Localization of effective actions in open superstring field theory

We consider the construction of the algebraic part of D-branes tree-level effective action from Berkovits open superstring field theory. Applying this construction to the quartic potential of massless fields carrying a specific worldsheet charge, we show that the full contribution to the potential localizes at the boundary of moduli space, reducing to elementary two-point functions. As examples of this general mechanism, we show how the Yang-Mills quartic potential and the instanton effective action of a Dp/D(p − 4) system are reproduced.

In this paper we focus on the construction of tree-level effective actions for the massless states of the open superstring.Our starting point will be the WZW-like action by Berkovits [2], in the large Hilbert space.In this regards, an analytic construction of the Yang-Mills quartic potential has been given some time ago by Berkovits and Schnabl [13] 3 .
Focussing on the algebraic couplings between the massless fields up to quartic order, we find out that their computation is fully captured by singular terms at the boundary of moduli space.More precisely, when we are computing the tree-level effective potential of a massless string field Φ A which can be decomposed in eigenstates of a world-sheet charge J 0 with eigenvalues ±1 (an assumption which includes the non-abelian gauge field and also many other interesting cases such as the instanton moduli on a Dp-D(p−4) system, or D-branes moduli on a Calabi-Yau)) we find out that, because of the failure of the propagator to truly invert the BRST charge, the whole quartic potential localizes at the boundary of moduli space.If not for this singular term, the amplitude would identically vanish.Concretely, we show that the full effective potential at quartic order reduces to simple two-point functions S eff (Φ A ) = 1 8 h (−−) h (++) + g (+−) g (−+) + (+) ↔ (−) . (1. 3) The (h, g) fields are the projection to level zero of the following star products A ], J 0 = 0, (1.6) A ], J 0 = 0, (1.7) where P 0 is the projector on the kernel of L 0 and [ , ] is the graded commutator with respect to Witten star product.
This mechanism is interesting per se, but a very practical advantage is that the (h, g) fields can be readily computed by leading order OPE and one doesn't need to know the full four-point function of the massless field Φ A to compute its quartic potential.
The paper is organised as follows.In section 2 we set up the construction of the treelevel effective action in Berkovits WZW-like open superstring field theory and we focus on the quartic potential of the massless fields.We show that picture changing at zero momentum is subtle and in general develops singular contributions at the boundary of moduli space, which can in fact account for the full effective action.In section 3 we give a proof that the effective action is indeed localized at the boundary, when the string field entering the quartic potential can be expressed as the sum of two fields of opposite R charge in the N = 2 description of the string background under consideration.Section 4 contains two non-trivial examples of this localization mechanism: the quartic Yang-Mills potential and the quartic action for the instanton-moduli of a D3-D(−1) system.We conclude in section 5 with some comments and future directions.Appendix A contains the detailed computation of the localization of the action and Appendix B is a collection of our conventions and of useful formulas needed in the main text.

Tree level effective action
We start with the classical action for Neveu-Schwarz open strings on a given D-brane system in the large Hilbert space [2,16] in terms of the connections Here is the BRST charge of the RNS superstring, and η 0 = dz 2πi η(z) is the zero mode of the η-ghost, [17].Φ is the dynamical open string field which is a generic ghost and picture number zero state in the Large Hilbert Space Tr is Witten integration [19] in the large Hilbert space and * product is always understood when string-fields are multiplied.Finally the BRST charge Q B and η 0 are mutually anti-commuting graded derivations with respect to the star product and the Witten integration of Q B /η 0 −exact terms is identically zero.
Since we will follow a perturbative approach, it is convenient to expand the action (2.1) in "powers" of the dynamical string field Φ.To do so we observe that, given a star algebra derivative D, its t-dependent connection A D (t) can be explicitly written as where the adjoint action is defined through a graded * -commutator Using this representation for the two connections A t and A Q and performing the dt integral in (2.1) we end up with where we have isolated the kinetic term from the infinite tower of interaction vertices.We are interested in computing the tree-level effective potential for the zero momentum part of a massless field.To do so we fix the standard gauge b 0 Φ = 0, (2.8) which means (2.13) A prototype example of the kind of massless field we are interested in is the zero momentum part of the gauge field living on the world-volume of a Dp-brane, which in our setting is represented by the string field where A µ are constant Chan-Paton matrices in U(N).Notice that ψ µ is a superconformal matter primary of weigth 1/2 and it is in fact part of a worldsheet N = 1 superfield ψ µ + θ(i √ 2∂X µ ).The string field Φ A is on-shell in the large Hilbert space ) are the physical vertex operators at picture −1 and 0 respectively, in the small Hilbert space.This choice of Φ A can be generalized to a string field of the form where V 1 2 is a grassmann-odd superconformal matter primary of weight 1/2 This means that there exists a grassmann-even world-sheet superpartner V 1 of weight 1 such that where T F (z) is the supercurrent of the matter N = 1 SCFT.This generic construction implies the physical condition and just as in the case of the zero-momentum gauge field.Notice in addition that Φ A is in the kernel of the total matter+ghosts L 0

Integrating out the massive fields
Our aim is to obtain an effective action for the spacetime fields which are encoded in the worldsheet field V 1 2 . To this end, following the notation by Berkovits and Schnabl [13], we split the string field as where R contains the massive fields.This decomposition can be performed using the projector on the kernel of L 0 which we denote P 0 and we obviously have As in standard field theory, the tree-level effective action for Φ A is given by solving the equation of motion for R in the form R = R(Φ A ) and then computing the classical action To obtain the equation of motion for R consider the variation of the action, which we schematically write as which implies that the R-equation is the projection outside the kernel of L 0 of the full equation of motion The explicit form of the string field "EOM" depends on the action (2.1) and for our purposes it can be written as (2.34) The operators appearing above are perturbatively defined as Notice that, because e ad Φ −1 ad Φ is an invertible operator on the star-algebra, we have that which is the well known equation of motion of the Berkovits OSFT.However the prefactor e ad Φ −1 ad Φ cannot be ignored when we are interested in the projection outside the kernel of L 0 , and the equation for R thus reads In principle one could search for an exact solution R(Φ A ) which seems however quite challenging.Therefore, as in standard field theory, we can resort to perturbation theory.We setup a perturbative approach by introducing a coupling constant g and we write (2.39) The R equations can now be solved iteratively by expanding in powers of g.The first non-trivial equations are given by where, since in this paper we will only deal with these equations at order g 2 , we do not write down the equation at order 4 and higher.These equations can all be solved by fixing the gauge ξ 0 = b 0 = 0.In particular at order g 2 we find Notice that the presence of the projector outside of the kernel of L 0 ensures that the action of b 0 /L 0 is well-defined.By plugging (2.43) into the classical action (2.1) we get the tree level effective action at order g 4 which explicitly reads5 (2.44) Notice that the effective action to this order contains the elementary quartic vertex of (2.1) plus a term with a propagator, which is the result of having integrated out the heavy fields.In case of the zero momentum gauge field (2.14) this quantity has been computed exactly by Berkovits and Schnabl and shown to reproduce the expected quartic Yang-Mills potential [13] Φ where we have distinguished the contributions from the propagator term and the elementary quartic vertex in (2.44).

Subtleties from picture changing
Although it is not apparent in the explicit expression (2.44), this quantity is affected by some subtleties.These subtleties are easiest to see by taking the explicit example of the zero momentum gauge field (2.14) but, as we will see later on, they are in fact fairly generic.Let's have a closer look at the propagator term in (2.44) Given this expression we can "change" the picture assignment integrating by parts the derivations Q B and η 0 , in a way analogous to [12].Taking advantage of η 0 Q B Φ A = 0, this is done as follows Notice that we have taken into account the nontrivial projector outside the kernel of L 0 that arises in the commutator Therefore we see that the performed manipulation (which is obviously related to picturechanging) gives rise to the singular term6 which we now analyze.We start by considering In the above equation we have represented the star product as the action of the U operators on the finite distance OPE, as discussed in [20].The action of P 0 only selects the level zero contribution from the above wedge with insertions.This means that only the level zero component of the U operator (that is the identity) gives contribution and, moreover, only the level zero contribution from the OPE (if present) participates.It is not difficult to check that for the explicit case of the gauge field (2.14) we get Notice that the matter primaries ψ and ∂X have a regular OPE and so their product doesn't contribute at level zero.Moreover the projector selects the terms where superghosts OPE closes on the identity in the form and because of the special role played by γ −1 , this is a large Hilbert space phenomenon.
In the same fashion we can check that From these simple considerations we see that the singular projector term (2.49), far from vanishing, is in fact proportional to the full Yang-Mills quartic potential Notice that this quantity involves the computation of a four-point function in a region of moduli space where two insertions are collapsed on one-another and replaced by their level zero contribition in the OPE.Equivalently we can think of P 0 as an infinitely long Siegel-gauge strip which creates a degenerated four-punctured disk at the boundary of moduli space.Taking into account this boundary term in the "picture-changed" propagator term (2.47), one can repeat (with the obvious modifications) Berkovits-Schnabl computation [13] and get the correct Yang-Mills quartic potential (2.45).

Localization of the action at quartic order
Given the fact that (2.53) is already proportional to the full answer, it is tempting to claim that the Yang-Mills quartic potential is fully localized at the boundary of moduli space.However it doesn't seem to be possible to prove this with the ingredients we have used up to now.Recently Sen has provided a similar mechanism for the heterotic string, in the small Hilbert space, [21] which is based on the existence of a conserved charge in the matter sector.In the sequel we will also assume an extra conserved charge in the matter sector and things will drastically simplify.

Conserved charge and N = 2
This charge, which we will call J, is understood as a U(1) R-symmetry of an N = 2 world-sheet supersymmetry in the matter sector.It is well known that the original N = 1 worldsheet supersymmetry is enhanced to an N = 2 in superstring backgrounds supporting space-time fermions (and thus space-time supersymmetry), see for example [18].In order for the N = 1 → N = 2 enhancement to happen, it must be possible to express the original N = 1 matter supercurrent T F as the sum of two supercurrents of opposite "chirality" so that an N = 2 super-Virasoro algebra can be realized T (z) T (±) The full matter SCFT has c = 15 but one may be interested in subsector of the full background (a flat 4-dimensional Minkowski space (c = 6) or a Calabi-Yau internal space (c = 9) are both examples with enhanced N = 2).We will be interested in the case where our original N = 1 superconformal primary V 1 2 (2.18) splits into the sum of two "short" N = 2 superconformal primaries such that (w) = regular. (3.10) The R-current J(z) defines a conserved charge and the short superconformal primaries V (±) are J 0 -eigenstates . (3.12) From (3.1,3.9)we see that the original matter field V 1 also decomposes as However, despite the notation, the super-descendents V ± 1 are not charged under J 0 , because the net J-charge in (3.9) is zero.In the matter SCFT only correlators with total vanishing J-charge are non-zero.This gives a selection rule that drastically simplifies the computation of the effective action (2.44)

Localization of the effective action
As anticipated above, we now assume that our physical string field Φ A decomposes in eigenstates with J charge equal to ±1 with Φ (±) . (3.15) Now we want to compute the effective action (2.44) in the presence of the above decomposition The details of the computations are shown in the appendix A. We just report here the final result which gives us a completely localized effective action where only projectortype terms remain which shows that the quartic effective action is entirely given by two-point functions of Fock space states which we now analyse.

Auxiliary fields
The basic fields which enter in the above expression for the effective action are7 A ] with J = ±2 (3.20) A ] with J = 0 (3.21) To determine the explicit form of the above string fields we only need to know the leading OPE between the matter superconformal primaries Where we have introduced the "auxiliary fields" H (±) 1 which are weight-one matter primaries with J charge equal to ±2 and H 0 which is proportional to the identity in the matter CFT and is neutral under J.Then, using the universal OPE's in the ghost/superghost sector we find by a direct computation analogous to (2.50) (3.28) Now all the contribution to the effective action just depends on the auxiliary fields (3.23), (3.24) and their two-point functions.
Notice in particular that in the case that the OPE structure of the matter fields is not precisely of the form (3.23,3.24)(which is the case for example at generic momenta), the amplitude (3.17) would identically vanish.

Examples
To simplify a bit our result we can compute the universal contributions from the ghosts as c∂c ξ e −2φ (z) c(w) = −(z − w) 2 , (4.1) remaining with purely matter two-point functions where tr is the trace in Chan-Paton space and the bracket is in the matter sector.This compact result is universal.The details of the H fields depend on the chosen matter SCFT, i.e. the string background in which we are interested.To illustrate how this mechanism works we now give two concrete examples.

Yang-Mills
Consider a system of N coincident D(2n) euclidean branes.Their worldsheet theory contains the ψ µ superconformal fields which we rearrange according to a U(n) ∈ SO(2n) decomposition with (,, j) = 1, ..., n.We can bosonize these fields with n free bosons h i such that The localizing R-charge can be taken to be8 With this choice we have We then write Therefore our matter superconformal primaries of definite J-charge are given by and the auxiliary fields are easily extracted from the leading term in the OPE where ψ ı (z) ≡: ψ ı ψ  : (z).The effective action (4.3) is then easily computed to be To recover a more familiar covariant expression, notice that the N × N matrices A  , A  are related to the original A µ 's by (4.4) where the only non-vanishing entries of τ and τ are We can easily check that where ǫ µν = ǫ µν is a block-diagonal anti-symmetric matrix whose only non-zero entries are given by Using these properties, together with the ciclicity of the trace, one can easily verify that the usual covariant form of the Yang-Mills potential is reproduced The quartic potential of the scalars transverse to the D(2n) branes and their interaction with the gauge fields can be obtained in the same way. 10

D3/D(−1) system
We now consider the low energy effective action of a system of N coincident (euclidean) D3 branes with k D(−1) branes sitting on the D3 world-volume.This system is known to give a string theory description of supersymmetric gauge theory instantons [22,23].
A direct string theory construction of the D3-D(−1) effective action to leading order in α ′ has been given in [24].Here we will be interested to show that the effective action of this system is also exactly localized at the boundary of moduli space.The presence of the D3 branes brakes SO (10) to the product of a Wick rotated Lorentz group SO(4) on the D3 branes and SO(6) along the transverse directions.Then we have three sectors of open strings: • D3-D3 strings.These are strings with both endpoints on the set of D3 branes.Their description in terms of string fields is exactly the same we have used in the previous sections.These string fields carry an N × N Chan-Paton factor.
• D(−1)-D(−1) strings.These are strings with both endpoints on the set of D(−1) branes.The corresponding string fields are exactly the same, up to the Chan-Paton factor that now is a k × k matrix.
• D3-D(−1) strings.These are stretched strings between the two types of Dbranes.The corresponding vertex operators must include twist fields (∆, ∆) that are necessary to change the boundary conditions from Neumann to Dirichlet (∆) and viceversa ( ∆) in the X µ sector.These are a pair of conjugated boundary primary fields of weight 1  4 .The bosonic twist fields ∆, ∆ must be dressed with four-dimensional spin fields S α (or S α in case of anti D(−1)'s) in order to change the boundary conditions of the ψ µ system.See [24] for further details (which are partly summarized in appendix B).The Chan-Paton factor carried by these string fields is a N × k or a k × N matrix.
It is important to notice that the composite fields ∆S α and ∆S α have analogous properties to the worldsheet fermion ψ, in particular they are superconformal primaries of weight 1  2 .Assembling things together, the total massless string field is then given by where ).This string field can be decomposed in a basis of eigenstates of the current J 0 as defined in (4.7) The diagonal matter vertex is easily decomposed as where using the same notation as in (4.4)The off-diagonal spin fields are defined through bosonization by the scalars h 1 , h 2 and we consider For more details on bosonization and spin fields, see appendix B. Finally we can decompose the off-diagonal part of where (4.40) where 11 Other choices for the localizing current are in principle possible as in the case of pure Yang-Mills.For example another possible choice of J along the D3 branes is the linear combination of ∂h i with a relative minus sign between ∂h 1 and ∂h 2 .However, while for the pure Yang-Mills all the possible choices are equivalent, the presence of the stretched strings makes a difference whether we have anti D(−1)'s rather than D(−1)'s.In presence of anti D(−1)'s the corresponding spin fields would have opposite chirality and they would be un-charged under J 0 defined in (4.31).So to localize their action one would choose a different J 0 with an opposite sign in front of ∂h 2 .

Φ (−)
Then using our general result (4.3) the quartic potential is given by the two-point functions of the matter auxiliary fields (3.23), (3.24) H 0 , H 1 .The computations are easily carried out using standard OPE's and for the sake of clarity we write the auxiliary fields as a sum of two components, the first one along the D3 branes directions and the second one along the transverse directions where repeated holomorphic and anti-holomorphic indices are summed.After a little algebraic manipulation it is easy to see that (4.45), (4.46) and (4.47) carries a SU(2) representation in terms of ladder ± and uncharged t'Hooft symbols (see appendix B for conventions and defintions) where we have defined (4.54) The covariant tensor T µν is given by Restricting our attention to the 4 dimensional worldvolume of the D3-branes, we readily get the quartic potential where contains (on the D(−1) slot) the ADHM constraints.
For completeness, the full quartic potential in covariant form is easily computed to be If we switch off the D3 degrees of freedom, this result is in agreement with [24] once that a suitable rescaling of the fields is carried out.We also get the algebraic couplings on the D3 which were not considered in [24] because they are α ′ suppressed wrt the couplings on the D(−1)'s, by dimensional analysis.It is interesting to note that the coupling A ω a ω (although not forbidden in principle) does not appear in the quartic potential (4.58) since there is not the corresponding auxiliary field.

Conclusions
In this paper we have found that there is an hidden localization mechanism at work on the worldsheet which accounts for the algebraic part of the D-branes massless effective action.These boundary contributions emerge in the process of picture changing, by taking into account that the propagator fails to truly invert the BRST charge.This localization mechanism can be considered as a rigorous justification to the use of the auxiliary-fields in the computation of certain superstring amplitudes [24,25], in a genuine zero-momentum setting.
It would be interesting to extend our analysis to higher orders O(g k>4 ) and to analyze the pattern of the involved, possibly new, auxiliary-fields.It would be also interesting to see if some couplings involving space-time fermions localize.These analysis could give new insights on the problem of α ′ corrections in non-abelian D-brane systems.
It should be noted that the same kind of localization at the boundary of moduli space that we discuss in this paper is at work in the topological string via the holomorphicanomaly [26,27] and it would be interesting to explore the connections.
From the string field theory perspective we would like to understand if there is an analogous (perhaps more closely related to [21]) localization mechanism in the A ∞ formulation in the small Hilbert space [11].This in particular could be useful for analyzing loop contributions, after having introduced the Ramond sector [7,8].
We hope that our observations could be a useful step to better understand the structure of string field theories and how they relate to the low energy effective world.

B Supersymmetry, bosonization and correlation functions
Supersymmetry Here we recollect our conventions on supersymmetry notation and the main properties of the t'Hooft symbols.The Euclidean Lorentz group SO(4) on the D3 branes system is realized on spinors in terms of the Pauli matrices τ a from which it is possible to construct the ordinary gamma matrices satisfying the euclidean Clifford algebra.It is well known that the self-dual and antiself-dual generators of SO( 4) are two indices gamma matrices (γ µν ) β α , (γ µν ) α β given in terms of the self-dual and antiself-dual t'Hooft symbol which maps the Wick rotated Lorentz group to SU(2) They are symmetric in the spinor indices and antisymmetric in the spacetime indices.
In this paper, since we are dealing with instantons we are using only the self-dual two-indices gamma matrices.Self-dual t'Hooft symbols are realized as and satisfy a a set of properties including which are useful for deriving our results in the main text.Spinor indices are raised and lowered as follows where the ǫ matrices are defined such that ǫ 12 = ǫ 12 = ǫ Bosonization Bosonization in ten dimensions allows us to introduce five commuting scalars h i , one for each complex dimension, and write all the content of the ψ, S α matter sector in terms of these scalars.Associated to each scalar we have a current proportional to ∂h i such that J i (z) e ±ikh j (w) ∼ ±k δ ij z − w e ±ikh j (w).(B.8) We can extend this current to all the spacetime with linear combinations of the five elementary currents.The existence of this current is fundamental to localize the effective action.We have equivalent choices in the full spacetime (for example when we analyze Yang-Mills on a D(2n) branes system as in section 4.1), but when we introduce spin fields on the worldvolume of the D3 branes system we have two inequivalent choices for the total current under which matter fields like ψ are generically charged up to a sign, while spin fields are divided in two families of opposite chiralities , S α = e ± i 2 (h 1 −h 2 ) . (B.10) The bosonized spin fields are charged under one current and uncharged under the other one, so that the choice of the chirality of the spin fields (and hence the choice between D(−1)'s or anti D(−1)'s) implies the choice of the localizing charge.
The bosonized spin fields makes explicit the subleading contribution from the OPEs which are required for the auxiliary fields (4.45), (4.46), (4.47): 2 ) (w) ∼ (z − w) 1 2 e i(h 1 +h 2 ) = (z − w) where an effective minus sign is present in the second two-point function to account for the correct odd grassmanality of the superconformal primary ∆S α .We refer to [24] for more details.In this work we also use the two following OPE's S α (z) S β (w) ∼ ǫ αβ (z − w) Notice that the first term in (B.17), proportinal to the identity, is responsible for the auxiliary field H 0 , while the second term, which is subleading, is responsible for the charged auxiliary fields H ± 1 .The other OPE (B.18) is responsible for the absence of an auxiliary field giving rise to an Awa w coupling, since when, properly dressed with a twist field and the ghosts, the weight zero contribution is not present.