B-branes and supersymmetric quivers in 2d

Journal of High Energy Physics, Feb 2018

Abstract We study 2d \( \mathcal{N} \) = (0, 2) supersymmetric quiver gauge theories that describe the low-energy dynamics of D1-branes at Calabi-Yau fourfold (CY4) singularities. On general grounds, the holomorphic sector of these theories — matter content and (classical) superpotential interactions — should be fully captured by the topological B-model on the CY4. By studying a number of examples, we confirm this expectation and flesh out the dictionary between B-brane category and supersymmetric quiver: the matter content of the supersymmetric quiver is encoded in morphisms between B-branes (that is, Ext groups of coherent sheaves), while the superpotential interactions are encoded in the A∞ algebra satisfied by the morphisms. This provides us with a derivation of the supersymmetric quiver directly from the CY4 geometry. We also suggest a relation between triality of \( \mathcal{N} \) = (0,2) gauge theories and certain mutations of exceptional collections of sheaves. 0d \( \mathcal{N} \) = 1 supersymmetric quivers, corresponding to D-instantons probing CY5 singularities, can be discussed similarly.

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B-branes and supersymmetric quivers in 2d

HJE B-branes and supersymmetric quivers in 2d Cyril Closset 0 1 2 5 Jirui Guo 0 1 2 3 Eric Sharpe 0 1 2 4 Topological Strings 0 850 West Campus Drive, Virginia Tech , Blacksburg, VA, 24061 U.S.A 1 220 Handan Road, Shanghai , 200433 China 2 Geneva 23 , CH-1211 Switzerland 3 Department of Physics, Fudan University 4 Department of Physics MC 0435 5 Theory Department , CERN We study 2d N = (0; 2) supersymmetric quiver gauge theories that describe the low-energy dynamics of D1-branes at Calabi-Yau fourfold (CY4) singularities. On general grounds, the holomorphic sector of these theories | matter content and (classical) superpotential interactions | should be fully captured by the topological B-model on the CY4. By studying a number of examples, we con rm this expectation and esh out the dictionary between B-brane category and supersymmetric quiver: the matter content of the supersymmetric quiver is encoded in morphisms between B-branes (that is, Ext groups of coherent sheaves), while the superpotential interactions are encoded in the A1 algebra satis ed by the morphisms. This provides us with a derivation of the supersymmetric quiver directly from the CY4 geometry. We also suggest a relation between triality of N = (0; 2) gauge theories and certain mutations of exceptional collections of sheaves. 0d N = 1 supersymmetric quivers, corresponding to D-instantons probing CY5 singularities, can be discussed similarly. D-branes; Field Theories in Lower Dimensions; Supersymmetry and Duality - 2.2 D1-brane on C 4 2.3 Triality acting on N = (0; 2) supersymmetric quivers Triality from mutation | a conjecture 4 D-instanton quivers and gauged matrix models D( 1)-brane on C 5 N = 1 gauged matrix model from B-branes at a CY5 singularity 4.2.1 4.2.2 4.3.1 4.3.2 1 Introduction 2 D1-brane quivers and 2d N = (0; 2) theories 2.1 N = (0; 2) quiver gauge theory from B-branes at a CY4 singularity From B-branes to quiver { i { 58 59 60 eld theories can be engineered on systems of branes in string theory. The string theory embedding often provides us with an elegant geometric understanding of eld theory phenomena. In particular, rich classes of eld theories, the supersymmetric quiver gauge theories, can be engineered by considering parallel Dp-branes at the tip of a conical local Calabi-Yau (CY) n-fold Xn, with p = 9 2n, in type IIB string theory. One obtains the following types of supersymmetric gauge theories in the open-string sector: 6d N = (0; 1) quiver theories on D5-branes at the tip of a CY2 cone. 4d N = 1 quiver theories on D3-branes at the tip of a CY3 cone. 2d N = (0; 2) quiver theories on D1-branes at the tip of a CY4 cone. 0d N = 1 quiver theories on D-instantons at the tip of a CY5 cone. All these quiver gauge theories consist of unitary gauge groups QI U(NI ) with matter elds in adjoint and bifundamental representations.1 The 6d case corresponds to D5-branes at the tip of an ADE singularity C2= , and the quiver gauge theories are the corresponding ADE quivers [1]. The 4d case has been thoroughly studied from various points of views | see e.g. [2{15] for a very partial list. It is a special and important case because the D3-branes admit a smooth near-horizon limit of the form AdS5 X5 [3, 4, 16] and the 4d quiver gauge theories ow to non-trivial 4d N = 1 superconformal xed points. The 2d and 0d cases had attracted less attention until more recently | see [17, 18] for some early work. A recent breakthrough was the introduction of \brane brick models" [19, 20], which gave an algorithm to determine the N = (0; 2) quiver gauge theory 1That is, elds XIJ in the fundamental of U(NI ) and in the antifundamental of U(NJ ). { 1 { corresponding to D1-branes probing a toric CY4 singularity, similarly to brane tiling methods for D3-branes at toric CY3 singularities [11{13]. The brane brick models were derived using mirror symmetry in [21]. There are also hints that a similar structure exists for D( 1)-branes at toric CY5 singularities.2 Note that this line of work (and the present paper) is only concerned with the classical structure of the N = (0; 2) gauge theory. In a parallel development, there has been some important progress in our understanding of two-dimensional N = (0; 2) gauge theories as full- edged quantum eld theories [23{25]. Incidentally, it was discovered that the simplest SQCD-like N = (0; 2) theories enjoy a beautiful triality [23] | an infrared \duality" of order three similar to Seiberg duality. Triality also seems to be a generic property of D1-branes quivers [26]. There has also been some interesting recent work on engineering N = (0; 2) models from F-theory [27{30]. See also [31{42] for related works on quantum aspects of N = (0; 2) theories. In this paper, we study 2d and 0d quivers from the point of view of B-branes on the CY n-fold Xn. A B-brane is simply a (half-BPS) D-brane in the topological B-model on Xn. The B-model is a gs = 0 limit of type II string theory which (somewhat trivially) captures all 0 corrections. It can thus be used to accurately describe the local physics of branes at a Calabi-Yau singularity. Since the B-model is independent of Kahler deformations, we can use any convenient limit, such as, for instance, the large volume limit of a resolved singularity, to study the quantities of interest. In this way, we loose a lot of important information | for instance, we do not keep track of the central charges of the branes, which determines their stability properties; yet, the B-model is su cient in order to extract all the information about the holomorphic sector of the low-energy open strings. That is, we can read o the matter spectrum and the superpotential interactions of the low-energy quiver gauge theories on Dp-branes from the B-branes alone.3 This approach was successfully carried out for D3-branes at CY3 singularities [6{ 10, 15]. What we present here is a straightforward extension of some of those earlier works. It provides a string theory derivation of some brane brick models results, without the need to rely on mirror symmetry. Our techniques are also more general, since they are valid beyond the realm of toric geometry.4 Mathematically, a B-brane E on Xn is an object in the (bounded) derived category of coherent sheaves of Xn: (1.1) (1.2) We can think of B-branes E as coherent sheaves; more generally they are chain complexes of coherent sheaves (up to certain equivalences). Given two B-branes E , F , we may compute their Ext groups: 2Very recently, these 2d and 0d quivers were also related to cluster algebras [22]. 3An important caveat is that we need to be given a particular set of B-branes, the \fractional branes", 4This is as a matter of principle. In this paper, all our examples will be toric geometries, somewhat by happenstance, and also so that we can compare our results to the brane brick model literature. E 2 Db(Xn) : ExtiXn (E ; F ) ; { 2 { which are the morphisms in the derived category. Physically, they encode the low-energy modes of the open strings stretched between the D-branes E and F [43{46]. to [47, 48] for comprehensive reviews of the derived category approach to B-branes. We refer D-branes quivers from B-branes. Consider a D(9 2n)-brane transverse to the Calabi-Yau singularity Xn. Away from the singularity, the brane is locally in at space. From the point of view of Xn, it is a point-like brane, which is described by a skyscraper sheaf Op at a point p 2 Xn. When at the singularity, it is expected that Op \fractionates" into marginally stable constituents. The resulting \fractional branes" fEI g realize a gauge group: Y U(NI ) I (1.3) HJEP02(18)5 on their worldvolume in the transverse directions. There are also massless open strings connecting the fractional branes among themselves, which realize bifundamental (or adjoint) matter elds XIJ . In this way, the low-energy open string sector at the singularity is described by a supersymmetric quiver gauge theory : to each fractional brane EI , we associate a node in the quiver, denoted by eI . The matter elds corresponds to various quiver arrows connecting the nodes: eI the matter elds, which we will discuss below. In all cases, the fractional branes are such that: ! eJ . There are also interaction terms among These Ext0 = Hom groups are identi ed with vector multiplets in d = 10 2n dimensions; a single vector multiplet is assigned to each node eI , realizing the gauge group U(NI ). The other Exti groups (with i = 1; gauge groups. Ext0Xn (EI ; EJ ) = IJ : (1.4) ; n 1) correspond to matter elds charged under the We should probably emphasize that, in this paper, we will be mostly interested in the supersymmetric quiver as an abstract algebraic object, consisting of nodes, arrows and relations. The assignment of particular gauge groups U(NI ) is part of the data of a quiver representation, and the gauge group ranks can vary depending on the physical setup (that is, which D-branes are we using to probe the singularity). In other words, our concept of supersymmetric quiver can encode many di erent supersymmetric theories with the same structure but distinct gauge groups.5 A crucial property of Ext groups on a Calabi-Yau variety Xn is the Serre duality relation: ExtiXn (EI ; EJ ) = ExtnXni(EJ ; EI ) ; i = 0; ; n : (1.5) This corresponds to the CPT symmetry of the d-dimensional quiver quantum eld theory. Generalizing some relatively well-known results for D3-branes, it is natural to propose the following identi cation of Ext groups with supersymmetry multiplets in various dimensions: 5Not all unitary gauge groups are allowed, however. Gauge anomalies provide strong constraints on the allowed quiver ranks. { 3 { where XIJ are 6d N = (0; 1) hypermultiplets in the bifundamental representation of U(nI ) U(nJ ). Note that the quiver link eI | eJ is unoriented since the hypermultiplet is non-chiral | this corresponds to the Serre duality Ext1(EI ; EJ ) = Ext1(EJ ; EI ) on X2. In this case, X2 must be an ADE singularity while the supersymmetric quivers are extended Dynkin diagrams. D3-brane quivers. For D3-branes on X3, we have: Ext1X3 (EJ ; EI ) , eI ! eJ , XIJ ; where XIJ are 4d N = 1 chiral multiplets in the bifundamental of U(nI ) U(nJ ), or in the adjoint of U(nI ) if I = J . The arrows are oriented. Therefore, such quiver gauge theories are generally chiral theories. More precisely, we denote by: 1 dIJ dim Ext1X3 (EJ ; EI ) the number of arrows from eI to eJ in the 4d N = 1 quiver. D3-brane quivers are \ordinary" quivers (with relations), consisting of nodes and arrows, of the type most studied by both physicists and mathematicians. D1-brane quivers. D1-branes on X4 lead to the richer structure of 2d N = (0; 2) quiver gauge theories. Those quivers have two distinct types of arrows, corresponding to (0; 2) chiral multiplets XIJ and (0; 2) fermi multiplets IJ , respectively. We propose the identi cation: , , , , , eI ! eJ , XIJ ; IJ : XIJ ; IJ : Note that the Ext2X4 (EI ; EJ ) = Ext2X4 (EJ ; EI ) by Serre duality. Thus the second type of arrow is unoriented. This corresponds to the self-duality of the fermi multiplet in such theories. We also de ne: 1 dIJ dim Ext1X4 (EJ ; EI ) ; 2 dIJ dim Ext2X4 (EJ ; EI ) : Here dI1J is the number of chiral multiplets from eI to eJ (in bifundamental representations 2 2 if I 6= J or adjoint representation if I = J ). Similarly, dIJ = dJI denotes the number of bifundamental fermi multiplets if I 6= J , while 12 dI2J is the number of adjoint fermi multiplets if I = J . D( 1)-brane quivers. Finally, we may consider D-instantons on X5, which results in a quiver with two types of oriented arrows: (1.6) (1.7) (1.8) (1.9) (1.10) (1.11) The corresponding N = 1 gauged matrix model contains two types of \matter" multiplets, the chiral and fermi multiplets [49]. In this case, the quantities: 1 dIJ dim Ext1X5 (EJ ; EI ) ; 2 dIJ dim Ext2X5 (EJ ; EI ) ; (1.12) give the number of arrows of either types from eI to eJ . We will brie y discuss these gauged matrix models in section 4. Elusive fractional branes. The above identi cations between Ext groups and supersymmetric multiplets in D-brane quivers are conjectures, that we may check in many explicit computations. The practical usefulness of these identi cations rely on identifying the fractional branes EI in the rst place, as distinguished objects in the B-brane category on Xn. To the best of our knowledge, this remains an open problem in general. In this note, we will deal with simple examples where we can describe the fractional branes explicitly. Interactions terms: product structure in the derived category. Importantly, the D-brane quivers have interactions terms, which are encoded in superpotentials in various dimensions. On D5-branes, the interactions are fully determined by supersymmetry, while D3-brane quivers have a non-trivial 4d N = 1 superpotential W (X). The 2d N = (0; 2) theories have two types of \superpotential" interactions, encoded in holomorphic functions J (X) and E(X) [50]. The 0d N = 1 matrix models also have two kinds of holomorphic \superpotentials", distinct from the 2d superpotentials, denoted by F (X) and H(X) [49].6 These interactions terms can be recovered from the fractional branes by considering the product structure between Ext groups. Let A denote the graded algebra Ext for a given set of fractional branes, where the grading is by the degree of the Ext groups. (It is also the ghost number of the B-model.) There exists multi-products: of degree 2 k, satisfying relations amongst themselves, that generate a minimal A1 structhe product obtained by composition. These multi-products correspond to disk correlators in the topological B-model. It is known that the A1 structure encodes the 4d N = 1 superpotential of D3-brane quivers [10, 52]. Following the same methods, we will be able to derive the 2d N = (0; 2) and 0d N = 1 quiver interactions. This paper is organized as follows. In section 2, we discuss the construction of 2d N = (0; 2) supersymmetric quiver gauge theories from the knowledge the B-branes on a CY fourfold. In section 3, we discuss triality of 2d N = (0; 2) quivers in this context, and relate triality to mutations of exceptional collections of sheaves. In section 4, we discuss the similar construction of 0d N = 1 quiver theories from B-branes on a CY vefold. A few complementary points are discussed in appendices. 6What we call F -term was called J-term in [49]. We choose this notation in order to distinguish between the 2d and 0d interactions. 7A minimal A1 structure is an A1 structure in which m1 = 0 [51]. { 5 { D1-brane quivers and 2d N = (0; 2) theories Two-dimensional gauge theories with N = (0; 2) supersymmetry are built out of three types of supermultiplets: vector, chiral and fermi multiplets [50]. In Wess-Zumino gauge, the vector multiplet (V; Vz) contains a gauge eld A , left-moving gaugini and an auxiliary scalar D, transforming in the adjoint of the Lie algebra g = Lie(G), with G the gauge group. The charged matter elds consist of chiral multiplets and fermi multiplets | and of their charge-conjugate multiplets, the anti-chiral multiplet e and the anti-fermi multiplet e, respectively. They satisfy the half-BPS conditions: the chiral multiplets . In components, the chiral super eld reads: D+ = 0 ; D+ = E( ) : = + + + with a complex scalar and + a right-moving fermion. The fermi super eld is given by: = +G +E ; with a left-moving fermion an G an auxiliary eld. The chiral and fermi multiplets are valued in some representations R and R of g, respectively. Consequently, the potential E( ) is valued in R as well. The canonical kinetic Lagrangian for the matter elds is: Lkin = Z d +d + i Dz ; (2.1) (2.2) (2.3) (2.4) (2.5) (2.6) with J the conjugate potential for the anti-fermi multiplet. This Lagrangian is supersymmetric provided that (2.5) is satis ed. The auxiliary elds G, G can be integrated out, which sets G = J and G = J . We then obtain the following Lagrangian for the fermi multiplets: Lfermi = 2i + EE + J J + + : (2.7) { 6 { with Dz the gauge covariant derivative, and with the trace over g kept implicit. A standard super-Yang-Mills term can also be constructed for the vector multiplet. To every fermi multiplet , we also associate an \N = (0; 2) superpotential" J = J ( ) transforming in the conjugate representation R , such that: R Z with the trace over g, Tr : R ! C. The interaction Lagrangian reads: Tr (EJ ) = 0 ; LJ = d + J ( ) Z d + J ( ) ; Note that there is a symmetry that exchanges fermi and anti-fermi multiplets: $ ; E $ J ; E $ J : In the presence of several fermi multiplets in distinct irreducible representations, each fermi multiplet can be \dualized" independently.8 (2.8) (2.9) ! eJ 2.1 N = (0; 2) quiver gauge theory from B-branes at a CY4 singularity Systems of D1-branes at CY4 singularities engineer a simple yet rich class of gauge theories with product gauge group: HJEP02(18)5 G = Y U(NI ) : I To each U(NI ) gauge group, one associates an N = (0; 2) vector multiplet, denoted by a node eI in a quiver diagram. The matter elds in chiral multiplets are in bifundamental representations NI NJ between unitary gauge groups. To each chiral multiplet XIJ in the fundamental of U(NI ) and antifundamental of U(NJ ), we associate a solid arrow eI in the quiver diagram. The matter elds in fermi multiplets are also in bifundamental representations. To each bifundamental fermi multiplet IJ , we associate the dashed link eI - - - eJ in the quiver diagram. While IJ denotes a fermi multiplet in the bifundamental NI NJ of U(NI ) U(NJ ), the associated link in the quiver is unoriented, re ecting the fermi duality (2.8).9 We may also have chiral and fermi multiplets in the adjoint representation of a single gauge group U(NI ), corresponding to a special case of the above with I = J . To each IJ , one associates an E-term and a J -term. Given that IJ transform in the bifundamental representation NI NJ , by convention, the potential E IJ transforms in NI NI NJ as well, while the potential J IJ transforms in the conjugate representation NJ . In other words, E IJ is given by a direct sum of oriented paths p (counted with complex coe cients) from eI to eJ in the quiver, travelled along chiral multiplet arrows, and J IJ is similarly a direct sum of oriented paths pe from eJ to eI : E IJ (X) = J IJ (X) = X paths p X paths peecIpeJ XJL1 XL1L2 where the sum is over all possible paths p and pe of lengths k and ek, respectively. The eld rede nitions) as part of the de nition of the N = (0; 2) suepersymmetric quiver. They must be such that the supersymmetry constraint (2.5) holds. This means that, for any closed loop P for chiral multiplets 8See [53] for a discussion of some subtleties in this symmetry in (0; 2) NLSMs. 9In practice, we still nd it convenient to write oriented dashed arrows for fermi multiplets, re ecting a choice of representation for the fermi multiplets (that is, which is and which is ). This is because such a choice is needed to write down the o -shell supersymmetric action. { 7 { in the quiver, we must have: X p;p e p+pe=P cp ecpe = 0 ; 8P ; where the sum is over all pairs of quiver paths p : eI ! based at fermi multiplets IJ such that the closed path p + pe coincides with P . ! eJ and pe : eJ ! 2.1.1 From B-branes to quiver Consider a D1-brane probing a local Calabi-Yau fourfold singularity X4. Away from the singularity, the D1-brane is described in the B-brane category as a skyscraper sheave Op at a point p 2 X4. At the singularity, we expect that the D1-brane fractionates into a nite number n of mutually-stable components: Op = E1 En : The fractional branes EI , with I = 1; ; n, are a distinguished set in the derived category of coherent sheaves on the local CY fourfold. If we normalize the central charge of the D1-brane to Z(Op) = 1, the fractional branes must be such that their central charge align at a special small-volume point | a \quiver point" | in the quantum Kahler moduli space of X4, with Z(EI ) 2 R>0 and PI Z(EI ) = 1. In the case of an orbifold of at space, X4 = C4= , the \quiver point" is the orbifold point, where perturbative string theory is valid, and the fractional branes are in one-to-one correspondence with the irreducible representations of [1]. We will not study stability issues at all in this work. We will only assume that we may identify (or guess) a suitable set of fractional branes. In general, there might be many allowable sets of fractional branes, some of which give the same quiver, and some of which give di erent quivers. This last possibility should correspond to eld theory dualities. We will comment on this point in section 3. Given the fractional branes: EI 2 Db(X4) ; as objects in the B-brane category, we may compute the morphisms between them. For EI and EJ given as coherent sheaves on X4, the morphisms are elements of the Ext groups: ExtiX4 (EI ; EJ ) ; i = 0; 1; 2; 3; 4 : These groups encode massless open strings stretched between fractional branes [45]. We should have: metric quiver: Ext0X4 (EI ; EJ ) = Hom(EI ; EJ ) = IJ C ; to obtain a physical quiver. This is because Ext0 is identi ed with the massless gauge eld in the open string spectrum. In our setup, we identify Ext0(EI ; EJ ) with the N = (0; 2) vector multiplet at the node eI of the quiver. The degree-one Ext groups are identi ed with the chiral multiplets in the supersym, XIJ ; { 8 { (2.11) ! eI (2.12) (2.13) (2.14) (2.15) (2.16) By Serre duality, we have Ext3X4 (EJ ; EI ) = Ext1X4 (EI ; EJ ), so that Ext3X4 (EI ; EJ ) is identi ed with the anti-chiral multiplets XIJ . This identi cation of chiral multiplets with Ext1 is well-known in the case of four-dimensional N = 1 quivers associated to D3-branes on a CY threefold [ 7, 44, 48, 54 ]. The new ingredient on a CY fourfold is that we also have independent degree-two Ext groups, with: Ext2X4 (EJ ; EI ) = Ext2X4 (EI ; EJ ) by Serre duality on X4. It is natural to identify these groups with the fermi multiplets IJ in the N = (0; 2) quiver: , , IJ : The self-duality relation (2.17) for Ext2 correspond to the fact that fermi and anti-fermi multiplet are indistinguishable. For each pair of distinct nodes I, J , we may pick the basis of the Ext2 vector spaces: f ; g 2 Ext2X4 (EJ ; EI ) ; f ; g 2 Ext2X4 (EI ; EJ ) ; where and correspond to fermi multiplets JI , respectively, and such that Serre duality exchanges with , and with . This choice of basis is completely convention-dependent, however. This corresponds exactly to the freedom (2.8) of labelling fermi and anti-fermi multiplets in the supersymmetric eld theory. For I = J , Ext2(EI ; EI ) is self-dual, and each pair of Serre-dual elements correspond to a pair of fermi and anti-fermi multiplets II , II in the adjoint representation of U(NI ). As a simple consistency check of these identi cations between Ext groups and N = (0; 2) super elds, it is interesting to look at the product variety X4 = X3 C, with X3 a CY threefold singularity. This non-isolated singularity preserves N = (2; 2) supersymmetry in two-dimension, and the 2d quiver should simply be the dimensional reduction of the N = 1 supersymmetric quiver for D3-branes on X3. Each 4d N = 1 vector multiplet decomposes into one N = (0; 2) vector multiplet and one adjoint fermi multiplet, and each 4d N = 1 chiral multiplet decomposes into one N = (0; 2) chiral multiplet and one fermi multiplet. In terms of Ext groups, this means that we should have: Ext0X4 (EI ; EJ ) = Ext0X3 (EI ; EJ ) ; Ext1X4 (EI ; EJ ) = Ext0X3 (EI ; EJ ) Ext2X4 (EI ; EJ ) = Ext1X3 (EI ; EJ ) Ext3X4 (EI ; EJ ) = Ext2X3 (EI ; EJ ) Ext4X4 (EI ; EJ ) = Ext3X3 (EI ; EJ ) : Ext1X3 (EI ; EJ ) ; Ext2X3 (EI ; EJ ) ; Ext3X3 (EI ; EJ ) ; This can be shown to be the case in general orbifolds C3= C | see appendix A. Note that (2.20) is consistent with Serre duality (1.5). One can similarly consider the decomposition X4 = X2 C2, which preserves 2d N = (4; 4) supersymmetry. { 9 { A comment on conventions. To avoid any possible confusion, let us note that we are using the physicist notation for the chiral multiplets in the N = (0; 2) superpotentials, and the mathematical notation of composition when discussing elements of Ext . For instance, we have: x y. When talking about the fractional branes, we write these maps as: y x E3 ! E2 ! E1 : On the other hand, we have chosen the convention that Ext(EJ ; EI ) corresponds to the chiral multiplet XIJ , so that the direction of the arrows in the quiver are ipped: a map EJ ! EI corresponds to a quiver arrow eI ! eJ . In our example (2.21), denoting by X and Y the chiral multiplets associated to the Ext group elements, we have: where on the right-hand-side we associated a gauge group U(NI ) to each node eI . In these conventions, we can write x y as the matrix product XY for the chiral multiplets. Anomaly-free condition and quiver ranks. Consider an N = (0; 2) quiver with nodes feI g and gauge group (2.9). For each U(NI ) factor, the cancellation of the non-abelian anomaly requires: X J6=I dI1J + dI3J 2 dIJ NJ + 2NI 1 + dI1I = 0 : (2.24) Here the rst sum is over the chiral and fermi multiplets in bifundamental representations, while the second term denote the contribution from the vector multiplet (with dI0I = 1) and from adjoint matter. Using Serre duality, this can be written as: (2.21) (2.22) (2.23) 4 J i=0 X X( 1)iNI dim ExtiX4 (EI ; EJ ) = 0 : (2.25) This condition imposes constraint on the allowed ranks NI in the quiver. If we consider a single D1-brane, the ranks NI should be xed from rst principle; however, the explicit dictionary between brane-charge basis and quiver-rank basis is not always known. The anomaly-free condition then provides a strong constraint. The solutions to (2.25), as a linear system for the positive integers NI , correspond to all stable D-brane con gurations at the singularity. In particular, the unique solution fNI g such that each NI is the smallest possible positive integer is expected to correspond to a single D1-brane. In the special case of toric Calabi-Yau singularities, we know from [19, 21, 26] that there exists \toric quiver" with equal ranks, NI = N , corresponding to N D1-branes. We should also mention that the abelian quadratic anomalies, from the U(1)I factors in U(NI ), do not vanish in general. Instead, they should be cancelled by closed string contributions a la Green-Schwarz [1, 17, 55]. 2.1.2 To complete the determination of the N = (0; 2) supersymmetric quiver from the fractional branes on X4, we need to discuss the E- and J -terms (2.10). It is convenient to package them into a gauge-invariant \(0; 2) superpotential" W de ned as:10 Here, the index I runs over all the fermi multiplets. This W can be computed by following the methods of [10], which studied 4d N = 1 quiver theories on D3-branes at CY3 singularities. On general ground, the superpotential coupling constants are encoded in open string correlation functions. Those can be described in the language of A1 algebra | see e.g. [56] and references therein. An A1 algebra is a (graded) algebra A together with a set of multiplications mk : such a way that there is an A1 map:11 for all integer n > 0. The rst relation states that (m1) 2 = 0, so one can think of m1 : A ! A as a di erential. The Ext group elements between B-branes, on the other hand, generate a minimal A1 algebra, for which m1 = 0. To compute the multi-products on the Ext algebra, we proceed as follows. Given an A1 algebra Ae, one de nes H (Ae) to be the cohomology of m1. If Ae has no multiplications beyond m2, then it has been shown [57] that one can de ne an A1 structure on H (Ae) in f : H (Ae) ! Ae ; (2.28) with f1 equal to a particular representation H (Ae) ,! Ae in which cohomology classes map to (noncanonical) representatives in Ae, and such that m1 = 0 in the A1 algebra on H (Ae). One can then use the consistency conditions satis ed by elements of an A1 map to solve algebraically for f1 mk. In terms of B-branes, the algebra Ae is the algebra of complexes of coherent sheaves, with chain maps between complexes. In that construction, m1 is essentially the BRST charge Q of the B-model. The \physical" open string states then live in the cohomology H (Ae), which gives us the derived category Db(X) | we refer to [48] for a pedagogical discussion. We can identify the minimal A1 algebra A H (Ae) with the Ext algebra we are interested in. Practically, in the examples discussed in this paper, each B-brane will be a single coherent sheaf, which can be represented in the derived category by a locally-free resolution. 10This expression is only formal. The N = (0; 2) superpotential that appears in the gauge theory Lagrangian is the usual Tr I JI (X)), since superspace treats and asymmetrically. This formal W rst appeared in [22]. It elegantly encodes the algebraic structure of the N = (0; 2) quiver relations. This point is further discussed in appendix C. 11That is, a family of maps satisfying certain consistency conditions [57]. Consider a correlation function of r boundary vertex operators ai 2 A on the open-string worldsheet. In the A1 language, this can be written as: ha1 ari = ha1; mr 1(a2; ; ar)i ; in terms of the higher-product mr 1 and the pairing (2.29) [10]. Each Ext elements x 2 A is dual to a \ eld" X in the supersymmetric quiver | see appendix C for further details. In the case of a 2d N = (0; 2) quiver describing B-branes on a CY4 geometry, we have the Ext algebra: ; , as in (2.19). The coupling constants cJ and cE appearing as: where the summands denote all Ext groups between the various fractional branes, of degree 0; X, and by ; ; 4. Let us denote by x 2 A the Ext1 elements corresponding to the chiral multiplets e 2 A the Ext2 elements corresponding to the fermi and anti-fermi multiplets cJ Tr( X1 Xr) + cE Tr( X10 Xr00 ) in the superpotential (2.26) can be computed as the open-string correlators: The Ext elements can then be represented by chain maps between resolutions, modulo chain homotopies. The m2 products in A are given by chain map composition. The higher products can then be computed by the procedure just described. We elaborate on this procedure in appendix C.3, and we illustrate the computation of the higher products, in a speci c example, in appendix D. All of the other examples below will actually have mk = 0 for k > 2. Open string correlators and A1 products. Let A denote the Ext algebra associated to a local Calabi-Yau n-fold. There exists a natural \trace map" of degree n, which we denote by : A ! C. Note that A is a graded algebra, with a of degree q if a 2 Extq. Serre duality de nes a natural pairing of degree n: Explicit formula for the E- and J -terms. We can now spell out the precise formula for the coupling constants appearing in (2.10). Consider a fermi multiplet IJ corresponding to to 2 Ext2(EJ ; EI ), and the charge-conjugate anti-fermi multiplet IJ corresponding 2 Ext2(EI ; EJ ). For each path p as in (2.10), we have the elements x 2 Ext1 corresponding to the chiral multiplets X. We thus have: for the E-term coe cients, and One can show that: and for the J -term coe cients. We can check this identi cation for a number of geometries previously studied by independent techniques, and we nd perfect agreement. Last but not least, we should note that, according to the dictionary (2.34){(2.35), the Tr(EJ ) = 0 constraint (2.11) translates into a very non-trivial relation amongst products of open string correlators. In appendix C, we give a general argument for why this constraint will hold for E and J de ned by the A1 algebra as above. In addition, we will check, in every example below, that the condition Tr(EJ ) = 0 indeed holds, thus providing an additional consistency check on our computations. It would be interesting to also understand the rst-principle origin of this constraint in the Calabi-Yau fourfold geometry. space, there is a single \fractional brane", the skyscraper sheaf Op, which corresponds to a single transverse D1-brane. Consider Op at the origin of C4, without loss of generality. Ext0(Op; Op) = Ext4(Op; Op) = C ; Ext1(Op; Op) = Ext3(Op; Op) = C4 ; Ext2(Op; Op) = C6 : From this result, we directly read o the N = (0; 2) supermultiplet content according to the general rules. We have a single vector multiplet, 4 chiral multiplets and 3 fermi multiplets. If there are N fractional branes at a point, all these elds are in the adjoint of a U(N ) gauge group. This reproduces the eld content of maximally supersymmetric N = (8; 8) Yang-Mills theory in 2d, as expected. To compute the interaction terms, we will need to describe the Ext algebra more explicitly. 2.2.1 An explicit basis for Ext (Op; Op) The Ext algebra can be computed from the Koszul resolution of Op, which reads: 0 ! O ! O D where:12 A = x y z w 0 y 0 1 0 C w C A z 0 x 0 0 w 0 0 x z 0 C x A 12Here and in the following, we denote a map M : Cn ! Cm by an m maps corresponds to matrix multiplication (for instance, A B = AB). n matrix, so that composition of Let us present explicit expressions for the generators of Ext . We will use the notation: Xji 2 Exti(Op; Op) ; j = 1; ; dim Exti(Op; Op) : (2.39) Every Ext element can be represented by a chain map between two copies of the Koszul resolution; the actual Ext element is given by the corresponding element in its cohomology, by the de nition of Ext as a derived functor. First of all, Ext0(Op; Op) is spanned by the single element: Secondly, Ext1(Op; Op) is spanned by maps of the form: A basis can be obtained by taking O ? 1? y O O D ! O ! O ! O ? 1? y 4 4 D D O y ? ? 4 D C ! O ! O ! O ! O ? 1? y 6 6 C C ? ? y 4 6 C B ! O ! O ! O ! O ? 1? y 4 4 B B ? ? y 6 4 B A 4 ? ? y ! O ! O A A ! O ? 1? y ! O A ! O A 0 C ; A 0 C ; A 0 C A 1 and demanding the diagram be anti-commutative. For example, when = (1; 0; 0; 0)t, we can take B 0 0 1 0 0 1 B 0 0 1 0 C = BBBB 00 00 00 01 CCC C C ; 0 0 0 0 0 0 0 1 0 0 0 1 = BBB 00 00 00 00 10 01 CC Similarly, Ext2(Op; Op) is spanned by maps of the form: = A ! O 6 ? !? y As before, we can choose ' to be one of the unit column vectors with six entries, and then make the diagram commutative. For example, ' = BBBB 10 CCC 0 1 0 0 0 1 A Ext3(Op;Op) is spanned by maps of the form: is one of the unit vectors with four entries and is such that A = D. For example, when = (0;0;0;1)t, = ( 1;0;0;0). Finally, Ext4(Op;Op) is spanned by: O D! O4 C ! O6 B ! O4 A ! O 2.2.2 Multiplication of maps The multiplication rule can be determined by composing these maps. For example, X31 X21 is computed by: D ! O4 C ! O6 B ! O4 A ! O D ! O4 C ! O6 B ! O4 A ! O D ! O4 C ! O6 B ! O4 A ! O ? 21?y ? 31?y ? 21?y with: O D ! O4 C ! O6 B ! O4 A 0 0 0 0 0 1 A C ; 21 = BBB 00 00 00 01 00 00 CCC ; 21 = 0 0 1 0 ; O ? 12?y ? 31?y O ? 13?y B 0 0 0 0 C 0 0 0 0 O ? 1? y ? 21?y ? 31?y 0 0 1 0 31 = BBB 01 CC C ; 31 = BBBB 00 A 0 C 1 0 0 C Proceeding in this way, we nd the multiplication rules: m2(Xi1; Xj1) = Xi1 Xj1 = BB 0 B B B B 0 X12 X22 X42 X12 0 X32 X52 The product X1 Xj1 is given by the matrix element ij in (2.40). One can also compute i the products: X62 X12 = X52 X22 = X42 X32 = X14 ; which commute. All other products between degree-two maps vanish. This shows that the Serre dual of X12; X22; X33 are X62; X52; X42 respectively. One can also show that the higher products vanish in this case | that is, mk = 0 if k > 2. Therefore, any nonzero correlation function can be reduced to one of the following: hX12 X31 X41i = 1 ; hX32 X11 X41i = 1 ; hX52 X11 X31i = 1 ; hX22 X21 X41i = hX42 X21 X31i = 1 ; hX62 X11 X21i = 1 : 1 ; 2.2.3 The C4 quiver: N = (8; 8) SYM on C 4 has the eld content of N From (2.36){(2.37), we see that the N = (0; 2) gauge theory corresponding to D1-branes = (8; 8) SYM. We can also verify that the product structure encoded in (2.41) reproduces the correct supersymmetric interactions. In N = (0; 2) notation, this theory consists of four chiral multiplets, denoted and three fermi multiplets a (a = 1; 2; 3), with the E and J terms: and a (a = 1; 2; 3), This is reproduced by our computation, with the identi cations: for the chiral multiplets, and Ea = [ ; a] ; J a = abc b c : = X41 ; a = (X31 ; X21 ; X11) ; a = (X62 ; X52 ; X42) ; a = (X12 ; X22 ; X32) (2.40) (2.41) (2.42) (2.43) (2.44) for the fermi multiplets, as one can easily check using (2.34){(2.35), and Ea = J a = X ij X ij h a; m2( i; j )i i j ; h a; m2( i; j )i i j ; Ext0(E0; E0) = C ; Ext0(E1; E1) = C ; Ext0(E0; E1) = 0 ; Ext0(E1; E0) = 0 ; Ext1(E0; E0) = 0 ; Ext1(E1; E1) = 0 ; Ext1(E0; E1) = C4 ; Ext1(E1; E0) = C4 ; Ext2(E0; E0) = C6 ; Ext2(E1; E1) = C6 ; Ext2(E0; E1) = 0 ; Ext2(E1; E0) = 0 ; for f ig the set of all chiral super elds | here, by abuse of notation, we identi ed the quiver elds with the corresponding Ext elements in the open-string correlators. Note that the condition (2.5) is satis ed, Tr(EaJ a) = 0. The interaction terms (2.42) display an SU(3) avory symmetry. On-shell, there is a larger SU(4) avor symmetry, with ( a; a) sitting in the 6 of SU(4). It will often be the case that the avor symmetry displayed by the N = (0; 2) quiver is smaller than the symmetry expected from the CY4 geometry. Those larger geometric symmetries can be thought to arise in the infrared of the gauge theory, as accidental symmetries [20]. The next simplest class of examples are supersymmetric orbifolds of at space. Consider the CY4 singularity C4= , with brane EI for each irreducible representation I of line bundle O with the corresponding given by: a discrete subgroup of SU(4). There exists one fractional [46]. We also denote by I the trivial -equivariant structure. The fractional branes are EI = I Op ; with Op the skyscraper sheaf supported at the origin. In the following, we consider a few examples with abelian, for simplicity. 2.3.1 Consider C4=Z2, where the generator of Z2 acts on the C4 coordinates (x; y; z; w) as: We have two fractional branes: (x; y; z; w) 7! ( x; y; z; w) : E0 = 0 Op ; E1 = 1 Op : for the trivial and non-trivial representation of Z2, respectively. The dimensions of the Ext groups can be computed following the methods of [46]. We have: (2.45) (2.46) (2.47) (2.48) a at each nodes, and two sets of four chiral multiplets, Ai = (Aa; A4) and Bi = (Ba; B4), in bifundamental representations. The quiver arrows have multiplicities equal to the number of distinct chiral or fermi multiplets. with the higher Ext groups determined by Serre duality. We can also recover this spectrum from the results of section 2.2. Let us replace X in (2.39) by a, b, c, d according to the following diagram: HJEP02(18)5 c 7 E0 l a b , E1 w d ; which encodes all possible Ext groups. From the Koszul resolution (2.38) and the fact that the maps A; B; C; D are all odd under Z2, we see that the superscript of a and b can only take values 1; 3, while the superscript of c and d can only take values 0; 2; 4, in agreement with (2.48). This gives us the N = (0; 2) quiver indicated in gure 1. The B-model correlation functions can be read o from (2.41). The N = (0; 2) superpotential immediately follows. Let 100; 200; 300; 111; 211; 311 denote the fermi superelds corresponding to c24; c25; c26; d24; d25; d26, respectively. Note that they are Serre dual to c23; c22; c21; d23; d22; d21. Let us also denote the chiral super elds corresponding to aj1; bj1 by Aj ; Bj . We then have, for instance: and so on and so forth. It is convenient to introduce the notation: J 1 = 00 E 100 = 0a0 ; X i;j X i;j 1a1 ; c24bi1aj1 c23bi1aj1 BiAj = B2A3 B3A2 ; BiAj = B1A4 B4A1 ; Ai = (Aa; A4) ; Bi = (Ba; B4) ; with the index a = 1; 2; 3, to emphasize an SU(3) avor symmetry. The interaction terms are given by: J 0a0 = abcBbAc ; E 0a0 = BaA4 B4Aa ; J 1a1 = AaB4 E 1a1 = abcAbBc : A4Ba ; This satis es Tr(EJ ) = 0, and it is in perfect agreement with the results of [19]. Note that, while the Lagrangian of the theory only has an SU(3) U(1) global symmetry, the E and J terms of either node, taken together, t into the 6 of SU(4), while the elds Ai and Bi each sit in the 4 of SU(4). This is the sign of an enhanced global symmetry in the infrared of the gauge theory, which can also be seen in the geometry. (2.49) (2.50) (2.51) = + F ; F = F ( ) : Here, the N = 1 superpotential F is an holomorphic function of the bosons in chiral multiplets. Given the chiral multiplets i and fermi multiplets a, one can write the supersymmetric action: as complex conjugate in the matrix integral, while there is a single fermion . The second type of multiplet is the fermi multiplet , with a single fermionic component , such that: SF = Z d F a( ) a = F a( )Fa( ) + (4.3) Another quadratic action in the fermions can be written in terms of an holomorphic potential Hab( ) = Hba( ): SH = Hab( ) a b : This is supersymmetric provided that HabFb = 0. The third type of sypersymmetry multiplet is the gaugino multiplet, which implements a gauge constraint on eld space. The gaugino multplet V consists of two components, the fermion and the real boson D, with: V = + D : Given a theory of chiral and fermi multiplets with some non-trivial Lie group symmetry, we can gauge a subgroup G (with Lie algebra g) of that symmetry by introducing an g-valued gaugino multiplet, with the action: (4.2) (4.4) (4.5) (4.6) (4.7) (4.8) with acting on in the appropriate representation, and an overall trace over the gauge group is implicit. Here is a 0d Fayet-Iliopoulos parameter. Integrating out D, we obtain: Sgauge = 2 i : where (schematically), which is the moment map (minus the \level" ) of the G action on the bosonic eld space. 4.1 N = 1 gauged matrix model from B-branes at a CY5 singularity D-instantons at CY5 singularities engineer precisely such gauged matrix models with gauge group Q I U(NI ). For each node eI in the 0d N = 1 quiver, we have a U(NI ) gaugino multiplet. The matter elds are either chiral or fermi multiplets, in adjoint or bifundamental representations. We have thus a quiver with two type of oriented arrows: eI ! eJ for chiral multiplets XIJ , and eI 99K eJ for fermi multiplets IJ . Finally, we also have the F and H-type interaction terms. To each fermi multiplet IJ , we associate the element FIJ , a direct sum over oriented paths p from eI to eJ , of length k: Sgauge = Z d 1 2 D i + i ; X FIJ (X) = cIpJ XIK1 XK1K2 similarly to (2.10), with given coe cients cIpJ . In addition, to every pair of fermi multiplets KL, we associate the H-term action SHIJ;KL , which is a sum over closed loops p from eI back to itself, which includes both IJ and KL, in addition to chiral multiplets X: X paths pe e SHIJ;KL = cIJ;KL Tr( IJ XJM1 p XMk 1K KL XLN1 XNk0 1I ) : Note that the closed path pe has length k + k0 + 2, including the two fermions. This quiver structure naturally arises from open strings between fractional D( 1)branes at a CY5 singularity, where each node eI corresponds to a fractional brane EI . As The non-vanishing Ext0 elements are identi ed with the gaugino multiplets. The degreeone Ext groups are identi ed with chiral multiplets: e (4.9) (4.10) (4.11) (4.12) (4.14) (4.15) , eI ! eJ , , XIJ ; IJ : in bifundamental (if I 6= J ) or adjoint (if I = J ) representations. Similarly, the degree-two Ext groups are identi ed with the fermi multiplets: Ext2X5 (EI ; EJ ). By Serre duality, we also have Ext4X5 (EJ ; EI ) = Ext1X5 (EI ; EJ ) and Ext3X5 (EJ ; EI ) = Interaction terms. The F -terms (4.3) and H-terms (4.4) also arise naturally in the Bmodel. As discussed in section 2.1.2, the Ext-group generators satisfy an A1 algebra with multi-products mk. Consider a fermi multiplet IJ corresponding to 2 Ext2(EJ ; EI ), and let us denote by 2 Ext3(EI ; EJ ) the Serre dual generator. For each path p as in (4.8), we have the elements x 2 Ext1 corresponding to the chiral multiplets X. We propose that: for the F -term coe cients in (4.8). Similarly, consider the fermi multiplets 2 Ext2(EL; EK ), respectively. We propose that the H-term coe cients in (4.9) are given by: p e cIJ;KL = D xJM1 xMk 1K xLN1 xNk0 1I E = m2( ; mk(xJM1 ; e ; xMk 1K ; ; xLN1 ; ; xNk0 1I )) ; with ek = k + k0 + 1. We will check this prescription in some examples below. Note that this corresponds exactly to computing the formal 0d N = 1 superpotential: which was recently introduced in [22]. 4.2 We can work out the very simplest case, a D( 1) brane on X5 = C5, exactly like in section 2.2. Consider the skyscraper sheaf Op at the origin of C5. We have: Ext0(Op; Op) = Ext5(Op; Op) = C ; Ext1(Op; Op) = Ext4(Op; Op) = C5 ; Ext2(Op; Op) = Ext3(Op; Op) = C10 : Using the above dictionary to N = 1 super elds, this reproduces the expected eld content of the maximally-supersymmetric N = 16 matrix model, as we will review below. Proceeding as before, the Koszul resolution of Op on C5 reads: 0 ! O ! O E ! O ! O 10 B ! O ! O ! Op ! 0; (4.16) (4.17) 0 u 1 B B w C y CA x (4.18) w 00 CCCC ; E = BBB z C : C C Similarly to section 2.2, we choose as bases of the Ext groups the commutative diagrams composition. The products m2(Xi1; Xj1) = X1 Xj1 are given by: i whose leftmost nonzero vertical map has 1 at an entry and 0 elsewhere. We denote them by Xi, following the same conventions. The multiplication rule is again determined by j 0 x 0 0 0 u 0 0 0 0 0 x X11 X21 X31 X41 X51 0 0 0 w 0 y 0 z 0 0 y 0 0 0 0 u z 0 0 X11 0 X12 X22 X42 X72 X21 X12 0 X32 X52 X82 X31 X22 X32 0 X62 X92 0 C C C u C w CA z w 0 C 0 w B B B B 0 BB y B B B B B 0 0 z 0 x 0 u 0 y 0 0 0 0 0 u z 0 u 0 0 y 0 0 z w 0 x 0 x 0 u C C C A w 0 0 u z 0 0 x u C w CC z C A y X41 X42 X52 X62 0 X120 X51 X72 X82 X92 X120 0 products m2(Xi1; Xj2) mapping Ext1 Ext2 to Ext3, according to: The elements in this table are the products of the elements in the rst column multiplied by elements in the rst row. (For example, X11 X21 = X12.) Similarly, we have non-zero X12 X22 X32 X42 X52 X62 X72 X92 X120 X31 X13 We also nd the following Serre dual elements to Xi2: Using the multiplication rule (and the cyclic property of the open-string correlators), we see that any nonzero correlation function can be computed in terms the following hX3X1X1itype correlators: and the following hX2X2X1i-type correlators: hX13X51X41i = 1 ; hX43X51X11i = 1 ; hX73X41X11i = 1 ; hX130X21X11i = 1 ; hX32X120X11i = 1 ; hX22X120X21i = 1 ; hX12X120X31i = 1 ; hX12X92X41i = 1 ; hX12X62X51i = 1 ; hX23X51X31i = 1 ; ; hX33X51X21i = 1 ; hX53X41X31i = 1 ; hX83X31X21i = 1 ; hX63X41X21i = 1 ; hX93X31X11i = 1 ; hX52X92X11i = 1 ; hX42X92X21i = 1 ; hX42X82X31i = 1 ; hX22X82X41i = 1 ; hX22X52X51i = 1 ; hX62X82X11i = 1 ; hX72X62X21i = 1 ; hX52X72X31i = 1 ; hX32X72X41i = 1 ; hX32X42X51i = 1: (4.19) (4.20) (4.21) (4.22) HJEP02(18)5 4.2.2 The C5 quiver: N = 16 SYM Consider the N = 16 supersymmetric GMM with gauge group U(N), corresponding to N D( 1)-branes in at space. Its eld content can be deduced from dimensional reduction of 2d N = (8; 8) SYM in section 2.2.3. In N = 1 language, we have a single U(N) gaugino multiplet, 5 chiral multiplets in the adjoint representation, and 10 fermi multiplets in the adjoint representation. It is convenient to denote the chiral and fermi multiplets by n and mn = nm, with n = 1; 5, since n and nm transform in the 5 and 10 of an SU(5) avor symmetry. This spectrum is reproduced by the Ext groups above. We identify the elds with the Ext elements according to Xn1 = n, n = 1; ; 5, and: X12 = 21 ; X62 = 43 ; X22 = 31 ; X72 = 51 ; X32 = 32 ; X82 = 52 ; X42 = 41 ; X92 = 53 ; X52 = 42 ; X120 = 54 : The interaction terms are determined by the F - and H-terms [49]: Fmn = m n n m ; Hmn;pq = mnpqr r : HJEP02(18)5 One can check that the open-string correlators (4.21){(4.22) precisely reproduce these interactions. Note that, to check that the H-term: SH = 4 1 mnpqr Tr( mn pq r) is supersymmetric, we need to use the Jacobi identity for U(N ). This is equivalent to the non-trivial condition HabFb = 0 mentioned above, which must always be realized by the B-brane correlators. Given the above results for C5, we can easily study various N = 1-preserving orbifolds Consider for instance C5=Z5, where Z5 acts as: (x; y; z; w; t) 7! (!x; !y; !z; !w; !t) ; ! = e 5 2 i on the C5 coordinates. We have ve fractional branes denoted by Ei, i = 0; weights for the sheaves in the Koszul resolution of Ei are given by: i + 1 BB i + 2 CC BB i + 3 CC BB i + 3 CC B i + 3 C C A i + 3 ! BB i + 1 CC D! BBB ii ++ 22 CCCC !C BBB ii ++ 33 CCCC !B BBB i + 4 CCC !A E B C B B i : 0 i + 4 1 B i + 4 C (4.23) (4.24) (4.25) (4.26) ; 4. The (a) C5=Z5(1; 1; 1; 1; 1) quiver. (b) C5=Z3(1; 1; 1; 1; 2) quiver. chiral and fermi multiplet arrows indicate their multiplicities. We then nd the spectrum: Ext[0C5=Z5](Ei; Ej) = Ext[1C5=Z5](Ei; Ej) = Ext[2C5=Z5](Ei; Ej) = 0 0 0 ( SpanCfX10g if j i mod 5 ; otherwise, ( SpanCfX11; X21; X31; X41; X51g if j + 1 i mod 5 ; otherwise, ( SpanCfX12; X22; X32; X42; X52; X62; X72; X82; X92; X120g if j + 2 i mod 5 ; otherwise. The higher Ext groups are obtained by Serre duality. The correlation functions can be read o from (4.21){(4.22). Let us introduce the chiral multiplets: In : eI with I an integer mod 5, m; n = 1; ; 5, and quiver is shown in gure 10a. The interaction terms are: I mn = Inm. The gauged matrix model F Imn = m n I I+1 In Im+1 ; H Imn; Ip+q2 = mnpqr Ir 1 : Note the obvious SU(5) avor symmetry. This quiver was discussed in [49, 67, 68]. 4.3.2 As a last example, consider the C5=Z3 orbifold: (x; y; z; w; t) 7! (!x; !y; !z; !w; !2t) ; ! = e 3 : 2 i (4.27) (4.28) (4.29) We have three fractional branes Ei, i = 0; 1; 2. The weights for the sheaves in the Koszul resolution of Ei are: i + 1 0 8> 0 The spectrum consists of: Ext[0C5=Z3](Ei; Ej ) = SpanCfX10g if j i mod 3 ; BB i + 2 CC i + 1 i ! BB i + 1 CC D! BBB ii ++ 22 CCCC !C BBB ii ++ 11 CCCC !B BBB i + 2 CCC !A E B C B B i : Ext[1C5=Z3](Ei; Ej ) = < SpanCfX21; X31; X41; X51g if j + 1 Ext[2C5=Z3](Ei; Ej ) = < >: SpanCfX11g 8> SpanCfX12; X22; X42; X72g >: SpanCfX32; X52; X62; X82; X92; X120g if j + 2 if j i mod 3 ; if j + 2 i mod 3 ; i mod 3 ; if j if j + 1 i mod 3 ; i mod 3 ; i mod 3 : (4.30) (4.31) (4.32) (4.33) The corresponding 0d N = 1 quiver is shown in gure 10b. The correlation functions can be read o from (4.21){(4.22). Taking advantage of the residual SU(4) avor symmetry, let us introduce the chiral multiplets: with a = 1; 4, and I an integer mod 3. Similarly, we de ne the fermi multiplets: AI : eI X!11 eI 1 ; BIa : eI X1a+1 interaction terms read: mn = nm are de ned as in (4.23). In this notation, the I F ab = BIaBIb+1 BIbBIa+1 ; H Ia; Ibc = abcdBId 1 F Ia = AI BIa 1 H Iab; Ic 1 = abcdBId 1 Many more N = 1 matrix models can be worked out in this way. It would also be instructive to study fractional branes on local Fano fourfold varieties, such as the resolution of the C5=Z5(1; 1; 1; 1; 1) to Tot(O( 5) ! P4). We leave this and many other related questions for future work. Acknowledgments We would like to thank P. Aspinwall, S. Franco, D. Ghim, C. Herzog, S. Katz, W. Lerche, I. Melnikov, T. Pantev, and R.K. Seong for useful conversations and comments. E.S. was partially supported by NSF grants PHY-1417410 and PHY-1720321. A Dimensional reductions Fourfolds versus threefolds. Let X be a Calabi-Yau orbifold [Xc=G] of complex dimension 3, with a set of fractional branes fEig supported at a point p 2 Xc, a xed point of the G-action. Let N3 denote the normal bundle Np=X . Let us build another Calabi-Yau orbifold Y = C X, which again has an isomorphic set of fractional branes fEig, supported at x 0 f g p 2 C X, of codimension four. Let N denote the normal bundle to x in Y , and 0 the structure sheaf with trivial G-equivariant Ext0Y (Ei; Ej ) = H0(x; i j ) G Ext1Y (Ei; Ej ) = H0(x; i Ext2Y (Ei; Ej ) = H0(x; i Ext3Y (Ei; Ej ) = H0(x; i Ext4Y (Ei; Ej ) = H0(x; i = Ext0X (Ei; Ej ) ; = Ext0X (Ei; Ej ) = Ext1X (Ei; Ej ) = Ext2X (Ei; Ej ) = Ext3X (Ei; Ej ) : j j j j j j (N3 ^2N3))G (^2N3 ^3N3))G ^3N3) G This directly con rms (2.20) in the case of an orbifold singularity. We conjecture that it holds more generally. Fourfolds versus twofolds. Similarly, we may consider X a Calabi-Yau orbifold [Xc=G] of complex dimension 2, with a set of fractional branes fEig supported at a point p 2 Xc, a xed point of the G-action. Let NX denote the normal bundle Np=X . Let us build another Calabi-Yau orbifold Y = C 2 X, which again has an isomorphic set of fractional branes fEig, supported at x f(0; 0)g p 2 C X, of codimension four. Let N denote the normal bundle to x in Y , and 0 the structure sheaf with trivial G-equivariant structure. Then, 2 0 N = NX ) 2 ^2NX ) 2 = NX NX2 ^2NX ; (^2NX ) 2 ; ^2NX = ^2NX : j j j ) G ( 02 NX ))G Ext0X (Ei; Ej ) Ext1X (Ei; Ej ) Ext1X (Ei; Ej ) Ext2X (Ei; Ej ) ; Ext2X (Ei; Ej ) j (NX ^2NX )G (NX ) 2 ^2NX ))G (^2NX ) 2))G This decomposition corresponds to the dimensional reduction of a 6d N = 1 quiver theory (or, equivalently, of a 4d N = 2 theory) to 2d, giving rise to an N = (4; 4) quiver theory. Each N = (4; 4) vector multiplet splits into one N = (2; 2) vector multiplet, two chiral multiplets and one fermi multiplet. Each N = (4; 4) hypermultiplet splits into two chiral and two fermi multiplets. This is precisely the decomposition seen here. B Fractional D3-branes on a local P 2 Consider the well-known case of fractional D3-branes on the Calabi-Yau threefold: Xe3 = Tot(O( 3) ! P2) ; (B.1) which is a crepant resolution of the orbifold singularity X3 = C3=Z3. The corresponding 4d N = 1 quiver gauge theory is very well studied | see e.g. [2, 3, 58, 69]. In this appendix, we review this 4d N = 1 quiver using the B-brane language. This will help to illustrate, in a more familiar context, the tools that we similarly use to study D1-brane quivers. B.1 Fractional branes and supersymmetric quivers Let us discuss two particular sets of fractional branes. Below, we will see how they are related by mutation of exceptional collections, providing a geometric realization of Seiberg duality [58]. rst set of fractional branes: theory (I) Fractional branes on the resolution (B.1) can be constructed from the data of a strongly exceptional collection on P2, as in section 3.3. Let us rst consider the exceptional collection: EI = f 2(2) ; The corresponding three fractional branes on Xe3 are: E0 = i O ; E1 = i E2 = i (B.2) (B.3) where i is the inclusion from P2 into Xe3. Let z0; z1; z2 be the homogeneous coordinates of P2 and Ui be the open set in which zi 6= 0. Denote the local coordinates in Ui by (xi; yi) and the coordinate of the O( 3) in Ui by wi. We have w1 = x30w0; w2 = y03w0 = y13w1. In the following we will take ber of Koszul resolutions: 0 ! O(k + 3) w0 (k + 3) w0 ! O(k) ! i OP2 (k) ! 0 (k) ! i P2 (k) ! 0 : It is straightforward to compute the Ext groups themselves. The Ext1 quiver reads: Ext1(i O( 1)[2]; i (1)[1]) is generated by ci 2 C0(Hom1(i O( 1)[2]; i (1)[1])): A basis of the Ext groups can be chosen as follows: Ext1(i O; i O( 1)[2]) is generated by ai 2 C2(Hom 1(i O; i O( 1)[2])): O(2) a1 = 1 x0y0 ! O( 1) 1 x20y0 ; a3 = 1 x0y02 : Ext1(i (1)[1]; i O) is generated by bi 2 C0(Hom1(i (1)[1]; i O)): i O o a 3 9 i (1)[1] 3 c ! O (1) O (1) b1 : (x0;y0)?y ( x0; y0)?y b2 : ( 1;0)??y c1 : (4) ! O( 1) y0 ?? x0 y (1) c2 : b3 : (0; 1)? c3 : (4) ? y O(3) O(2) 1 ?? 0 y (4) (1) O ! O( 1) 10 ??y (1) (a) C3=Z3 quiver. (b) A Seiberg dual quiver. multiplicity 3, while the arrow M has multiplicity 6. The generator of Ext3(i O; i O) can be chosen to be t 2 C2(X; Hom1(i O; i O)) with One can then compute: tjU0 = 1 x0y0 : (m2(ai; m2(bj ; ck))) = ijk : Note that there is a GL(3) symmetry inherited from P2, and a corresponding SU(3) avor symmetry in the N = 1 gauge theory. The N = 1 quiver gauge theory is the one shown in gure 11a, with a gauge group U(N ) U(N ) U(N ). The bifundamental chiral multiplets Ai; Bi; Ci correspond to the Ext1 elements ai; bi; ci, and the product structure (B.4) leads to the N = 1 superpotential: W = Tr( ijkAiBj Ck) : This quiver can also be obtained by orbifold projection from 4d N = 4 theory [2, 3]. B.1.2 A second set of fractional branes: theory (II) Consider another strongly exceptional collection on P2: EII = fO( 1) ; O ; O(1)g : The corresponding fractional branes are: E0 = i O[1] ; E1 = i O(1) ; E2 = i O( 1)[2] : We repeat the same analysis as before. The Ext1 quiver reads: (B.4) (B.5) (B.6) (B.7) The corresponding N = 1 quiver is shown in gure 11b. x a0 3 f 3 b0 6 d0 / i O(1) b01 : O(2) O(3) O(3) O(4) Ext1(i O[1]; i O(1)) is generated by b0i 2 C0(Hom1(i O[1]; i O(1))): 1? y O O ! O(1) a02 : b02 : O(2) d01 = d04 = 1 1 x0y0 x20y02 O(2) O(3) O(3) ! O( 1) A0i = ai ; B0i = bi ; O O ! O(1) ! O(1) a03 : b03 : 1 1 ; d03 = d06 = 1 1 x20y0 x0y03 ; : 0d01 d03 d021 Mij = BBBd03 d05 d04CC : C Ext1(i O(1); i O( 1)[2]) is generated by d0n 2 C2(Hom 1(i O(1); i O( 1)[2])): The eld theory is shown in gure 11b. The elds A0i, B0i are both in the 3 of the SU(3) avor symmetry, while the elds Mij = Mji span the 6 of SU(3). They are identi ed with the Ext1 elements according to: O(2) O(3) O(3) O(4) O O y ! O(1) (B.8) (B.9) One can then derive the superpotential: be U(2N ) duality. B.2 Seiberg duality as mutation W = A0iMij A0j : Moreover, due to the non-abelian anomaly-cancellation condition, the gauge group must U(N ) U(N ). This is also what is obtained from the usual rules of Seiberg The two N = 1 quiver theories of gure 11 are related by a Seiberg duality on node e0. Consider for instance the \Theory (I)". A Seiberg duality at node e0 reverses the arrows Ai and Bj while generating the new mesons Mfij , with the identi cation Mfij = AiBj across the duality. The superpotential dual to (B.5) reads: W = ijkMfij Ck + A0iMfij B0j : (B.10) This contains a mass term for Ci and the antisymmetric part of Mfij . Integrating those elds out, we are left with \Theory (II)", including the superpotential (B.9). Similarly, if we start from Theory (II) and perform a Seiberg duality at node e0, we ip the arrows A0i, B0j , and generate the dual mesons N ij = A0iB0j , with the superpotential: W = Mij N ji + BiN ij Aj : (B.11) Integrating out the massive elds | Mij and the symmetric part of N ij | we recover Theory (I) and (B.5), with the identi cation N ij = ijkCk. Mutation of exceptional collection. It was proposed in [58] that Seiberg duality could be realized as mutation on exceptional collections of sheaves. Start with Theory (II) and the corresponding exceptional collection EII (B.6). Using the left mutation: LOO(1) = 1(1) (B.12) (B.13) (C.1) (C.2) (C.3) (C.4) on P2, we see that a left mutation of the collection EII at the second sheaf precisely gives the collection EI in (B.2): fO( 1) ; O ; O(1)g fO( 1) ; Therefore, the Seiberg duality at node e0 of Theory (I) is indeed realized by a mutation of the underlying sheaves. This observation has been generalized to a number of other cases [8]. C A1 structure and N = (0; 2) quiver In this appendix, we discuss the A1 structure of the Ext algebra, and how it is related to the structure of the N = (0; 2) quiver. This discussion is a straightforward generalization of a similar discussion for 4d N = 1 quivers by Aspinwall and Katz [10]. See also [51, 68, 70]. C.1 An algebraic preliminary Let V be a graded vector space, and let T (V ) be the associated graded tensor algebra: Let d be an derivative operator of degree 1 acting on T (V ), satisfying the graded Leibniz rule: d(A B) = dA B + ( 1)jAjA dB ; with A; B 2 T (V ), and jAj denoting the degree of A. We also require that: Using the Leibniz rule, the action of d on T (V ) is determined by its action on V itself. Let us decompose d as: d V = d1 + d2 + with dk : V ! V k : T (V ) = 1 M V n : n=1 d2 = 0 : denote the corresponding map of degree A as being the dual of V [1]: together with the multi-products: Let V [1] denote the vector space V with all degrees decreased by one, and let s : V ! V [1] 1. Given this data, we can de ne an A1 algebra given by the dual of the map s k dk s 1 : V [1] ! V [1] k. The nilpotency condition (C.3) is equivalent to the following A1 relation on the multi-products: X ( 1)r+stmn+1 s(1 r ms 1 t) = 0 ; 8n > 0 ; where the sum is over all r; t 0, s > 0, such that r + s + t = n [51]. (C.5) (C.6) (C.7) (C.8) (C.9) (C.10) (C.11) (C.12) Ext algebra and N = (0; 2) quiver In our physical setup, the vector space A is spanned by the various Exti groups (i = ; 4) among the fractional branes on a CY4 singularity. Schematically: A = Ext0 Ext1 Ext2 Ext3 Ext4 : The grading of A is given by the degree i of Exti. Any z 2 A of degree q is associated to a local vortex operator in the B-model, with the degree identi ed with the ghost number. Given z 2 A, let z(1) denote the corresponding one-form descendant. The one-form operators can be used to deform the B-model according to [10, 70]: S ! S + X Zi zi(1) : i The coupling Zi is identi ed with a \quiver eld" in the space-time (D1-brane) theory. Note that Zi has degree 1 qi if zi has degree qi. The quiver elds are elements of the vector space V , in the notation of subsection C.1. Let us denote by z 2 A the Serre dual of z 2 A, with the Ext algebra A given by (C.8). Let us then choose a basis of A according to: fzig = fe0 ; x ; I ; I ; x ; e0g ; C.2 0; with: e0 2 Ext0 ; x 2 Ext1 ; I ; I 2 Ext2 ; x 2 Ext3 ; e0 2 Ext4 : As discussed in the main text, the choice of basis for Ext2 is arbitrary. Any given choice introduces a distinction between the elements and the Serre dual elements , which is a matter of convention. The dual vector space V spans the \quiver elds". We choose a basis of V : fZig = fe ; X ; I ; I ; X ; eg ; dual to (C.10). The element e correspond to the vector multiplets, while X and correspond to the chiral and fermi multiplets, respectively. Note the degrees: degree: e 1 X 0 I 1 X In particular, the chiral multiplets have degree 0. Given this explicit basis of V , we de ne a derivative d as follows. First, let us introduce the N = (0; 2) \superpotential": with JI (X) and EI (X) some arbitrary functions of the chiral multiplets X . This W is an arbitrary gauge-invariant function of degree 1 that is independent of e, except that we need to impose the constraint: Let us also de ne the derivatives: Tr(EI J I ) = 0 : by left derivation on W | that is, we use the cyclic property of the trace to write (C.13) with X is de ned as the sum of all possible forms of W de ne the degree-one derivative d on V as:24 with X in front, with X removed. Given the superpotential, we d2X = 0 ; W = Tr I J I (X) + EI (X) ; (C.13) (C.14) (C.15) (C.16) (C.17) (C.18) de = e dX = X e ; e d I = EI (X) d I = J I (X) dX de = X e X e I I X X X ; e e e e + X I ; I e ; I X I + I I e e e e : By direct computation, one can check that d2 = 0. The relations: d2e = 0 ; d2X = 0 ; d 2 I = 0 ; d 2 I = 0 are obvious.25 The key relation is: which holds true if and only if the non-trivial constraint (C.14) is satis ed. This is nothing but the requirement that the N = (0; 2) superpotential be properly supersymmetric. Since we explicitly displayed a nilpotent derivative d on the vector space V spanned by the quiver 24This is the analogue of equations (30) and (39) of [10]. 25To check the last two relations, one uses that: dF (X) = F (X) e e F (X) ; for any degree-zero holomorphic function F (X), which follows from the second line in (C.16). elds, it follows from the general discussion above that the multi-products mk acting on the Ext vector space A satisfy the A1 relations (C.7). In this way, we see clearly that the A1 relations on a CY4 are intimately related to the supersymmetry constraint (C.14). We should also note that the di erential d de ned in (C.16) has: d1 = 0 ; (C.19) mk = 0 for k given by m2. where dk is de ned as in (C.4), if and only if the potentials EI and J I do not contain any linear terms in X . In such a case, we have m1 = 0 in the dual Ext algebra, which gives us a minimal A1 structure. Linear terms in EI or J I are mass terms, and the corresponding elds can always be integrated out, as discussed in examples in section 3. Therefore, (C.19) always holds for the low-energy quiver. Similarly, we see from (C.4) that there exists non-zero higher products mk for k = 2; ; kmax, with kmax the highest order in the elds X that appear in the potentials EI , J I . In the simplest case when EI , J I are all quadratic in the chiral multiplets, we have 3, and the A1 algebra reduces to an associative algebra with a product C.3 General procedure to compute the higher products Let us discuss in more detail the procedure to compute the higher products of the Ext A1 algebra [10], which we outlined in section 2.1.2. Consider an A1 algebra Ae and the A1 map: Let the rst map: f1 = i : H (Ae) ! Ae ; (C.20) (C.21) reads: be the inclusion map de ned by picking representatives of cohomology classes, and let d = me1 : Ae ! Ae denote the di erential on Ae. The rst A1 constraint on the maps fk i m2( ; ) = i( ) i( ) + df2( ; ) : We can compute i( ) i( ), and use the result to de ne m2( ; ) and f2( ; ). The next A1 constraint is of the form: i m3( ; ; ) = f2( ; m2( ; )) f2(m2( ; ); ) + i( ) f2( ; ) f2( ; ) i( ) + df3( ; ; ): Using the previously-computed m2 and f2, this expression allows us to compute m3 and f3. Proceeding inductively in this fashion, one can construct mk and fk to any order k. D Higher products on a local P 1 P 1 In this appendix, we spell out the computation of the higher products on the local P1 P 1 geometry of section 2.5, using the procedure summarized in appendix C.3. HJEP02(18)5 sition of the chain maps, one nds the products: m2(b1; d1) = 10 ; m2(a1; e1) = 10 ; m2(b1; d2) = 20 ; m2(a2; e1) = 20 ; m2(b2; d1) = 30 ; m2(a1; e2) = 30 ; m2(b2; d2) = 40 ; m2(a2; e2) = 40 : It follows that f2(b; d) = f2(a; e) = 0. De ne the 1-cochains and as follows: ( )01 = ( )02 = ( )03 = ( )12 = ( )13 = 0 ; ( )23 = x 1u 1 ; One can compute where the chain maps are de ned by d1 c1 = d 1 ; d2 c2 = d 1 ; d1 c2 = d 2 ; d2 c2 = d 2 ; 1 = 1 = 0 0 ) ; ) ; 2 = 2 = 0 ! 0 ! ( ; 0) ; ( ; 0) ; f2(d1; c1) = f2(d2; c1) = 1 ; f2(d1; c2) = 1 ; f2(d2; c2) = c1 a1 = d 1 ; c2 a1 = d 2 ; c1 a2 = d 1 ; c2 a2 = d 2 : f2(c1; a1) = f2(c1; a2) = 1 ; f2(c2; a1) = 1 ; f2(c2; a2) = 2 ; 2 : 2 ; 2 : m3(d1; c1; a1) = 0 ; m3(d1; c2; a1) = 0 ; m3(d2; c1; a1) = m3(d2; c2; a1) = 1 ; 2 ; m3(d1; c1; a2) = 1 ; m3(d1; c2; a2) = 2 ; m3(d2; c1; a2) = 0 ; m3(d2; c2; a2) = 0 : between the corresponding complexes. This implies m2(d; c) = 0 and: Thus, m2(c; a) = 0 and: Plugging these results into the A1 map constraint: im3(d; c; a) = f2(d; m2(c; a)) f2(m2(d; c); a) + d f2(c; a) f2(d; c) a + df3(d; c; a) ; (D.2) (D.3) Plugging these results into: f2(e1; c1) = 1 ; f2(e1; c2) = 2 ; f2(e2; c1) = 1 ; f2(e2; c2) = 2 ; f2(c1; b1) = f2(c1; b2) = 1 ; f2(c2; b1) = 1 ; f2(c2; b2) = 2 ; 2 ; 1 = 1 = 0 0 2 = 2 = 0 0 ( ; 0) ; ( ; 0) : m3(e1; c1; b1) = 0 ; m3(e1; c2; b1) = 0 ; m3(e2; c1; b1) = 10 ; m3(e2; c2; b1) = 20 ; m3(e1; c1; b2) = m3(e1; c2; b2) = m3(e2; c1; b2) = 0 ; m3(e2; c2; b2) = 0 : 0 ; 1 0 ; 2 Similarly, if we de ne and we get m2(e; c) = 0; m2(c; b) = 0 and ( )01 = ( )03 = ( )13 = ( )23 = 0 ; ( )02 = ( )12 = x 1u 1 ; ( )01 = ( )02 = ( )12 = ( )23 = 0 ; ( )03 = ( )13 = x 1 ; (D.4) (D.5) im3(e; c; b) = f2(e; m2(c; b)) f2(m2(e; c); b) + e f2(c; b) f2(e; c) b + df3(e; c; b) ; This completes the computation of the three-product m3. 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Cyril Closset, Jirui Guo, Eric Sharpe. B-branes and supersymmetric quivers in 2d, Journal of High Energy Physics, 2018, 51, DOI: 10.1007/JHEP02(2018)051