New supersymmetric index of heterotic compactifications with torsion

New supersymmetric index of heterotic compactifications with torsion

Abstract

We compute the new supersymmetric index of a large class of heterotic compactifications with torsion, corresponding to principal two-torus bundles over warped K3 surfaces with H-flux. Starting from a UV description as a (0,2) gauged linear sigma-model with torsion, we use supersymmetric localization techniques to provide an explicit expression of the index as a sum over the Jeffrey-Kirwan residues of the one-loop determinant. We finally propose a geometrical formula that gives the new supersymmetric index in terms of bundle data, regardless of any particular choice of underlying two-dimensional theory.

1 Introduction

Heterotic compactifications play a fundamental role in building phenomenologically relevant models for particle physics; in this perspective, some realistic Calabi-Yau (CY) models from the point of view of the field content and the interactions were built, see  [1]. However remains the problem of stabilizing the massless moduli characterizing the Calabi–Yau manifold and the stable holomorphic vector bundle. Although a suitable choice of gauge bundle can stabilize a significant fraction of the complex structure moduli [2], torsional compactifications, in which a non-trivial Kalb-Ramond H-flux is turned on, constitute an essential approach towards solving completely this moduli problem.

The supersymmetry conditions at order for compactifications to four dimensions with H-flux have been known for almost thirty years [3]. Nevertheless our knowledge of solutions of these equations, known as Strominger’s system, is very limited. Indeed the compactification manifold is conformally balanced instead of Kähler, and the Bianchi identity, see eq. (138d), is notoriously hard to solve as it is non-linear in the flux.1

There are two pitfalls that await any attempt to construct compactifications with torsion from a low-energy perspective. First, the Bianchi identity implies that, if H-flux is present at leading order in , there exists no large-volume limit of the compactification in general (see [4] for a recent discussion). Second, the underlying non-linear sigma-model is generically destabilized by worldsheet instantons [5]. A promising approach that was developed recently is to obtain the worldsheet theory as the infrared fixed point of a gauge theory, generalizing the well-known Calabi-Yau gauged linear sigma-models (GLSMs) [6] to torsion gauged linear sigma-models (TGLSMs) [7, 8, 9, 10, 11, 12, 13]. Worldsheet instanton corrections may indeed cancel for theories with a UV (0,2) GLSM description [14].2

There exists a single well-known class of compactifications with torsion, given by principal two-torus bundles over a warped K3 base together with the pullback of a stable holomorphic vector bundle over K3; following the general usage, they will be named Fu-Yau (FY) compactifications thereafter. These solutions were first obtained by Dasgupta, Rajesh and Sethi from type IIB orientifolds by S-duality [16], and subsequently studied geometrically by Goldstein and Prokushkin in [17], where their structure was made explicit. Fu and Yau managed to solve the Bianchi identity in [18], using the Chern connection (with a sequel [19] discussing more physical aspects), while a different choice of connection was put forward in [20]. These compactifications lead to or supersymmetry in space-time. The first class of torsion GLSM that was obtained by Adams and collaborators [7] was especially designed to give a worldsheet theory for the former.

The microscopic description of Fu-Yau compactifications as torsion GLSMs provides some evidence for their consistency at the quantum level, beyond the supergravity regime.3 This approach was also used in [9] to compute their massless spectra using Landau-Ginzburg methods, and in [21] to obtain exact statements about their duality symmetries. A very interesting aspect of the latter work, which will play an important role in the present paper, was that covariance of the theory under perturbative dualities along the two-torus fiber imposes that its moduli are those of a rational conformal field theory.

It is natural to ask whether there are other important results regarding heterotic compactifications with torsion that can be obtained in this GLSM framework. Typically, the non-conformal gauged linear sigma-models allow one to compute exactly quantities that are invariant under the RG flow on the worldsheet. A good example of this is the elliptic genus [22] which was indeed obtained, for CY compactifications, using their formulation as a GLSM [23, 24, 25] and supersymmetric localization [26].

In the case of torsion GLSMs for Fu-Yau compactifications that interest us in this work, the elliptic genus vanishes since they have too many fermionic zero modes. We will consider instead their new supersymmetric index [27], which contains important information about four-dimensional physics. It counts the BPS states in space-time [28], and allows to compute the one-loop threshold corrections to the gauge and gravitational couplings of heterotic compactifications, see [29, 30] and  [31, 32] for subsequent work. While these are well-established results for compactifications, our main motivation is to extend this analysis to the more general case of Fu-Yau geometries.4

In this article, we will derive the new supersymmetric index directly from torsion GLSMs corresponding to Fu-Yau compactifications with supersymmetry, using supersymmetric localization. Several steps of the derivation are similar to the computation of the elliptic genera for ’ordinary’ gauged linear sigma-models [23, 24, 25]. There are however important new aspects related to the presence of gauge anomalies canceled against classically non gauge-invariant interactions. With the choice of supercharge appropriate to the problem, the action of the torsion multiplet, representing the torus fiber, is not -exact, and the measure in field space is not -invariant; as we will demonstrate, supersymmetric localization makes sense nonetheless for the full theory.

Independently of physics, the elliptic genus of a holomorphic vector bundle over a compact complex manifold is obtained as the holomorphic Euler characteristic of a formal power series with vector bundle coefficients; whenever suitable topological conditions are met (essentially the tadpole conditions), it gives a weak Jacobi form. The twining partition function that we define as an intermediate step in the computation of the new supersymmetric index, see eqns (34) and (35), is the natural (non-holomorphic) generalization of the Calabi-Yau elliptic genus to Fu-Yau geometries. More generally, it provides a non-holomorphic genus for principal two-torus bundles over CY -folds, which transforms as a weak Jacobi form, and whose topological nature follows from quantization of the torus moduli. We will present a definition of this quantity in geometrical terms, independently of any GLSM or other worldsheet formulation, in eq. (90); a proof of this statement will be provided for an example based on the quartic.

The plan of this article is as follows. In section 2 we review the construction of torsion GLSMs. Then in section 3 we present the new supersymmetric index, and proceed in section 4 to its path integral computation using localization. In section 5 we generalize the results to higher rank gauge groups, provide an anomaly-free charge assignment for a large class of models and illustrate our results with an explicit example. In section 6 we provide the geometrical formula for the new supersymmetric index, and finally we summarize the work exposed in this article in section 7 and give directions for future work. Conventions for superspace are gathered in appendix A, some results about theta functions and modular forms can be found in appendix B, and a summary of Fu-Yau geometry is given in appendix C.


Conventions:

  • In the following, we set , which means that the self-dual radius is one.

  • The real part (resp. imaginary part) of any complex quantity is denoted by an index 1 (resp. 2).

  • .

  • left-moving holomorphic.

  • One defines and .

  • The volume of the worldsheet torus is .

2 Gauged linear sigma-models with torsion

We review in this section the construction of torsion gauged linear sigma-models proposed in [7]. These are gauge theories in two-dimensions with supersymmetry which are expected to flow in the infrared to non-linear sigma-models whose target space corresponds to Fu-Yau compactifications; a brief presentation of these non-Kähler heterotic solutions is given in appendix C.

As the first step of this construction, one considers a standard (0,2) gauged linear sigma-model for the K3 base; generically such model suffers from gauge anomalies, that, in the usual case of Calabi-Yau GLSMs, should be made to vanish by a suitable choice of field content hence of gauge bundle in space-time. In the present case, one cancels instead the anomalous variation of the functional measure against a classically non-gauge-invariant Lagrangian for a torsion multiplet modeling the principal bundle, thereby realizing the Green-Schwarz mechanism on the worldsheet.

2.1 Anomalous gauged linear sigma-model for the base

For simplicity of the discussion, we restrict ourselves in the following discussion to the case of a gauge group on the worldsheet; the generalization to higher rank gauge groups is rather straightforward and will be briefly mentioned in section 5. The conventions we use for superfields, as well as the components Lagrangian, are given in appendix A.

A gauged linear sigma-model for a complete intersection Calabi-Yau manifold in a weighted projective space [6] contains first a set of chiral multiplets , as well as a set of Fermi multiplets , interacting through the superpotential

(1)

where the are quasi-homogeneous polynomials of the appropriate degree to preserve gauge invariance at the classical level and, geometrically, to obtain a hypersurface of vanishing first Chern class. This Calabi-Yau hypersurface corresponds then to the complete intersection .

Second, the holomorphic vector bundle is described, in the simplest examples, by adding a set of Fermi multiplets , a single chiral multiplet and the superpotential

(2)

where the are again quasi-homogeneous polynomials. Let us denote the gauge charges of the different superfields as (with and negative, the other ones positive):

(3)

In the geometrical “phase”, where the real part of the Fayet-Iliopoulos coupling is taken large and positive, one expects that the model flows to a non-linear sigma-model on the CY hypersurface with left-handed fermionic degrees of freedom transforming as sections of a rank holomorphic vector bundle , determined by the short exact sequence5

(4)

As the multiplets contain chiral fermions there are potentially gauge anomalies on the worldsheet that should be canceled. The model should also contain a non-anomalous global right-moving symmetry which corresponds in the infrared to the symmetry of the superconformal algebra, and a global left-moving symmetry, used to implement the left-moving GSO projection.

The variation of the effective Lagrangian under a super-gauge transformation of chiral parameter writes

(5)

with the field strength superfield, and the anomaly coefficient

(6)

which measures the difference between the second Chern character of the tangent bundle of the base manifold and the second Chern character of the vector bundle over the latter. If one considers a model with , then the theory is at this point ill-defined quantum mechanically.

2.2 Two-torus principal bundle and anomaly cancellation

In the original work of Adams and collaborators [7], the two-torus bundle over the K3 base is built up by first constructing a non-compact bundle, and then changing complex structure in field space, allowing to discard the decoupled non-compact part from the bundle, while preserving supersymmetry.

To start, one introduces two extra chiral multiplets and , whose (imaginary) shift symmetry is gauged as

(7)

The compact bosonic fields will ultimately parametrize the torus fiber.

A generic two-torus is characterized by a complex structure and a complexified Kähler modulus , such that the metric and Kalb-Ramond field are given by

(8)

The Lagrangian for and , corresponding to a complexification of this two-torus, reads [21]:

(9)

The couplings between the chiral superfields and the field strength superfield contain field-dependent Fayet-Iliopoulos (FI) terms (last line) that are classically non-invariant under (super)gauge transformations:

(10)

This gauge variation should be such that it compensates the one-loop anomaly (6) of the base GLSM; this can be viewed as a worldsheet incarnation of the Green-Schwarz mechanism. Finally, in order for the action to be single-valued under in any instanton sector, the couplings should be integer-valued.

Moduli quantization

In order to restrict the non-compact fibration described above to a fibration while maintaining worldsheet supersymmetry, one has to define a complex structure in field space that allows for a decoupling of the real part of these multiplets. This is compatible with supersymmetry provided that the couplings between the gaugini and the fermionic components of the superfields vanish [7]. It amounts to the following relations between the Fayet-Iliopoulos parameters and the charges [21] \cref@addtoresetequationparentequation

(11a)
(11b)

with the complex charge defined as

(12)

Using these relations the gauge-variation of the field-dependent Fayet-Iliopoulos term reads:

(13)

which should be cancelled against the gauge anomaly from the chiral fermions in order to get a consistent quantum theory. One obtains the condition

(14)

reproducing the tadpole condition from the integrated Bianchi identity in Fu-Yau compactifications [19], see app. C.

The torus moduli and are partially quantized by the pair of supersymmetry conditions (11); in a model with worldsheet gauge group one obtains one such condition for each complex charge , hence the moduli are generically fully quantized. As was shown in [21], covariance of the theory under T-duality symmetries along the fiber provides another way of understanding quantization of the torus moduli. Under the transformation in , each complex charge is mapped to . For consistency this charge should belong to the same lattice as the original one, namely .

Demanding that this property holds for every topological charge in the model is actually a non-trivial statement. Generically, this is true if and only if the elliptic curve admits a non-trivial endomorphism

(15)

which is known as complex multiplication. This property holds if and only if both and are valued in the same imaginary quadratic number field with the discriminant of a positive definite integral quadratic form:

(16)

Crucially, conformal field theories with a two-torus target space are rational iff their and moduli satisfy these conditions [36, 37]; this property will play an important role in section 4.

One could also consider incorporating in the torsion GLSM terms corresponding to extra Abelian gauge bundles over the total space (that would be Wilson lines along the torus in the case), which are indeed allowed by the space-time supersymmetry constraints [19]. We leave this generalization of the models for future work.6

Torsion multiplet

Whenever the supersymmetry conditions (11) are met, the non-compact real parts of decouple and one can reorganize their imaginary parts into a torsion multiplet , with

(17)
(18)

shifted as under supergauge transformations, with the complex charge defined in eq. (12). The Lagrangian is given (omitting temporarily the topological B-field term for simplicity) by

(19)

where the superspace covariant derivative reads .

As usual going to Wess-Zumino gauge is convenient in order to exhibit the physical degrees of freedom; in the present situation one should not forget nevertheless that the theory is not classically gauge invariant, hence such gauge choice only makes sense in the path integral of the full quantum theory, including the base GLSM, as will be clear below when supersymmetric localization will be put into action.

In this gauge the torsion multiplet contains a compact complex boson coupled chirally to a gauge field and a free right-moving Weyl fermion. After going to Euclidean signature7 and some rescaling of the fields, one has

(20)

with and , and where denotes the field strength of the gauge field. After integrating by parts, one gets the following Lagrangian

(21)

where the left-moving currents and are coupled to the gauge fields, but not the right-moving ones.

Since we are working in Wess-Zumino gauge, the appropriate supersymmetry transformations are given by

(22)

where refers to the supergauge transformation which is needed to restore Wess-Zumino gauge after the supersymmetry transformation, corresponding to the chiral parameter . The transformation properties of the different component fields are listed in appendix A, eq. (115); the Lagrangian for the torsion multiplet is not invariant under this transformation, but its variation is precisely such that it compensates the variation of the effective action of the base GLSM under the gauge transformation back to WZ gauge.

To summarize, a consistent torsion gauged linear sigma-model is given by a base K3 GLSM whose gauge anomaly is canceled by a torsion multiplet, provided that the tadpole condition (14) holds. Finally one has to choose the and charges of all the multiplets in order to cancel the global anomalies, and to obtain the correct central charges of the IR superconformal algebra and rank of the vector bundle. The global charges of the torsion multiplet, proportional to their gauge charge , correspond naturally to charges under a shift symmetry. Consistent choices of global charges will be given in section 5.

3 New supersymmetric index of compactifications

We consider compactifications to four dimensions of the heterotic string theory. For any superconformal field theory with corresponding to the ’internal’ degrees of freedom of such compactification, the new supersymmetric index is defined as the following trace over the Hilbert space in the right Ramond sector:

(23)

where is the right-moving fermion number and the zero-mode of the right-moving R-current, which is part of the (right-moving) superconformal algebra. In general, this index is independent of D-term deformations, while it is sensitive to F-term deformations [27].

It was observed in [29, 30] that the threshold corrections to the gauge and gravitational couplings of heterotic string compactifications on are easily obtained in terms of the new supersymmetric index (23). Furthermore Harvey and Moore showed in [28] that it counts the four-dimensional BPS states as

(24)

One of the goals of this paper is to extend this analysis to Fu-Yau geometries. Formula (24) was proven using representation theory of the superconformal algebra underlying the CFT. As was explained in [38], non-linear sigma-models with a Fu-Yau target space are invariant under the action of the generators of a superconformal algebra, at the classical level, hence we expect that a similar reasoning holds in the present case.

3.1 New supersymmetric index of K3 T compactifications

We first review the computation of the new supersymmetric index in the familiar case of compactifications of the heterotic string, without Wilson lines for simplicity. We emphasize the role of the left-moving GSO projection and the formulation of the index as a chiral orbifold in order to facilitate the generalization to Fu-Yau compactifications in the next subsection.

We assume that the gauge bundle lies in the first only. More specially, we consider a gauge bundle with the embedding . The internal CFT is then the tensor product of a theory with and a theory corresponding to the second factor.

Using the factorization of the CFT in the two-torus and K3 factors, hence the decomposition of the corresponding superconformal algebra into the direct sum , we split the right-moving R-current as follows:

(25)

It allows to expand the superconformal index into the sum of two terms. For the second one, we get

(26)

for two different reasons. First, the fermionic partners of the have a pair of fermionic zero modes of opposite fermion numbers, hence the trace over the two-torus Hilbert space vanishes. Second the K3 SCFT has superconformal symmetry, hence the eigenvalues of , which are twice the eigenvalues of the Cartan current of the R-symmetry, come in pairs of opposite sign [28].

In order to trace over the internal Hilbert space of the theory we have to define a left-moving GSO projection corresponding to the first factor. We assume the existence of a left-moving symmetry, acting on the SCFT describing the K3 surface as on the remaining free left-moving Weyl fermions of the first .

We consider the following twining partition function in the RR sector, with a chemical potential for this symmetry:

(27)

with the left-moving current, and where the trace is over the Hilbert space of the superconformal field theory with . Then, using standard orbifold formulæ, see  [39], the new supersymmetric index is obtained as a sum over the sectors of the chiral quotient corresponding to the left-moving GSO projection:

(28)

where the modular form comes from the contribution of the second factor, see appendix B.

The partition function over the two-torus degrees of freedom is straightforward. For a torus with complex and Kähler moduli and the soliton sum is given by

(29)

Then the contribution to the partition function (27) reads

(30)

Finally one needs to compute the trace over the Hilbert space of the theory with a K3 target space. Let us consider the case of the standard embedding of the spin connection in the gauge connection, enhancing the supersymmetry of the K3 SCFT to . Then plugging back the expression (30) into equation (28), and tracing over the Hilbert space of the 6 free Weyl fermions with twisted boundary conditions, one gets finally the index in terms of the K3 elliptic genus [28]:

(31)

where the elliptic genus of K3 is defined by

(32)

As the elliptic genus is a topological invariant [22], it can be computed anywhere in the moduli space of K3 compactifications, for instance using Landau–Ginzburg orbifolds [40, 41, 39], or toroidal orbifolds [42]. It was shown recently [23, 24, 25] how to compute the elliptic genus directly at the level of the gauged linear sigma-model, using supersymmetric localization [26]. As we shall see in the next section, this localization method can be generalized to compute the new supersymmetric index of Fu-Yau compactifications.

The new supersymmetric index of compactifications is actually universal, independent of the choice of gauge bundle, as was shown in [43], and reviewed recently in [44]. The quantity should be a non-holomorphic modular form of weight -2, with a pole at the infinite cusp (this will remain valid in the case of Fu-Yau compactifications). Factorizing the index as

(33)

one can show that should be a holomorphic modular form of weight 10, hence proportional to . Due to the relation (24) the space-time anomaly cancellation condition fixes the coefficient to one. Hence the expression (31), obtained from the standard embedding with worldsheet supersymmetry, extends to any compactification; it means in particular that the “Mathieu moonshine” is a property of the new supersymmetric index regardless of the choice of gauge bundle [44]. Determining whether this property extends to Fu-Yau compactifications is one of the motivations for the present work.

3.2 New supersymmetric index of Fu-Yau compactifications

We now consider the main topic of this work, the computation of the new supersymmetric index of Fu-Yau compactifications based on their worldsheet formulation as torsion gauged linear sigma-models.

The starting point of the computation is the same as for compactifications. However in the case of torsion GLSMs one cannot split the worldsheet theory as a tensor product of the and the K3 factors, as none of them makes sense as a quantum theory in isolation. We assume as before that the gauge bundle over the total space, which is the pullback of a Hermitian-Yang-Mills gauge bundle over the K3 base, is embedded as in the first factor.

In analogy with the case, we decompose the new supersymmetric index in terms of a twining partition function for the torsion GLSM as follows,

(34)

where we have defined

(35)

the trace being taken into the Hilbert space of the superconformal theory obtained as the infrared fixed point of the torsion GLSM.

A crucial point at this stage is that the right-moving fermions associated with the factor, that belong to the torsion multiplet, are free in the Wess-Zumino gauge, see the Lagrangian (21), in particular not coupled to the components of gauge multiplet; this is the feature of the theory that eventually leads to supersymmetry in space-time. The right-moving R-current of the superconformal algebra, whose zero-mode appears in the trace (35), is of the form

(36)

where the ellipsis stands for a term in , as the bottom component of the torsion multiplet can have a shift R-charge, the contributions of the chiral and Fermi multiplets and -exact terms, where is the localization supercharge (see next section), relating the exact R-current to the Noether one defined in the UV theory. Because there are two right-moving fermionic zero-modes and that need to be saturated in the path integral, and that there are no interactions involving these fermionic fields in the Lagrangian, we do not have to care about these extra terms in any case, as their contribution to the path integral vanishes.

In summary, the new supersymmetric index of Fu-Yau compactifications follows from the twisted partition function (35) that can be formulated as a path integral. Considering the theory on a two-dimensional Euclidean torus of complex structure , the quantity to compute can be schematically written as

(37)

where we have included a background gauge field for the global symmetry

(38)

in order to implement the twisted boundary conditions.8 The torsion multiplet will be coupled chirally to this flat connection, in the same way as it couples to the dynamical gauge field, see eq. (21).

The left- and right-moving fermions have periodic boundary conditions along both one-cycles of the worldsheet torus. We have also included for latter convenience coupling constants and in front of the chiral and Fermi multiplets actions, respectively and , besides the usual factor in front of the vector multiplet action and in front of the Fayet-Iliopoulos term . Finally denotes the torsion multiplet action.

To take care of the gauge redundancy one should in principle introduce a gauge-fixing procedure and the corresponding Faddeev-Popov ghosts; however it does not really impact the computation of the path integral through supersymmetric localization that will follow, see [46] for details.

Having set the calculation in functional language will allow us to deal with it using localization techniques. In this formulation one sees that the insertion of the operator only contributes through the free right-moving fermion which is part of the torsion multiplet, and this insertion appears as a prescription to deal with the fermionic zero modes. This will be important in proving that the supersymmetric localization method is valid in this context, as we shall explain below.

4 New supersymmetric index through localization

In this section we obtain the twining partition function of Fu-Yau compactifications, defined by eq. (35), allowing to compute their new supersymmetric index using eq. (34). In this section we consider the case of a worldsheet gauge group; the main result is given by equation (70). The generalization to higher rank will be provided in the next section.

4.1 Justification of the supersymmetric localization method

Supersymmetric localization techniques have been successfully applied to compute the elliptic genera of ordinary (0,2) gauged linear sigma-models, see [23, 24, 25]. Our goal is to extend these results to the new supersymmetric index of Fu-Yau compactifications using the torsion gauged linear sigma-models.

One immediate objection to this project is that, as mentioned above, the contribution of the torsion multiplet to the action is not invariant under the supersymmetry transformations (22); furthermore, the operator insertion in the path integral (37) is not supersymmetric. As we will see below, these two obstacles can be successfully overcome.

The GLSM corresponding to the base contains a vector multiplet together with chiral and Fermi multiplets (conventions related to superspace are gathered in appendix A). Define the supercharge, cf. equation (22):

(39)

As was noticed in [24], the Lagrangian describing the dynamics of these multiplets, including the superpotential term and the Fayet-Iliopoulos term , are actually exact with respect to the transformation introduced above. One finds

(40)

In an ordinary GLSM, this would imply immediately that the path integral is independent of the coupling constants , and and of the FI parameter .

To understand what happens in the present situation, let us write the contribution of the base and of the vector multiplet to the TGLSM as where the first term is the vector multiplet action, written as a -exact term, and denotes the (-exact as well) contribution of the chiral and Fermi multiplets and of the constant FI term. The functional integral we aim to compute is of the schematic form

(41)

One considers then the derivative with respect to :

(42)

As mentioned above, the operator insertion has the effect of saturating the fermionic zero modes present in the measure over the torsion multiplet. Hence its variation under the action of the supercharge , while non-zero, leads to terms which do not saturate the fermionic zero modes anymore, and thus do not contribute to the path integral.

Since the supersymmetry transformation we are considering contains a supergauge transformation of chiral parameter , see eq. (22), there is a non-trivial transformation of the functional measure over the chiral and Fermi multiplets due to the gauge anomaly. At the same time, the torsion multiplet action is not classically invariant under the action of the supercharge, see eq. (13). Whenever the tadpole condition (14) is satisfied, these two variations cancel each other:

(43)

In conclusion, whenever the quantum anomaly of the base GLSM is canceled against the classical contribution from the torsion multiplet, we get as in more familiar examples

(44)

using an analogue of Stokes’ theorem in field space. The result of the path integral is then independent of the gauge coupling, allowing to take a free-field limit . The same reasoning allows to take the limit and in the chiral and Fermi multiplets actions respectively. By rescaling the superfields and one sees that the superpotential couplings do not contribute to the path integral which is localized in the free-field limit of the theory, as far as the base GLSM is concerned.

A similar argument regarding the dependence of the path integral on the torsion multiplet couplings would fail, as the torsion multiplet action is not -exact, being not even -closed.9 Nevertheless, this action is Gaussian hence the path integral can be performed exactly. As expected, it implies that the result of the path integral computation does depend on the moduli of the principal two-torus bundle in the Fu-Yau geometry.

As in [24] the localization locus contains the following zero-modes, that should be integrated over:

  • Gauge holonomies on the worldsheet two-torus, parametrized by , being defined on the torus of complex structure to avoid gauge redundancy,

  • The zero-mode of the auxiliary field in the gauge multiplet,

  • The gaugino zero-modes , .

Setting aside the contribution of the torsion multiplet, most of the steps that go into the derivation of the elliptic genus by Benini, Eager, Hori and Tachikawa [24], especially the reduction of the integral over the gauge holonomies into a contour integral of the one-loop determinants, carry over to the present situation without significant modifications. We refer the reader to this article for a detailed account of the computation and provide below justifications of this statement.

In order to saturate the gaugino zero-modes, the contribution from the chiral multiplets at one-loop is of the form, in the limit ,

(45)

Since the torsion multiplet has no coupling to the gaugini, as was explained in section 2, it is not involved in the saturation of their zero-modes; hence this part of the derivation is unchanged. Furthermore, the torsion multiplet has no coupling to the auxiliary -field (by supersymmetry); as a consequence, the -dependence of the one-loop determinant lies entirely in the contribution from the chiral multiplets of the base, allowing to reduce the integral over the -plane to a contour integral as in [24].10 Then, the singularities that arise in the limit after integrating over the zero-mode , as well as the contour deformation in the -plane leading to the contour prescription, remain the same as in the aforementioned computation.

For a rank-one gauge group, the formula for the twining partition function (35) is then of the form:

(46)

where

(47)

the various factors in the above formula being the one-loop contributions of the various multiplets around the localization locus, and denoting a contour around the singularity . and form a partition of the set of poles of the product of chiral multiplet determinants and are described in detail in [24]. These poles correspond to ’accidental’ bosonic zero-modes and occur whenever

(48)

(resp ) being the gauge (resp ) charge of the multiplet. The set of poles is then split into two sets according to or . Notice that the choice of or in (47) give the same result since the sum of the residues of a meromorphic function on the torus vanishes.

In the case where the gauge group has an arbitrary rank, the formula generalizes using a notion of residue in higher dimensions, the Jeffrey-Kirwan residue [47]; one obtains the following expression for the twining partition function in terms of the one-loop determinant [25]:

(49)

with now a meromorphic -form, and the order of the Weyl group. The sum does not depend on the choice of co-vector , in the dual of the Cartan subalgebra, per singular locus .

4.2 Contribution of the K3 base

As we have noticed previously, the contributions from the chiral and Fermi multiplets corresponding to the base, as well as from the gauge multiplets, are similar to those appearing in the elliptic genus computed in [24]. However, since in the present context issues of gauge invariance are crucial, we need to be a little bit more careful regarding the definition of the chiral fermionic determinants. In the end, taking into account the contribution of the torsion multiplet, the tadpole condition will translate into a cancellation of the prefactors in these expressions.

In order to define the determinant of a chiral Dirac operator coupled to a (background) flat gauge field, one has to specify a way to split the determinant of the self-adjoint operator into a ’holomorphic’ part and an ’anti-holomorphic’ part. According to Quillen’s theorem [48], the zeta-regularized determinant of the former is given by (see  [49] for a discussion in a similar context):

(50)

where is here a compact notation which takes into account both the gauge field and the background . Splitting , one can define the chiral determinant as:

(51)

modulo an overall factor independent of ; other definitions can be interpreted as corresponding to different choices of local counterterms.

With this prescription, as was argued by Witten in [50] in a related context, the gauge functional obtained after the path integral over the fermionic degrees of freedom can be viewed as a holomorphic section of a holomorphic line bundle over the space of gauge connections. The determinant is indeed annihilated by the covariant derivative (restricted to its zero-mode part in the present situation). It turns out that this choice, besides its nice geometrical interpretation, is naturally compatible with the contribution from the torsion multiplet Lagrangian, see eq. 63 below, leading to an expression without modular anomalies.

Equipped with this result, one can express the contribution of a chiral multiplet of gauge charge and charge