructive, and useful for our later implementation, to consider explicitly the h = 2 case. Let the i = 0, 1, 2 borders of the world-sheet be on the D-branes labelled by A, B and C, respectively. As mentioned above, it is useful in this special h = 2 case to define the i = 2 component d 2 I with the opposite sign with respect to Eq. (3.1), so we have d 1 By so doing, we can then define an additional (redundant) quantity, describing the displacement between the D-branes attached to the i = 1 and i = 2 borders, as d 3 I = Y B I − Y C I . Now the three distances d i I for i = 1, 2, 3 are on an equal footing, reflecting the symmetry of the world-sheet topology, and we have d 1 I +d 2 I +d 3 I = 0 (see Fig. 4). One may easily verify that the product over the n s transverse directions in Eq. (3.9) evaluates to a function of the squared masses m 2 i , defined as in Eq. (3.2). One finds Finally, let us turn to the contribution of the world-sheet fields X µ , ψ µ along the worldvolume direction of the D-branes. In absence of magnetic fields, the result can be found in Ref. [30] and it reads Figure 4: The double annulus world-sheet, with three boundaries labeled with i = 0, 1, 2 attached to three D-branes, with Chan-Paton factors A, B, C. The relative positions and background field strengths of branes B and C with respect to brane A determine the masses and the twisted boundary conditions, as described in the text.
In the presence of constant background gauge fields, F gl gets modified, since string coordinates along the D-branes are sensitive to such backgrounds. The relevant modification to the bosonic theory was derived in Ref. [15]. Using the techniques described in Refs. [33,15,67], it is possible to generalize this construction to the Neveu-Schwarz spin structure of the RNS superstring [11]. The result is that switching on the background fields amounts to multiplying F (0) gl by a factor, as where, assuming the background fields to be non-zero only in one plane, we have R(µ, ) = e −iπ ·τ · det (Im τ ) det (Im τ ) The matrix τ is the supersymmetric analogue of the twisted (or Prym) period matrix, the bosonic version of which was computed with the sewing method in [33,35]. Its calculation in outlined in Appendix B.2. Inspecting Eq. (3.13), we see that F ( ) gl can be factorized as the product of a term F ( ) , capturing the contribution along the magnetized plane, times an -independent term F ⊥ arising from the unmagnetized directions. In the field theory limit, F ( ) will generate the contributions of gluons polarized in the plane of the background field, while F ⊥ will give rise to gluons polarized in the transverse directions. Explicitly, we have (3.14) 1 + e iπ(2 ·τ + ς)· Nα k n−1/2 α 1 + e −iπ (2 ·τ + ς)· Nα k n−1/2 α 1 − e 2πi ·τ · Nα k n α 1 − e −2πi ·τ · Nα k n α .
Focusing now on the h = 2 case, we can use super-projective invariance to fix three bosonic and two fermionic moduli. A convenient gauge choice in the super Schottky formalism is to specify the positions of the fixed points, given in terms of homogeneous coordinates 6 on CP 1|1 , as 16) with (0 < u < 1), which leads to Implementing this projective gauge fixing in Eq. (3.3), we can finally express the h = 2 partition function as 19) in terms of the bosonic super-projective invariant built out of four points, (z 1 , z 2 , z 3 , z 4 ), see Eq. Eq. (A.13).
4 Taking the field theory limit

Expanding in powers of the multipliers
We are interested in computing the α → 0 limit of the integrand of the superstring amplitude. In this limit, we expect massive string states to decouple, so that one is left with the massless spectrum. Possible contributions from the tachyon ground state cancel after GSO projection in the superstring case, or should be discarded by hand in the bosonic case. It is in principle non trivial to take this limit before integration over (super) moduli, since this requires constructing a map between the dimensionless moduli of the (super) Riemann surface and the dimensionful quantities that arise in the computation of field theory Feynman diagrams. This task is considerably simplified in the Schottky parametrization, where, as discussed for example in Ref. [11], the contributions of individual string states can be identified by performing a Laurent expansion of the integrand of the string partition function in powers of the multipliers. One finds a correspondence between the order of expansion and the mass level of the string, and furthermore, within each mass level, one can track individual states by tracing the origin of each term to a specific factor in the string integrand. The main difference between the bosonic string and the RNS superstring is that for the latter, which we discuss here, the expansion is in powers of k 1/2 i rather than k i , as is already apparent from our discussion in Section 3. More precisely, since the measure of integration contains a factor k −3/2 i , a term proportional to k (n−3)/2 i corresponds to a contribution from a state belonging to the n-th mass level circulating in the i-th string loop (where n = 0 corresponds to the tachyonic ground state). Therefore, all terms with n > 1 acquire a positive mass squared, m 2 = (n − 1)/(2α ), and decouple in the limit α → 0. We conclude that it is necessary to expand the various factors in the integrand of Eq. (3.18) only up to terms of order k i 1/2 , in order to get the complete massless field theory amplitude.
This task is made possible by the fact that the multipliers of only finitely many super-Schottky group elements contribute at order k 1/2 1 k 1/2 2 . The reason is that the leadingorder behaviour of the multiplier k α = k(T α ) is related in a simple way to the index N i α introduced in Section 3: one may verify that unless of course the left-most factor of T α is S ∓1 i . Thus, for every super Schottky group element T α not in the primary class of an element in the set {S 1 , S 2 , S 1 S 2 , S −1 1 S 2 }, the multiplier k   Next, we compute F ⊥ , defined in Eq. (3.14). Using the expressions for the multipliers k 1/2 (S −1 1 S 2 ) and k 1/2 (S 1 S 2 ), given in Eq. (A.25), we find The expansion of the determinant of the super period matrix is given in Eq. (A. 33), and, substituted here, leads to the factor det (Im τ ) Notice that logarithmic dependence on (super) moduli must be retained exactly: indeed, as shown in Ref. [11] and discussed here in Section 4.3, it will turn into polynomial dependence on Schwinger parameters in the field theory limit. The expansion of the factor F ( ) , also given in Eq. (3.14), is more intricate, as well as more interesting, because of the dependence on the external fields. Writing where R is the background-field dependent factor of the infinite product appearing in Eq. (3.15), we see that we can separately expand the three factors. The determinant of the twisted super period matrix is by far the most intricate contribution. It is discussed in Appendix B, and a complete expression with the exact dependence on the fields, through , is very lengthy. We will see in Section 4.3, however, that in the field theory limit we must expand in powers of the components of as well: at that stage, we will be able to write a completely explicit expression also for det (Im τ ). The exponential factor in the numerator of Eq. (4.5) can be computed using the expression for τ in Eq. (A.32), and is given by Finally, the remaining factor in F ( ) is given by where we defined the factors The last required ingredient is F The remaining, mass-independent, factor in F  Figure 5: Two types of homology cycles on the double annulus.
This completes the list of the factors in Eq. (3.18). It is now straightforward to combine them, and expand the resulting polynomial in k 1/2 i to the relevant order. Before proceeding, however, we must consider more carefully our choice of variables in view of the field theory limit.

A parametrization for the symmetric degeneration
In order to go beyond the specification of the mass states circulating in the string loops, and identify the contribution of individual Feynman diagrams in the field theory limit, we must refine our parametrization of (super) moduli space. Let us now, in particular, concentrate on Feynman diagrams with the symmetric topology depicted in the first two lines of Fig. 1. While individual Feynman diagrams will not be symmetric under the exchange of any two lines when the propagating states are different, we expect that, when summing over all states at a given mass level, the result should be fully symmetric, since there are no features distinguishing the three propagators at the level of the world-sheet geometry. This symmetry requirement will guide our choice of parametrization for the region of moduli space close to this degeneration, along the lines already discussed in Ref. [11].
It is clear that the parametrization in terms of the bosonic moduli k 1/2 1 , k 1/2 2 and u ≡ 1 − y + θφ will not be sufficiently symmetric, since the first two chosen moduli are multipliers of super-Schottky group generators, while the third one is a cross-ratio of the fixed points. To present the integration measure in a sufficiently symmetric way, we must parametrize it to be symmetric under permutations of the super-Schottky group elements S 1 , S 2 and S −1 1 S 2 . The reason for this is that the homology cycles b 1 , b 2 and (b −1 1 · b 2 ) lift to these super-Schottky group elements on the covering surface CP 1|1 − Λ, and any two of b 1 , b 2 and (b −1 1 · b 2 ) (along with the appropriate choice of a-cycles) constitute a good canonical homology basis (see Fig. 5a). Our choice of S 1 and S 2 as the generators is arbitrary, so, in order to preserve modular invariance, the measure must be parametrized to display the symmetry under permutations of S 1 , S 2 and S −1 1 S 2 . To reinforce this point, note that any other homology cycle built out of b cycles will intersect itself, as is the case for example for the (b 1 · b 2 ) cycle, depicted in Fig. 5b.
A natural way to symmetrize the measure is to use the multiplier of S −1 1 S 2 as the third bosonic modulus, instead of u. Defining − e iπς 3 k 1/2 3 to be the eigenvalue of S −1 1 S 2 with the smallest absolute value, so that k 3 is the multiplier of that super Schottky group element, one can compute k 1/2 3 using Eq. (A.24). It is related to y implicitly through In these definitions, ς 3 is the spin structure around the b 3 ≡ b −1 1 · b 2 homology cycle, and therefore it is given simply by ς 3 = ς 1 + ς 2 (mod 2). k 1/2 3 is then positive, just as k 1/2 1 and k 1/2 2 are. As discussed in Ref. [11], the field theory limit becomes particularly transparent if one factors the three multipliers k i in order to assign a parameter to each section of the Riemann surface that will degenerate into an individual field theory propagator. This is done by defining where √ p i is defined to be positive. In analogy to the discussion of Ref. [11], each p i will be interpreted, in the field theory limit, as the logarithm of the Schwinger proper time associated to a propagator.
For bosonic strings, the discussion leading to Eq. (4.12) was sufficient to construct a symmetric measure of integration, prepared for the symmetric degeneration in the field theory limit. In the present case, instead, one must also worry about fermionic moduli: our current choice of θ and φ as moduli will not yield a symmetric measure, since they are super-projective invariants built out of the fixed points of S 1 and S 2 only. In order to find the proper Grassmann variables of integration, we take advantage of the fact that we are allowed to rescale θ and φ with arbitrary functions of the moduli, since such a rescaling automatically cancels with the Berezinian of the corresponding change of integration variables. Such a rescaling of course leaves the integral invariant, but it can be used to move contributions between the various factors of the integrand, in such a way that individual factors respect the overall exchange symmetry of the diagram, as we wish to do here. In order to find a pair of odd moduli invariant under permutations of S 1 , S 2 , S −1 1 S 2 , we proceed as follows. Definê for (ij) = (12), (23), (31). For the factors c ij we make the choice with c 23 and c 31 obtained by permuting the indices (123), and where u 3 and v 3 are the fixed points of the transformation S −1 1 S 2 . In Eq. (4.14), we have introduced the symbols q i , i = 1, 2, 3, defined by 7 Note that the spin structures of q 1 and q 2 are swapped compared with what one might expect. This, however, is reasonable, because the q i defined in this way factorize the NS sewing parameters e iπςi k 1/2 i as follows: e iπς1 k With this choice for c ij , one can check that e iπς 3 dθ 12 dφ 12 = e iπς 1 dθ 23 dφ 23 = e iπς 2 dθ 31 dφ 31 , (4.16) so that the Grassmann measure of integration has the required symmetry. It is not difficult to rewrite the various objects computed in Section 4.1 in terms of the new variables, and expand the results to the required order in p i . In order to do so, we use as well as With these results, it is straightforward to verify the symmetry of the full string integrand.
In particular, we find that the product of the two-loop measure of integration times the ghost factor is given by where the contribution of the spin structure to dµ 2 in Eq. (3.6) has been absorbed in dθ 12 dφ 12 . Similarly, the contribution of the orbital modes defined in Eq. (4.7) becomes Here, and in the rest of this section, we understand 'cyclic permutations' to mean cyclic permutations of the indices (1, 2, 3) for The indices ofθ 12φ12 , on the other hand, are not permuted.
In order to reconstruct the full contribution of fields in the directions parallel to the magnetized plane, we still need the other factors appearing in Eq. (4.6). The exponential factor takes the form The last factor in Eq. (4.6) is the twisted determinant det (Im τ ), whose calculation is described in Appendix B. The result for generic values of u is a lengthy combination of hypergeometric functions, which however simplifies drastically in the limit we are considering here, where u, proportional to p 3 , is small. In this limit (B.40) reads Next, we need the contribution of the untwisted gluon sector, given in Eq. (4.3). In the current parametrization it reads where the determinant of the period matrix, given by Eq. (A.33), becomes det(Im τ ) = 1 4π 2 log p 1 log p 2 + log p 2 log p 3 + log p 3 log p 1 (4.23) Finally, we need the ingredients for the scalar sector, given above in Eq. (4.9) and Eq. (4.10). The mass contribution takes the form while the mass-independent factor is given by This completes the list of ingredients needed for the analysis of the symmetric degeneration of the surface. We now turn to the calculation of the α → 0 limit.

Mapping moduli to Schwinger parameters
The last, crucial step needed to take the field theory limit is the mapping between the dimensionless moduli and the dimensionful quantities that enter field theory Feynman diagrams. This α -dependent change of variables sets the space-time scale of the scattering process and selects those terms in the string integrand that are not suppressed by powers of the string tension. The basic ideas underlying the choice of field theory variables have been known for a long time (see for example Ref. [68]), and were recently refined for the case of multi-loop gluon amplitudes in Ref. [11]. The change from bosonic strings to superstrings does not significantly affect those arguments: in the present case we will see that integration over odd moduli will simply provide a more refined tool to project out unwanted contributions, once the Berezin integration is properly handled. Following Ref. [11], we introduce dimensionful field-theoretic quantities with the change of variables These definitions make it immediately obvious that terms of the form p c i i must be treated exactly, as we have done. On the other hand, terms proportional to high powers of i are suppressed by powers of α in the field theory limit, which is the source of further simplifications in our final expressions.
For completeness, we give here the results for the various factors in Eq. (3.3) as Taylor expansions powers of q i (that is, in half-integer powers of p i ), but with the field-and mass-dependent coefficients of the leading terms worked out exactly. Beginning with the contribution of gluon modes perpendicular to the magnetic fields, F ⊥ (µ), we find which we recognize as the first Symanzik polynomial [69] of graphs with the topology of those in the first two lines of Fig. 1, expressed in terms of standard Schwinger parameters. The result for the contribution of gluon modes parallel to the magnetic field is more interesting, as one begins to recognize detailed structures that are known to arise in the corresponding field theory. Multiplying Eq. (4.19) by Eq. (4.20), and dividing by Eq. (4.21), one finds Using the fact that B 1 + B 2 + B 3 = 0, one can verify that ∆ B can be understood as the charged generalization of the first Symanzik polynomial for this graph topology, and indeed for vanishing fields ∆ B tends to ∆ 0 . It is then easy to see that F ( ) (µ), in the same limit, reproduces F ⊥ (µ) with the replacement d − 2 → 2, as expected.
The contribution from the D-brane world-volume scalars can be obtained combining Eq. (4.24) and Eq. (4.25). One finds where one recognizes the exponential dependence on particle masses, each multiplied by the Schwinger parameter of the corresponding propagator, which is characteristic of massive field-theory Feynman diagrams. Note that the masses m 2 i appearing in Eq. (4.31) arise via symmetry breaking from the distance between D-branes, and therefore represent classical shifts of the string spectrum: below, for brevity, we will often call 'massless' all string states that would be massless in the absence of symmetry breaking.
The final factor, including the integration measure and the contribution from the ghosts, can be read off Eq. (4.18), and can be organized as The complete integrand of Eq. (3.3) is the product of F ⊥ from Eq. (4.27), F from Eq. (4.29), F scal from Eq. (4.31) and dµ 2 F gh from Eq. (4.32). In the proximity of the symmetric degeneration, it can be organized in a power series in terms of the variables q i , as we have done for individual factors. We write It is now straightforward to extract the contribution of massless states, which is contained For bosonic strings, one had to discard the contribution of the tachyonic ground state by hand: in this case, one can simply implement the GSO projection and observe the expected decoupling of the tachyon. We now turn to the analysis of this point.

The symmetric degeneration after GSO projection
Starting with the expression in Eq. (4.33), we can now describe more precisely the connection between the powers of the multipliers and the mass eigenstates circulating in the loops. For the symmetric degeneration, we now see that the power of p i corresponds to the mass level of the state propagating in the i-th edge of the diagram. Indeed one observes that dp i and one recognizes that dt i e − n−1 2α t i is a factor one would expect to see in a Schwingerparameter propagator for a field with squared mass m 2 = n−1 2α . In particular, if n = 0, then the state propagating in the i-th edge will be a tachyon, and will have to be removed by the GSO projection.
A cursory look at F in Eq. (4.29), F ⊥ in Eq. (4.27), F scal in Eq. (4.31) and dµ 2 F gh in Eq. (4.18), would suggest that tachyons can propagate simultaneously in any number of edges of the diagram: indeed, we can find terms proportional to i dp i /q 3 i times 1, q 1 , q 1 q 2 , q 1 q 2 q 3 , . . ., which correspond respectively to three, two, one or no edges with propagating tachyons. A closer inspection shows, however, that the nilpotent objectθ 12φ12 multiplies only terms with an odd number of factors of q i , a property which is preserved when we multiply terms together. Since the Berezin integral over dθ 12 dφ 12 picks out the coefficient ofθ 12φ12 , it follows that, after carrying out the Berezin integration, each term must contain an odd number of factors of q i .
As a consequence, after Berezin integration and truncation of the integrand to O(p 0 i ), Eq. (4.33) will be written as a sum of four terms, proportional to 3 i=1 dp i /p 3/2 i multiplied by the factors The first three terms in Eq. (4.35) carry the contributions of tachyons propagating in loops: since we wish to excise tachyons from the spectrum, we need to implement a GSO projection in such a way that these three terms vanish. This is achieved by simply averaging the amplitude over the four spin structures (ς 1 , ς 2 ) ∈ (0, 0), (1, 0), (0, 1), (1, 1) ; one clearly sees that the first three terms in Eq. (4.35) vanish while the fourth term is independent of ς and thus unaffected. Therefore the GSO-projected amplitude is free of tachyons while the massless sector is intact, as desired.
We are now in a position to take the field theory limit for the symmetric degeneration. The only missing ingredient is the normalization factor N ( ) h introduced in Eq. (3.3). It is given by where C h is the normalization factor for an h-loop string amplitude in terms of the ddimensional Yang-Mills coupling g d , calculated in Appendix A of Ref. [68]. For h = 2 it is given by (4.37) The denominator in Eq. (4.36) arises from the Born-Infeld contribution to the normalization of the boundary state (see for example Ref. [70]). It does not contribute to the field theory limit, since cos(π 1 ) cos(π 2 ) = 1 + O(α 2 ), after expressing the twists i in terms of the background field strengths via Eq. (4.26).
Applying the GSO projection to Eq. (4.33), and using dp i /p i = −dt i /α , we finally find that the QFT limit of the partition function can be represented succinctly by where the limit on the second line is finite after Berezin integration. In order to see that, and in order to give our results as explicitly as possible, we define (for simplicity we omitt the arguments of the functions f ) and similarly for f With our definitions, one easily sees that f Performing the Berezin integration is then a simple matter of combinatorics, and one finds We can read off the various terms in the integrand by picking the coefficients of the appropriate factors of q i from F in Eq. (4.29), F ⊥ in Eq. (4.27), F scal in Eq. (4.31) and dµ 2 F gh in Eq. (4.18), and then selecting the coefficient ofθ 12φ12 , divided by α . We find The other terms in the integrand can be obtained from the above by cyclic symmetry. We conclude by noting that Eq. (4.41) does not give the complete 2-loop contribution to the vacuum amplitude with this topology, since the string theory calculation distinguishes the three D-branes where the world-sheet boundaries are attached. This is reflected in the integration region over the Schwinger parameters t i , already discussed in Ref. [15]: they are not integrated directly in the interval 0 < t i < ∞, as would be the case in field theory, but they are ordered, as 0 < t 3 < t 2 < t 1 < ∞. In order to recover the full amplitude, with the correct color factors and integration region, one must sum over all possible attachments of the string world-sheet to the D-branes, effectively summing over the different values of the background fields B i and masses m 2 i . In the absence of external fields, this sum amounts just to the introduction of a symmetry and color factor; for non-vanishing B i , it reconstructs the correct symmetry properties of the amplitude under permutations.

The incomplete degeneration
In the last three sections, 4.2, 4.3 and 4.4, we have given the tools to compute the field theory limit of the partition function in the vicinity of the symmetric degeneration, see Fig. 6a: our final result is summarized in Eq. (4.41). The field theory two-loop effective action, however, includes also the Feynman diagrams with a quartic vertex depicted in the last two lines of Fig. 1.
The main feature of vacuum graphs with a four point vertex, which drives the corresponding choice of parametrization for the neighborhood of moduli space depicted in (c) Figure 6: The symmetric (Fig. 6a), incomplete (Fig. 6b), and separating (Fig. 6c) degenerations of the two-loop vacuum amplitude. Fig. 6b, is the fact that such graphs have only two propagators, each one encompassing a complete loop, and furthermore they are symmetric under the exchange of the two loops. It is natural therefore to associate to each propagator a Schwinger parameter linked to the Schottky multiplier of the corresponding string loop. The fact that there are no further Schwinger parameters implies also that the third bosonic modulus must be integrated over its domain except for a small region around each boundary. We therefore call the configuration depicted in Fig. 6b the incomplete degeneration. To compute the field theory limit for the incomplete degeneration, we must retrace our steps back to Section 4.1, where the various factors in the partition function were expressed in terms of k i and u (or y). We then relate the multipliers to Schwinger parameters as and replace i according to Eq. (4.26).
As may be expected from the simplicity of the target graph, the string partition function simplifies drastically when the α → 0 limit is taken in this way. One finds for example that the determinant of the (twisted) period matrix reduces to det (Im τ ) while the hyperbolic functions appearing in the field theory limit arise in a direct way from combinations like The resulting expressions are very simple because, with no Schwinger parameter associated to u, factors of the form u ± i do not contribute to the field theory limit. This is what makes it possible to perform the integration over the third bosonic modulus, and over the two fermionic moduli: indeed, in the parametrization considered here, the entire partition function can be written explicitly, in the α → 0 limit, in terms of just three simple integrals over the non-degenerating coordinates of super-moduli space. After GSO projection, one finds As the notation suggests, Eq. (4.46) in principle contains contributions from both the incomplete and the separating degenerations, and we now turn to the problem of disentangling them. We also see that in order to complete the calculation one just needs to determine three numerical constants, given by the following integrals over the nondegenerating super-moduli, To determine the domain of integration M 1|2 in Eq. (4.47), and to identify the different degeneration limits, note that the separating, symmetric and incomplete degenerations all come from the region of super-moduli space in which the two Schottky multipliers k 1 and k 2 are small. In this limit, we can think of super moduli space as a 1|2-dimensional space parametrized by (u|θ, φ). The separating degeneration corresponds to the limit y → 0, while the symmetric degeneration corresponds to the limit u → 0, and the incomplete degeneration comes from the region of super moduli space interpolating between the two limits. As pointed out in Refs. [71,21], however, this simple characterization must be made more precise, in particular with regards to the choice of parameters near the two degenerations.
First of all, let us briefly consider the first term in braces on the right-hand side of Eq. (4.46). This dominates in the limit y → 0, and we expect it to represent the contributions of the one-particle reducible (1PR) Feynman diagrams, which we neglect. We can now concentrate on the evaluation of the integrals relevant for our purposes, which are I 2 and I 3 in Eq. (4.47). They can be calculated using Stokes' theorem for a super-manifold with a boundary (see section 3.4 of [71]), since the integrands are easily expressed as total derivatives. We write These expressions mean that the corresponding integrals are localized on the boundary of M 1|2 , which consists of two loci associated with the two distinct ways in which the double annulus world-sheet can completely degenerate: the symmetric degeneration of fig. 6a, and the separating degeneration of fig. 6c. To use Stokes' theorem, it is important to characterize precisely the 0|2-dimensional boundary of the super-moduli-space region over which we are integrating. More precisely, we need to find bosonic functions of the worldsheet moduli ξ i (u, θ, φ), defined near the boundaries of M 1|2 , such that the vanishing of ξ i defines a compactification divisor D i . Such functions are called canonical parameters in section 6.3 of [21]. It is important to note that, for singular integrands such as those of I 1 and I 3 , it is not sufficient to define the canonical parameter ξ up to an overall factor, which may include nilpotent terms. For example, if we attempt to rescale ξ = (1 + θφ)ξ , then log ξ = log ξ + θφ, so that the Berezin integral dθ dφ log ξ does not coincide with dθ dφ log ξ .
In the small-u region, the proper choice of the canonical parameter ξ sym is dictated by our parametrization of the symmetric degeneration: we must take ξ sym = p 3 , as defined in Eq. (4.12), in order to properly glue together the two regions. Although p 3 and the crossratio u vanish at the same point, they are related by a non-trivial rescaling at leading order in the multipliers. Indeed which affects the Berezin integral of Eq. (4.48), as discussed above. Note that, not having introduced a parametrization for the separating degeneration, we would not have a similar guideline in the small-y region. Furthermore, the fact that the corresponding field theory diagram needs to be regulated 9 in order to make sense of the vanishing momentum flowing in the intermediate propagator introduces an ambiguity also in the field theory result. With this choice of parametrization, we can now use Stokes' theorem to determine the values of I 2 and I 3 . Taking ξ sep = y as a canonical parameter for the separating degeneration, we find where we used dθ dφ θφ = −1. Similarly Inserting these results into Eq. (4.46), discarding the separating degeneration, and introducing the overall normalization given in Eq. (4.37), we obtain our final expression for the contribution of diagrams with a four-point vertex to the field-theory effective action. It is given by In order to identify the contributions of individual Feynman diagrams to Eq. (4.52), we can retrace the steps of the calculation and assign each term in our result to the appropriate world-sheet conformal field theory, as we did for the symmetric degeneration in Eq. (4.41). We find that we can rewrite Eq. (4.52) as where here the superscripts denote the powers of k 1/2 i (i = 1, 2) from which the coefficients were extracted, and we have omitted the arguments of the functions f for simplicity. The precise identification is A few remarks are in order. First of all we note that f 11 gh vanishes; this corresponds to the fact that, in the infinite product in F gh (k i , η) in Eq. (3.11), n ranges from 2 to ∞, not from 1 to ∞ as in the case of F gl and F scal . As a consequence, there is no term proportional to k 1/2 (S 1 S 2 ) in the partition function for the ghost systems. We will see that this corresponds to the fact that there is no quartic ghost vertex in the associated Yang-Mills theory. Next, we note that all terms associated with the four-point vertex diagram are not factorizable into the product of two contributions, proportional to k 1/2 1 and k 1/2 2 respectively. If, on the other hand, we had traced the origin of the terms associated with the separating degeneration, and proportional to the integral I 1 , we would have found that the factor multiplying 1/y in Eq. (4.46) can be written as This means that no contributions arise from the Schottky group elements S 1 S 2 and S −1 1 S 2 , which would imply a genuine correlation between the two loops. Rather, as expected, these terms are simply the product of factors rising from individual disconnected loops. Finally, we note that the result I 3 = 0 is crucial in order to recover the correct field theory limit: indeed, as will be verified in the next section and shown in Appendix C, no field theory diagram yields hyperbolic functions with the parameter dependence displayed on the last line of Eq. (4.46). We see once again that the field theory limit, once the contributions of individual diagrams have been identified, provides non-trivial checks of the procedures used to perform the integration over super-moduli.

Yang-Mills theory in the Background Field Gervais-Neveu gauge
In order to make a precise comparison between string theory and field theory at the level of individual Feynman diagrams, as was done in a simple case in Ref. [11], we need a precise characterization of the field-theory Lagrangian we are working with, including gauge fixing and ghost contributions. In principle, this presents no difficulties, since our target is a U (N ) Yang-Mills theory, albeit with a rather special gauge choice. There are however a number of subtleties, ranging from the special features of the background field framework, to issues of dimensional reduction, and to the need to break spontaneously the gauge symmetry in order to work with well-defined Feynman diagrams in the infrared limit, which altogether lead to a somewhat complicated and unconventional field theory setup. We will therefore devote this section to a detailed discussion of the field theory Lagrangian which arises from the field theory limit of our chosen string configuration. The first layer of complexity is due to the fact that the string theory setup naturally corresponds to a field theory configuration with a non-trivial background field. In general, such a background field breaks the gauge symmetry: in our case, since we are working with mutually commuting gauge fields with constant field strengths, and we have a string configuration with separated D-brane sets, one will generically break the U (N ) gauge symmetry down to U (1) N . We will have to adjust our notation to take this into account. Notice also that our background fields break Lorentz invariance as well, since only certain polarizations are non-vanishing. As a consequence, the polarizations of the quantum field will also be distinguished as parallel or perpendicular to the given background.
Furthermore, it is interesting to work in a generic space-time dimension d, and we will find it useful to work with massive scalar fields giving infrared-finite Feynman diagrams. We will therefore work with a d-dimensional gauge theory obtained by dimensional reduction from the dimension D > d appropriate to the string configuration. This yields n s = D − d adjoint scalar fields minimally coupled to the d-dimensional gauge theory, and we will choose our background fields such that these fields acquire a non-vanishing expectation value, giving mass to some of the gauge fields.
Finally, as suggested originally in Ref. [45], and recently confirmed by the analysis of Ref. [11], covariantly quantized string theory picks a very special gauge in the field theory limit: a background field version of the non-linear gauge first introduced by Gervais and Neveu in Ref. [16]. This gauge has certain simplifying features: for example at tree level and at one loop it gives simplified color-ordered Feynman rules which considerably reduce the combinatoric complexity of gauge-theory amplitudes [45]. Only at the twoloop level, however, the full complexity of the non-linear gauge fixing becomes apparent. One effect, for example, is that the diagonal U (1) 'photon', which ordinarily is manifestly decoupled and never appears in 'gluon' diagrams, in this case has non-trivial, gaugeparameter dependent couplings to SU (N ) states, and the decoupling only happens when all relevant Feynman diagrams are summed.
In what follows, we adopt the following notations: we use calligraphic letters for matrix-valued u(N ) gauge fields, and ordinary capital letters for their component fields; we use M, N, . . . = 1, . . . , D for Lorentz indices in D-dimensional Minkowski space before dimensional reduction, and µ, ν, . . . = 1, . . . , d for Lorentz indices in the d-dimensional reduced space-time; finally, I, J, · · · = 1, . . . , n s indices enumerate adjoint scalars, and A, B, . . . = 1, . . . , N indices enumerate the components of u(N ) vectors and matrices. In this language, A M will denote the D-dimensional classical background field, Q M the corresponding quantum field, while C and C are ghost and anti-ghost fields.
We will now proceed to write out the quantum lagrangian (including gauge-fixing and ghost terms) in terms of matrix-valued fields. We will then comment on the form taken by various terms in component notation, which is more directly related to the vertices appearing in diagrammatic calculations.

The u(N ) Lagrangian
We begin by constructing the D-dimensional Yang-Mills Lagrangian, which, in the presence of a background gauge field, depends on the combination A M + Q M . The fieldstrength tensor F M N can be expressed in terms of the covariant derivative of the quantum field with respect to the background field, where F M N (A) is the field strength tensor for the background field only, while D A+Q M is the covariant derivative with respect to the complete gauge field. The classical Lagrangian for the quantum gauge field Q can then be written as where Tr here denotes the trace over the u(N ) Lie algebra, and we have removed terms independent of Q, as well as terms linear in Q, because they are not relevant for our effective action calculation. In anticipation of the string theory results, we now wish to fix the gauge using a background field version of the non-linear gauge condition introduced by Gervais and Neveu in Ref. [16], setting where γ is a gauge parameter. The gauge-fixing Lagrangian L gf is then given by Notice that the overall covariant gauge-fixing parameter which would appear in front of Eq. (5.4) has been set equal to one. Note also that this gauge fixing modifies not only the gluon propagator, as expected, but also the three-and four-gluon vertices. In particular, the symmetric nature of the quadratic term in the gauge-fixing function G will generate Feynman rules involving the symmetric u(N ) tensors d abc , which in turn will induce spurious couplings between gluons and u(1) photons. Finally, we need the Lagrangian for the Faddeev-Popov ghost and anti-ghost fields, C and C. It is defined as usual in terms of the gauge transformation of the gauge-fixing function, as using C as parameter of the gauge transformation. The result is This completes the construction of the pure Yang-Mills Lagrangian in D dimensions; next, we want to dimensionally reduce it to d dimensions. The reduction splits the Ddimensional gauge fields (both classical and quantum) into a d-dimensional field and n s ≡ D − d adjoint scalars, according to with µ = 0, . . . , d−1 and I = 1, . . . , n s , and we have assumed that the classical background scalars take on constant values M I , which we will use to spontaneously break the gauge symmetry and give masses to selected components of the gauge field. Similarly, since we are neglecting the dependence of the fields on the reduced coordinates, the covariant derivative splits into a d-dimensional covariant derivative and a pure commutator with the background scalar fields, as Indeed, the D-dimensional d'Alembertian differs from the d-dimensional one by a mass term: for any field X, Notice that in this section we work with the metric η = diag(+, −, . . . , −). However, when summing over reduced dimensions, our summation convention does not include the negative signature of the metric, and must be understood simply as a summation over flavor indices I. With these conventions, the gauge condition in Eq. (5.3) becomes When these further changes are implemented in the Lagrangian, a number of non-trivial interaction vertices are generated. It is then useful to organize the dimensionally-reduced Lagrangian as a sum of terms with different operator content. One can write As is typical in cases of broken symmetry, the Lagrangian in Eq. (5.11) displays a variety of interactions, and is considerably more intricate than the combination of Eqns. (5.2), (5.4) and (5.6). In order to compute Feynman diagrams, and to compare with the string theory calculation, it is useful to write down an expression for the Lagrangian in terms of component fields as well. In order to do so, we now assume that the matrices A µ and M I are all mutually commuting: we can then pick a basis of u(N ) in which they are diagonal.
In this basis, we write Similarly, we write the quantum matrix fields as all satisfying X AB = (X BA ) * , since u(N ) matrices are Hermitian; the factors of 1/ √ 2 ensure that the matrix element fields are canonically normalized. Notice that, thanks to diagonal form of the classical field A µ , the covariant derivative D µ does not mix matrix entries. Indeed, defining one can write where indices on the right-hand side are not summed. In particular, the covariant derivative of diagonal entries reduces to the ordinary derivative. Motivated by this, we can define a covariant derivative D µ acting directly on matrix entries, as opposed to u(N ) elements. Suppressing the A, B indices on the derivative symbol, we write Note that D µ is a derivation, obeying the Leibnitz rule where again on the right-hand side the indices A and B are fixed and not summed. As an example, the term quadratic in Φ in Eq. (5.11) can be written in component notation as which is the correctly normalized quadratic part of the Lagrangian for N massless real scalar fields φ AA , and 1 2 N (N − 1) complex scalars φ AB , A < B, with mass |m AB |. To give a second example, the gauge-fixing condition in component notation reads where C is summed over but there is no summation over A or B. Note that after dimensional reduction and spontaneous symmetry breaking the gauge fixing has become more unconventional from the d-dimensional point of view, involving scalar fields as well as gauge fields, and mass parameters. We conclude this section by giving the explicit expression for the background field that we will be working with. We choose it so that, for each A, the abelian field strength (1) magnetic field in the {x 1 , x 2 } plane. A possible choice, already employed in Ref. [11], is where we defined the antisymmetric tensor We now turn to the evaluation of selected two-loop vacuum diagrams, contributing to the effective action, which we can then compare with the corresponding expressions derived from string theory. Preliminarily, we collect useful expressions for the relevant coordinatespace propagators in the presence of the background field.

Propagators in a constant background field
The quantum field theory objects that we wish to compute, in order to compare with string theory results, are two-loop vacuum diagrams contributing to the effective action, and computed with our chosen background field, Eq. (5.22). At two loops, these diagrams can be computed in a straightforward manner in coordinate space, directly from the path integral definition of the generating functional, where J µ , J I , η and η are matrix sources for the fields in the complete Lagrangian L. The only non-trivial step is the computation of the quantum field propagators in the presence of the background field: diagrams are then simply constructed by differentiating the free generating functional with respect to the external sources. For a background field of the form of Eq. (5.22), the solution is well-known for the scalar propagator (see, for example, Ref. [15]): we briefly describe it here, and discuss the generalization to vector fields. For scalar fields, the propagator in the presence of the background in Eq. (5.22) can be expressed in terms of a heat kernel as In Eq. (5.26), we have introduced the tensor where the projectors η µν and η µν ⊥ identify components parallel and perpendicular to the background field, and are given by The propagator G AB (x, y) in Eq. (5.25) satisfies where we noted explicitly the variable on which the derivatives act. In fact, covariant derivatives act on a propagator with color indices (AB) as For real scalar fields, or vanishing backgrounds, one recovers the well-known expression for the scalar propagator as a Schwinger parameter integral, Ghosts fields are scalars, and they share the same propagator. For gluons, on the other hand, the background field strength F µρ enters the kinetic term, given in the first line of Eq. (5.11). The propagator must then satisfy To diagonalize this equation, one can introduce the projection operators It is then easy to show that the function satisfies Eq. (5.32), provided the functions G AB ± (x, y) satisfy Eq. (5.36) simply gives a scalar propagator with a mass shifted by the appropriate background field. It's easy therefore to write the solution for the complete gluon propagator explicitly as (5.37) Note that this can be written also in the more compact and elegant form where S αβ 1 are the Lorentz generators in the spin one representation appropriate for gauge bosons, The propagator in Eq. (5.38) naturally generalizes to other representations of the Lorentz group, simply changing the form of the generators. The spin one-half case, where S αβ 1/2 = i[γ α , γ β ]/4, will be useful for example when studying the gluino contribution to the effective action in the supersymmetric case.

Selected two-loop vacuum diagrams
We will now illustrate the structure of the field theory calculation of the effective action by outlining the calculation of a selection of the relevant two-loop diagrams. A complete list of the result for all 1PI diagrams depicted in Fig. 1 is given in Appendix C.
We begin by considering the ghost-gluon diagram given by the sum of Eq. (C.10) and Eq. (C.11), which we denote by H b (B AB , m AB ). The relevant interaction vertex, involving ghost, antighost and gluon fields, arises from the sixth line in Eq. (5.11), and may be written explicitly in component language using Eq. (5.13). Upon integrating by parts, it can be rewritten as Sewing two copies of this vertex together to obtain the desired diagram, one first of all observes that terms linear in the gauge parameter γ, which involve double derivatives of scalar propagators, cancel out upon contracting color indices. Next, one notices that some of the color contractions lead to a non-planar configuration, which would correspond to an open string diagram with only one boundary. We are not interested in these contributions, since the corresponding diagram is built of propagators which are neutral with respect to the background field, and does not contribute to the effective action. Furthermore, we do not expect to obtain this diagram from our string configuration, since we start with a planar worldsheet. Discarding non-planar contributions, one finds that the remaining planar terms can be written as where Σ(gBt) is defined in Eq. (5.27). The integrand in Eq. (5.41) is then proportional to the product of three heat kernels, which we write as .
In Eq. (5.43) we have simplified the notation by using a single index i = 1, 2, 3 in place of the pairs of color indices (AB), (BC), (CA), respectively. Furthermore, we have defined Σ µν = 3 i=1 Σ µν (gB i t i )/t i , and we have taken advantage of the fact that the complex phases in each K i (x, y; t i ) cancel due to the fact that 3 i=1 B i = 0. At this point one sees that the integrand in Eq. (5.41) is translationally invariant, depending only on the combination z = x − y. We can then, for example, replace the integral over x with an integral over z while the integral over y gives a factor of the volume of spacetime, which we will not write explicitly. One needs finally to evaluate the gaussian integral Note that taking the inverse of Σ µν is trivial because it is a diagonal matrix, which can be written as , (5.45) where ∆ 0 and ∆ B were defined in Eq. (4.28) and Eq. (4.30) respectively, while η ⊥ µν and η µν are given in Eq. (5.28). One finds then Putting together all these ingredients, one may evaluate Eq. (5.41). Using the symmetry of the Schwinger parameter integrand under the exchange t 1 ↔ t 2 one can write which can be directly matched to the string theory result. We conclude this section by briefly describing the calculation of two further Feynman diagrams, which arise in our theory because of the pattern of symmetry breaking and dimensional reduction.
Labeling this diagram as H d (B AB , m AB ), we find for it the coordinate space expression where we relabeled double indices as was done for Eq. (5.47). Finally, we briefly consider a diagram with a quartic vertex: the figure-of-eight scalar self-interaction shown in Eq. (C.23), which we label E i (B k , m k ). The relevant interaction term in the Lagrangian comes from L Φ 4 in Eq. (5.11) and can be written as which immediately gives Contracting flavor indices we get, as expected, the product of two one-loop integrals, The diagrams in Eqs. 5.1 -C.22 can be calculated similarly. One easily sees that all these results, and those for the remaining diagrams, given in Appendix C, are directly comparable with the ones obtained from the field theory limit of the string effective action.
6 Discussion of results

Comparison between QFT and string theory
We have now assembled all the results that we need to establish and verify a precise mapping between the degeneration limits of the string world-sheet and the 1PI Feynman diagram topologies in the field theory limit. Furthermore, as announced, we can trace the contributions of individual string states propagating in each degenerate surface, and these can be unambiguously mapped to space-time states propagating in the field theory diagrams. This diagram-by-diagram mapping allows us in particular to confirm that covariantly quantized superstring theory 10 naturally selects a specific gauge in the field theory limit, and the gauge condition is given here in Eq. (5.3). More precisely, our string theory results for the symmetric degeneration are given in Eq. We finally note that we can also characterize the contributions to all 1PI Feynman diagrams according to the Schottky multiplier they originate from. As an example, consider the infinite product over the Schottky group which arises from the determinant of the non-zero modes of the Laplacian, appearing in Eq. (3.11). Tracing the gluon contributions to different Feynman diagrams back to that product, one may verify that all terms appearing in 1PI diagrams originate from at most one value of the index α in the product. More precisely, gluon contributions in Eq. (4.41) come from T α = {S 1 , S 2 , S −1 1 S 2 }, and if, say, a factor of k 1/2 1 = √ p 1 √ p 3 comes from the infinite product, then the necessary factor of √ p 2 must come from elsewhere in the amplitude. On the other hand, all terms in Eq. (4.53) come from T α = S 1 S 2 . This is not surprising from a world-sheet point of view: in fact, one may recall from Fig. 5 that S 1 S 2 corresponds to a homology cycle which passes around both handles with a self-intersection between them, and furthermore this is the only Schottky group element with this property which survives in the field theory limit. We see that this homological property is directly related to the graphical structure of the resulting Feynman diagram.

Comparison with bosonic string theory
It is interesting to compare our results with the field theory limit of the bosonic string effective action which was studied in Refs. [35,15,67]. This comparison was discussed also in [11], but we are now in a position to make a more detailed analysis. Bosonic strings are clearly a simpler framework, since the world-sheet is an ordinary two-dimensional manifold, and not a super-manifold: one can then use the techniques applying to ordinary Riemann surfaces, and specifically the (purely bosonic) Schottky parametrization discussed in detail for example in Ref. [11]. At two loops, one can use the SL(2, R) invariance of the amplitude to choose the fixed points of the two Schottky group generators as The two-loop partition function can then be written as where µ denotes the set of bosonic moduli, µ = {k 1 , k 2 , u}, and one may compare with the corresponding two-loop superstring expression, given in Eq. (3.18). Note that the integration variable u is equal to the gauge-fixed value of a projective-invariant cross ratio of fixed points, u = η 1 −η 2 The various factors in Eq. (6.2), already discussed in [15,11], are given by Here τ is the period matrix of the Riemann surface, whose expression in the Schottky parametrization can be found, for instance, in Eq. (A.14) of [31]. Similarly, τ is the twisted period matrix, the bosonic equivalent of τ , computed here in Appendix B. The most obvious difference between the measures in Eq. (3.18) and Eq. (6.2) is the occurrence of half-integer powers of the multipliers in the former. In the bosonic string, the mass level of states propagating in the i-th loop increases with the power of k i , whereas in the superstring it increases with the power of k 1/2 i . Necessarily, the propagation of a massless state must correspond to terms of the form dk i /k i = d log k i in the integrand, so tachyons propagating in loops correspond to terms of the form dk i /k 2 i in the bosonic theory and dk i /k 3/2 i in the superstring, as seen explicitly in Eq. (6.2) and in Eq. (3.18), respectively. These tachyonic states must be removed by hand in the bosonic theory, whereas they are automatically eliminated from the spectrum of the superstring upon integrating over the odd moduli and carrying out the GSO projection.
The identification of the symmetric degeneration proceeds in the same way for the two theories: in particular, the symmetry of Fig. 6a leads to the choice of the parameters p i , defined by Eq. (4.12). The cross-ratio u can then be written as 4) and the integration measure takes the symmetric form It is interesting to note that in the field theory limit a number of contributions arise in slightly different ways in the two approaches. As an example, let us consider the twisted determinant of the period matrix for the bosonic string. To lowest order in k i , it is given by a combination of hypergeometric functions with argument u, in a manner similar to what happens for its supersymmetric counterpart. In the neighborhood of the symmetric degeneration, the hypergeometric functions can be expanded in powers of p 3 , and the bosonic string determinant reduces to where ∆ B is defined in Eq. (4.30). We note that the term proportional to p 3 in Eq. (6.6) receives a contribution from the series expansion of the hypergeometric functions, and contributes to Feynman diagrams with a gluon polarized parallel to the magnetic field propagating in the leg parametrized by t 3 .
For the superstring, the situation changes: one needs to keep terms only up to order q i , which implies that all the hypergeometric functions appearing in the expression for the supersymmetric twisted determinant can be replaced by unity. Since the first-order term in the expansion of the hypergeometric functions is crucial in order to get the correct coefficient of p 3 in Eq. (6.6), and in turn to match the field theory diagrams, it is necessary that terms proportional to q 3 arise from the nilpotent contributions to det (Im τ ). This is indeed what happens: expanding the supersymmetric twisted determinant in powers of q i one finds To be precise, we note that terms proportional to p 3 and q 3θ12φ12 in det (Im τ ) and det (Im τ ), respectively, also receive contributions from sources other than the ones we have discussed, specifically from factors of the form u n i i /2 , with n i integers. It is easy to see, however, that these contribute in the same way in the two cases, since in the bosonic case we have u n i i /2 = p while in the superstring case we get As a consequence, and as required, when all of the other factors are inserted, the coefficient of p 1 p 2 p 3 in the bosonic string measure is the same as the coefficient of q 1 q 2 q 3θ12φ12 in the superstring measure, and the same field theory amplitude is obtained for the massless sectors of the bosonic and supersymmetric theories. The terms computed in section 4.5, which correspond to field theory diagrams with the topology of the diagrams in the bottom two rows of Fig. 1 as well as 1PR graphs, also appear in the bosonic theory. In fact, one gets once more an expression of the form of Eq. (4.46) in the field theory limit, but the integrals I 1 , I 2 and I 3 get replaced bỹ (6.10) When using the bosonic string, the integralĨ 3 has to be discarded by hand, either by arguing that it corresponds to tachyon propagation, or by explicitly matching to the field theory result. In the case of the superstring, on the other hand, the correct result emerges automatically, provided a consistent integration procedure in super-moduli space is followed. The complete answer for the four-point vertex diagrams emerges in both cases from the terms proportional to I 2 =Ĩ 2 = 1. As in the superstring case, the contributioñ I 1 is related to the sepating degeneration.
The group of transformations which preserves the skew-symmetric quadratic form is OSp(1|2), which can be realised by GL(2|1) matrices of the form where the five even and four odd variables are subject to the two odd and two even constraints, so that the group has dimension 3|2. We can define a map from homogeneous coordinates to superconformal coordinates by Then, any other map of the form f • S, with S an OSp(1|2) matrix, also defines superconformal coordinates. Recall [75] that two C 1|1 charts (z|θ) and (ẑ|θ) belong to the same superconformal class whenever D θẑ =θ D θθ , where is the super derivative which satisfies D 2 θ = ∂ z . In particular, we can cover CP 1|1 with two superconformal charts z 1 = f (z 1 , z 2 |θ) t and z 2 = (f • I) (z 1 , z 2 |θ) t , where I is the OSp(1|2) matrix In general, one can find an OSp(1|2) matrix taking two given points |u = (u 1 , u 2 |θ) t and |v = (v 1 , v 2 |φ) t to |0 ≡ (0, 1|0) t and |∞ ≡ (1, 0|0) t ∼ I|0 respectively; one such matrix is We can further stipulate that a point |w = (w 1 , w 2 |ω) t be mapped to a point equivalent to (1, 1|Θ uwv ) t , where now there is no freedom in choosing the fermionic co-ordinate, which is therefore a super-projective invariant built out of the triple {|u , |v , |w }. The image of |w under Γ uv is then A general dilatation of the superconformal coordinates corresponds to the OSp(1|2) matrix 12 which has |0 and |∞ as fixed points. Note that for |ε| < 1, |0 is an attractive fixed point and |∞ is a repulsive fixed point.
We may use such a dilatation to scale the bosonic coordinates of Γ uv |w as desired, obtaining for example which gives us an explicit expression for the odd super-projective invariant Θ z 1 z 2 z 3 , as where z i = (z i |ζ i ), as in Eq. (3.4) of [64] and Eq. (3.222) of [20]. As with projective transformations, super-projective transformations preserve crossratios of the form one must keep in mind, however, that the simple relations between the three possible cross ratios that can be constructed with four points are modified by nilpotent terms. For example, one finds that which can be checked quickly by noting that the left-hand side is OSp(1|2)-invariant, so that one can fix 3|2 parameters, for example by choosing |z 1 = |0 , |z 2 = |∞ , and |z 4 = (1, 1|φ) t . With these ingredients, it is now easy to construct a super-projective transformation with chosen fixed points and multiplier: using Γ uv to map a pair of points |u and |v to |0 and |∞ respectively, one easily verifies that the transformation has |u as an attractive fixed point and |v as a repulsive fixed point. Here k, for which we take |k| < 1, is called the multiplier 13 of the super-projective transformation S.
The bracket notation has the benefit of allowing us to write S as

A.2 The super Schottky group
Taking the quotient of CP 1|1 by the action of S, defined in Eq. (A. 16), is equivalent to the insertion of a pair of NS punctures at |u and |v , which are then sewed with a sewing parameter related to k. To see this, recall that sewing of NS punctures at P 1 and P 2 is defined by taking two sets of superconformal coordinates, say (x|θ) and (y|ψ), which vanish respectively at the two points, (x|θ)(P 1 ) = (0|0) = (y|ψ)(P 2 ), and then imposing the conditions [9] xy = − ε 2 , yθ = εψ , xψ = − εθ , θψ = 0 . where P and I are defined in Eq. (A.10) and Eq. (A.7), respectively. Then, by acting on both sides with f, we get (x|θ) ∼ (−ε 2 /y εψ/y), which can easily be found to satisfy Eq. (A.18). Let us take (z|ζ) to be a superconformal coordinate on CP 1|1 , with (z|ζ)(P 1 ) = f|u and (z|ζ)(P 2 ) = f|v . Recall that the super-projective transformation Γ uv defined in Eq. (A.8) simultaneously maps |u and |v to |0 and |∞ , respectively. Then if |x = Γ uv • f −1 • (z|ζ) and |y = I −1 • Γ uv • f −1 • (z|ζ), we have that (x|θ) = f|x and (y|ψ) = f|y are local superconformal coordinates which vanish at P 1 and P 2 respectively, since I −1 |∞ = |0 and f|0 = (0|0). As a consequence, we can perform a NS sewing by making the identification in Eq. (A.19) using these expressions for |x and |y , and we find that we need to impose an equivalence relation on (z|ζ): This is what we wanted to show, with S matching the definition in Eq. (A. 16), as long as we identify ε = −e iπς k 1/2 , so the NS sewing parameter is directly related to the Schottky group multiplier. Topologically, this sewing has the same effect (at least on the reduced space CP 1 ) as cutting out discs around u and v and identifying their boundaries, so this quotient adds a handle to the surface, increasing the genus by one.
To build a genus-h SRS, we may repeat this sewing procedure h times, choosing h pairs of attractive and repulsive fixed points u i = (u i |θ i ), v i = (v i |φ i ), and h multipliers k i , for i = 1, . . . , h. The super-Schottky group S h is the group freely generated by We then subtract the limit set Λ (the set of accumulation points of the orbits of S h ) from CP 1|1 , and we quotient by the action of the super Schottky group. this leads to the definition Note that the fixed points must be sufficiently far from each other, and the multipliers sufficiently small, to allow for the existence of a fundamental domain with the topology of CP 1|1 with 2h discs cut out. The fixed points u i , v i and the multipliers k i are moduli for the surface, but for h ≥ 2 we can use the OSp(1|2) symmetry to fix 3|2 of these: in our conventions, we take |u 1 = |0 , |v 1 = |∞ , |v 2 = |1, 1|Θ u 1 v 2 v 1 , so the super-moduli space M h has complex dimension 3h − 3|2h − 2.
To build multi-loop open superstring world-sheets in a similar way, we should start with the super-disc D 1|1 which can be obtained by quotienting CP 1|1 by the involution (z|θ) → (z * |θ), so that RP 1|1 becomes the boundary of the disk. A super-projective map will be an automorphism of D 1|1 if it preserves RP 1|1 , so we should build the super Schottky group from super-projective transformations whose fixed points u i , v i are in R 1|1 and whose multipliers k i are real. If we quotient D 1|1 − Λ by h of these, then we will get a SRS with (h + 1) borders and no handles. α . We can find k 1/2 α by setting the spin structure around the b-cycles to zero, ς = 0, then using the cyclic property of the supertrace 14 . This leads to a quadratic equation, with roots one root being the inverse of the other. We then pick k 1/2 α to be the root whose absolute value satisfies |k where y was defined in Eq. (3.19). Note that k 1/2 (S −1 1 S 2 ) can be obtained from k 1/2 (S 1 S 2 ) by swapping the attractive and repulsive fixed points of S 1 in the cross-ratio, as might be expected.

A.2.2 The super period matrix
The super abelian differentials are an h-dimensional space of holomorphic volume forms, i.e. sections of the Berezinian bundle, defined on a genus-h SRS. They are spanned by Ω i , i = 1, . . . , h, which can be normalized by their integrals around the a-cycles, according to while their integrals around the b-cycles define the super period matrix Here a i and b i are closed cycles on the SRS which are projected to the usual homology cycles on the reduced space. The Ω i 's can be expressed in terms of the super Schottky parametrization as in Eq. (21) of Ref. [30]. In our current notation where dz = (dz | dψ), the sum (i) α is over all elements of the super-Schottky group which do not have S ±1 i as their right-most factor, D ψ is the superconformal derivative D ψ = ∂ ψ + ψ∂ z , and finally Φ is the matrix The matrix Φ has the property that, if f |z = (z|ψ), then for any w|, and furthermore for |w = (w, 1|ω) t and |z = (z, 1|ψ) t the map (z|ψ) → w|z | w|Φ|z is superconformal. The super period matrix can be computed as Figure 7: Recall that the double of a Riemann surface Σ is defined by taking two copies of Σ, replacing the charts on one copy with their complex conjugates, and identifying corresponding points on the boundaries of the two copies [24].
The sum is over all elements of the super Schottky group which do not have S ±1 j as their left-most element or S ±1 i as their right-most element. It is not difficult to compute the leading terms of the super period matrix in the small-k i expansion. For h = 2, using the fixed points in Eq. (3.16), we find This completes our review of the super Schottky parametrization. Our next task is to introduce twisted boundary conditions corresponding to external background gauge fields.

B Appendix B B.1 The twisted determinant on a Riemann surface
The worldsheet theory of strings becomes 'twisted' in a number of contexts: for example, on orbifolds [76], in electromagnetic fields [77,13,14] or when an open string is stretched between a pair of D-branes which have a velocity [78] or are at an angle [79] with respect to each other. If we appropriately pair up the string spacetime coordinate fields X µ as complex coordinates (e.g. in our case, by setting Z ± = (X 1 ± iX 2 )/ √ 2), then in these backgrounds the worldsheet fields ∂Z ± are described by non-integer mode expansions on the upper-half-plane, as in Eq. (2.5). This means that on the double of the worldsheet (the double of the upper-half-plane is the complex plane; see Fig. 7), ∂Z ± (z, z) is no longer a single-valued field but rather it has a monodromy, changing by a factor of e ±2πi as it is transported anti-clockwise around z = 0.
Computing multi-loop amplitudes in these backgrounds is complicated because it is not easy to use the sewing procedure when states propagating along plumbing fixture belong to a twisted sector. We must use, instead, the approach of [67]. This takes advantage of the fact that although the ∂Z ± fields have non-trivial monodromies along the a i -cycles of the double worldsheet, the monodromies along the b j -cycles are trivial. Therefore, the idea is to build the double worldsheet by sewing along the b i cycles, and then to perform the modular transformation swapping the a j and b i cycles with each other, in order to obtain the partition function expressed in terms of the Schottky moduli which are the appropriate ones for the worldsheet degeneration we are interested in.
From a more physical point of view, we are using the fact that in a different region of moduli space, the string diagram can be described as a tree-level interaction between three closed strings being emitted or absorbed by the D-branes. In terms of the closed string moduli, the string partition function is given by [67] The overall factor is just the Born-Infeld lagrangian for the background fields on the D-branes, divided by √ G because all of the background-field independent factors are included in the measure [dZ] cl h . The factor R h (q i , ), which is dependent on both the worldsheet moduli and on the background field strengths, has a simple form so long as it is expressed in terms of the closed string Schottky group moduli, in other words, in terms of the multipliers of a Schottky group whose 2h Schottky circles are homotopic to the b i cycles of the worldsheet instead of the a j cycles which we have been using. Let us denote the multipliers of the elements T α of this Schottky group as q α , then we have where the notation α has the same meaning as for the super Schottky group case, defined after Eq. (3.7). The modular transformation that swaps the a i -and b j -cycles, necessary to switch between the open string and closed string channels, acts non-analytically on the Schottky group multipliers. 15 We need to rewrite Eq. (B.2) in terms of the open string moduli, so the following strategy is used: R h (q i , ) is re-expressed in terms of functions which transform in simple ways under modular transformations, the modular transformations are then carried out, and finally the results are re-expressed in terms of the open string Schottky moduli, allowing us to investigate the field theory limit. This analysis was performed in [33,35] and the results are summarized in section 2 of [15]. Assuming without loss of generality that the h'th twist is nonzero, h = 0, the result is that where R h (k α , ·τ ) is the same as in Eq. (B.2) but with the closed string channel multipliers q α replaced with open string channel multipliers k α , and the twists replaced with · τ , (τ ij ) being the period matrix computed in the open string channel. τ is the twisted period matrix, defined by (Eq. (3.24) of [67]) The Prym differentials Ω i appearing in the integrand in the first line of Eq. (B.4) are (h − 1) 1-forms with trivial monodromies along the a i cycles and twists along the b icycles, i.e. they obey Ω i (S j (z)) = S j Ω i (z) ; (B.6) and they are regular everywhere. 16 Assuming without loss of generality that h = 0, they can be expressed as (Eq. (3.11) of [67]): In Eq. (B.7) the ζ i are a basis of h 1-forms which are holomorphic everywhere except some arbitrary base point z 0 , which can be computed in terms of the Schottky group as (Eq. (3.15) of [67]) where the first sum is over all Schottky group elements which don't have S ± i as their right-most factor and the second sum is over all Schottky group elements. η i and ξ i are the attractive and repulsive fixed points of the generator S i , respectively. Note that the dependence on z 0 cancels out when the ζ i are combined as in Eq. (B.7). Also in Eq. (B.8), The other object appearing in the integrand in the first line of Eq. (B.4), ∆ i (z), is the vector of Riemann constants or Riemann class; it can be expressed in the Schottky parametrization as (Eq. (A.21) of [31]) where the second sum where ω i are the abelian differentials (Eq. (A.10) of [31]), it is easy to check that the integrand of the first line of Eq. (B.4) has twists along the a i cycles and trivial monodromies along the b i -cycles (therefore the integrand does not depend on the starting point w). For simplicity, from now on we focus only on the case h = 2, which yields det(Im τ ) = 1 2πi w Ω ·τ (z)e 2πi · ∆(z) , (B.12) where Ω ≡ Ω 1 is the sole component of the Prym form. Instead of explicitly evaluating the integral over z in Eq. (B.12), it is possible to find an alternative expression for det(Im τ ) in the following way. First of all, we recall the object D( ) ij defined in Eq. (3.14) of [67]. For each i, j = 1, . . . , (h − 1), D ij ( ) is a spacetime rotation matrix; the i, j indices refer to worldsheet homology cycles. In the complex spacetime coordinates in which the background fields are diagonalized, D ij ( ) is diagonal with two non-trivial entries D(± ) ij . For h = 2, the i, j indices can take only one value, so we only have one independent object D 11 (± ) ≡ D(± ). It is given by where γ P ≡ a 2 a 1 a 2 −1 a 1 −1 is the Pochhammer contour shown in Fig. 9, and Ω is the Prym differential which has trivial monodromies around the b i homology cycles and monodromies S i around the a i homology cycles. Ω can be expressed with the Schottky group a 1 a 2 a 1 a 2 γ P Figure 9: The Pochhammer contour γ P = a 2 a 1 a 2 −1 a 1 −1 .
γ P η 1 η 2 Figure 10: Our Pochhammer contour (Fig. 9) can be deformed arbitrarily close to four copies of the line interval [η 1 , η 2 ] ⊆ R, with each copy on a different branch of the Prym form Ω .
thanks to a relation derived in [67], given by Eq. (3.28) of that reference. In the case h = 2, it becomes simply where the second equality is just writing (τ ) 11 in terms of det(Im τ ) with Eq. (B.4). Note that γ P crosses each boundary of the worldsheet once in each direction so it starts and ends on the same branch of Ω and the integral in Eq. (B.13) is well-defined. Now we take the formulae Eq. (B.7), Eq. (B.8) and Eq. (B.10) to get an expression for Ω (z) via Eq. (B.14) and interpret them as defining a one-form not on the worldsheet but on the complex plane, with poles at the Schottky group limit set. If we expand Ω (z) as a power series in k i , we see that at leading order it has poles only at the Schottky fixed points, so at leading order we are free to deform the Pochhammer contour through the Schottky circles and arbitrarily close to the line interval [η 1 , η 2 ] as in Fig. 10. In this way, we can write the Pochhammer integral as four copies of a real integral taking care to account for the different orientations and branches; it turns out that we get

(B.21)
Now we need to define the objects appearing in Eq. (B.21). The Prym differentials Ω i we used to compute det(Im τ ) are holomorphic one-forms; the natural analogues on SRSs are holomorphic volume forms: sections of the Berezinian bundle. Just as holomorphic differentials can be written locally as dz ∂ z f (z), sections of the Berezinian can be written locally as dz D ψ f (z|ψ), the combination being invariant under change of superconformal coordinates [21]. We note that we can write equation Eq. (B.8) for ζ i as , so to find the corresponding SRS volume forms we replace the expressions inside the logarithms with their natural superconformal analogues and replace dz ∂ z → dz D ψ . This yields Then we can write down a basis of (h − 1) holomorphic volume forms Ω j (z) with the expected monodromies along the homology cycles using the analogue of Eq. (B.7), noting that the dependence on the base point |z 0 cancels out: We can calculate Ω j (z|ψ) as a series expansion in k 1/2 i . Truncating to finite order, we only need to sum Eq. (B.24) over finitely many terms of the super-Schottky group, because if the contribution from T α is O(k α 1/2 ) and the left-most factor of T α is not S ±1 i , then the contribution from S ± i T α is O(k i /2 k α 1/2 ). Restricting ourselves to h = 2, this means that if we only want to compute to order k i 1/2 then we only need to sum over the super-Schottky group elements T α ∈ {Id, S ±1 1 , S ±1 2 , (S 1 S 2 ) ±1 , (S −1 1 S 2 ) ±1 , (S 1 S −1 2 ) ±1 , (S 2 S 1 ) ±1 } . (B.27) Using the fixed points given in Eq. (3.16), we obtain the following expression for Ω (z) ≡ Ω 1 (z): (1 − S 1 ) 2 (1 + S 1 )(1 − u)θφψ S 1 (1 − S 2 )u(1 − z) 2
Note that the gauge choice γ 2 = 1 gives many of these diagrams a much simpler form, for example, the second lines of Eq. (C.7) and Eq. (C.9), the third line of Eq. (C.13) and the third and fourth lines of the Eq. (C.18) all vanish in this gauge. In fact, the last example is a special case of the fact that both propagators in the diagrams with quartic vertices must have the same polarization precisely when γ 2 = 1, which corresponds to the fact that k 1/2 1 and k 1/2 2 must be taken from the same CFT in string theory.