B-branes and supersymmetric quivers in 2d

We study 2d $\mathcal{N}=(0,2)$ supersymmetric quiver gauge theories that describe the low-energy dynamics of D1-branes at Calabi-Yau fourfold (CY$_4$) 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 CY$_4$. 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 $A_\infty$ algebra satisfied by the morphisms. This provides us with a derivation of the supersymmetric quiver directly from the CY$_4$ 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 CY$_5$ singularities, can be discussed similarly.


Introduction
Many supersymmetric quantum field theories can be engineered on systems of branes in string theory. The string theory embedding often provides us with an elegant geometric understanding of field theory phenomena. In particular, rich classes of field 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 X n , 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 CY 2 cone.
• 4d N = 1 quiver theories on D3-branes at the tip of a CY 3 cone.
• 0d N = 1 quiver theories on D-instantons at the tip of a CY 5 cone.
In this paper, we study 2d and 0d quivers from the point of view of B-branes on the CY n-fold X n . A B-brane is simply a (half-BPS) D-brane in the topological B-model on X n . The B-model is a g s = 0 limit of type II string theory which (somewhat trivially) captures all α 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 Kähler 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 sufficient in order to extract all the information about the holomorphic sector of the low-energy open strings. That is, we can read off 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 CY 3 singularities [6][7][8][9][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 X n is an object in the (bounded) derived category of coherent sheaves of X n : 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: which are the morphisms in the derived category. Physically, they encode the lowenergy modes of the open strings stretched between the D-branes E and F [44][45][46][47]. We refer to [48,49] for comprehensive reviews of the derived category approach to B-branes.

D-branes quivers from B-branes
Consider a D(9 − 2n)-brane transverse to the Calabi-Yau singularity X n . Away from the singularity, the brane is locally in flat space. From the point of view of X n , it is a point-like brane, which is described by a skyscraper sheaf O p at a point p ∈ X n . When at the singularity, it is expected that O p "fractionates" into marginally stable constituents. The resulting "fractional branes" {E I } realize a gauge group: 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 fields X IJ . In this way, the low-energy open string sector at the singularity is described by a supersymmetric quiver gauge theory: to each fractional brane E I , we associate a node in the quiver, denoted by e I . The matter fields corresponds to various quiver arrows connecting the nodes: e I −→ e J . There are also interaction terms among the matter fields, which we will discuss below. In all cases, the fractional branes are such that: These Ext 0 ∼ = Hom groups are identified with vector multiplets in d = 10 − 2n dimensions; a single vector multiplet is assigned to each node e I , realizing the gauge group U (N I ). The other Ext i groups (with i = 1, · · · , n − 1) correspond to matter fields charged under the gauge groups. 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 (N I ) 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 different supersymmetric theories with the same structure but distinct gauge groups. 5 A crucial property of Ext groups on a Calabi-Yau variety X n is the Serre duality relation: Ext i Xn (E I , E J ) ∼ = Ext n−i Xn (E J , E I ) , i = 0, · · · , n . (1.5) This corresponds to the CPT symmetry of the d-dimensional quiver quantum field theory. Generalizing some relatively well-known results for D3-branes, it is natural to propose the following identification of Ext groups with supersymmetry multiplets in various dimensions: D5-brane quivers. For D5-branes on X 2 , we have: (1.6) where X IJ are 6d N = (0, 1) hypermultiplets in the bifundamental representation of U (n I )×U (n J ). Note that the quiver link e I -e J is unoriented since the hypermultiplet is non-chiral-this corresponds to the Serre duality Ext 1 (E I , E J ) ∼ = Ext 1 (E J , E I ) on X 2 . In this case, X 2 must be an ADE singularity while the supersymmetric quivers are extended Dynkin diagrams.
D3-brane quivers. For D3-branes on X 3 , we have: where X IJ are 4d N = 1 chiral multiplets in the bifundamental of U (n I ) × U (n J ), or in the adjoint of U (n I ) if I = J. The arrows are oriented. Therefore, such quiver gauge theories are generally chiral theories. More precisely, we denote by: the number of arrows from e I to e J 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 X 4 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 X IJ and (0, 2) fermi multiplets Λ IJ , respectively. We propose the identification: Note that the Ext 2 X 4 (E I , E J ) ∼ = Ext 2 X 4 (E J , E I ) 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 define: (1.10) Here d 1 IJ is the number of chiral multiplets from e I to e J (in bifundamental representations if I = J or adjoint representation if I = J). Similarly, d 2 IJ = d 2 JI denotes the number of bifundamental fermi multiplets if I = J, while 1 2 d 2 IJ is the number of adjoint fermi multiplets if I = J. D(−1)-brane quivers. Finally, we may consider D-instantons on X 5 , which results in a quiver with two types of oriented arrows: ( 1.11) The corresponding N = 1 gauged matrix model contains two types of "matter" multiplets, the chiral and fermi multiplets [50]. In this case, the quantities: give the number of arrows of either types from e I to e J . We will briefly discuss these gauged matrix models in section 4.

Elusive fractional branes
The above identifications between Ext groups and supersymmetric multiplets in Dbrane quivers are conjectures, that we may check in many explicit computations. The practical usefulness of these identifications rely on identifying the fractional branes E I in the first place, as distinguished objects in the B-brane category on X n . 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) [51]. 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) [50]. 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 A ∞ structure. 7 In particular, if a ∈ Ext • (E 1 , is the product obtained by composition. These multi-products correspond to disk correlators in the topological B-model. It is known that the A ∞ structure encodes the 4d N = 1 superpotential of D3-brane quivers [10,53]. Following the same methods, we will be able to derive the 2d N = (0, 2) and 0d N = 1 quiver interactions. We note that similar ideas were previously reported in [54]. 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 fivefold. A few complementary points are discussed in Appendices.
potential E(Φ) is valued in R Λ as well. The canonical kinetic Lagrangian for the matter fields is: with D z 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 Λ, we also associate an "N = (0, 2) superpotential" J = J(Φ) transforming in the conjugate representationR Λ , such that: with the trace over g, Tr : R Λ ⊗R Λ → C. The interaction Lagrangian reads: withJ the conjugate potential for the anti-fermi multiplet. This Lagrangian is supersymmetric provided that (2.5) is satisfied. The auxiliary fields G,Ḡ can be integrated out, which sets G =J andḠ = J. We then obtain the following Lagrangian for the fermi multiplets: Note that there is a symmetry that exchanges fermi and anti-fermi multiplets: In the presence of several fermi multiplets in distinct irreducible representations, each fermi multiplet can be "dualized" independently. 8 2.1 N = (0, 2) quiver gauge theory from B-branes at a CY 4 singularity Systems of D1-branes at CY 4 singularities engineer a simple yet rich class of gauge theories with product gauge group: To each U (N I ) gauge group, one associates an N = (0, 2) vector multiplet, denoted by a node e I in a quiver diagram. The matter fields in chiral multiplets are in bifundamental representations N I ⊗N J between unitary gauge groups. To each chiral multiplet X IJ in the fundamental of U (N I ) and antifundamental of U (N J ), we associate a solid arrow e I −→ e J in the quiver diagram. The matter fields in fermi multiplets are also in bifundamental representations. To each bifundamental fermi multiplet Λ IJ , we associate the dashed link e I ---e J in the quiver diagram. While Λ IJ denotes a fermi multiplet in the bifundamental N I ⊗N J of U (N I ) × U (N J ), the associated link in the quiver is unoriented, reflecting the fermi duality (2.8). 9 We may also have chiral and fermi multiplets in the adjoint representation of a single gauge group U (N I ), corresponding to a special case of the above with I = J.
To each Λ IJ , one associates an E-term and a J-term. Given that Λ IJ transform in the bifundamental representation N I ⊗N J , by convention, the potential E Λ IJ transforms in N I ⊗N J as well, while the potential J Λ IJ transforms in the conjugate repre-sentationN I ⊗ N J . In other words, E Λ IJ is given by a direct sum of oriented paths p (counted with complex coefficients) from e I to e J in the quiver, travelled along chiral multiplet arrows, and J Λ IJ is similarly a direct sum of oriented paths p from e J to e I : where the sum is over all possible paths p and p of lengths k and k, respectively. The numerical coefficients c IJ p , c IJ p are to be given (up to field redefinitions) as part of the definition of the N = (0, 2) supersymmetric quiver. They must be such that the supersymmetry constraint (2.5) holds. This means that, for any closed loop P for chiral multiplets in the quiver, we must have: where the sum is over all pairs of quiver paths p : e I → · · · → e J and p : e J → · · · → e I based at fermi multiplets Λ IJ such that the closed path p + p coincides with P .

From B-branes to quiver
Consider a D1-brane probing a local Calabi-Yau fourfold singularity X 4 . Away from the singularity, the D1-brane is described in the B-brane category as a skyscraper sheave O p at a point p ∈ X 4 . At the singularity, we expect that the D1-brane fractionates into a finite number n of mutually-stable components: The fractional branes E I , 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(O p ) = 1, the fractional branes must be such that their central charge align at a special small-volume point-a "quiver point"-in the quantum Kähler moduli space of X 4 , with Z(E I ) ∈ R >0 and I Z(E I ) = 1. In the case of an orbifold of flat space, X 4 ∼ = C 4 /Γ, 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 Γ [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 different quivers. This last possibility should correspond to field theory dualities. We will comment on this point in section 3. Given the fractional branes: as objects in the B-brane category, we may compute the morphisms between them. For E I and E J given as coherent sheaves on X 4 , the morphisms are elements of the Ext groups: (2.14) These groups encode massless open strings stretched between fractional branes [46]. We should have: to obtain a physical quiver. This is because Ext 0 is identified with the massless gauge field in the open string spectrum. In our setup, we identify Ext 0 (E I , E J ) with the N = (0, 2) vector multiplet at the node e I of the quiver. The degree-one Ext groups are identified with the chiral multiplets in the supersymmetric quiver: By Serre duality, we have Ext 3 is identified with the anti-chiral multipletsX IJ . This identification of chiral multiplets with Ext 1 is well-known in the case of four-dimensional N = 1 quivers associated to D3branes on a CY threefold [7,45,49,56]. The new ingredient on a CY fourfold is that we also have independent degree-two Ext groups, with: by Serre duality on X 4 . It is natural to identify these groups with the fermi multiplets Λ IJ in the N = (0, 2) quiver: The self-duality relation (2.17) for Ext 2 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 Ext 2 vector spaces: where α and β correspond to fermi multiplets Λ IJ and Λ JI , whileᾱ andβ correspond to anti-fermi multipletsΛ IJ andΛ JI , respectively, and such that Serre duality exchanges α withᾱ, and β withβ. This choice of basis is completely convention-dependent, however. This corresponds exactly to the freedom (2.8) of labelling fermi and antifermi multiplets in the supersymmetric field theory. For I = J, Ext 2 (E I , E I ) is self-dual, and each pair of Serre-dual elements correspond to a pair of fermi and anti-fermi multiplets Λ II ,Λ II in the adjoint representation of U (N I ).
As a simple consistency check of these identifications between Ext groups and N = (0, 2) superfields, it is interesting to look at the product variety X 4 ∼ = X 3 × C, with X 3 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 X 3 . 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: Ext 0 (2.20) This can be shown to be the case in general orbifolds C 3 /Γ × C-see Appendix A.
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: where x · y ≡ x • y. When talking about the fractional branes, we write these maps as: On the other hand, we have chosen the convention that Ext(E J , E I ) corresponds to the chiral multiplet X IJ , so that the direction of the arrows in the quiver are flipped: a map E J → E I corresponds to a quiver arrow e I → e J . 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 (N I ) to each node e I . 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 {e I } and gauge group (2.9). For each U (N I ) factor, the cancellation of the non-abelian anomaly requires: Here the first sum is over the chiral and fermi multiplets in bifundamental representations, while the second term denote the contribution from the vector multiplet (with d 0 II = 1) and from adjoint matter. Using Serre duality, this can be written as: This condition imposes constraint on the allowed ranks N I in the quiver. If we consider a single D1-brane, the ranks N I should be fixed from first 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 N I , correspond to all stable D-brane configurations at the singularity. In particular, the unique solution {N I } such that each N I 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, N I = N , corresponding to N D1-branes. We should also mention that the abelian quadratic anomalies, from the U (1) I factors in U (N I ), do not vanish in general. Instead, they should be cancelled by closed string contributionsà la Green-Schwarz [1,17,57].
2.1.2 A ∞ structure and N = (0, 2) superpotential To complete the determination of the N = (0, 2) supersymmetric quiver from the fractional branes on X 4 , 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 defined as: 10 W = Tr Λ I J I (X) +Λ I E I (X) . (2.26) 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 CY 3 singularities. On general ground, the superpotential coupling constants are encoded in open string correlation functions. Those can be described in the language of A ∞ algebra-see e.g. [58] and references therein. An A ∞ algebra is a (graded) algebra A together with a set of multiplications m k : A ⊗k → A that satisfy the A ∞ relations: r+s+t=n (−1) r+st m n+1−s (a 1 , · · · , a r , m s (a r+1 , · · · , a r+s ), a r+s+1 , · · · , a n ) = 0 , (2.27) for all integer n > 0. The first relation states that (m 1 ) 2 = 0, so one can think of m 1 : A → A as a differential. The Ext group elements between B-branes, on the other hand, generate a minimal A ∞ algebra, for which m 1 = 0.
To compute the multi-products on the Ext • algebra, we proceed as follows. Given an A ∞ algebra A, one defines H • ( A) to be the cohomology of m 1 . If A has no multiplications beyond m 2 , then it has been shown [59] that one can define an A ∞ structure on H • ( A) in such a way that there is an A ∞ map: 11 with f 1 equal to a particular representation H • ( A) → A in which cohomology classes map to (noncanonical) representatives in A, and such that m 1 = 0 in the A ∞ algebra on H • ( A). One can then use the consistency conditions satisfied by elements of an A ∞ map to solve algebraically for f 1 • m k . In terms of B-branes, the algebra A is the algebra of complexes of coherent sheaves, with chain maps between complexes. In that construction, m 1 is essentially the BRST charge Q of the B-model. The "physical" open string states then live in the cohomology H • ( A), which gives us the derived category D b (X)-we refer to [49] for a pedagogical discussion. We can identify the minimal A ∞ algebra A ≡ H • ( A) 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. The Ext elements can then be represented by chain maps between resolutions, modulo chain homotopies. The m 2 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 specific example, in Appendix D. All of the other examples below will actually have m k = 0 for k > 2.
Open string correlators and A ∞ 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 γ : A → C. Note that A is a graded algebra, with a of degree q if a ∈ Ext q . Serre duality defines a natural pairing of degree −n: a, b ≡ γ (m 2 (a, b)) . (2.29) Consider a correlation function of r boundary vertex operators a i ∈ A on the openstring worldsheet. In the A ∞ language, this can be written as: a 1 · · · a r = a 1 , m r−1 (a 2 , · · · , a r ) , (2.30) in terms of the higher-product m r−1 and the pairing (2.29) [10]. Each Ext elements x ∈ A is dual to a "field" 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 CY 4 geometry, we have the Ext algebra: where the summands denote all Ext groups between the various fractional branes, of degree 0, · · · , 4. Let us denote by x ∈ A the Ext 1 elements corresponding to the chiral multiplets X, and by α, α ∈ A the Ext 2 elements corresponding to the fermi and anti-fermi multiplets Λ,Λ, as in (2.19). The coupling constants c J and c E appearing as: c J Tr(ΛX 1 · · · X r ) + c E Tr(ΛX 1 · · · X r ) (2.32) in the superpotential (2.26) can be computed as the open-string correlators: 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 Λ IJ corresponding to α ∈ Ext 2 (E J , E I ), and the charge-conjugate anti-fermi multipletΛ IJ corresponding toᾱ ∈ Ext 2 (E I , E J ). For each path p as in (2.10), we have the elements x ∈ Ext 1 corresponding to the chiral multiplets X. We thus have: for the E-term coefficients, and for the J-term coefficients. We can check this identification for a number of geometries previously studied by independent techniques, and we find 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 defined by the A ∞ 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 first-principle origin of this constraint in the Calabi-Yau fourfold geometry.

D1-brane on C 4
To illustrate our methods, we start by considering the simplest case, X 4 = C 4 . In flat space, there is a single "fractional brane", the skyscraper sheaf O p , which corresponds to a single transverse D1-brane. Consider O p at the origin of C 4 , without loss of generality. One can show that: and From this result, we directly read off 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 fields are in the adjoint of a U (N ) gauge group. This reproduces the field 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.

An explicit basis for Ext
The Ext algebra can be computed from the Koszul resolution of O p , which reads: where: 12 Let us present explicit expressions for the generators of Ext • . We will use the notation: 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 definition of Ext as a derived functor. First of all, Ext 0 (O p , O p ) is spanned by the single element: is spanned by maps of the form: A basis can be obtained by taking and demanding the diagram be anti-commutative. For example, when α = (1, 0, 0, 0) t , we can take is spanned by maps of the form: As before, we can choose ϕ to be one of the unit column vectors with six entries, and then make the diagram commutative. For example, is spanned by maps of the form: Here ρ is one of the unit vectors with four entries and τ is such that

Multiplication of maps
The multiplication rule can be determined by composing these maps. For example, with: from which we see that β 1 3 · α 1 2 = −ϕ 3 , and so on and so forth, so that X 1 3 · X 1 2 = −X 2 3 . Proceeding in this way, we find the multiplication rules: The product X 1 i · X 1 j is given by the matrix element ij in (2.40). One can also compute the products: , which commute. All other products between degree-two maps vanish. This shows that the Serre dual of X 2 1 , X 2 2 , X 3 3 are X 2 6 , −X 2 5 , X 2 4 respectively.
One can also show that the higher products vanish in this case-that is, m k = 0 if k > 2. Therefore, any nonzero correlation function can be reduced to one of the following: , we see that the N = (0, 2) gauge theory corresponding to D1-branes on C 4 has the field content of N = (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 Σ and Φ a (a = 1, 2, 3), and three fermi multiplets Λ a (a = 1, 2, 3), with the E and J terms: This is reproduced by our computation, with the identifications: for the chiral multiplets, and for the fermi multiplets, as one can easily check using (2.34)-(2.35), and for {φ i } the set of all chiral superfields-here, by abuse of notation, we identified the quiver fields with the corresponding Ext elements in the open-string correlators. Note that the condition (2.5) is satisfied, Tr(E a J a ) = 0. The interaction terms (2.42) display an SU (3) flavory symmetry. On-shell, there is a larger SU (4) flavor symmetry, with (Λ a ,Λ a ) sitting in the 6 of SU (4). It will often be the case that the flavor symmetry displayed by the N = (0, 2) quiver is smaller than the symmetry expected from the CY 4 geometry. Those larger geometric symmetries can be thought to arise in the infrared of the gauge theory, as accidental symmetries [20].

Orbifolds C 4 /Γ
The next simplest class of examples are supersymmetric orbifolds of flat space. Consider the CY 4 singularity C 4 /Γ, with Γ a discrete subgroup of SU (4). There exists one fractional brane E I for each irreducible representation ρ I of Γ [47]. We also denote by ρ I the trivial line bundle O with the corresponding Γ-equivariant structure. The fractional branes are given by: with O p the skyscraper sheaf supported at the origin. In the following, we consider a few examples with Γ abelian, for simplicity.
where the generator of Z 2 acts on the C 4 coordinates (x, y, z, w) as: We have two fractional branes: for the trivial and non-trivial representation of Z 2 , respectively. The dimensions of the Ext groups can be computed following the methods of [47]. We have: 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: 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 Z 2 , 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 Figure 1. The B-model correlation functions can be read off from (2.41). The N = (0, 2) superpotential immediately follows. Let Λ 1 00 , Λ 2 00 , Λ 3 00 , Λ 1 11 , Λ 2 11 , Λ 3 11 denote the fermi superfields corresponding to c 2 4 , c 2 5 , c 2 6 , d 2 4 , d 2 5 , d 2 6 , respectively. Note that they are Serre dual to Let us also denote the chiral superfields corresponding to a 1 j , b 1 j by A j , B j . We then have, for instance: and so on and so forth. It is convenient to introduce the notation: with the index a = 1, 2, 3, to emphasize an SU (3) flavor symmetry. The interaction terms are given by: This satisfies 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, fit into the 6 of SU (4), while the fields A i and B i 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.
Consider C 4 /Z 3 with the orbifold action: (2.52) As before, we denote by ρ i (i = 1, 2, 3) the trivial line bundle with equivariant structure i, in conventions in which ρ 0 has the trivial equivariant structure, and ρ * 1 = ρ 2 . The possible Ext groups can be organized in the following diagram: They are of the form: with i defined mod 3, and the higher Ext groups determined by Serre duality. From the orbifold weights (2.52) on the coordinates, we can determine the weights for the sheaves in the Koszul resolution of E i . The result is: Thus the spectrum is given explicitly by: for X = a, e, k , Let us denote by Λ 01 , Λ 12 , Λ 20 the fermi multiplets corresponding to b 2 1 , f 2 1 , h 2 1 , respectively, with the charge-conjugate fermi multiplets Λ 01 , Λ 12 , Λ 20 corresponding to the Serre dual elements a 2 6 , e 2 6 , k 2 6 ; let us denote by Λ 1 00 , , respectively. We also denote by A i , B j , · · · the chiral multiplets associated to a 1 i , b 1 j , · · · , so that we have the 12 chiral multiplets: in the spectrum. The corresponding N = (0, 2) supersymmetric quiver is shown in Figure 2. We can directly compute the superpotential terms from (2.41). One finds: ( 2.55) and One can check that Tr(EJ) = 0. This again agrees with the results of [19]. Note that this quiver theory has only a U (1) 3 (toric) flavor symmetry, though there is an expected enhancement to SU (2) 2 × U (1) in the infrared. (2.57) We have four fractional branes, E i , i = 0, 1, 2, 3. In this case, the weights for the sheaves in the Koszul resolution of E i are given as follows: The Ext groups can be summarized by the diagram: x x q q q q q q q q q q q q q Here, the number attached to each arrow is the degree, and we omitted the degree-0 and degree-4 operators from one sheaf to itself, which also survive the orbifold projection and correspond to N = (0, 2) vector multiplets. We can similarly compute the interaction terms. They will be presented in section 2.4 below, after we reconsider the same quiver in a different guise. Any other supersymmetry-preserving orbifold of C 4 can be worked out similarly.

Fractional branes on a local P 3
Another interesting class of examples are given by Calabi-Yau fourfold singularities X 4 that admit a crepant resolution: One of the simplest such singularity is the C 4 /Z 4 orbifold (2.57), which admits a crepant resolution as the total space of the canonical line bundle over P 3 : For a Calabi-Yau threefold the total space of the canonical line bundle over a Fano surface, nice bases of fractional branes can be found in terms of strongly exceptional collections [7,9,15,60,61]. We can similarly construct a well-behaved set of sheaves on X 4 starting from what is known as a strongly exceptional collection of sheaves on P 3 . We will discuss this procedure in section 3.3. In the rest of this section, we will just postulate the sets of fractional branes, without further explanation. We will discuss two distinct sets of fractional branes on (2.59), which give rise to two distinct supersymmetric quivers. In section 3, we will show that those two quivers are related by a field theory infrared duality, and by a mutation of the corresponding exceptional collections. We should emphasize that these two quiver gauge theories are only two relatively simple examples among an infinite number of dual theories for D1branes probing the same CY 4 geometry. We refer to Appendix B for a review of the simpler case of a CY threefold.

Fractional branes and Ext algebra (I)
Consider the following strongly exceptional collection on P 3 : (2.60) Let i denote the embedding i : P 3 → X 4 . The four fractional branes are identified with , namely: One can compute the Ext groups explicitly. One finds: 13 where I, J = 0, 1, 2, 3. The corresponding quiver diagram for the Ext groups reads: Here the arrows stand for elements of Ext 1 and the dashed lines stand for elements of Ext 2 , with the multiplicities indicated. This coincides with the [C 4 /Z 4 ] orbifold quiver in section 2.3.3.
Interestingly, the Ext groups fill out irreducible representations of GL(4), which are induced from the underlying GL(4) symmetry of P 3 . The precise representations can be worked out from the Bott-Borel-Weil theorem [62,63]. The Ext 1 elements a, b, c, d naturally span the 4 or 4 (fundamental or fully anti-symmetric representations), while the Ext 2 elements ψ and λ fall into 6's (anti-symmetric representations) of GL(4). 14 An explicit basis for Ext • . Let us compute the Ext generators explicitly. 15 We take (x i , y i , z i ) to be the coordinates on the patch U i such that the i-th homogeneous coordinate of P 3 is nonzero, i = 0, 1, 2, 3. We also take w i to be the coordinate of the fiber of O(−4) over U i . A sheaf of the form i * E has a Koszul resolution: Every state in the Ext quiver can be represented by a chain map between the corresponding locally-free resolutions of sheaves, as follows: : : : : : : : : Note that, since the maps given by: are exact, ψ 4 , ψ 5 , ψ 6 can be equivalently represented by: respectively.
We denote the Serre dual of λ by λ , with λ n ∈Č 3 (X, Hom −1 (i * Ω(1) [1], i * Ω 3 (3)[3])): Note that, since the maps given by ( , are exact, λ 4 , λ 5 , λ 6 can be equivalently represented by And similarly for the other two Ext 4 generators. From this data, we determine the multiplication rules m 2 (x, y) by composition. One finds: and with all other products Ext 1 · Ext 1 vanishing. (All higher products also vanish.) It is convenient to define the basis: (a 1 , a 2 , a 3 , a 4 ) ≡ (a 3 , −a 2 , a 1 , a 4 ) , such that the matrices (a i · b j ), (b i · c j ), (c i · d j ) and (d i · a j ) are all antisymmetric. This is simply a manifestation of the GL(4) symmetry mentioned above.

Supersymmetric quiver (I)
From the above results, we have a complete description of the 2d N = (0, 2) supersymmetric quiver for D1-branes on the C 4 /Z 4 singularity. The chiral multiplets are identified with the Ext 1 group elements according to: with i, j, k, l ∈ 1, · · · , 4, and the elements a, b, c, d defined in (2.65). As expected, the quiver theory has an SU (4) global symmetry, with the fields (2.66) in the 4 of SU (4). The fermi multiplets Λ n 02 ∼ ψ n and Λ m 13 ∼ λ m naturally fit in the 6 of SU (4), which we denote by Λ n = Λ ij = −Λ ji . We define the fermi multiplets in terms of the elements of Ext 2 according to: and similarly for the (Serre dual) anti-fermi multiplets. The supersymmetric quiver is displayed in Figure 3. The SU (4)-preserving interactions terms encoded in (2.63)-(2.64) take the simple form: This satisfies Tr(EJ) = 0. It again agrees with the results of [19] for the C 4 /Z 4 orbifold.

Supersymmetric quiver (III)
To present the final result for the N = (0, 2) quiver theory associated to the fractional branes (2.69), it is convenient to take advantage of the SU (4) global symmetry. Let us introduce the chiral multiplets: which are identified with the Ext 1 elements a , c as indicated. We also introduce the fields M ij = M ji ∼ b and D ij = −D ji ∼ d in the 10 and 6 of SU (4), respectively: We define the fermi multiplets Λ i 03 ∼ θ and Γ ijk ∼ γ in the 4 and 20 of SU (4), respectively. We have: in terms of the Ext 2 elements θ . The fields Γ ijk are such that: We choose the explicit set of 20 fields: (2.75) Figure 4: Another C 4 /Z 4 (1, 1, 1, 1) supersymmetric quiver, which we dub "Theory (III)".
which are identified with the Ext 2 elements as indicated (the fields Γ 231 , Γ 124 , Γ 134 , Γ 234 are redundant). The supersymmetric quiver is shown in Figure 4. It is convenient to introduce the notation: The interaction terms for Λ 03 are given by: The E-terms for Γ read: To write down the J-terms, it is more convenient to use the explicit choice of 20 components as in (2.75). We find: and therefore Tr(EJ) = 0, as required. Interestingly, this supersymmetric quiver cannot be realized as a brane brick model [19]. This is an example of a "non-toric quiver" (even though the CY 4 geometry is itself toric, in this case). Since both quiver theories (I) and (III) appear to describe the low-energy dynamics of D1-branes at the [C 4 /Z 4 ] singularity, we expect that these two gauge theories are related by an infrared duality. It is indeed the case, as we will discuss in section 3.
2.5 Fractional branes on a local P 1 × P 1 As our last example, we consider a toric singularity which is not an orbifold. Let X 4 be the real cone over the seven-manifold known as Q 1,1,1 : This singularity was also discussed in [19]. In order to describe fractional D1-branes on X 4 , we will consider the following crepant resolution, the local P 1 × P 1 geometry: We can construct a set of fractional branes on (2.83) in terms of a strongly exceptional collection on P 1 × P 1 , similarly to the local P 3 example. We choose the collection {O(−1, −1), O(0, −1), O(−1, 0), O} on P 1 × P 1 . The corresponding fractional branes on the resolved singularity (2.83) are then given by: As before, i denotes the embedding i : , from which we can compute: (2.85) The corresponding Ext 1,2 quiver diagram reads: where the solid lines represent Ext 1 elements, the dashed lines represent Ext 2 elements, and the number labeling each line is the corresponding degeneracy.

2.5.1
The Ext algebra on a local P 1 × P 1 Let us compute the A ∞ structure satisfied by the Ext group elements. If we denote by x 0 , x 1 the homogeneous coordinates on the first P 1 and y 0 , y 1 the homogeneous coordinates on the second P 1 , then X 4 can be covered by four open sets U ij , i, j = 0, 1, defined by We also define local coordinates x = x 1 /x 0 , w = x 0 /x 1 , u = y 1 /y 0 , v = y 0 /y 1 in the corresponding open sets, and define y ij , z ij to be the coordinates of the fibers in U ij . Thus, we have the transition functions y 01 = uy 00 , z 11 = xuz 00 , and so forth. We have the following Koszul resolutions of the fractional branes: where all the bundle maps are written on coordinate patch U 00 . Every state in the Ext quiver diagram can be represented by a chain map between two of the above complexes. Let us introduce the notation: The representatives for the Ext 1 elements d 1,2 and e 1,2 are given by same maps as in a 1,2 and b 1,2 , respectively. −1))), i = 1, 2: with: The Serre dual elements to α 1,2 , denoted by α 1,2 , can be defined in the following way: ) is generated by: Closedness requires: If the two sides of the above identity were both zero, c would be exact. We deduce that one of (c) 013 and (c) 123 is ±x −1 u −1 and the other is zero, and similarly for (c) 023 and (c) 012 . Different choices only differ by exact terms and sign convention. In the following, we will fix: [2]) is generated by the elements β 1 , · · · , β 4 : Again, different choices do not affect the cohomology class they represent. β 2 , β 3 and β 4 are defined similarly with x −1 u −1 replaced by x −2 u −1 for β 2 , x −1 u −2 for β 3 and x −2 u −2 for β 4 . The Serre dual elements are given by β 1 , · · · , β 4 defined by The generator of Ext 4 at each node has the following form: It can be shown that α 1 is Serre dual to α 2 , α 2 is Serre dual to −α 1 and β i is Serre dual to β i : From the composition of the chain maps, one can compute the products: m 2 (a 1 , e 1 ) = β 1 , m 2 (a 2 , e 1 ) = β 2 , m 2 (a 1 , e 2 ) = β 3 , m 2 (a 2 , e 2 ) = β 4 , (2.86) amongst the Ext 1 elements. In addition, this model also has non-zero higher products, whose computation is rather more technical [10,59]. We discuss it in Appendix D. One finds the non-zero products: and: with all other products amongst the Ext 1 elements vanishing.

2.5.2
The local P 1 × P 1 quiver Given the above result, it is straightforward to write down the corresponding quiver gauge theory, shown in Figure 5. From the geometric structure (2.82), one would expect that the corresponding supersymmetric quiver theory has an SU (2) 3 global symmetry. However, the A ∞ structure only preserves the minimal "toric" flavor symmetry U (1) 3 , which is the apparent symmetry of the quiver gauge theory. The N = (0, 2) quiver has four pairs of chiral multiplet, which are identified with the above Ext 1 elements according to: with k, n, i ∈ 1, 2. The k and n index are related to the SU (2)×SU (2) induced from the P 1 × P 1 geometry; however, the interaction terms break this symmetry to its maximal torus. The quiver has the fermi multiplets: which are identified with the Ext 2 elements as indicated. From the A ∞ product structure discussed above, we find the interaction terms: and

Triality and mutations of exceptional collections
For some D3-brane quiver theories, it was proposed long ago that Seiberg duality in the gauge theory can be understood in terms of mutations of the underlying branes [60]. More precisely, for a singularity X 3 whose crepant resolution X 3 is the total space of the canonical line bundle over a del Pezzo surface B 2 , we can construct the fractional branes on X 3 in terms of an exceptional collection of B-branes on B 2 [6,7,60], and Seiberg dualities can be realized as mutations of the exceptional collection [8]. (See Appendix B for an explicit example.) We may consider the Calabi-Yau fourfold analogue of this setup, which involves the singularity X 4 whose crepant resolution is X 4 = Tot(K → B 3 ), with B 3 a Fano threefold and K its canonical line bundle. The fractional branes on X 4 can be similarly constructed from the data of an exceptional collection {E} of sheaves on B 3 , in principle. In the previous section, we considered the simplest possible example, B 3 = P 3 . A mutation of the exceptional collection gives another exceptional collection {E }, and we can again consider the corresponding N = (0, 2) quiver gauge theory. It is natural to suspect that the geometric operation amounts to a field theory duality between the different N = (0, 2) quiver gauge theories. A well-studied example 16 of an N = (0, 2) gauge theory duality is the triality of Gadde, Gukov and Putrov (GGP) [23]. We will show, in the simplest example of local P 3 , that indeed mutation is triality. This obviously deserves further study, which we leave for future work.

Triality acting on N = (0, 2) supersymmetric quivers
Let us first review GGP triality and its action on quiver gauge theories [23]. The triality transformation can be formulated as a local operation at a single node e 0 of an N = (0, 2) supersymmetric quiver without adjoint matter fields, as depicted in Figure 6. The central node e 0 is a U (N 0 ) gauge group, while the nodes e 1 , e 2 , e 3 realize a "flavor" group U (N 1 ) × U (N 2 ) × U (N 3 ) from the point of view of U (N 0 ). 17 In the "original" theory, shown in Figure 6(a), we have chiral multiplets Φ i in the fundamental representation of U (N 0 ), chiral multiplets Φ k in the antifundamental representation of U (N 0 ), and fermi multiplets Λ n in the fundamental representation of U (N 0 ). (The flavor indices i, k, n run over i = 1, · · · , N 1 ; k = 1, · · · , N 2 ; n = 1, · · · , N 3 .) We must have: to cancel the non-abelian gauge anomaly. The theory can also have non-trivial interaction terms. Let Ξ and X denote any additional fermi and chiral multiplets, respectively, distinct from Λ and Φ, Φ, in any larger N = (0, 2) quiver in which Figure 6(a) might be embedded. We have: which must be such that Tr(EJ) = Tr The "triality move" can be described as follows: Given the above Theory (i) with gauge group U (N 0 ), we obtain Theory (ii) as shown in Figure 6(b). The dual gauge 16 Other two-dimensional dualities are also known amongst N = (2, 2) and N = (0, 2) gauge theories, see for example [65,66]. 17 For simplicity, we write down a single arrow e 0 → e i (i = 1, 2, 3) for the matter fields of the U (N 0 ) gauge group at node e 0 . In general, the "effective flavor group" U (N i ) at the node e 0 corresponds to a combination of both quiver gauge groups and actual flavor symmetries, which may be broken explicitly by interaction terms. We choose the slightly schematic depiction of Figure 6 to avoid clutter.   group is U (N 0 ) with dual rank given by the number of antifundamental chiral multiplets minus N 0 : The dual charged matter fields in chiral and fermi multiplets, denoted by Φ k , Φ n and Λ i , transform under the "flavor" group as indicated on the Figure. In addition, the new theory also contains some "mesonic fields" M k i and Γ k n . Those fields are identified with the following U (N 0 )-invariant combinations of matter fields in Theory (i): To fully specify the new theory, we need to determine the new interaction terms. Given that the original theory has interaction terms (3.2), the interaction terms for any "spectator" fermi multiplet Ξ are obtained by substituting ΦΦ = M inside E Ξ and J Ξ : In addition, the interaction terms of the new fermi multiplets Λ and Γ are given by: 18

6)
18 These transformation rules were left implicit in most of the literature. They where recently studied explicitly in [22]. as holomorphic functions of X, Φ and Φ . One easily sees that the constraint Tr(EJ) = 0 is again satisfied (given that it is satisfied in the original theory). Theory (iii), shown in Figure 6(c), is similarly obtained from Theory (ii) by the same triality operation. The new gauge group is U (N 0 ) with: The new matter fields Λ k , Φ i and Φ n are as indicated. We also have the new mesons M and Γ , which are identified with the U (N 0 )-invariant combinations: in Theory (ii). Applying the rules above for the interaction terms, one finds that: in particular. This implies that Γ and M are both massive, and can be integrated out by imposing the linear relation J Γ = 0. We are left with the mesons M and Γ only, as shown in Figure 6(c). Finally, one can check that another triality move, starting from Theory (iii), gives a theory which is identical to Theory (i) after integrating out all the massive fields. Thus, we confirm that the triality operation is indeed a "duality" of order three. More precisely, this is the case if we act repeatedly on a single node of a given N = (0, 2) quiver. If we act subsequently on different nodes, one uncovers very rich, infinitedimensional "triality trees".

Triality and the C 4 /Z 4 quiver
Let us now discuss an example of the triality operation on a full-fledged D1-brane quiver. Consider the C 4 /Z 4 singularity with crepant resolution the local P 3 geometry. Two distinct quiver gauge theories were derived in section 2.4, which we dubbed "Theory (I)" and "Theory (III)." They are reproduced in Figure 7(a) and 7(c), respectively.
From Theory (I) to (II). It is straightforward to apply the "triality operation" of subsection 3.1 to these quiver gauge theories. For definiteness, conside "Theory (I)," whose field content is shown in Figure 8(a). The interaction terms read:   1, 1, 1). The numbers denote the multiplicities of the arrows.  A triality operation on the node e 0 (lower left) leads to the quiver shown in Figure 8(b), with the chiral and fermi multiplets as indicated. In particular, we have the mesonic fields M and Γ , which are given in terms of the elementary fields of Theory (I) by: By contruction, the mesons M ij sit in the 4⊗4 of SU (4), which decomposes into 10⊕6. Similarly, the fermionic fields Γ sit in the 4 ⊗ 6 ∼ = 20 ⊕ 4 . From the matter content shown in Figure 8(b), we see that the 6 component of M ij and the 4 components of Γ ijk can become massive by pairing with Λ ij 13 and C i , respectively. To see that this indeed happens, we simply look at the interaction terms, which are obtained by applying the triality rules (3.5)-(3.6). In particular, from (3.10) we find: 12) which states that the antisymmetric part of M ij is massive, and can be set to zero in the low-energy theory. Let us denote by M ij = 1 2 ( M ij + M ji ) the remaining light mesons, which span the 10 of SU (4). Similarly, it follows from (3.6) and (3.10) that the fields C i are massive. The corresponding constraint reads: This sets the 4 (fully antisymmetric) component of Γ ijk to zero. Let us denote by: the remaining fields, spanning the 20 of SU (4). Here and in the following, the notation {X ijk } denotes the projection of the three-tensor X ijk with two antisymmetrized indices onto the 20 of SU (4). We are thus left with the quiver shown in Figure 8(c). The interaction terms are given explicitly by: One can again verify that Tr(EJ) = 0.
From Theory (II) to (III). Starting from Theory (II) with the interaction terms (3.15), we can again perform a triality operation on node e 0 . The process of integrating out massive fields is similar, as depicted in Figure 9. At the intermediate step (Figure 9(a)), we have the new mesons N and Γ, which are identified with the fundamental fields of Theory (II) according to: We see from (3.15) that the 20 part of N couple with Γ to form a mass term J Γ ijk = { N ijk }. Setting { N ijk } to zero, we are left with chiral fields in the 4 of SU (4), which we denote by C i , defined such that: Similarly, we have the following mass term for B i :  Integrating out the massive fields, we obtain Theory (III), shown in the middle. Another triality move on Theory (III) gives the theory on the right, which is equivalent to Theory (I).
Integrating out B i , we are left with the 20 component of Γ ij k . It is convenient to define the new fields: We then obtain Theory (III) shown in Figure 9(b), with the interaction terms: From Theory (III) to (I). Finally, we can close this triality cycle by performing a triality operation on node e 0 of Theory (III). The intermediate step is shown in Figure 9(c). We have the new mesons K and Γ , which are identified with the fundamental fields of Theory (III) as: It follows from J Γ ijk = −{K kij } that the 20 component of K kij is massive. The remaining light fields, denoted by B i , are defined by: Similarly, we have the term: which gives a mass to M ij and the symmetric part of Γ ij . If we define the new fermi multiplets: we precisely reproduce Theory (I) in Figure 8(a), with the interaction terms (3.10).
These three N = (0, 2) quiver gauge theories are thus related by a triality cycle. Note that the quiver ranks of Theory (I) are (N, N, N, N ), while the quiver ranks of both Theory (II) and (III) are (3N, N, N, N ). In each case, this is the only rank assignment compatible with the non-abelian anomaly-free condition. (Abelian anomalies are not cancelled; they are expected to be cancelled by the contribution of bulk modes in string theory.) Theories (II) and (III) are examples of "non-toric" quivers.

Triality from mutation-a conjecture
We expect that the triality relations of N = (0, 2) quiver gauge theories are realized in string theory in the same way that all known Seiberg-like dualities are realized: by a change of "brane basis". This intuition was realized in the type IIB mirror picture in [21], where triality was related to certain permutations of Lagrangian 4-cycles. We would like to understand the analogous notion in the B-model.
Fractional branes from strongly exceptional collections. Following previous work [7][8][9]15], we consider the local Fano setup. We focus on B 3 = P 3 , although we expect that most of the following is valid more generally. 19 Let us denote by E k the sheaves on B 3 . A sheaf E is called exceptional if Ext i B 3 (E, E) = δ i,0 C. A strongly exceptional collection: on B 3 is a collection of exceptional objects such that 20 In particular, each sheaf in E is exceptional. To describe fractional branes, we also need our collection to be "maximal" in some appropriate sense. Let b n = dim H n (B 3 , R) denote the Betti numbers of B 3 . We call the strongly exceptional collection E complete if it contains n = 2 + b 2 + b 4 sheaves-physically, this corresponds to the most general D-brane wrapping the 0-, 2-, 4-and 6-cycles [8]. We have n = 4 on P 3 . 19 Including more general local geometries, such as the local P 1 × P 1 of section 2.5. 20 An exceptional collection E is such that Ext i B3 (E k , E l ) = 0 for k > l, ∀i. In this section, we consider the stronger condition of strong exceptionality, following [8].
Given a complete strongly exceptional collection (3.25) on B 3 , we propose that there exists a good set of fractional branes on X 4 = Tot(K → B 3 ) given by: From the strong exceptionality condition on E, it follows that the sheaves E I ≡ E n−I [I] are ordered such that Ext 1 ( E I , E J ) is non-vanishing only if I = J + 1. Thus, we have: where the arrows denote the Ext 1 B 3 ( E I , E J ) elements. The pushforward to X 4 will "close the quiver," by adding additional Ext groups due to the contribution of the embedding.
As an example, consider the following strongly exceptional collections on P 3 : , Ω 2 (2) , Ω(1) , O}. The intermediate quiver (3.28) reads: with the dimension of the Ext 1 groups indicated over the arrows. The corresponding fractional branes were discussed in section 2.4.1.
Triality and mutations. A natural geometric operation on these fractional branes is provided by mutations of exceptional collections [67]. Consider the strongly exceptional collection (3.25). A mutation at position k, with k < n, is a braiding operation on the exceptional collection: (E 1 , · · · , E k , E k+1 , · · · E n ) (E 1 , · · · , L E k E k+1 , E k , · · · · · · E n ) . Here, the new sheaf L E k E k+1 at position k is given by a left mutation. Note that a left mutation of an exceptional pair of sheaves (E, F ) produces another exceptional pair (L E F, E). The precise definition of L E F can be found in [67]. For our purposes here, we just note the properties L E[1] F = L E F and L E (F [1]) = (L E F ) [1] under the translation functor. The effect of (3.30) on the fractional branes may also be called a mutation at node e I , with I = n − k > 0. Given the ordered fractional branes (3.27), a mutation at e I corresponds to: Here we defined the new fractional brane: 32) by abuse of notation. We conjecture that mutations of a strongly exceptional collection which preserve the strongly exceptional condition realize the field-theory triality operation of section 3.1. 21 This proposal passes some obvious sanity checks. First of all, note that the pair (E I−1 , E I ) involved in the mutation has: According to our general rules, n a is the number of incoming arrows at node e I -in the language of section 3.1, the number of antifundamental chiral multiplets Φ under U (N 0 ) is n a N A , and we have: The condition that the new collection is strongly exceptional leads to: in the new quiver. 22 This means that we now have outgoing arrow from e I to e I−1 in the supersymmetric quiver: e I n A −→ e I−1 . This matches the fact that the antifundamental multiplets are dualized to fundamental multiplets under triality ( Φ Φ ). We also see that: These relations imply that the fundamental chiral multiplets of the original theory are dualized to fermi multiplets (Φ Λ ), and the fermi multiplets are dualized to antifundamental chiral multiplets (Λ Φ ). In this way, we elegantly reproduce the simplest aspects of the triality, as summarized in Figures 6(a) and 6(b).
Examples: Consider B 3 = P 3 , as discussed above. For definiteness, we start from the strongly exceptional collection: (3.37) The corresponding fractional branes were discussed in section 2.4.3-they were dubbed  38) and the full Ext 1,2 quiver on the local CY 4 reads: This gives the supersymmetric quiver that we called "Theory (III)" above. Now, consider a mutation at E 0 , which is a mutation at the third position in (3.37), at E 3 = O. It is a well-known result that: on P 3 . Therefore, the new strongly exceptional collection is given: The corresponding Ext 1,2 quiver on the CY fourfold reads: A triality operation at node e 0 of Theory (I) gives theory (II). Unfortunately, we cannot directly realize it by mutation, because E 0 corresponds to the last sheaf in the exceptional collection (3.40). However, remark that Theory (I) has a Z 4 symmetry that rotates the four nodes of the quiver. Therefore, a triality at any node of Theory (I) gives Theory (II), up to a rotation of the nodes. We can then consider any other mutation of adjacent sheaves in (3.40) to obtain Theory (II). Consider a mutation at position 1. One can show that: We thus obtain the new strongly exceptional collection on P 3 : The corresponding fractional branes are: One can again compute the Ext 1,2 quiver. It reads: , we precisely reproduce the "Theory (II)" quiver shown in Figure 7(b).

D-instanton quivers and gauged matrix models
Zero-dimensional "gauge theories"-gauged matrix models (GMM)-naturally arise as the low-energy description of D-instantons in type IIB string theory [68]. In particular, gauged matrix models with N = 1 supersymmetry can describe D-instantons at Calabi-Yau fivefold singularities [50].
Since 0d N = 1 superspace is spanned by a single Grassmanian coordinate θ, any superfield is of the form X = x + θy, with x, y some variables of opposite Grassmann parity. 23 The single supersymmetry is generated by Q = ∂ θ . A generic N = 1 GMM can be conveniently described using three elementary supermultiplets. The N = 1 chiral multiplet (Φ,Φ) consists of a complex boson φ,φ and a fermionψ. In superspace, we have: The chiral multiplet Φ has a single component, with Qφ = 0, while the anti-chiral multipletΦ has two components, with Qφ =ψ and Qψ = 0. The bosons φ and φ should be considered as complex conjugate in the matrix integral, while there is a single fermionψ. The second type of multiplet is the fermi multiplet Λ, with a single fermionic component λ, such that: Here, the N = 1 superpotential F λ is an holomorphic function of the bosons φ in chiral multiplets. Given the chiral multiplets φ i and fermi multiplets λ a , one can write the supersymmetric action: Another quadratic action in the fermions can be written in terms of an holomorphic potential H ab (φ) = −H ba (φ): This is supersymmetric provided that H ab F b = 0. The third type of sypersymmetry multiplet is the gaugino multiplet, which implements a gauge constraint on field space. The gaugino multplet V consists of two components, the fermion χ and the real boson D, with: 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: with χ acting on φ in the appropriate representation, and an overall trace over the gauge group is implicit. Here ξ is a 0d Fayet-Iliopoulos parameter. Integrating out D, we obtain: where µ ≡φφ − ξ (schematically), which is the moment map (minus the "level" ξ) of the G action on the bosonic field space.
4.1 N = 1 gauged matrix model from B-branes at a CY 5 singularity D-instantons at CY 5 singularities engineer precisely such gauged matrix models with gauge group I U (N I ). For each node e I in the 0d N = 1 quiver, we have a U (N I ) gaugino multiplet. The matter fields are either chiral or fermi multiplets, in adjoint or bifundamental representations. We have thus a quiver with two type of oriented arrows: e I → e J for chiral multiplets X IJ , and e I e J for fermi multiplets Λ IJ . Finally, we also have the F -and H-type interaction terms. To each fermi multiplet Λ IJ , we associate the element F IJ , a direct sum over oriented paths p from e I to e J , of length k: similarly to (2.10), with given coefficients c IJ p . In addition, to every pair of fermi multiplets Λ IJ and Λ KL , we associate the H-term action S H IJ,KL , which is a sum over closed loops p from e I back to itself, which includes both Λ IJ and Λ KL , in addition to chiral multiplets X: (4.9) Note that the closed path p has length k + k + 2, including the two fermions. This quiver structure naturally arises from open strings between fractional D(−1)branes at a CY 5 singularity, where each node e I corresponds to a fractional brane E I . As before, we must have: Ext 0 The non-vanishing Ext 0 elements are identified with the gaugino multiplets. The degree-one Ext groups are identified with chiral multiplets: 11) in bifundamental (if I = J) or adjoint (if I = J) representations. Similarly, the degreetwo Ext groups are identified with the fermi multiplets: By Serre duality, we also have Ext 4 Interaction terms. The F -terms (4.3) and H-terms (4.4) also arise naturally in the B-model. As discussed in section 2.1.2, the Ext-group generators satisfy an A ∞ algebra with multi-products m k . Consider a fermi multiplet Λ IJ corresponding to α ∈ Ext 2 (E J , E I ), and let us denote byᾱ ∈ Ext 3 (E I , E J ) the Serre dual generator. For each path p as in (4.8), we have the elements x ∈ Ext 1 corresponding to the chiral multiplets X. We propose that: for the F -term coefficients in (4.8). Similarly, consider the fermi multiplets Λ IJ and Λ KL corresponding to α ∈ Ext 2 (E J , E I ) and β ∈ Ext 2 (E L , E K ), respectively. We propose that the H-term coefficients in (4.9) are given by: with k = k + k + 1. We will check this prescription in some examples below. Note that this corresponds exactly to computing the formal 0d N = 1 superpotential: 15) which was recently introduced in [22].
We can work out the very simplest case, a D(−1) brane on X 5 = C 5 , exactly like in section 2.2. Consider the skyscraper sheaf O p at the origin of C 5 . We have: (4.16) Using the above dictionary to N = 1 superfields, this reproduces the expected field content of the maximally-supersymmetric N = 16 matrix model, as we will review below.

The Ext algebra of C 5
Proceeding as before, the Koszul resolution of O p on C 5 reads: where: Similarly to section 2.2, we choose as bases of the Ext groups the commutative diagrams whose leftmost nonzero vertical map has 1 at an entry and 0 elsewhere. We denote them by X i j , following the same conventions. The multiplication rule is again determined by composition. The products m 2 (X 1 i , X 1 j ) = X 1 i · X 1 j are given by: and the following X 2 X 2 X 1 -type correlators: 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 Φ n and Λ mn = −Λ nm , with n = 1, · · · 5, since Φ n and Λ nm transform in the 5 and 10 of an SU (5) flavor symmetry. This spectrum is reproduced by the Ext groups above. We identify the fields with the Ext elements according to X 1 n = φ n , n = 1, · · · , 5, and: , X 2 5 = λ 42 , X 2 6 = λ 43 , X 2 7 = λ 51 , X 2 8 = λ 52 , X 2 9 = λ 53 , X 2 10 = λ 54 . (4.23) The interaction terms are determined by the F -and H-terms [50]: is supersymmetric, we need to use the Jacobi identity for U (N ). This is equivalent to the non-trivial condition H ab F b = 0 mentioned above, which must always be realized by the B-brane correlators.
The higher Ext groups are obtained by Serre duality. The correlation functions can be read off from (4.21)-(4.22). Let us introduce the chiral multiplets: with I an integer mod 5, m, n = 1, · · · , 5, and Λ mn I = −Λ nm I . The gauged matrix model quiver is shown in Figure 10(a). The interaction terms are: Note the obvious SU (5) flavor symmetry. This quiver was discussed in [50,69,70].

C 5 /Z 3 (11112)
As a last example, consider the C 5 /Z 3 orbifold: (4.29) We have three fractional branes E i , i = 0, 1, 2. The weights for the sheaves in the Koszul resolution of E i are: The spectrum consists of: otherwise, (4.30) The corresponding 0d N = 1 quiver is shown in Figure 10(b). The correlation functions can be read off from (4.21)-(4.22). Taking advantage of the residual SU (4) flavor symmetry, let us introduce the chiral multiplets: where the Ext 1 elements λ mn = −λ nm are defined as in (4.23). In this notation, the interaction terms read: (4.33) 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 C 5 /Z 5 (1, 1, 1, 1, 1) to Tot(O(−5) → P 4 ). We leave this and many other related questions for future work.
Let us build another Calabi-Yau orbifold Y = C×X, which again has an isomorphic set of fractional branes {E i }, supported at x ≡ {0} × p ∈ C × X, of codimension four. Let N denote the normal bundle to x in Y , and ρ 0 the structure sheaf with trivial G-equivariant structure. Then, We then have: This directly confirms (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 [X c /G] of complex dimension 2, with a set of fractional branes {E i } supported at a point p ∈ X c , a fixed point of the G-action. Let N X denote the normal bundle N p/X . Let us build another Calabi-Yau orbifold Y = C 2 × X, which again has an isomorphic set of fractional branes {E i }, supported at x ≡ {(0, 0)} × p ∈ C 2 × X, of codimension four. Let N denote the normal bundle to x in Y , and ρ 0 the structure sheaf with trivial G-equivariant structure. Then, One can then compute: γ(m 2 (a i , m 2 (b j , c k ))) = ijk .

(B.4)
Note that there is a GL(3) symmetry inherited from P 2 , and a corresponding SU (3) flavor symmetry in the N = 1 gauge theory. The N = 1 quiver gauge theory is the one shown in Figure 11(a), with a gauge group U (N )×U (N )×U (N ). The bifundamental chiral multiplets A i , B i , C i correspond to the Ext 1 elements a i , b i , c i , and the product structure (B.4) leads to the N = 1 superpotential: 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 P 2 : The corresponding fractional branes are: We repeat the same analysis as before. The Ext 1 quiver reads: 3 a x x q q q q q q q q q q 6 d f f w w w w w w w w w w The corresponding N = 1 quiver is shown in Figure 11 The field theory is shown in Figure 11(b). The fields A i , B i are both in the 3 of the SU (3) flavor symmetry, while the fields M ij = M ji span the 6 of SU (3). They are identified with the Ext 1 elements according to: One can then derive the superpotential: Moreover, due to the non-abelian anomaly-cancellation condition, the gauge group must be U (2N ) × U (N ) × U (N ). This is also what is obtained from the usual rules of Seiberg duality.

B.2 Seiberg duality as mutation
The two N = 1 quiver theories of Figure 11 are related by a Seiberg duality on node e 0 . Consider for instance the "Theory (I)". A Seiberg duality at node e 0 reverses the arrows A i and B j while generating the new mesons M ij , with the identification M ij = A i B j across the duality. The superpotential dual to (B.5) reads: This contains a mass term for C i and the antisymmetric part of M ij . Integrating those fields 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 e 0 , we flip the arrows A i , B j , and generate the dual mesons N ij = A i B j , with the superpotential: Integrating out the massive fields-M ij and the symmetric part of N ij -we recover Theory (I) and (B.5), with the identification N ij = − ijk C k .
Mutation of exceptional collection. It was proposed in [60] that Seiberg duality could be realized as mutation on exceptional collections of sheaves. Start with Theory (II) and the corresponding exceptional collection E II (B.6). Using the left mutation: on P 2 , we see that a left mutation of the collection E II at the second sheaf precisely gives the collection E I in (B. Therefore, the Seiberg duality at node e 0 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. A ∞ structure and N = (0, 2) quiver In this Appendix, we discuss the A ∞ 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 [52,70,72].

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: with A, B ∈ T (V ), and |A| 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: where the sum is over all r, t ≥ 0, s > 0, such that r + s + t = n [52].
C.2 Ext algebra and N = (0, 2) quiver In our physical setup, the vector space A is spanned by the various Ext i groups (i = 0, · · · , 4) among the fractional branes on a CY 4 singularity. Schematically: The grading of A is given by the degree i of Ext i . Any z ∈ A of degree q is associated to a local vortex operator in the B-model, with the degree identified with the ghost number. Given z ∈ A, let z (1) denote the corresponding one-form descendant. The one-form operators can be used to deform the B-model according to [10,72]: The coupling Z i is identified with a "quiver field" in the space-time (D1-brane) theory. Note that Z i has degree 1 − q i if z i has degree q i . The quiver fields are elements of the vector space V , in the notation of subsection C.1. Let us denote byz ∈ A the Serre dual of z ∈ A, with the Ext algebra A given by (C.8). Let us then choose a basis of A according to: with: e 0 ∈ Ext 0 , x α ∈ Ext 1 , α I ,ᾱ I ∈ Ext 2 ,x α ∈ Ext 3 ,ē 0 ∈ Ext 4 . (C.11) As discussed in the main text, the choice of basis for Ext 2 is arbitrary. Any given choice introduces a distinction between the elements α and the Serre dual elementsᾱ, which is a matter of convention. The dual vector space V spans the "quiver fields". We choose a basis of V : {Z i } = {e , X α , Λ I ,Λ I ,X α ,ē} , (C. 12) dual to (C.10). The element e correspond to the vector multiplets, while X α and Λ I correspond to the chiral and fermi multiplets, respectively. Note the degrees: e X α Λ IΛ IX αē degree: 1 0 −1 −1 −2 −3 In particular, the chiral multiplets have degree 0. Given this explicit basis of V , we define a derivative d as follows. First, let us introduce the N = (0, 2) "superpotential": W = Tr Λ I ⊗ J I (X) +Λ I ⊗ E I (X) , (C. 13) with J I (X) and E I (X) some arbitrary functions of the chiral multiplets X α . This W is an arbitrary gauge-invariant function of degree −1 that is independent of e, except that we need to impose the constraint: Tr(E I ⊗ J I ) = 0 . (C.14) Let us also define the derivatives: by left derivation on W-that is, we use the cyclic property of the trace to write (C.13) with X α on the left, and the derivative with respect to X α is defined as the sum of all possible forms of W with X α in front, with X α removed. Given the superpotential, we define the degree-one derivative d on V as: 24 are obvious. 25 The key relation is: which holds true if and only if the non-trivial constraint (C.14) is satisfied. 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 fields, it follows from the general discussion above that the multi-products m k acting on the Ext vector space A satisfy the A ∞ relations (C.7). In this way, we see clearly that the A ∞ relations on a CY 4 are intimately related to the supersymmetry constraint (C.14). We should also note that the differential d defined in (C.16) has: where d k is defined as in (C.4), if and only if the potentials E I and J I do not contain any linear terms in X α . In such a case, we have m 1 = 0 in the dual Ext algebra, which gives us a minimal A ∞ structure. Linear terms in E I or J I are mass terms, and the corresponding fields 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 m k for k = 2, · · · , k max , with k max the highest order in the fields X α that appear in the potentials E I , J I . In the simplest case when E I , J I are all quadratic in the chiral multiplets, we have m k = 0 for k ≥ 3, and the A ∞ algebra reduces to an associative algebra with a product given by m 2 .

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 • A ∞ algebra [10], which we outlined in section 2.1.2. Consider an A ∞ algebra A and the A ∞ map: be the inclusion map defined by picking representatives of cohomology classes, and let d = m 1 : A → A denote the differential on A. The first A ∞ constraint on the maps f k reads: i • m 2 (α, β) = i(α) • i(β) + df 2 (α, β) .
Using the previously-computed m 2 and f 2 , this expression allows us to compute m 3 and f 3 . Proceeding inductively in this fashion, one can construct m k and f k to any order k.