hep-th/0306254

ITP–UH–03/03

String Field Theory Vertices for Fermions of Integral Weight

Alexander Kling and Sebastian Uhlmann

[5mm] Institut für Theoretische Physik

Universität Hannover

Appelstraße 2, 30167 Hannover, Germany

[2mm] E-mail: kling,

Abstract:

We construct Witten-type string field theory vertices for a fermionic
first order system with conformal weights in the operator formulation
using delta-function overlap conditions as well as the Neumann function
method. The identity, the reflector and the interaction vertex are treated in
detail paying attention to the zero mode conditions and the charge
anomaly. The Neumann coefficients for the interaction vertex are shown to be
intimately connected with the coefficients for bosons allowing a simple proof
that the reparametrization anomaly of the fermionic first order system
cancels the contribution of two real bosons. This agrees with their
contribution to the central charge. The overlap equations for the
interaction vertex are shown to hold. Our results have applications in
N=2 string field theory, Berkovits’ hybrid formalism for superstring field
theory, the -system and the twisted -system used in bosonic
vacuum string field theory.

## 1 Introduction

The study of nonperturbative physics in string theory has enjoyed a lively interest in recent years. String field theory, especially in the formulation of [1], has proven to be a useful tool for developing qualitative as well as quantitative results. In particular, Sen’s conjectures on tachyon condensation [2, 3, 4] have set the stage for putting string field theory to work (see, for instance, the reviews [5, 6, 7, 8] and references therein). In this framework it should be possible to describe the result of the condensation process, the closed string vacuum, as a solution to its equation of motion. However, no plausible candidate for a solution in open string field theory has been found yet.

In order to gain some insight into the structure of solutions describing
D-branes, Rastelli, Sen and Zwiebach proposed to expand all string fields
around a (unknown) closed string vacuum solution. This vacuum
string field theory [9, 10] is
described as a certain singular limit, in which the kinetic operator
consists only of ghosts.
In this limit, the equation of motion of bosonic string field theory
factorizes into a matter and a ghost part, which can be solved independently;
the matter part of the string field has to be a projector of the star algebra.
This separation, however, does not pertain to open string field theory, where
the kinetic operator mixes matter and ghost sectors. Much work has been done
in the context of bosonic string theory to classify these solutions in
terms of projectors of the star algebra [11, 12, 13, 14, 15, 16],
most prominently the sliver. They have been identified with (multiple)
D-branes of various dimensions, and it was shown that they reproduce the
correct ratio of tensions. Their similarity to noncommutative solitons has
been investigated [17, 18]. In such considerations
a major technical simplification is achieved by taking advantage of the map
from Witten’s star product to a continuous^{1}^{1}1Originally, the discrete
version of this map has been proposed in [19] and further
developed in [20]. Moyal product [21].

Much less is known for superstrings. Different generalizations of bosonic vacuum string field theory to non-BPS and - brane systems have been proposed [22, 23, 24] using Berkovits’ formulation for a nonpolynomial superstring field theory [25] as well as cubic open superstring field theory [26, 27]. Although superstring field theories applied to the problem of tachyon condensation perform well [28, 29, 30], it is considerably harder to make sense of the conjectured versions around the tachyon vacuum. Despite considerable efforts [31, 32, 33, 34] up to now no satisfactory solutions to vacuum superstring field theory have been given. Even worse, recent results using the numerical technique of level truncation seem to indicate that the pure ghost ansatz for the kinetic operator fails to describe the theory around the tachyon vacuum [35]. Therefore it is necessary to gain more insight into the structure of solutions to string field theory with more general kinetic operators mixing different sectors of the theory. This is, of course, a technically demanding task.

Due to these difficulties it seems worthwhile to take an apparent sidestep and to consider alternative approaches to tackle the problems stated above. In order to study the properties of string field theory solutions with these mixing properties, one may study a different string field theory containing world-sheet fermions instead of reparametrization ghosts. Namely, string field theory for N=2 strings [36] shares its internal structure with Berkovits’ proposal for a nonpolynomial superstring field theory. Its action and equation of motion include two BRST-like generators and (corresponding to the BRST charge and the field from the fermionization of the world-sheet superghosts in the N=1 case), both mixing world-sheet bosons and fermions (instead of matter and ghost fields for N=1 strings). The main advantage of this model is its simplicity; no ghosts are needed in addition to the matter fields. The equality of the structure and the simplicity of the field realization of the BRST-like operators turn this theory into a viable candidate for studying the intricacies which general solutions to the equation of motion for nonpolynomial string field theory bring about. Clearly, this equation of motion contains the star product. Whereas the operator formulation [37, 38] and the diagonalization [19, 21, 39] of the bosonic vertex may be transferred literally to N=2 strings, the fermionic part differs from the N=1 case since (after twisting, see section 2) the fermions here have conformal weights 0 and 1, respectively. Therefore, we commence the investigation of the operator formulation of the fermionic part of the star product in N=2 string field theory, which is lacking in the present literature. This also has applications to Berkovits’ hybrid formalism [25], where the compactification part of the theory is described by a twisted N=2 superconformal algebra. Moreover, this first order system is isomorphic to the ghost system of N=1 strings. Both, Witten’s superstring field theory as well as Berkovits’ nonpolynomial string field theory involve this ghost system in a nontrivial manner; the former features insertions of picture changing operators while the latter is formulated in the large Hilbert space. Therefore, a solid understanding of the structure of the star product in this sector is of interest and can easily be gleaned from the results presented here. Eventually, we point out that this fermionic system is also equivalent to the twisted system (cf. end of section 2). In the context of bosonic vacuum string field theory this auxiliary boundary conformal field theory was used to find solutions to the ghost equations of motion from surface state projectors constructed in the twisted system [40]. This BCFT is obtained by twisting the energy momentum tensor with the derivative of the ghost number current. The ghost fields of the twisted theory have conformal weights and thus correspond to the fermionic first order system of N=2 string field theory. However, it is unclear to us whether this equivalence can be traced to any deeper interrelation.

In this paper we construct the vertices needed to formulate N=2 string field theory in the twisted fermionic sector from scratch using the operator language. In particular we pay attention to the anomaly of the current contained in the N=2 superconformal algebra. Together with the overlap equations for the zero-modes this fixes the choice of vacuum for -string vertices when one avoids midpoint insertions. For the identity vertex and the reflector the construction is accomplished using -function overlap conditions. The reflector is shown to implement BPZ conjugation as a graded antihomomorphism. To obtain the explicit form of the interaction vertex we have to invoke the Neumann function method. Supplemented with the above mentioned conditions on the vacuum the vertex is fixed. The Neumann coefficients are expressed in terms of coefficients of generating functions. We find an intimate relationship between the coefficients for the fermions and those for bosons allowing us to employ known identities from the boson Neumann matrices. Resorting to this relationship we show that the contribution of the system to the reparametrization anomaly cancels those of two real bosons. This is in accordance with their contribution to the central charge. Finally, we explicitly check that the overlap equations for the interaction vertex are fulfilled.

The paper is organized as follows. In the next section we briefly review the nonpolynomial string field theory for N=2 strings. The embedding of the N=2 into a small N=4 superconformal algebra is described and the representation of the generators in terms of N=2 fields are given. Eventually, this small N=4 superconformal algebra is twisted, leading to a topological theory [41]. After twisting all fields have integral conformal weights. In section 3 the identity vertex is constructed. As a starting point -function overlap conditions for arbitrary -string vertices are considered. After deducing the form of the identity vertex from the corresponding overlap equations its symmetries are discussed in detail with particular emphasis on the anomaly of the current. In section 4 the 2-string vertex is considered. Starting from general -string overlap equations formulated in terms of -Fourier-transformed fields we discuss constraints on the vacua arising from the zero-mode overlap conditions. Avoiding midpoint insertions these conditions fix the vacuum on which the -string vertex is built. The reflector is discussed as an application of the tools described in this section. A detailed discussion of BPZ conjugation as implemented by the 2-vertex completes this section. We define BPZ conjugation and its inverse via the the bra-reflector and the ket-reflector, respectively, and their compatibility is shown. The interaction vertex is constructed in section 5. The Neumann coefficients of the -string vertex are expressed in terms of generating functions constructed out of the conformal transformations which map unit upper half-disks into the scattering geometry of the vertex. The so obtained Neumann matrices are shown to be closely related to the Neumann matrices for bosons. Therefore, identities for the bosonic Neumann matrices entail corresponding identities for the fermionic ones. In this way, the anomaly of midpoint preserving reparametrizations is shown to cancel the contribution of two real bosons, which is in agreement with conformal field theory arguments. Finally, the overlap conditions for the interaction vertex are checked explicitly. Parts of this calculation are relegated to the appendix where also formulas for the bosonic vertices and Neumann coefficients are collected. The paper is concluded with a short summary and a discussion of possible applications and further developments.

## 2 Twisting the world-sheet action

Nonpolynomial string field theory for N=2 strings. The nonpolynomial string field theory action [36] coincides for N=1 (Neveu-Schwarz) and N=2 strings and contains in its Taylor expansion a kinetic term and a cubic interaction similar to Witten’s superstring field theory action. In a notation suitable for both cases, it reads

(2.1) |

here, is defined via Witten’s midpoint gluing prescription (for convenience, all star products in this action are omitted; denotes the identity string field) and is a string field carrying Chan-Paton labels with an extension interpolating between and . The BRST-like currents and are the two superpartners of the energy-momentum tensor in a twisted small N=4 superconformal algebra with positive -charge. The action of and on any string field is defined in conformal field theory language as taking the contour integral, e. g.

(2.2) |

with the integration contour running around . The corresponding equation of motion reads

(2.3) |

where contour integrations are implied again. We set out to define unambiguously the star product and the integration symbol in (2.1).

Small N=4 superconformal algebra. Just as in bosonic and in superstring theory, the critical dimension for N=2 string theory can be determined from anomaly considerations. It turns out that such a theory has propagating on-shell degrees of freedom only in signature (2,2); thus, the (Kähler) spacetime is naturally parametrized by two complex bosons ,

(2.4) |

Their complex conjugates are denoted by . The metric on flat is taken to be with nonvanishing components . To obtain N=2 world-sheet supersymmetry, the four real bosons have to be supplemented by four Dirac spinors which may be combined into

(2.5) |

For open strings, we apply the doubling trick throughout this paper so that all fields are (holomorphically) defined on the double cover of the disk, i.e., on the sphere. In superconformal gauge, the world-sheet action for the matter fields now reads (on a Euclidean world-sheet with double cover )

(2.6) |

The action is normalized in such a way that the operator product expansions are the ones which should be expected from the transition from real to complex coordinates:

(2.7) |

The matter part of the constraint algebra for this theory is an N=2 superconformal algebra with generators

(2.8) | |||

In , the central charge is (as required from the ghosts, which we will, however, not introduce). Note that the superscripts on each quantity label the charge under the current . These currents can, in principle, be defined on general -dimensional Kähler manifolds for any .

In , we can extend the N=2 superconformal into a small N=4
superconformal algebra with additional generators^{2}^{2}2On a -dimensional manifold, and have to be replaced
by (the components of) nondegenerate - and -forms,
respectively. (Untwisted) N=4 supersymmetry requires a hyperkähler
spacetime manifold. [42]

(2.9) |

using the constant antisymmetric tensor with . The currents , and form an affine Kac-Moody algebra of level 2.

In order to obtain an algebra with central charge zero, we can twist the small N=4 superconformal algebra by shifting , i.e., reducing the weight of a field by one half of its charge. After twisting, all fields will have integral weights; in particular, and as fields of spin 1 may subsequently serve as BRST-like currents. We will show later that the bosonic and fermionic contributions to the anomalies of all currents in the small N=4 superconformal algebra on -vertices cancel for all even . Of course, the definition of requires at least two complex dimensions, i. e., .

Twisted action. With respect to the twisted energy-momentum tensor , and have weights 0 and 1; this suggests that they are no longer complex conjugates in the sense of eq. (2.5). Indeed, they constitute a first order system with Euclidean world-sheet action

(2.10) |

which is real after a Wick back-rotation to Minkowski space
for hermitean fields .^{3}^{3}3Here and in
the following, we sometimes omit the spacetime labels on if the
statement refers to any of the . That this action for
the fermionic part of the twisted theory is indeed the correct one is
corroborated by the fact that the full action is invariant under the
symmetries generated by all currents in the small twisted N=4 superconformal
algebra.

As fields of integral weight, both and are integer-moded. In particular, the spin 0 field has a zero-mode on the sphere. In analogy to the -system there are thus two vacua at the same energy level: the bosonic -invariant vacuum and , ; its fermionic partner, is annihilated by the Virasoro modes

Translation to the and twisted
system. All methods to construct string field theory
vertices used in this paper can be formulated in terms of conformal field
theory data. In particular, they only depend on the conformal weights and
the world-sheet statistics of the fields. The system as
well as the system from the fermionization of the world-sheet
superghosts and the twisted system of [40] are
fermionic first order systems with fields of conformal weights 1 and 0.
Therefore, all formulas in this paper remain valid upon the
substitutions^{4}^{4}4These substitutions can be used to compare part of our
results on the 3-vertex with those obtained in [65], a
preprint which appeared on the same day.

(2.11a) | |||

(2.11b) |

The factors of take care of the unusual normalization of the two-point function, which can be read off from eq. (2.7). For example, eq. (2.10) translates via the dictionary given above into the action for the system . Furthermore, states built from are in the small Hilbert space whereas states constructed from are in the large Hilbert space of [43].

## 3 Identity vertex

The identity vertex defines the integration in eq. (2.1); it is an element of the one-string Hilbert space corresponding to the identity string field . The identity vertex glues the left and right halves of a string together; therefore it can be defined via the corresponding overlap equations.

Overlap equations. In general, the overlap equations for an -vertex can be determined from conformal field theory arguments [44]: On the world-sheet of the -th string (), a strip, we introduce coordinates . The strip can be mapped into an upper half-disk with coordinates ; the upper half-disks are then glued together in the scattering geometry in a such a way that

(3.1) |

This is achieved by the conformal map

(3.2) |

where the phases have been chosen so as to give a symmetric configuration when mapping back to the upper half-plane.

A primary field of conformal weight in the boundary conformal field theory on the strip is glued according to

(3.3) |

In the last two lines we have used (3.1). This equality is required to hold when applied to the -string vertex . If we insert the open string mode expansion for , , we obtain a condition on the modes. For , the above condition extends to , so that one can take advantage of the orthogonality of the cosine to obtain the diagonal condition . Instead, we will impose the stricter condition . For , the overlap equations in general mix all modes.

Construction of the identity vertex. For the -system, we demand the stricter conditions

(3.4a) | |||||||

(3.4b) |

from which the gluing conditions (3.3) follow. The conditions on are compatible since . The obvious solution to eqs. (3.4) reads

(3.5) |

where the -invariant vacuum is annihilated by .

Symmetries of the vertex. Applying the gluing conditions (3.3) to the complex spin 1 fields

(3.6) |

we obtain

(3.7) |

Together with (3.4), this entails that the gluing conditions for the BRST-like spin 1 currents and ,

(3.8) |

are satisfied. In general, anomalies can only appear if the current contains pairs of conjugate oscillators. Thus, it is clear that the spin 2 currents , and are anomaly-free, just like the spin 0 current . More interesting are the (twisted) energy-momentum tensor and the current (when treated as primary fields).

The modes of the twisted energy-momentum tensor can be written as

(3.9) |

According to (3.3) these modes have to satisfy^{5}^{5}5
Treating the energy-momentum tensor as a primary field is justified iff
the central
charge vanishes. In this sense, one can understand eq. (3.10) as
a condition on the central charge. Note that the -string variant
of eq. (3.10)
does not follow from eq. (3.3) for .

(3.10) |

for the vertex to be reparametrization invariant. In complex dimensions, the contribution of the bosons to the left hand side of eq. (3.10) can be easily shown to be

(3.11) |

which is canceled by the fermionic contribution

(3.12) |

These contributions arise from terms and in and , respectively. Due to the absence of such terms, the are automatically anomaly-free.

Before considering the current , let us first recall the discussion in [1] of the -anomaly of -vertices: If the current is bosonized as , the action for this boson reads

(3.13) |

The operator product expansion is that of the free action, . The energy-momentum tensor for reads , where is the background charge, i. e., the coefficient of the third order pole in the operator product expansion . For the -system in complex dimensions, .

In a general gluing geometry the curvature is concentrated in one point, namely the midpoint of the string (). On such surfaces the term linear in contributes an anomalous factor of

(3.14) |

in the path integral^{6}^{6}6In the integral of the Ricci scalar over this
surface we have used ..
This integral measures the deficit angle of
this surface when circumnavigating the curvature singularity at the string
midpoint and contributes for an -string
vertex. Hence, the factor (3.14) produces a -anomaly of
in the path integral. Since the -charge^{7}^{7}7The charge of a bra vector is measured by ; we use
conventions where of is , an -vertex
constructed from -vacua requires

Therefore, we do not expect the -current to be anomaly-free;

(3.15) |

in general. Since its zero-mode measures the fermion number of the vertex, we instead expect . This relation holds trivially. For in eq. (3.15), one obtains .

## 4 Reflector

In this section we construct the 2-string vertex for the fermionic system . It is convenient to introduce -Fourier-transformed fields as a tool to diagonalize general -string overlap equations. The overlap equations fix the zero-mode part of the 2-vertex up to a sign. A discussion of BPZ conjugation motivates our choice for this sign.

-transforms. Introducing the combinations

(4.1) |

of left and right movers, the conditions imposed on the fermions following from the -function overlap of strings are

(4.2a) | ||||

(4.2b) |

For and similar equations have to be fulfilled. The conditions (4.2) are easily diagonalized if we introduce -Fourier-transformed fields [37],

(4.3a) |

where . Note that now form canonically conjugate pairs (the upper index is taken modulo ). We choose the following ansatz for the -vertex in terms of -transformed oscillators

(4.4) |

with Neumann matrices and a vacuum state . The vacuum state will be determined below from the zero-mode overlap conditions; the summation range of should then be adjusted in such a way that only creation operators w. r. t. this vacuum appear in the exponential.

In application to , eqs. (4.2) now read

(4.5a) | ||||

(4.5b) |

As already discussed in section 3, the overlap conditions will only contain a sum of two oscillators (rather than infinitely many), if after inserting the mode expansions the cosines can be integrated over . This is obviously possible also for if . Therefore, and, if is even, appear in the vertex (4.4) with Neumann matrices and , respectively. Here, denotes the twist matrix with components .

Before we turn to the 2-string vertex, let us briefly discuss the overlap conditions for the zero-modes of the -transformed oscillators. It is consistent with (4.5) to demand

(4.6a) | ||||

(4.6b) |

Note that eqs. (4.6a) entail that no (for ) may appear in the exponential of the vertex (4.4). The appearance of is forbidden by eq. (4.6b) since is diagonal. In terms of the original one-string oscillators, this means that no occurs in the exponential of the vertex.

It is easy to see that the conditions on the vacuum (4.6) are
solved by^{9}^{9}9Here one has to use the fact that is
Grassmann even while is Grassmann odd, i.e., the
bra-vacua have opposite
Grassmannality compared to the corresponding ket-vacua.
This is a consequence of the odd background charge (cf. the end of
section 2).

(4.7) |

The subscripts indicate in which string Hilbert space the corresponding vacuum state lives. The vacuum (4.7) already features the charge required by the -anomaly, namely . This choice allows us to avoid midpoint insertions.

Overlap equations for the reflector. Expressed in terms of -transforms, the overlap conditions for the reflector simply become

(4.8a) | ||||||

(4.8b) |

which can be rewritten in terms of modes acting on as

(4.9a) | ||||||

(4.9b) |

for the nonzero-modes. The conditions for the zero-modes read

(4.10) |

The zero-modes and put no restrictions on the vertex. Along the lines of [37], one finds

(4.11a) | ||||

(4.11b) |

as a solution to eqs. (4.9). Since no zero-modes appear in the vertex, the vacuum has to be annihilated by and in order to satisfy eq. (4.10). Thus the vacuum is a symmetric combination of up- and down-vacua in the two-string Hilbert space,

(4.12) |

This is consistent with eq. (4.7). In the last expression it is understood that the first entry corresponds to string , while the second corresponds to string . The overall sign is determined by requiring that implements BPZ conjugation.

BPZ conjugation. On a single field BPZ conjugation acts as with ; since inverts the time direction, it is suggestive that on a product of fields, BPZ conjugation should reverse the order of the fields. This statement will be put on a more solid ground below. The action of BPZ on fields induces an action on states: defines the out-state which is created by . In terms of modes this prescription yields

(4.13) |

for a field of conformal weight . To fix the choice of vacuum in (4.11), recall that is an element of the tensor product of two dual string Hilbert spaces and thus induces an odd linear map from to , which is nothing but BPZ conjugation [45],

(4.14) |

In order to be compatible with the usual definitions of BPZ conjugation, we demand in particular that the invariant vacuum is mapped into under BPZ conjugation. Therefore we fix the vacuum to be

(4.15) |

Note that with this choice of vacuum and using eq. (4.14) one finds

Now consider the corresponding ket state . Observe that the conformal transformation maps to . Therefore, the overlap equations for for a field of conformal weight , , transform into . This implies that the overlap equations for the and vertices are invariant under BPZ conjugation for fields of integral conformal weight. Indeed, this can be verified for (4.9) using (4.13) on the level of modes, and we can immediately write down the solution

(4.16a) | ||||

(4.16b) |

It is easy to see that eqs. (4.9), now taken to act on the ket vertex, are fulfilled. Eventually we have to fix our choice of vacuum. In order to fulfill the zero-mode overlap equations (4.10), has to be an antisymmetric combination of up and down vacua

(4.17) |

We fix the overall sign of to be a plus sign by requiring

(4.18) |

where denotes the Grassmannality of the state . Moreover one finds

which can be checked using (4.11). Eq. (4)
is the statement that BPZ conjugation acts as a *graded antihomomorphism*
on the algebra of modes. To emphasize the gradation we explicitly kept
the sign stemming from the anticommutation of the modes. Note that
there is no problem in commuting the modes since after acting on the
vertex they belong to different Hilbert spaces, so the only effect is
an additional sign. Finally, it is straightforward to check that

(4.20) |

by using standard coherent state techniques (cf. [46, 47, 48]) and eq. (4.11). One can then check that . This completes the construction of the reflector state from the overlap equations.

## 5 Interaction vertex

In this section, we set up the Neumann function method [49, 50, 51, 52, 53, 54, 55, 56] for general -string vertices, since even in terms of the -Fourier-transforms the overlap equations are not directly soluble for . In the case of the 3-string vertex, the Neumann coefficients are computed explicitly in terms of generating functions. The observation that they are intimately related to the well-known bosonic Neumann coefficients helps us to show that the -anomaly of the (bosonic and fermionic) 3-vertex vanishes in any even dimension . Furthermore, it will be shown that the 3-vertex for the system satisfies its overlap equations.

Neumann function method. The Neumann function method is based on the fact that the large time transition amplitude is given by the Neumann function of the scattering geometry under consideration. To find the Fock space representation of the interaction vertex one makes an ansatz quadratic in the oscillators,