Elliptic genera of 2d (0,2) gauge theories from brane brick models

Journal of High Energy Physics, Jun 2017

We compute the elliptic genus of abelian 2d (0, 2) gauge theories corresponding to brane brick models. These theories are worldvolume theories on a single D1-brane probing a toric Calabi-Yau 4-fold singularity. We identify a match with the elliptic genus of the non-linear sigma model on the same Calabi-Yau background, which is computed using a new localization formula. The matching implies that the quantum effects do not drastically alter the correspondence between the geometry and the 2d (0, 2) gauge theory. In theories whose matter sector suffers from abelian gauge anomaly, we propose an ansatz for an anomaly cancelling term in the integral formula for the elliptic genus. We provide an example in which two brane brick models related to each other by Gadde-Gukov-Putrov triality give the same elliptic genus.

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Elliptic genera of 2d (0,2) gauge theories from brane brick models

Received: March Elliptic genera of 2d (0,2) gauge theories from brane brick models Sebastian Franco 1 3 5 7 8 Dongwook Ghim 1 3 6 8 Sangmin Lee 1 2 3 4 6 8 Rak-Kyeong Seong 0 1 3 8 Open Access 1 3 8 c The Authors. 1 3 8 0 Department of Physics and Astronomy, Uppsala University 1 365 Fifth Avenue , New York, NY 10016 , U.S.A 2 Center for Theoretical Physics, Seoul National University 3 160 Convent Avenue , New York, NY 10031 , U.S.A 4 College of Liberal Studies, Seoul National University 5 The Graduate School and University Center, The City University of New York 6 Department of Physics and Astronomy, Seoul National University 7 Physics Department, The City College of the CUNY 8 SE-751 08 Uppsala , Sweden probing a toric Calabi-Yau 4-fold singularity. We identify a match with the elliptic genus of the non-linear sigma model on the same Calabi-Yau background, which is computed using a new localization formula. The matching implies that the quantum e ects do not drastically alter the correspondence between the geometry and the 2d (0; 2) gauge theory. In theories whose matter sector su ers from abelian gauge anomaly, we propose an ansatz for an anomaly cancelling term in the integral formula for the elliptic genus. We provide an example in which two brane brick models related to each other by Gadde-Gukov-Putrov triality give the same elliptic genus. brick; models; Conformal Field Theory; D-branes; Duality in Gauge Field Theories; Super- 1 Introduction 2 Review of 2d (0,2) gauge theories and brane brick models 6.1 6.2 6.3 3 Elliptic genus from gauge theory 4 Elliptic genus from geometry 5 Abelian anomaly and its cancellation 5.1 General discussion 5.2 Anomaly cancelling factor | an ansatz 6 Orbifold models C4=Z2(0; 0; 1; 1) C4=Z2(1; 1; 1; 1) C4=Z3(1; 1; 2; 2) C4=Z2 Z2(0; 0; 1; 1)(1; 1; 0; 0) 6.5 C4=Z4(1; 1; 1; 1) 7 Non-orbifold models H4 | anomaly and triality 8 Discussion A Theta functions and their identities A.1 Theta functions A.2 Proving theta function identities B Degenerate poles and the Je rey-Kirwan residue Introduction Geometry has played a key role in the study of supersymmetric gauge theories and their dynamics. Comparing the moduli space of vacua has led to the discovery and veri cation of dualities. In three or higher dimensions, it is possible to examine how quantum e ects modify the classical moduli space of vacua. Depending on the number of supersymmetries and other factors, the moduli space can be (partially) lifted or its geometry can signi cantly deviate from the classical one. In two dimensions, under suitable conditions, gauge theories with classical moduli space of vacua may ow to a non-linear sigma model whose target space is the quantum corrected version of the classical moduli space. The pioneering work [1] on gauged linear sigma models (GLSM) shows that non-linear sigma models and Landau-Ginzburg (LG) theories can appear as di erent phases of the same gauge theory, thereby establishing a connection between the two. Non-abelian gauge theories may exhibit even richer structures with various phases of the quantum moduli space [2]. led to exciting discoveries. While gauged linear sigma models with (0; 2) SUSY have been studied for many years with heterotic model building in mind (see, e.g., [3, 4]), the study of 2d (0; 2) non-abelian gauge theories has been limited in its scope until recently. One of the most interesting recent breakthroughs is Gadde-Gukov-Putrov (GGP) triality [5, 6], which identi es three seemingly unrelated quiver gauge theories that ow to the same superconformal eld theory at low energies. There are various ways to realize 2d (0; 2) gauge theories in string and M-theory [7{20]. We will focus on brane brick models [10{14], which arise from D1-branes probing noncompact toric Calabi-Yau 4-fold singularities (CY4). A brane brick model is a type IIA brane con guration of D4-branes suspended between a NS5 brane that wraps a holomorphic given by the probed Calabi-Yau geometry. The general structure and construction of brane brick models was rst spelled out in [10]. The connection between the CY4 geometry and the gauge theory through brane brick models was elaborated in [11]. How triality is realized in terms brane brick models was explained in [12]. Finally, in [13] it was shown how the results of [10{12] can be recast geometrically from a mirror CY4 perspective. This paper addresses yet another aspect of brane brick models. Our main goal is to compute the elliptic genus of abelian brane brick models as a means to probe their infrared dynamics. Given the geometric origin of brane brick models, the most naive candidate for the infrared theory is a non-linear sigma model whose target space is the CY4 associated with the gauge theory. The naive guess turns out to be correct. In a number of examples, we show that the gauge theory computation and the sigma model computation of the elliptic genus agree perfectly. To the extent the elliptic genus can di erentiate theories, the infrared behavior of the gauge theory is the same as that of the sigma model. There are several technical aspects of our computation that make the comparison between the gauge theory and the geometry non-trivial. For gauge theories, the supersymmetric localization for the elliptic genus was carried out in depth in [21{23]; see also [24] for a recent review and further references. Localization reduces the path integral to a nite dimensional contour integral over gauge fugacity variables, supplemented by the Je reyKirwan (JK) residue prescription [25]. In simple examples, the contour integral can be performed explicitly and the result is a function of the modular parameter and avor fugacity variables. For the non-linear sigma model we obtain a simple geometric formula by combining elements from related works in the literature [26, 27]. For any triangulation of the toric diagram of the CY4, the geometric formula expresses the elliptic genus as a sum over tetrahedra in the triangulation. As expected, the sum is independent of the speci cs of the triangulation. The gauge theory computation is further complicated by the fact that, in some theories, the matter sector produces non-vanishing abelian gauge anomalies. Since the gauge theories that we consider have a clear string-theoretic origin, the anomaly should be cancelled through an interaction between open string modes and closed string modes. We have not been able to derive the precise anomaly cancelling mechanism from string theory. Instead, assuming the existence of a canceling mechanism, we have found an ansatz for the anomaly cancelling factor in the JK integral formula, which works for a large class of examples. Our ansatz is valid for theories in which the total number of chiral elds is greater than the number of Fermi elds in such a way that the anomaly polynomial can be written as a sum of squares with positive coe cients. As an application of these computations, we will check GGP triality for brane brick models. Brane brick models di er from the SQCD-like theories considered in the original papers on triality [5, 6] in that they correspond to quivers without avor symmetry nodes. Triality has been proven in [5] by the use of an elliptic genus computation for SQCDlike theories and it is reasonable to expect that these calculations extend to more involved theories. However, a systematic study of more complicated theories, such as quiver theories with only gauge nodes, has been lacking due to the increasing complexities of the required JK residue computations. For brane brick models, so far, triality has been shown to leave the CY4 target space invariant by using the underlying geometry of the brane brick model construction [12]. In this paper, we will verify that brane brick models connected via triality share the same elliptic genus, as expected from the results in [12]. This paper contains examples of the elliptic genus computation for simple brane brick models. The rest of this paper is organized as follows. In section 2, we brie y review 2d (0; 2) theories and brane brick models [10{12] and set up our notation. Section 3 reviews how to compute the elliptic genus from the gauge theory following [21{23]. In section 4, we propose a geometric formula that computes the elliptic genus from a triangulation of the toric diagram of the CY4. In section 5, we discuss the general form of anomalies that are present in abelian brane brick models. We propose an ansatz for an anomaly cancelling factor that works in a large class of examples. In section 6, we compute the elliptic genus of some orbifold models and nd perfect agreement between the gauge theory and geometric computations. In section 7, we con rm the agreement of the two computations in two nonorbifold models. In one of the examples, we also con rm the expectation that two gauge theories that are related by triality share the same elliptic genus. Section 8 concludes the paper with a discussion on future directions. Review of 2d (0,2) gauge theories and brane brick models 2d (0,2) gauge theories. We now brie y review basic aspects of 2d (0; 2) gauge theories to establish our notation. For more thorough reviews, we refer to [1, 5, 8]. There are three types of supermultiplets in 2d (0; 2) gauge theories. We will use super eld formalism with superspace coordinates (x ; + ). All component elds are assumed to be complex-valued unless speci ed otherwise. The multiplets are: Chiral multiplet i The physical component elds are a boson and a right-moving Fermion Fermi multiplet The only physical eld is the left-moving fermion , which is a supersymmetry singlet in the free theory limit. Besides, the super eld contains an auxiliary G and a coupling to a holomorphic function E( ) of chiral super elds through a deformed chirality condition. Vector multiplet V It contains the real gauge boson v , complex gaugini , and a real auxiliary eld D. They couple to matter elds minimally through a supersymmetric completion of the gauge-covariant derivative. For each Fermi multiplet a, in addition to the holomorphic Ea-term mentioned above, it is possible to introduce another holomorphic term called J a( ). The (0; 2) supersymmetry requires that J - and E-terms satisfy an overall constraint: Integrating out the auxiliary elds D , we obtain a familiar looking D-term potential (and its fermionic partner). For abelian theories, the potential takes the form X tr [Ea( i)J a( i)] = 0 : VD = VF = where t are complexi ed Fayet-Iliopoulos (FI) parameters. Integrating out the auxiliary elds Ga, we obtain what may be called an F -term potential, as well as Yukawa-like interactions between scalars and pairs of fermions. Brane brick models. We can represent the 2d (0,2) quiver gauge theory that lives on the worldvolume of D1-branes probing a toric CY4 by a brane brick model [10{13]. When we T-dualize the D1-branes at the CY4 singularity, we obtain a Type IIA brane con guration of D4-branes wrapping a 3-torus T 3 and suspended from an NS5-brane that wraps a holomorphic surface intersecting with T 3. This Type IIA brane con guration, which we call the brane brick model, is summarized in table 1. The holomorphic surface encodes the geometry of the probed toric Calabi-Yau 4-fold and originates from the zero locus of the Newton polynomial of its toric diagram. Brane Brick Model Oriented face between bricks i and j Unoriented square face between bricks i and j Gauge Theory Gauge group Bifundamental chiral eld from node i to node j Bifundamental Fermi eld between nodes i and j Quiver diagram Oriented (black) arrow from node i to node j Unoriented (red) line between nodes i and j a J - or an E-term Interaction by J - or E-term Plaquette encoding The brane brick model encodes all the data needed to write down the full Lagrangian of the gauge theory. Moreover, it combines geometric and combinatorial data in a powerful way that enables us to analyze various properties of the gauge theory. Sometimes, it is more convenient to work with the periodic quiver, which is the graph dual of the brane brick model. Being graph dual, they contain exactly the same information. The dictionary between the brane brick model (or equivalently the periodic quiver) and the gauge theory is summarized in table 2. Non-compact target space and its regularization. We are dealing with theories whose target spaces are non-compact. Such theories may contain an in nite number of states along at directions, and the elliptic genus may not be well-de ned. In order to regulate this, we will use three of the four global U(1) isometries of the toric CY4 in order to re ne the elliptic genera. The remaining U(1) in the CY4 is identi ed with the R-symmetry of the gauge theory. It cannot be used as a re nement since it does not commute with the supercharges. Instead, it will be used to saturate the fermionic zero mode from the decoupled U(1) gaugino in the path integral; see section 3. A comment on the central charge cR and the R-charge. The right-moving central charge cR and the R-charge assignments of a 2d (0; 2) SCFT are closely related via a brane brick models. This cannot be true as long as the theory is a non-trivial unitary CFT. A similar breakdown of c-extremization has been reported in the theory of a free (0; 2) chiral multiplet in [28]. This failure is presumably due to the non-compactness of the corresponding target space. The non-compactness makes the vacuum non-normalizable and allows for an additional non-holomorphic current whose two-point function with the R-current might not vanish, violating an assumption of the extremization principle. A remedy to this breakdown will be the subject of a future investigation. Elliptic genus from gauge theory Recently, several groups [21{23] independently derived a localization formula for computing the elliptic genus of a 2d (0; 2) gauge theory. The elliptic genus has become a powerful tool for studying the dynamics of these theories. For example, it has been recently used to verify GGP triality [5]. This section summarizes how to compute the elliptic genus following [22, 23]. The elliptic genus is de ned by the trace over the Ramond-Ramond (R-R) sector, in which fermionic elds satisfy periodic boundary condition, as follows:1 I(q; xi) = TrRR( 1)F qHL qHR Y xaKa ; Given a charge vector , we have x = Q ensures that the q-dependence drops out of (3.1). where Ki are the Cartan generators for the global avor symmetry group. The parameter = e2 i awa . Note that (0; 2) supersymmetry For a 2d (0; 2) GLSM, (3.1) can be evaluated in terms of a contour integral of a meromorphic (r; 0)-form Z1-loop, I(q; xi) = where r is the rank of gauge group G. Z1-loop is de ned on the moduli space M of at connections of G over T 2, where the contour C in (3.2) is an r-dimensional cycle in M. Each chiral, Fermi, vector multiplets, respectively. The one-loop determinants are given by are the weights for the representation R of the gauge and avor groups in which the chiral and Fermi multiplets transform. Note that for the vector multiplet contribution evaluated over ui. The de nitions of the functions 1(q; y) and (q) in (3.3) are reviewed in appendix A.1. The contour integral in (3.2) is evaluated by following the Je ery-Kirwan (JK) residue prescription [25]. The physical motivation for the prescription is given in [22, 23]. In the end, the prescription gives a formula for the elliptic genus in (3.2): I(q; xi) = JK-Res (Qju ; ) Z1-loop(q; u; ai) ; where jW j is the order of the Weyl group W of the gauge group G. In addition, Qju is the covector of a singular hyperplane. is a generic charge covector that selects a set of poles u that contribute to the JK residues in (3.4) depending on their covectors Qju . 1One can use the NS-NS boundary condition to de ne the elliptic genus [21], which is di erent from the R-R boundary condition we use here. For 2d (2; 2) theories, spectral ow can be used to compare the results from di erent boundary conditions. This is not the case for 2d (0; 2) theories. We will focus on the R-R boundary condition, which makes it easier to compare with the geometric formula in section 4. A pole is called non-degenerate when it is determined by the intersection of exactly r JK-Res (Qj0; ) 2 Cone(Qj1 , where Cone(Qj1 , ,Qjr ) is a subspace of Rr spanned by Qj1 , ,Qjr with positive coe cients. Let us make two important observations regarding the role of . First, it determines which poles contribute to the index. Second, the nal answer to the integral is independent of the choice of . In other words, individual poles contributing to the index depend on the choice of , but the nal sum over all residues is independent of the choice. corresponding poles degenerate. In appendix B, we present the so-called ag method [23] for resolving the JK residue for degenerate poles. The ag method generalizes the JK residue formula in (3.5) for any type of pole and arbitrary high rank r of the gauge group. In fact, the computational complexity of the JK residue formula increases extremely fast with the rank r. For the present paper, this is not an issue since we focus on abelian theories with gauge group U(1)r for small values of r. U(1) decoupling and the modi ed elliptic genus. Abelian gauge theories from brane mental or adjoint representations. As a result, the overall diagonal U(1) decouples from the rest of the theory, leaving us with U(1)r 1. This decoupling can be easily implemented by the following rede nition of gauge holonomy variables, u00 = u0j = uj ur (i = 1; We may discard the decoupled U(1) vector multiplet at the classical level and compute decoupled U(1) as elaborated below. A naive inclusion of the decoupled U(1) makes the elliptic genus vanish. From the path integral point of view, the vanishing is due to the gaugino zero modes. In the contour integral formula (3.4), we have du00 in the measure, but the integrand is independent of u00, leading to a vanishing result. In order to avoid the trivially vanishing result, we modify the de nition of the index following the spirit of [29]. The key idea is to include the R-symmetry fugacity in the = 0 (b = 1) gives the original index, which we call I0(q; xi). Because the R-charge does not commute with superHowever, we can consider its rst derivative which under suitable conditions has a chance to become a supersymmetric index. Generally, checking whether I1(q; xi) quali es as a supersymmetric index is challenging. In our case, however, the derivative in (3.7) can be directly associated to the U(1) decoupling. Since the free U(1) decouples from the interacting part, any twisted partition function should factorize as follows I(q; xi; b) = Ifree(q; b) The free part, Ifree(q; b), is exact and q-independent for arbitrary values of , since the theory is free. This exactness does not rely on supersymmetry. Supersymmetry does imply considerably renormalized. Going back to the rst derivative in (3.7), we set = 0 to obtain I1(q; xi) = Iint(q; xi; b = 1) + Ifree(q; b = 1) @Iint(q; xi; b) term quali es as a supersymmetric index. In the rest of this paper, we will loosely call the rst derivative I1(q; xi) the elliptic genus (or the index) and denote it by I(q; xi) without any subscript. Canonical example: C4. We present here the elliptic genus for the simplest abelian brane brick model corresponding to C4 [8, 10].2 The theory has a single U(1) gauge group. Its toric and quiver diagrams are shown in gure 1. The full global symmetry is U(1)R, where we assign fugacities x, y, z to each of the U(1) factors in the Cartan 2The elliptic genera of its SU(N ) or U(N ) generalizations were thoroughly studied in [30]. Table 3. Global charges of matter elds in the C4 theory. subalgebra of SU(3).3 Table 3 summarizes the global symmetry charges carried by the chiral and Fermi elds of the theory. The one-loop integrand from the matter sector is given by Z1-loop = i (q) 1(q; x) 1(q; y) 1(q; z) The gaugino from the decoupled U(1) contributes since the elliptic genus of the free fermion reads = (q)2 ; Ifree = i 1(q; b) Following the prescription in (3.9), we have IC4 (q; x; y; z) = i (q)3 1(q; x) 1(q; y) 1(q; z) i (q)3 1(q; x1) 1(q; x2) 1(q; x3) : 1(q; s1) 1(q; s2) 1(q; s3) 1(q; s4) In the second line, for later convenience, we introduced the shorthand notation, x1 = x ; s1 = px=yz ; x2 = y ; s2 = py=zx ; x3 = z ; s3 = pz=xy ; s4 = pxyz : Poles in fugacity variables. 3It is important to note that the global symmetry of a 2d (0; 2) gauge theory depends not only on its quiver, but also on its J- and E-terms. For brevity, throughout the paper we will often provide only quiver diagrams, but the full theories are taken into account in our computations. Unless explicitly noted, the complete information about the theories we consider can be found in our earlier works [10{14]. IC4 (q; x; y; z) = F (q; x; y; z) = Qi4=1(1 F (q; x; y; z) ; 1 + X(si + si 1) q2 + O(q3) and are Laurent polynomials in si. We can show that the poles are absent in Fk for all and taking the limit ! 0, we obtain IC4 j !0 = 1(q; x) 1(q; y) 1(q; e2 i =xy) where we used the identity (A.9). Comparing (3.15) and (3.16), we deduce that all Fk 1 belong to the O( ) part of (3.16), thereby proving the absence of the pole. Note that this proof relies crucially on the (q)3 factor in the numerator which originates from the de nition of the modi ed elliptic genus (3.9). In the next section, we will show that the absence of poles for Fk 1 generalizes to all toric CY4. Elliptic genus from geometry In this section, we propose a geometric formula for computing the elliptic genus. This formula only depends on the toric diagram of the Calabi-Yau 4-fold. The proposal for such a geometric formula is motivated by two relevant results known in the literature. The rst comes from the equivariant localization approach to the computation of the elliptic genus for non-linear sigma models in [26]. The second is based on the computation of the equivariant index, which counts holomorphic functions on a Calabi-Yau cone, in [27]. This index is the Hilbert series of the coordinate ring formed by the holomorphic functions and has been shown to relate to the volume function of the base Sasaki-Einstein manifold. toric diagram as a convex polytope in Z3. Martelli-Sparks-Yau formula for Hilbert series. We begin with a brief review of the geometric formula of [27] derived by Martelli-Sparks-Yau (MSY), specialized to a CY4; The toric diagram of X is de ned by a collection of integer valued vectors the toric diagram. The Calabi-Yau condition makes it possible to work in an SL(4; Z) basis The toric diagram also de nes a solid cone 4X X is a U(1)4 bundle over 4X . Geometrically, the Hilbert series for X enumerates lattice points on the solid cone 4X , HX (t) = fmg i=1 It was shown in [27] that the normalized volume of the base Y as a function of the Reeb vector bi can be derived from the Hilbert series via = lim h 4HX (ti = e bi )i b4=4 The minimum of this function gives the volume of the Sasaki-Einstein base: = VY (b ) : (vI i) = BBB00 10 01 11CC HC4 (t) = t4=t1t2t3) VS7 (b) = b = (1; 1; 1) : =) VS7 (b ) = 1 : The Hilbert series is The volume function, is minimized at Up to an SL(4; Z) basis change, t1 = s1 ; t2 = s2 ; t3 = s3 ; t4 = s1s2s3s4 ; the Hilbert series in (4.5) agrees with the standard formula for C where each si independently counts holomorphic monomials of C. The key idea of the MSY formula for the Hilbert series is to triangulate the toric diagram by a set of minimal tetrahedra, treat each tetrahedra as a C4 and compute H for it, and sum all these individual HCn (s) = Y contributions. Concretely, consider a triangulation TX consisting of minimal tetrahedra, hva1 ; va2 ; va3 ; va4 i = ijklvai1 vaj2 vak3 val4 = 1 : Introduce a dual vector for each face of a minimal tetrahedron: The set of dual vectors gives a formula for the Hilbert series: (wap )i = HX (t) = fag2TX p=1 1 The rigorous derivation of this formula, which is explained in [27], is an application of the Duijstermaat-Heckman localization formula [32]. Elliptic genus from NLSM. On general grounds, we expect that the abelian GLSM's under consideration ow to NLSM's with CY4 target spaces. As shown in [26] in a similar but di erent context, it is possible to write down the NLSM and derive a formula for the elliptic genus from the path integral via localization. Let us sketch the derivation of the elliptic genus from the NLSM, leaving the details for a future work [33]. The eld content of the NLSM of our interest is as follows. The bosonic elds i describe the tangent bundle. The left-moving a (1 = 1; : : : ; 6) describe a vector bundle in the 6 (real) representation of the SU(4) holonomy group of the CY4. Finally, there are left-moving singlets , , which are the NLSM counterpart of the decoupled U(1) gaugini in the GLSM. The classical action of the NLSM contains suitably covariantized kinetic terms and a ) curvature term. To compute the elliptic genus via path integral, one separates the zero modes and the quantum uctuation around the zero modes. Supersymmetry ensures that the one-loop determinants, which capture the leading quantum become exact. The nal result is a nite-dimensional integral over bosonic and fermionic zero modes, where the integrand is product of one-loop determinants over uctuations. In computing the one-loop determinants, both the kinetic terms and the curvature term contribute. The dependence on the zero-modes is cancelled in an intermediate step, so that the nal result is a function of Ri| 0i 0|, where Ri| is a contracted version of the curvature tensor. As usual, the Fermion zero modes 0 are interpreted as di erential forms d , d . Hence, the elliptic genus becomes a (q-dependent) characteristic class integrated over the manifold. So far, we have sketched how to compute the un avored elliptic genus of the NLSM. To compute the avored elliptic genus, we should deform the NLSM to include terms that depend on the Killing vectors for the U(1)3 isometry of the toric CY4. The path integral then localizes on the xed points of the Killing vectors. The localization is similar to that used in the MSY formula we reviewed above. A crucial point is that the triangulation of the toric diagram amounts to dividing the CY4 into a number of C4 patches, and that the localization simply collects contributions from each patch. avored elliptic genus from triangulation. Combining the elements reviewed above, we are now ready to present the geometric formula for the elliptic genus of a The elliptic genus of C 4 serves as the building block of the whole construction. For C4, the GLSM and the NLSM are equivalent and we can copy the result (3.13): IC4 = i (q)3 1(q; x) 1(q; y) 1(q; z) To implement the SL(3; Z) basis change in the triangulation, we rewrite this as JC4 (t) = 1(q; t1) 1(q; t2) 1(q; t3) 1(q; t4=t1t2t3) Given a triangulation T (X) of the toric diagram for an arbitrary X, we rst compute the \pre-index" The arguments in the denominator are the same as for the Hilbert series: JX (t) = i (q)3 Qe3=1 1 q; zefag(t) Q4p=1 1 q; ypfag(t) The arguments in the numerator are z1fag(t) = z2fag(t) = z3fag(t) = At the nal stage, we translate ti into fugacities and turn o the R-symmetry fugacity: IX (x; y; z) = JX ti = where xa = (x; y; z) as in (3.14). The exponents mia in (4.19) is determined by the requirement that the bosonic part of the chiral ring matches between the gauge theory and the geometry. For this purpose, we can go back to the Hilbert series reviewed earlier in this section. H(t) = while the Molien sum gives The two results agree if we change the variables as Recall that for C4, we expressed ti in terms of si in (4.8). For orbifolds of C4, to be discussed in section 6, we can similarly rewrite ti in terms of si by comparing the Hilbert series computed from triangulation with the Hilbert series computed from the Molien sum triangulation gives (see gure 2) t4=t1t2t3) 1=t3)(1 t3t4=t1t2) H(s) = (1 + s3)(1 + s4) t1 = s1 ; t2 = s2 ; t3 = s3=s4 ; t4 = 1 : The same principle applies to all orbifolds. rings using the methods explained in [10, 11]. For non-orbifolds, the Molien sum is not available, but we can still compare the chiral Index theory and xed point formula. Our discussion leading to the geometric formula (4.19) relied heavily on the toric nature of the target space. Here we brie y digress to understand the formula from the standard index theory in a way less dependent on toric geometry. The elliptic genus of a general (0; 2) sigma model was derived in [34]. The sigma model consists of a d dimensional Kahler target space X equipped with a rank r holomorphic vector bundle E. Anomaly cancellation gives restrictions on the rst and second Chern classes of E and those of the tangent bundle T . If we use the splitting principle to write formally, the elliptic genus turns out to be [34] X i=1 ZX;E = P ( ; z) = Relating this to our formula takes a few steps. The most crucial step is to apply the stan xed point formula to the characteristic class above by means of the toric isometry. Then, the integral over the target space is replaced by the sum over xed points, and the curvature eigenvalues vi and wj are replaced by our fugacity variables xi and sj . Additional care should be taken to incorporate the decoupled Fermi multiplets. A detailed derivation along this line will be given in [33]. Triangulation with subtraction. The geometric localization is based on a triangulation of the toric diagram. There are toric diagrams of CY4 which do not admit simple triangulation, i.e. triangulations that only use the points in the toric diagram. For instance, integer lattice points. But, as we will see in section 6.2, it is possible to add up three tetrahedra and subtract one to construct the desired toric diagram. The geometric formula based on triangulation including subtraction was used in the computation of the Hilbert series in [35]. In this paper, we will apply the subtraction method to the geometric formula for the elliptic genus and nd results compatible with other computations. Poles and zero modes. Consider the q-expansion of the index obtained from the geometric formula, I = P In the previous section, for the index of C4, we observed that F0 has codimension 1 simple poles in (x; y; z) but all Fk 1 have no such poles and are Laurent polynomials in (x; y; z). Combining that observation and the fact that the geometric formula sums up contributions from triangulation, we deduce that, for all toric CY4, the codimension 1 poles are present only in F0 and absent from all Fk 1. Physically, from the NLSM point of view, the poles stem from the fact that the target space is non-compact. Without the fugacities, the \center of momentum" degree of freedom moving in the non-compact direction will cause a divergence. The fugacities regulate the divergence. It is comforting to notice that all \oscillator" degrees of freedom for k not a ected by the divergence. Abelian anomaly and its cancellation General discussion In 2d (0; 2) gauge theories, the di erence in the spectrum of left-moving and right-moving fermions can potentially lead to anomalies. As explained in [10{12], gauge theories associated to brane brick models are automatically free of non-abelian gauge anomalies. But, depending on the particulars of matter multiplets, they may appear to su er from abelian gauge anomalies. These gauge theories can be embedded in string theory [11{13]. So, there must exist an anomaly cancelling mechanism involving open string modes on branes and closed string modes away from branes. Although we have not identi ed the precise anomaly cancelling mechanism, we have found an ansatz for an anomaly cancelling factor in the contour integral formula. To set the stage for the anomaly cancelling factor, we should recall the relation between abelian gauge anomaly and modularity. In a theory with abelian gauge symmetry U(1)r, the anomaly matrix is de ned by Aij = Trchiral(QiQj ) TrFermi(QiQj ) (i; j = 1; The same information is encoded in the anomaly polynomial de ned by A(u) = 1( 1= jz= ) ( 1= ) where ui are the gauge holonomy variables. In the context of the elliptic genus, the abelian anomaly is tied to the modularity. We recall the modular properties of the functions in the additive notation: For the gauge theories under consideration, the elliptic variable z denotes gauge holonomy variables or avor fugacities. It follows that the abelian gauge anomaly of the matter sector of the gauge theory is re ected in the modular property of the one-loop determinant up a multiplicative factor whose exponent is proportional to the anomaly polynomial (5.2). The contour integral for the elliptic genus (3.2) is well-de ned only if the theory is anomaly-free. If we naively integrate an anomalous integrand, the result fails to exhibit de nite modularity. So, we must \cure" the anomaly before the integration. In the next subsection, we will present an anomaly cancelling factor that works for some class of gauge theories. Since one of our main results in this paper is to compare the gauge theory and the geometric computations, let us brie y digress to discuss the potential anomaly of the geometry formula In the additive notation, each term in the geometric formula takes the form where the fugacities satisfy the relation, The anomaly polynomial, re ected in the modular property, is i (q)3 1( jz1) 1( jz2) 1( jz3) ; 1( jy1) 1( jy2) 1( jy3) 1( jy4) za = y3) + ya (a = 1; 2; 3) : y12 + y22 + y32 + y42 z32 = (y1 + y2 + y3 + y4)2 : This factor vanishes as long as the triangulation of the toric diagram lives entirely in the CY hyperplane. We cannot compare the elliptic genus of the gauge theory with the geometric formula before determining how to cancel the anomaly of the gauge theory. Anomaly cancelling factor | an ansatz Our ansatz works for theories in which the net contribution of chiral elds is greater than that of Fermi elds in such a way that the anomaly polynomial can be written as a sum of squares with unit positive coe cients. To illustrate the point, let us consider a one-parameter family of orbifolds denoted by C4=Zn(1; 1; 1; 1): (z1; z2; z3; z4) (!nz1; !nz2; !n 1z3; !n 1z4) ; e2 i=n : The anomaly polynomial of an arbitrary Zn orbifold was computed in [7]. The result for the C4=Zn(1; 1; 1; 1) is A(u) = 4 X(ui Here, the sum runs from 1 to n and a cyclic identi cation mod n is understood. The change of variables from ui to u~i is not one-to-one, so rewriting A(u) in terms of u~ is not equivalent to the standard diagonalization of a real symmetric matrix. Wn(ui; v) = i 1(q; v=u~i) : This factor has a few peculiar features. First, it has its own \anomaly" which cancels precisely against the anomaly from the matter sector (5.8). Second, it depends on an auxiliary variable v. Remarkably, once we integrate over the u variables, the v-dependence completely disappears. Third, since the u-variables appear only in the numerator, the pole structure of the elliptic genus, which depends on the avor fugacities, is not a ected by the insertion of the anomaly cancelling factor. Fourth, the normalization of Wn is such that when we expand the elliptic genus in a power series of q, the leading term is not a ected by the insertion of Wn. We discovered the factor Wn in (5.9) \experimentally" while working on the orbifold section 6. But, further experiments revealed that it can be applied to a much larger class of theories. We conjecture that it works for all theories in which the anomaly polynomial admits the rewriting A(u) = where u~i are linear combinations of ui with integer coe cients. We can easily generalize the ansatz (5.9) to a larger class of orbifolds that include is labeled by integers (a1; a2; a3; a4) satisfying 0 Aij = 2 ij A(u) = X 4uiui+b + 2uiui+b a + 2uiui b a = X (ui ui+b + ui+a+b) is given by [36] There is yet another large class of orbifolds to which the anomaly cancelling factor (5.9) ; ng) to label gauge nodes and their holonomy variables. The anomaly matrix A = 41 m 1 n After multiplying by mn holonomy variables fu(i;j)g, we can reorganize the anomaly polynomial as follows, A(u) = X h4u(2i;j) 2u(i+a;j)u(i;j) 2u(i;j+b)u(i;j) +u(i+a;j b)u(i;j) + u(i a;j+b)u(i;j) + u(i a;j b)u(i;j) + u(i+a;j+b)u(i;j) = X(u(i;j) In the next section, we will show how the anomaly cancelling factor works in concrete examples with small values of m; n. Orbifold models In this section, we compute the elliptic genera of a few orbifold models. We nd perfect agreement between the geometric computation and the gauge theory computation, even when the latter includes the anomaly cancelling factor. The results also agree with an independent computation using the standard orbifold CFT method. The orbifold CFT method expresses the elliptic genus in terms of a sum over twisted theta functions: b1 = a1 + a4 ; b2 = a2 + a4 ; b3 = a3 + a4 : and recall the de nition of xa and si from (3.14). To incorporate the twisted boundary conditions, it is convenient to use the generalized [ ](q; y) = q = e2 i ; y = e2 iz : 1 = 2 = h1=2i ; 3 = 4 = 1=2 : Additional information on the theta functions are collected in appendix A.1. The orbifold form of the elliptic genus is given by h11==22i ; IC4=Zn(ai) = n k;l=0 Nk;l(ba; xa) = i (q)3 Y h11==22++bbaa((lk==nn))i (q; xa) ; Dk;l(ai; si) = Y h11==22++aaii((lk==nn))i (q; si) : where the numerator and denominator are The phase factors ck;l in (6.4) are xed by requiring that the index should have a de nite modular property and quasi-periodicity in shift of the fugacity variables according to the orbifold action. Barring the possibility of discrete torsion, these requirements should x ck;l uniquely, as we verify in a number of examples. We will not discuss discrete torsion in C4=Z2(0; 0; 1; 1) This is the simplest orbifold in the sense that the GLSM has two gauge nodes and that the gauge anomaly is absent. The toric diagram for the orbifold and the quiver diagram for the GLSM are shown in gure 2. Geometric formula. To apply the geometric formula, we have to specify how to triangulate the toric diagram. For orbifold models, the toric diagram is a tetrahedron with a non-minimal volume. We assign labels A, B, C, D to the four external vertices and call the whole toric diagram T (ABCD). The orientation is important here. Any even permutation of (ABCD) is equivalent to (ABCD), but an odd permutation implies an orientation reversal, which would ip the sign of the index. geometric formula of the elliptic genus. Consider, instead, naively applying the geometric formula pretending as if the three terms on the right-hand side of (6.34) were minimal tetrahedra. The pre-index from this (unjusti ed) process is Je(t) = 3=2=t31) 1(q; t43=2=t32) 1(q; t32t34=2=t31t3) 1(q; t31=t32) 1(q; t32=t3) 1(q; t34=t31) 3=2=t32) 1(q; t43=2=t31) 1(q; t31t34=2=t32t3) 1(q; t32=t31) 1(q; t31=t3) 1(q; t34=t32) 3=2=t31) 1(q; t3t4 3=2=t32) 1(q; t31t32=t49=2) # 1(q; t34=t31) 1(q; t34=t32) 1(q; t31t32=t3t4) 3 Upon the change of the variable (6.35), the pre-index (6.36) gives Ie = 1(q; y3) 1(q; s31) 1(q; s32=x3) 1(q; x3) 1(q; y3=x3) 1(q; s32) 1(q; x3) 1(q; s31=y3) 1(q; s32) 1(q; y3) 1(q; x3=y3) 1(q; s31) 1(q; x3y3) 1(q; s31) 1(q; s32) 1(q; x3) 1(q; y3) 1(q; s33) Remarkably, although the intermediate step (6.36) is not justi ed, the nal result (6.37) turns out to agree perfectly with the nine term expression we mentioned below (6.34). While we do not fully understand why (6.37) gives the correct result, we suspect that it has something to do with orbifold singularities. Despite the appearance, the three terms in (6.36) do not match the three non-minimal tetrahedra in (6.34). Presumably, the discrepancy is related to twisted sectors. Orbifold singularities away from the origin 4 are encoded by the vertices along the edges of the toric diagram. In gure 4, the vertices lie on the interval CE. Now, the subtraction (6.34) is done in such a way that origin. Hence, it is conceivable that the discrepancies associated with the twisted sectors have been cancelled out so as to produce the correct elliptic genus. Single term formula. Just as in (6.18), we nd a remarkably simple single term expression for the index: I = i (q)3 1(q; x3) 1(q; y3) 1(q; z3) 1(q; s31) 1(q; s32) 1(q; s23) 1(q; s34) : Orbifold CFT. I = i (q)3 1(q; x) 1(q; y) !k h11==22 22lk==33i (q; z) h11==22++lk==33i (q; s1) h11==22++lk==33i (q; s2) h11==22 lk==33i (q; s3) h11==22 lk==33i (q; s4) It is straightforward to verify that (6.39) agrees perfectly with (6.37) and (6.38). The quiver diagram for this orbifold model is given in gure 4. When we decouple the overall U(1), we choose a basis for the u variables such that u3 is decoupled, and u13 and u23 remain as independent variables. The anomaly polynomial for this theory is A(u) = 3(u122 + u223 + u231) = (u12 u~22 + u~23 + u~21 : As discussed in section 5, the anomaly cancelling factor for this orbifold is W(3)(ui; v) = In the multiplicative notation, the u~ variables are de ned as Since the overall U(1) has decoupled, W(3) is a function of two independent variables, say, W function we encountered earlier, W(3)(ui; v)ju13=u=1=u23 = W (u3; v) : Z1-loop = i(2 )2 (q)7 1(q; x)3 1(q; y)3 1(q; zu21) 1(q; zu32) 1(q; zu13)W(3)(ui; v) to be independent. For contributes up to four residues, some of which may vanish. There are three pairs of charge vectors for the chiral multiplets. They split the charge plane into six regions as depicted in gure 5. Symmetries of the quiver diagram guarantees that all six regions are equivalent. In the notation of gure 5, we can paraphrase the general prescription of section 3 as follows. Any choice of is equally good, and we choose = (1; 1). Then we look for pairs of charge vectors, which we call \cones", whose positive span contain . For = (1; 1), we nd three cones, A : (1; 0)&(0; 1) ; B : (1; 0)&( 1; 1) ; C : (1; 1)&(0; 1) : The cones determine the poles from which we take the residues. For example, cone A picks residues in a similar way. In the end, the elliptic genus can be written as I = RA + RB + RC ; where RA;B;C denote the partial sum of residues coming from the cones in (6.45). Since every charge vector appears exactly twice in the denominator in (6.44), each partial sum in (6.46) can contribute up to four residues. It turns out that RA contributes four terms, RA = 1(x=y) 1(xy) 1(x)W(3)(s1; s3) 1(s13y) 1(s33y) 1(y)W(3)(s1; s4) 1(s13=x) 1(s43=x) 1(y)W(3)(s2; s3) 1(s23x) 1(s33x) 1(x)W(3)(s2; s4) 1(s23=y) 1(s43=y) whereas RB and RC contribute two non-vanishing terms each, RB = RC = 1(x=y) 1(s1=z)W(3)(s1; s21) 1(s13=x) 1(s13y) 1(s13) 1(s3z)W(3)(s23; s3) 1(s33x) 1(s33y) 1(s33) 1(s2=z)W(3)(s2; s22) # 1(s23=y) 1(s23x) 1(s23) 1(s43=x) 1(s43=y) 1(s43) In (6.47) and (6.48), we suppressed the q-dependence of 1 to simplify the expressions. We residues is independent of v and equal to (6.37), (6.38) and (6.39). C4=Z2 Z2(0; 0; 1; 1)(1; 1; 0; 0) Geometric formula. If we choose not to use any subtraction, there is a unique way to triangulate the toric diagram. In the notation of gure 6, the triangulation reads The pre-index (with t4 = 1) is J (t) = xy-parity i (q)3 1(q; 1=t3) 1(q; t2=t1t3) 1(q; 1=t1t2) 1(q; t1) 1(q; t2) 1(q; t3=t2) 1(q; 1=t1t3) where we de ned the \xy-parity sum" as Upon substitution, I = 1(q; x2) 1(q; y2=z) 1(q; xy) 1(q; x=y) 1(q; y=xz) 1(q; z=xy) 1(q; x2z) 1(q; y2) 1(q; y=xz) 1(q; xyz) 1(q; x2=z) 1(q; y2) 1(q; x=yz) 1(q; z=xy) 1(q; x2) 1(q; y2z) 1(q; x=yz) 1(q; xyz) Some symmetries of the index are manifest from the formula: I(x; y; z) = I(1=x; y; z) = I(x; 1=y; z) = I(x; y; 1=z) = +I(y; x; z) : Orbifold CFT. The orbifold CFT method gives the index in the form ca;b a b(q; x) a b(q; y) a;b=1 a(q; s1) a(q; s2) b(q; s3) b(q; s4) The phase factors ca;b and the labels a b for the theta functions in the numerator are I = ca;b = BBB++ We can regard the C4=Z2 model. See (6.10) and (6.19). It is straightforward (but tedious) to prove that (6.53) and (6.55) agree perfectly. GLSM. The anomaly polynomial is A(u) = 4(u1 The anomaly cancelling factor is given by V (u; v1; v2) = W (u; v1)W (u; v2) ; W (u; v) = 1(q; vu) 1(q; v=u) 1(q; xu24) 1(q; xu42) 1(q; yu24) 1(q; yu42) 1(q; xu13) 1(q; xu31) 1(q; yu13) 1(q; yu31) V (u1u3=u2u4; v1; v2) : (6.59) The charge plane contains eight vectors (u12; u23; u34; u41) with the obvious constraint eight, we can make 32 cones. Some of the cones overlap with each other; see gure 7. We will say that a choice of is generic if it does not lie on the boundary of any of the cones. For any generic choice, four cones contribute to the JK-calculus. Up to symmetries of the gauge theory, there are two inequivalent choices of . Let us call them branch A and branch B. On both branches, two out of the four cones give vanishing contributions to the JK-integral. So the e ective number of cones is two. On branch A, all non-vanishing residues come from degenerate poles with four planes intersecting. The two cones, which look di erent a priori, both land on the same degenerate The end result is that there are four non-vanishing residues, which match the geometric formula (6.53) term by term. At the four poles, the V gives a trivial contribution, V (u = 1; v1; v2) = 1. On branch B, there are 16 non-vanishing residues, all with non-trivial contributions from the V function. The 16 residues can be divided into four groups of four terms. Each group matches a term in the geometric formula (6.53). A key step in proving the equality can be written as 1(y=x) 1(x)V (y) 1(y)V (x) 1(yz) 1(y=z) 1(y) 1(xz) 1(x=z) 1(x) 1(xz) 1(yz) 1(xy=z) 1(x=z) 1(y=z) 1(x) 1(y) We can verify this identity by plugging in the proposed form of V (6.58) and using the simpler identity (6.30) repeatedly. C4=Z4(1; 1; 1; 1) Geometric formula. As shown in gure 8, the toric diagram has four external vertices at (1; 0; 0), (0; 1; 0), (0; 0; 1), ( 1; 1; 1), and an internal vertex at (0; 0; 0). It can be triangulated in the usual way, so the geometric formula works well without using the subtraction method. The pre-index (with t4 = 1 inserted) is J (t) = i (q)3 1(q; 1=t1t2) 1(q; 1=t2t3) 1(q; 1=t3t1) 1(q; t1) 1(q; t2) 1(q; t3) 1(q; 1=t1t2t3) i (q)3 1(q; t21=t2t3) 1(q; t21=t2) 1(q; t21=t3) 1(q; 1=t1) 1(q; t2=t1) 1(q; t3=t1) 1(q; t31=t2t3) where the cyclic sum was de ned in (6.15). The relation between ti and (x; y; z) for this The geometric index can be summarized as I = tetra i (q)3 1(q; x2yz) 1(q; xy2z) 1(q; xyz2) 1(q; xy) 1(q; yz) 1(q; zx) 1(q; x2y2z2) where the tetrahedral sum was de ned in (6.24). Orbifold CFT and single term formula. If we apply the orbifold CFT formula (6.4) We can take an alternative route to nd something simpler. The key idea is that of the latter satis es an interesting identity (see (6.18) and (6.19)), I = i X4 ( 1)a+1 (q)3 1(q; x) 1(q; y) 1(q; z) a(q; s1) a(q; s2) a(q; s3) a(q; s4) i (q)3 1(q; x2) 1(q; y2) 1(q; z2) 1(q; s21) 1(q; s22) 1(q; s23) 1(q; s24) : In words, we may say that Z2(1; 1; 1; 1) orbifolding resulted in squaring the fugacity variables. By applying the same orbifolding once again, we can deduce that I = i X4 ( 1)a+1 (q)3 1(q; x2) 1(q; y2) 1(q; z2) a(q; s21) a(q; s22) a(q; s23) a(q; s24) i (q)3 1(q; x4) 1(q; y4) 1(q; z4) 1(q; s41) 1(q; s42) 1(q; s43) 1(q; s44) : and applying further Z2 orbifolding, we derived a new single term formula (6.66) for The anomaly polynomial is A(u) = 4(u12 + u34) Since it has some negative terms, we cannot use the ansatz for the anomaly cancelling factor proposed in section 5. Taking a leap of faith, we propose a new anomaly cancelling factor for this example, U(ui; v1; v2) = The one-loop integrand is (uij = ui=uj ) 1(q; v1u13) 1(q; v1=u13) 1(q; v2u24) 1(q; v2=u24) Z1-loop = The charge vectors from the chiral multiplets are depicted in gure 9. By choosing three out of four vectors, we can form four di erent cones. For any generic choice of , only one cone contains inside it. The four cones are related by the tetrahedral symmetry of the toric diagram, so it is guaranteed that the result will be independent of the choice of . mal and 24 are degenerate. A closer look reveals that the residues from the 24 degenerate poles all vanish, while all but four residues from the 40 normal poles also vanish. Interestingly, the anomaly cancelling factor (6.68) becomes 1 at the four surviving poles. The nal result is a sum of four terms, which match the geometric formula (6.64) term by term. Non-orbifold models Geometric formula. The toric diagram of the D3 theory is shown in gure 10. There are six di erent ways to triangulate it without subtraction. If we choose to triangulate it as = (1; 1; 1). the pre-index with t4 = 1 is J (t) = We set the other fugacities to be 1(q; t1t2) 1(q; t1t3) 1(q; t2t3) 1(q; t1) 1(q; t2) 1(q; t3) 1(q; t1t2t3) 1(q; t2) 1(q; t3) 1(q; t21t2t3) 1(q; t1) 1(q; t1t2) 1(q; t1t3) 1(q; t1t2t3) 1(q; t1) 1(q; t3=t2) 1(q; t1t2t3) 1(q; t2) 1(q; t3) 1(q; t1t2) 1(q; t1t3) t1 = x ; t2 = y ; t3 = z ; U(1)z +1=3 +1=3 +1=3 so that we have I = 1(q; xy) 1(q; xz) 1(q; yz) 1(q; x) 1(q; y) 1(q; z) 1(q; xyz) 1(q; y) 1(q; z) 1(q; x2yz) 1(q; x) 1(q; xy) 1(q; xz) 1(q; xyz) 1(q; x) 1(q; z=y) 1(q; xyz) 1(q; y) 1(q; z) 1(q; xy) 1(q; xz) The symmetries of the toric diagram are re ected in the index. The S3 subgroup is generated by the two elements: (x; y) ! (y; x) ; (x; y) ! (y; 1=xy) : The q-expansion of the index reads I = 1 + x + y + x2y + xy2 + x2y2 As expected from the discussion in section 4, the leading term contains codimension 1 poles, whereas the next term is a Laurent polynomial in (x; y; z). Curiously, both terms are independent of z. In fact, we can prove that the index is completely z-independent. Then we may set z to any convenient value to simplify the answer. For instance, setting I = 1(q; xy) 1(q; x=y) 1(q; x)2 1(q; y2) 1(q; y)2 1(q; x2) GLSM. The quiver diagram of the D3 theory is shown in gure 10. The theory consists of three adjoint chirals, six bifundamental chirals and six bifundamental Fermis. Their charges under the global symmetry are summarized in table 4. After the overall U(1) decoupling, we have the integrand Z1-loop = 1(q; x) 1(q; y) 1(q; 1=xy) a= 1 1(q; (u1z 1=3)apy) 1(q; (u12z1=3)a=pxy) 1(q:(u2z1=3)aypx) 1(q; (u2z1=3)apx) Figure 11 shows the charge covectors of six singularity lines. When we take (1; 1), three out of twelve singularities contribute to the elliptic genus. By adding the to the index: fX13; X23g, fX13; X21g and fX12; X23g. residues from three cones fX13; X23g, fX13; X21g and fX12; X23g, we obtain I = i (q)3 1(q; z) 1(q; xy) 1(q; xz=y) 1(q; x) 1(q; y) 1(q; y=z) 1(q; xz) 1(q; x) 1(q; y=z) 1(q; xyz) 1(q; y) 1(q; z) 1(q; xy) 1(q; xz) 1(q; y) 1(q; xz) 1(q; xy=z) 1(q; x) 1(q; z) 1(q; xy) 1(q; y=z) t1 = x ; t2 = y t3 = 1=z ; t4 = 1 ; It can be shown to be equal to (7.4). Note that the charge vectors in gure 11 divide the plane into six regions. There is a one-to-one map between the six regions and six di erent ways to triangulate the toric diagram (without subtraction) in gure 10, such that the equality between the GLSM result and the geometric result become manifest without any change of variables or theta function identity. H4 | anomaly and triality Let us now consider an example that has not previously appeared in the literature. The toric diagram for this geometry is shown in gure 12 and we refer to it as H4. Geometric formula. In the notation of gure 12, we use the triangulation With the change of variable, I = 1(x=y) 1(x) 1(x=yz) 1(y=z) 1(y) 1(z) 1(z=x) + 1(x) 1(z) 1(xz=y2) 1(y) 1(y=z) 1(xz=y) 1(y) 1(y=xz) 1(x=z) 1(x) 1(z) 1(z=y) 1(y) 1(z) 1(yz=x2) 1(x) 1(x=z) 1(yz=x) Its lowest terms in the q-expansion are I = y)2z(1 + z) GLSM. The H4 model has several toric phases. Figure 13 shows the periodic quivers for two of them and how they are related by triality. For a detailed discussion on how triality is realized on the periodic quiver, we refer the reader to [10{12]. Phase A. This phase has 13 chiral elds and 9 Fermi elds. The plaquettes in the periodic quiver correspond to The global charges of the elds are given in tables 5 and 6. The anomaly polynomial is A(u) = 2(2u1 The charge vectors in phase A of the H4 model is depicted in gure 14. The JK residue computation with = ( 2; 3; 1) leads to IA = 1(x=y) 1(x) 1(x=yz) 1(y=z) 1(y) 1(z) 1(z=x) 1(x) 1(z) 1(xz=y2) 1(y) 1(y=z) 1(xz=y) 1(y) 1(y=xz) 1(x=z) 1(x) 1(z) 1(z=y) 1(y) 1(z) 1(yz=x2) 1(x) 1(x=z) 1(yz=x) which agrees with (7.12) term by term. For this particular choice of , the W -function becomes trivial at each of the four contributing poles. = ( 2; 3; 1) and = ( 6; 2; 3). For a di erent choice, say, IA = 1(x=y) 1(v)2 1(vx) 1(v=x) 1(y=xz) 1(z=y) 1(xz) 1(vy) 1(v=y) 1(x=yz) 1(z=x) 1(yz) 1(vz) 1(v=z) 1(vx=yz) 1(vyz=x) 1(x=yz) 1(z=x) 1(yz) 1(vy=xz) 1(vxz=y) # 1(y=xz) 1(z=y) 1(xz) Despite its appearance, it can be shown to be independent of v and agree with (7.12). Phase B. This phase has 10 chiral elds and 6 Fermi elds. The plaquettes in the periodic quiver correspond to D32 X24 D41 Z12 D41 X13 D32 D21 The global charges of the elds are given in table 7. The anomaly polynomial is A(u) = 2(u1 so we can use the W -function as the anomaly cancelling factor. eld D21 D32 D43 D41 X13 1=2 1=2 1=2 1=2 1=4 1=4 1=2 1=2 1=2 1=2 1=4 1=4 1=4 1=2 1=2 1=2 1=2 3=4 3=4 1=4 1=4 5=4 1=4 3=4 3=4 1=4 1=4 3=4 3=4 1=4 1=4 3=4 The charge vectors in phase B of the H4 model is depicted in gure 15. The JK residue computation with the choice = (2; 3; 6) gives IB = 1(x=y) 1(x) 1(x=yz) 1(y=z) 1(y) 1(z) 1(z=x) + 1(y) 1(z=x) 1(xz=y) 1(x) 1(z) 1(y=z) + 1(x) 1(z) 1(xz=y2) 1(y) 1(y=z) 1(xz=y) 1(y) 1(z) 1(x2=yz) 1(x) 1(z=x) 1(x=yz) which agrees with (7.12) term by term. Again, the particular choice of makes the W function trivial at each of the four contributing poles. If we instead choose = ( 7; 1; 5), we obtain IB = 1(x=y) 1(v)2 1(vx) 1(v=x) 1(y=xz) 1(x)2 1(z=y) 1(xz) 1(vx=yz) 1(vyz=x) 1(x=yz) 1(z=x) 1(yz) 1(x=y) 1(xyz) 1(xy) 1(xz) 1(yz) i (q)3 1(xy) 1(xz) 1(yz) 1(x) 1(y) 1(z) 1(xyz) 1(vy) 1(v=y) 1(x=yz) 1(y)2 1(z=x) 1(yz) 1(vy=xz) 1(vxz=y) 1(y=xz) 1(z=y) 1(xz) 1(vz) 1(v=z) 1(vxyz) 1(v=xyz) Again, it can be shown to be independent of v and agree with (7.12). We computed the elliptic genera for a class of 2d (0; 2) gauge theories that arise on the worldvolume of D1-branes probing toric CY4 singularities. These quiver gauge theories are e ciently described by T-dual Type IIA brane con gurations of D4-branes suspended from NS5-branes known as brane brick models. The elliptic genera were computed using two independent methods. First, we calculated the elliptic genus using an integral formula based on the GLSM quiver description known from our previous work on brane brick models. Then, we proposed a new formula for the elliptic genus based on the CY4 target space geometry that the NLSM is expected to describe in the IR. For both methods, we regulated divergences arising from the non-compactness of the target space by introducing appropriate U(1)3 global symmetry fugacities. It was shown in several examples that the two methods lead to the same result. We hence conclude that quantum e ects, which are expected in the IR, do not drastically alter the correspondence between the target space geometry and this class of 2d (0; 2) gauge theories. For 2d (0; 2) gauge theories that naively su er from abelian gauge anomalies, we introduced an anomaly cancelling term in the integral formula for the elliptic genus. Using this extra term, we veri ed the agreement with the computation based on the target space We also provided further evidence for triality of 2d (0; 2) theories by matching the elliptic genera of brane brick models with the same target space geometry. For brane brick models, the target space of the NLSM is the classical mesonic moduli space of the corresponding 2d (0; 2) quiver gauge theory. Our results suggest that the mesonic moduli space is a good IR observable for the examples considered in this paper. This is far from being the case for more general 2d (0; 2) theories unrelated to brane brick models. It would be desirable to expand our investigation to more examples. A natural question, which we leave for future work, is whether the elliptic genus formula proposed in this paper based on the triangulation of the toric diagram can be extended to more general geometries. For instance, non-toric CY4 form a rich class of geometries yet to be studied in the context of 2d (0; 2) theories. In addition, it would be desirable to obtain a stringy explanation for the anomaly cancelling factor in the GLSM computation of the elliptic genus. We expect that there is a stringy anomaly in ow mechanism analogous to the Green-Schwarz mechanism that cancels the anomaly. This would be in line with our derivation of the anomaly cancelling term in the integral formula for the elliptic genus, which relies on its modular properties and holomorphy. Given that throughout this paper we have restricted ourselves to abelian brane brick models, it would be interesting to study their non-abelian extensions. Following [37], one can expect that the elliptic genus of such a non-abelian theory relies on a symmetric product of the target space of the corresponding abelian theory. However, a recent study of 2d maximal super-Yang-Mills with C4 as its target space showed that the naive symmetric product of C4 is not enough to obtain the elliptic genus of the non-abelian theory [30]. We hope to return to non-abelian brane brick models in the near future. Finally, it would be interesting to nd observables that are more re ned than the elliptic genus, such as the ones proposed in [38], for brane brick models. Acknowledgments We would like to thank M. Romo, P. Putrov, P. Yi and C. Vafa for enjoyable and helpful discussions. The work of S.F. is supported by the U.S. National Science Foundation grant PHY-1518967 and by a PSC-CUNY award. The work of D.G. is supported by POSCO TJ Park Science Foundation. The work of S.L. is supported by Samsung Science and Technology Foundation under Project Number SSTF-BA1402-08 and by the IBM Einstein Fellowship of the Institute for Advanced Study. The work of S.L. was also performed in part at the Aspen Center for Physics, which is supported by National Science Foundation grant PHY-1066293. R.-K.S. is supported by the ERC STG grant 639220 \Curved SUSY". Theta functions and their identities Theta functions To make the paper self-contained, we collect some well-known properties of eta and theta functions. We follow the standard conventions shared by many physics textbooks, including [39, 40] and consistent with [23]. The Dedekind eta function is Its modular properties are and a product representation, They have quasi-periodicity in as well as quasi-periodicity in z, ( ) = q1=24 Y1 (1 qn) ; q = e2 i ; Im( ) > 0 : ( + 1) = e i=12 ( ) ; = p The Jacobi theta functions with arbitrary twists have a sum representation, When ; 1( jz) = 2( jz) = 3( jz) = [ 00 ]( jz) = 4( jz) = [10=2]( jz) = where y = e2 iz. genus often makes use of the following identity, As mentioned in section 3, the de nition and computation of the (modi ed) elliptic h11==22i ( jz) = qk)(1 + yqk)(1 + y 1qk) ; = 2 Throughout the main body of this paper, we mostly use the multiplicative (exponential) notation with variables q and y. Sometimes, when we discuss the anomaly polynomial, it is more convenient to use the additive (logarithmic) notation with variables Switching between the two notations is straightforward: a( jz) = a(q = e2 i ; y = e2 iz) : Hopefully, which notation we are using is always clear from the context. Proving theta function identities We have encountered a number of theta function identities in the main text. Here we sketch a proof for them. Consider a function F ( jz) holomorphic in in z. Suppose F ( jz) is quasi-periodic in z in the following sense: and meromorphic F ( jz + 1) = F ( jz) ; F ( jz) = ei ( ) 2 i zF ( jz) ; where ( ) is some linear function of , and is an integer. For example, a( jz)2, a( j2z) periodic meromorphic function. When two or more functions have the same periodicity, their linear combinations are also quasi-periodic. Suppose we want to prove an identity of the type FL( jz) = FR( jz) ; F at some z. number. Intuitively, the torus, with multiplicity. where FL and FR are known to have the same periodicity. A standard way to prove this is to show that FL and FR have the same sets of zeros is strictly periodic ( = 0 = ) on the torus z z + . An elementary theorem in complex analysis asserts that a holomorphic function on a compact connected manifold some convenient value of z. For the identities in this paper, the poles and residues of FL;R are easy to nd, but the zeros are not. Suppose we have shown that the poles and residues match. Then we F = FL FR is holomorphic. If F is strictly periodic ( = 0 = ), then the theorem mentioned above guarantees that F is a constant, which can be determined by Let us make a slight digression. When 6= 0, F ( jz) is not really a function but a section of a holomorphic line bundle over the torus. The integer is the rst Chern counts the number of zeros minus the number of poles of F ( jz) on Returning to the main problem, if FL and FR have < 0, it follows that be a holomorphic section which has more poles than zeros. To avoid a contradiction, it must be that = 0 = < 0. Hence this proof applies to all of them. The following identity between 1 and 4, in the multiplicative notation, is used many times in the main text: 1(q; ab) 1(q; a=b) 4(q; 1)2 = 1(q; a)2 4(q; b)2 charge vectors Qu Qu and order them to form Degenerate poles and the Je rey-Kirwan residue Additional ideas are necessary for computing the JK residue in (3.5) in the presence of degenerate poles. Such poles appear when, for gauge group U(1)r, more than r hyperplanes corresponding to the charge vectors of the JK integral meet at a point. Degenerate poles can be regulated using ags as we review in this section. The calculation of the JK residue using the ag method was explained in detail in [23]. A degenerate pole u is associated to l r singular hyperplanes identi ed by the = fQ1; ; Qlg. We rst choose a subset of r vectors from B(F ) = fQi1 Let Fk be the vector space whose basis is given by the rst k vectors in B(F ). The vector spaces Fk, with k = 1; : : : ; r, form a set F = [F0 = f0g Fr = Cr] ; dim(Fk) = k : This is known as a ag F , whose basis is given by the ordered set B(F ). B(F ) is not necessarily unique for a degenerate pole associated to Qu . Hence, we have a nite set of all possible ags F for a given Qu , which we call F L(Qu ). Once we construct F L(Qu ), we de ne an ordered set F for each ag F 2 F L(Qu ) as After imposing the strong regularity condition on a trigger [23], we nally determine the For ags satisfying the positivity condition with respect to F , the JK residue reads F = < k = 1; where F L+(Qu ; ) = fF 2 F L(Qu )j associated to the ag F ; (F ) = sgn det( 1F ; 2 Cone( 1F ; is a sign factor ; rF ) . 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Sebastian Franco, Dongwook Ghim, Sangmin Lee, Rak-Kyeong Seong. Elliptic genera of 2d (0,2) gauge theories from brane brick models, Journal of High Energy Physics, 2017, DOI: 10.1007/JHEP06(2017)068