Quadrality for supersymmetric matrix models

Journal of High Energy Physics, Jul 2017

We introduce a new duality for \( \mathcal{N} \) = 1 supersymmetric gauged matrix models. This 0d duality is an order 4 symmetry, namely an equivalence between four different theories, hence we call it Quadrality. Our proposal is motivated by mirror symmetry, but is not restricted to theories with a D-brane realization and holds for general \( \mathcal{N} \) = 1 matrix models. We present various checks of the proposal, including the matching of: global symmetries, anomalies, deformations and the chiral ring. We also consider quivers and the corresponding quadrality networks. Finally, we initiate the study of matrix models that arise on the worldvolume of D(-1)-branes probing toric Calabi-Yau 5-folds.

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Quadrality for supersymmetric matrix models

Received: March Quadrality for supersymmetric matrix models Sebastian Franco 1 2 4 6 8 9 10 Sangmin Lee 1 2 3 4 5 7 9 10 Rak-Kyeong Seong 0 1 2 4 9 10 Cumrun Vafag 1 2 4 9 10 0 Department of Physics and Astronomy, Uppsala University 1 Seoul 08826 , Korea 2 365 Fifth Avenue , New York NY 10016 , U.S.A 3 Department of Physics and Astronomy, Seoul National University 4 160 Convent Avenue , New York, NY 10031 , U.S.A 5 College of Liberal Studies, Seoul National University 6 The Graduate School and University Center, The City University of New York , USA 7 Center for Theoretical Physics, Seoul National University 8 Physics Department, The City College of the CUNY 9 Cambridge , MA 02138 , U.S.A 10 SE-751 08 Uppsala , Sweden We introduce a new duality for N = 1 supersymmetric gauged matrix models. This 0d duality is an order 4 symmetry, namely an equivalence between four di erent theories, hence we call it Quadrality. Our proposal is motivated by mirror symmetry, but is not restricted to theories with a D-brane realization and holds for general N = 1 matrix models. We present various checks of the proposal, including the matching of: global symmetries, anomalies, deformations and the chiral ring. We also consider quivers and the corresponding quadrality networks. Finally, we initiate the study of matrix models that arise on the worldvolume of D(-1)-branes probing toric Calabi-Yau 5-folds. Brane Dynamics in Gauge Theories; D-branes; Supersymmetric Gauge The- 1 Introduction 2 Supersymmetric gauged matrix models Mirror symmetry and D-branes at toric singularities Quadrality from mirror symmetry Local geometry for order n dualities Conclusions B Review: C3=Z3 A The Hilbert series for C3=Z3, C4=Z4 and C5=Z5 { 1 { Quadrality Checks The quadrality dual The meaning of quadrality in 0d Abelian avor anomalies Periodicity Deformations Chiral ring Quadrality networks D-brane theories C 5 Local CP4 Local (CP1)4 The chiral ring and the probed CY5 Towards quadrality from compacti cation 4 5 6 7 8 9 2.1 2.2 2.3 2.4 3.1 3.2 3.3 3.4 4.1 4.2 5.1 5.2 5.3 5.4 7.1 7.2 7.3 7.4 Introduction Duality, the equivalence between seemingly di erent theories, is one of the most fascinating phenomena in quantum eld theory. Some of the best-understood examples of dualities involve supersymmetry, since it provides an enhanced control of the theories. Seiberg duality, an equivalence between two 4d N = 1 gauge theories in the IR limit, is the prototypical example of a supersymmetric duality [1]. More recently, Gadde, Gukov and Putrov (GGP) discovered that 2d N = (0; 2) gauge theories exhibit triality [2]. This is an IR equivalence, analogous to Seiberg duality, but of order 3. This means that triality relates three di erent theories and that the original theory is recovered after three consecutive transformations. The existence of triality makes it clear that the space of dualities is far richer than naively suspected and that, in general, it involves order n dualities. The realization of quantum eld theories in terms of branes is an extremely fruitful approach for understanding and uncovering dualities. In a recent paper [3], building on [4], we showed that mirror symmetry provides a geometric uni cation of dualities in di erent dimensions. Mirror symmetry naturally explains why (10 2n)-dimensional quantum theories, which are associated with CY n-folds, exhibit duality symmetries of order n eld 1. Our work not only explained 4d Seiberg duality and 2d GGP triality in a uni ed framework, but also led us to conjecture an order 4 duality for N = 1 gauged matrix models in 0d, which we called quadrality.1 The main goal of this paper is to introduce quadrality and to perform various checks of the proposal. Even though we use mirror symmetry to motivate quadrality, our claim is much stronger: we postulate it is a property of general N = 1 gauged matrix models. Quadrality applies to matrix models with potentials, which are ubiquitous in physics. N = 1 supersymmetry can be regarded as an appropriate dressing by fermions that allows for additional control of the theory. The organization of this paper is as follows. Section section 2 introduces the basics of N = 1 gauged matrix models. Section section 3 motivates quadrality by considering the 0d N = 1 theories on D(-1)-branes probing toric Calabi-Yau (CY) 5-folds and mirror symmetry. Section section 4 introduces quadrality for general N = 1 gauged matrix models and elaborates on its physical meaning. Various checks of the proposal are presented in section section 5. Section section 6 discusses how theories connected by sequences of quadrality transformations can be organized into quadrality networks. Section section 7 considers theories on the worldvolume of D(-1)-branes probing toric CY 5-folds. We also show, in the local CP4 example, how the chiral ring of the gauge theory reproduces the coordinate ring of the probed geometry and remains invariant under quadrality. Section section 8 discusses the possible derivation of quadrality via compacti cation. We present our conclusions in section section 9. In two appendices we review the application of Hilbert series to the chiral rings of gauge theories on D-branes. 2 In this paper we study N = 1 supersymmetric gauged matrix models. This section presents the basic properties of these theories. Despite the fact that the theories carry only one 1We can regard matrix models as quantum eld theories in 0d. { 2 { HJEP07(21)53 supercharge Q, some useful concepts in supersymmetric theories such as holomorphy and R-symmetry are still valid. The main reason is that in matrix models, unlike in quantum mechanics or QFT, we do not have to consider Hermitian conjugates of operators. In the matrix integral, all Fermi elds are regarded as independent holomorphic variables without Hermitian conjugates. The basic multiplets of the supersymmetric matrix models are as follows: 1. Gaugino multiplet V : 2. Chiral multiplet Xi: 3. Fermi multiplet a: [Q; i] = i ; fQ; ig = 0 ; [Q; i] = 0 : fQ; ag = Ga ; [Q; Ga] = 0 ; fQ; a g = G ; [Q; Ga] = 0 : a Each line presents the components of a supermultiplet and how they transform under the action of the supercharge. We have adopted a notation that resembles that of 2d (0; 2) gauge theories.2 The superalgebra Q2 = 0 holds trivially in all multiplets. As far as the superalgebra is concerned ( ; ) are independent of and ( ; G) are independent of ( ; G). However, in view of the structure of interaction terms, it is convenient to regard ( ; ; ) as a single multiplet and ( ; G; ; G) as another one. We now describe the general structure of N = 1 gauged matrix models, focusing on the abelian case. The non-abelian extension is straightforward. At least locally in eld space, both the D-term and J -term contributions are Q-exact, namely fQ; g = D ; [Q; D ] = 0 : Expanding in components and integrating out auxiliary elds, we obtain SD = ' X X 1 2 2 1 D2 + D X q i i i i X q i i i 2 i t t i X q i i i ; X q i i i i 2This resemblance may cause some confusion. See section 2.2 for precise relations. { 3 { We have xed the overall sign of SD and SJ such that their bosonic terms are positive de nite when and satisfy the reality condition ( i) = i. This condition is necessary in order for the matrix integral R D( ; ; ; )e S( ; ; ; ) to converge. In contrast, as mentioned earlier, we do not impose any reality condition for fermions. HJEP07(21)53 2.1 Novel interaction terms N = 1 matrix models allow for yet another type of interaction, which we call H-terms: F X 1 2 a;b=1 SH = Hab( ) a b ; fQ; SH g = X Hab( )Ja( ) b : a;b F X Hab b=1 Jb = 0 for each a : where Hab = Hba are antiholomorphic functions that depend exclusively on the 's. Unlike SD and SJ , SH is not locally exact. Using the fact that fQ; ag = Ga and that Ga = Ja( ) on-shell, we obtain Since every a is independent, SH is supersymmetric if and only if SJ = ' a X a Ga(Ja( ) Ga) + GaJ a( ) + X Ja( )J a( ) + X i a (2.6) (2.7) (2.8) (2.9) (2.10) (2.11) J -term mass: H-term mass: J = m X ; H 1; 2 = m : X = = + + ; G : X = ; Superspace notation and products. With 0d N = 1 SUSY, it is possible to introduce a Grassmann coordinate and package the di erent components into super elds. For the matter supermultiplets we have We refer to this condition as the H-constraint. Mass terms. The interactions that we just described give rise to two types of mass terms as summarized in gure 1: a chiral-Fermi mass via a J -term and a Fermi-Fermi mass via an H-term. They take the following form The basic distinction between them resides in the commutation properties of their lowest components. Let us now consider products of super elds. { 4 { Let us now discuss how 2d (0; 2) gauge theories are dimensionally reduced down to 0d N = 2 matrix models, which we express in 0d N = 1 language. This class of theories provides a concrete illustration of the structures that we introduced in the previous section. Upon dimensional reduction (followed by a Wick rotation), every multiplet of the 2d (0; 2) gauge theory splits into two di erent 0d N = 1 multiplets as follows: { 5 { 1. 2d gauge ! 0d gaugino + 0d chiral. 2. 2d chiral ! 0d chiral + 0d Fermi. 3. 2d Fermi ! a pair of 0d Fermi's. Let us denote the 2d gauge, chiral and Fermi multiplets The 0d eld content is A ( = 1; : : : ; G2d) ; I (I = 1; : : : ; C2d) ; A ; A (A; A = 1; : : : ; F2d) : fXig = fXI ; Y g ; f ag = f A; A ; I g : Despite using a similar notation for both 2d and 0d super elds, the distinction and correspondence between them should be clear from their subindices. From (2.17), it follows that the number of 0d multiplets of each type is given by G0d = G2d ; C0d = C2d + G2d ; F0d = 2F2d + C2d : In the special case of 2d theories on D1-branes probing toric CY 4-folds, the relation C2d G2d F2d = 0 holds [5]. The 0d matrix models obtained by dimensionally reducing such theories then satisfy F0d 3C0d + 5G0d = 0 : Returning to general 2d (0; 2) theories, we can derive the interaction terms of the 0d theory from those of the 2d parent. The J -terms are 0d a J a J A A A = A E A = EA 2d I=( ) Y X X Y In the last column, we used the standard quiver notation for bifundamental and adjoint elds.3 Below we will use this notation whenever it is helpful. We also adopted a convention in which a barred superscript is the same as an unbarred subscript and vice versa. The H-terms of the 0d theory are Y ) A=( ) : (2.21) The H-constraint (2.9) for I is automatically satis ed due to the trace condition of the 2d theory X(JAH AI + JAH AI ) = A X A J A X ! = 0 : (2.22) 3Of course there can be more than one eld for every pair of subindices. We leave this possibility implicit in order to simplify the notation. (2.16) (2.17) (2.18) (2.19) (2.20) HJEP07(21)53 The H-constraints for ( A; A) schematically read as follows X HAI JI + X H ABJB I B ! A=( ) 2E ( Y X X Y ) ( Y gure 2 shows how the two partial sums cancel against each other. 2.3 Gauge/global symmetries and anomalies A 0d theory contains no derivatives. So, we cannot gauge a symmetry by turning an ordinary derivative into a covariant derivative. It may seem unclear whether/how we can distinguish a gauge symmetry from a global symmetry. Guided by dimensional reduction from higher dimensions, we distinguish gauge symmetries from global symmetries in three ways. First, we assign a gaugino multiplet for each factor in the gauge symmetry. There is no multiplet associated to the global symmetry. Second, we require that all observables (more on this below) be gauge invariant, but may be charged under global symmetry. Third, we require that the \anomaly" for the gauge symmetry vanishes, but allow for non-vanishing global anomaly. By anomaly we mean any non-invariance of the integration measure under the symmetry action. In the presence of an anomaly, gauge or global, the integration over the symmetry orbit forces the partition function (i.e. the expectation value of the identity operator) to vanish. Since all observables are gauge invariant, a gauge anomaly implies that the theory is completely trivial. In contrast, a global anomaly can be cancelled by computing correlation functions of charged observables. In 0d, anomalies are linear. We will use the matching of abelian in the spirit of 't Hooft anomaly matching in higher dimensions, when we argue for the equivalence of several theories. The integration measure should carry a de nite charge, i.e. anomaly, under global symmetries. If two matrix models are dual, as a necessary condition, their global anomaly should be the same. The superalgebra Q2 = 0 suggests that the observables of the theory are cohomology classes of Q. We begin with the ring of all gauge invariant products of elementary elds and study the cohomology of Q. The resulting observables, (2.24) still form a ring. By a slight abuse of language, we may call it the \chiral ring". Suppose we enumerate all possible Oi's. In principle, solving the theory completely means calculating arbitrary correlation functions among them, Z hOi1 Oi2 Oin i = D( elds)Oi1 Oi2 Oin e S : Here, the classical action, the integral measure and the observables are all independently gauge invariant. As for the global symmetries, we leave the possibility that the integral measure has some abelian anomaly, which can be cancelled by the global charges of the observables. In summary, if we want to establish duality between two theories, we should check the following things: (1) global anomaly, (2) spectrum of observables (including their global charge and R-charge), (3) correlation functions. 3 Geometric motivation for quadrality In this section we use the geometric engineering of 0d N = 1 theories in terms of D(-1)branes probing toric CY 5-folds to motivate a new duality. This duality turns out to be of order 4, so we will refer to it as quadrality. 3.1 D-branes probing Calabi-Yau manifolds The (10 2n)-dimensional gauge theories that live on the worldvolume of D(9 2n)branes probing toric CY n-folds have been studied in great detail for n = 3 and 4. For toric CYs, the connection between gauge theory and geometry is considerably simpli ed by T-dual brane con gurations: brane tilings for CY3 [6, 7] and brane brick models for CY4 [5, 8, 9]. It is natural to continue with this sequence as summarized in table 1 and consider n = 5, which corresponds to D(-1)-branes probing toric CY 5-folds. The theories on their worldvolume are 0d N = 1 gauged matrix models of the type discussed in section section 2. T-duality relates D(-1)-branes probing toric CY 5-folds to Type IIA con gurations analogous to brane tilings and brane brick models, to which we refer as brane hyperbrick models. A brane hyperbrick model consists of an NS5-brane wrapped over a holomorphic { 8 { 4d N = 1 whole con guration lives on a T 4. Very much like their lower dimensional cousins, we expect brane hyperbrick models to provide valuable tools for connecting geometry to gauge theory. However, they are beyond the scope of this paper and we leave their development for future work. 3.2 Mirror symmetry and D-branes at toric singularities In order to set up the stage for the discussion in the coming section, we present a lightning review of the mirror approach to D-branes probing toric CY singularities. We refer the reader to [3, 10, 11] for details. in Z complex W plane A toric CYn M is speci ed by its toric diagram V , which is a convex set of points n 1. Its mirror geometry [12, 13] is an n-fold W given by a double bration over the with u; v 2 C and x 2 C , = 1; : : : ; n 1. P (x1; : : : ; xn 1) is the Newton polynomial (3.1) (3.2) (3.3) W = P (x1; : : : ; xn 1) W = uv P (x1; : : : ; xn 1) = X c~v xv11 : : : xvnn 11 ; ~v2V { 9 { where the c~v are complex coe cients and we sum over points ~v in the toric diagram. By rescaling the x variables, it is possible to set n of the coe cients to 1. The critical points of P are (x1; : : : ; xn 1) such that P (x1; : : : ; xn 1) (x1;:::;xn 1) = 0 = P (x1; : : : ; xn 1). The number of critical points is equal to the number of gauge nodes in the eld theory [10]. The two bers are a holomorphic (n 2) complex dimensional surface W coming from P (x1; : : : ; xn 1) and a C bration associated with the uv piece. The resulting Sn 2 S1 bration over a straight vanishing path that stretches between W = 0 and W = W hence gives rise to an Sn.4 The S1 ber vanishes at W = 0. The structure of the gauge theory is determined by how the surviving Sn 2's intersect on the vanishing locus W 1(0) : P (x ) = W = 0. The 4Vanishing paths can be curved. We refer the reader to [3] for a discussion of this possibility. This subtlety does not a ect our conclusions. associated to W0 . We indicate the type of avor contributed by each cycle: chiral (Cin and Cout) or Fermi (Fin and Fout). (b) Corresponding quiver diagram. We refer to this theory as T . geometry of the Sn 2's can be e ciently described using tomography, which was introduced in [11] and further developed in [3]. The x -tomography is the projection of the Sn 2 spheres at W = 0 onto the x -plane. In summary, we can obtain a detailed description of the con guration of Sn in the mirror geometry by combining the con guration of vanishing paths on the W -plane with the x -tomographies, = 1; : : : ; n. The discussion in the coming section will build on ideas introduced in [3]. It will just use basic properties of vanishing paths on the W -plane, which is a universal ingredient for any CY dimension. 3.3 Quadrality from mirror symmetry Following the discussion in the previous section, mirror symmetry maps D(-1)-branes probing a toric CY5 to a collection of Euclidean D4-branes wrapping 5-cycles. Let us consider some elementary features of this mirror con guration. Given a vanishing cycle C0, the branes wrapping the cycles that intersect with it can be regarded as avor branes. These avor cycles can be classi ed into four groups, depending on the type of matter elds they contribute to C0: fundamental and antifundamental chirals and fundamental and antifundamental Fermis. Following the quiver representation of these elds, we refer to fundamental and antifundamental representations as in and out, respectively. Extending the ideas introduced in [3], we expect that for any C0 the vanishing paths that are associated to the four types of avors are organized cyclically on the W -plane as shown in gure 3. This conjecture is a natural generalization of what occurs for CY 3- and 4-folds and is also based on the expected symmetry between quadrality and inverse quadrality. An important lesson from [3, 4] is that mirror symmetry not only provides a geometric uni cation of dualities for a xed dimension, but also uni es seemingly di erent QFT equivalences across dimensions. gure 4.a and b show that the mirror realizations of 4d them in terms of the same type of transformation. Seiberg duality and 2d triality are basically identical. This observation naturally leads to conjecture that the transformation in gure 4.c gives rise to an equivalence between N = 1 gauged matrix models that we call quadrality. Let us denote Qi the D-brane charge vector for the cycle Ci. The intersection matrix hCi; Cj i = hQi; Qj i of a CY5 is antisymmetric. The sign of hQi; Qj i determines the orientation of the bifundamental(s) connecting nodes i and j. The distinction between chiral and Fermi elds requires additional information regarding the intersection. Without loss of generality, let us assume that the only non-vanishing intersections of cycles are hQ1; Q0i = 1 hQ3; Q0i = 1 hQ2; Q0i = hQ4; Q0i = 1 1 In our discussion, any multiplicities of di erent types of avors are absorbed into the ranks of the corresponding avor nodes. Introducing more general intersection numbers hQi; Q0i 6= 1 or splitting the avor nodes into collections of multiple nodes with di erent brane charges is straightforward. Non-vanishing intersections between the avor nodes in the initial theory, corresponding to additional matter elds, can also be incorporated without a ecting our analysis. Quadrality on node 0 corresponds, as shown in gure 4, to shrinking the cycle C0 to zero size and reemerging on the W -plane on the wedge between C1 and C2 with a reversed orientation. The brane charges transform as follows: Q00 = Q0 Q02 = Q2 Q03 = Q3 Q04 = Q4 Q01 = Q1 + hQ1; Q0iQ0 = Q1 + Q0 (3.4) (3.5) by =2, while keeping the avor nodes xed. Inverse quadrality is obtained by moving C0 in the opposite direction to the wedge between C3 or C4, or by applying quadrality three times. Remarkably, even without a more detailed analysis of the mirror, we can draw important conclusions about what this transformation implies for quadrality. Rank of the gauge group. The transformation of the rank of the gauge group follows from conservation of the total brane charge. Initially, we have Since the ranks of the avor nodes remain constant, we have QT = X Ni Qi : 4 i=0 Conservation of brane charge QT = Q0T implies Dual avors. The avor vanishing paths divide the W -plane into four wedges. Quadrality corresponds to moving the cycle associated to the dualized gauge group to a neighboring wedge. This implies that, in the quiver, the transformation of avors is a =2 rotation of the corresponding arrows, while keeping the avor nodes xed. This is shown in gure 5 This is yet another manifestation of the geometric uni cation of QFT equivalences in di erent dimensions The transformations of avors in Seiberg duality and triality are also given by 2 =k rotations, with k = 2; 3, respectively [3, 4]. Mesons. The modi cation of Q1 in (3.5) leads to non-vanishing intersections between node 1 and nodes 2, 3 and 4 in the dual. These new intersections give rise to mesons. Furthermore, hQ01; Q2i = hQ0; Q2i ! hQ01; Q3i = hQ0; Q3i ! hQ01; Q4i = hQ0; Q4i ! 12 (Fermi) 31 (Fermi) M14 (chiral) (3.6) (3.7) (3.8) (3.9) The chirality and type of eld for each meson is thus determined by those of the original avors connecting node 0 to the global symmetry nodes. We conclude the dual contains the following mesons: 12 (Fermi), 31 (Fermi) and M14 (chiral). This expectation will be con rmed by a eld theory analysis in section section 4.1. The quiver. Summarizing our previous analysis, we conclude that the quiver diagram for the quadrality dual of theory T , which we call T 0, is the one shown in gure 6. Throughout the paper, we will often adopt a notation in which X and indicate chiral and Fermi avors, from the perspective of the dualized gauge group, respectively. Similarly, M and correspond to chiral and Fermi mesons, i.e. gauge singlets. Periodicity. As we mentioned, it is possible to argue that quadrality is an order 4 duality based on the fact that the avor vanishing paths divide the W -plane into four wedges. Here we provide a more explicit proof of the periodicity. For all i; j = 0; : : : ; 4, we have hQ0i000; Q0j000i = hQi + hQi; Q0iQ0; Qj + hQj ; Q0iQ0i = hQi; Qj i + hQi; Q0ihQ0; Qj i + hQj ; Q0ihQi; Q0i + hQi; Q0ihQ0; Q0i (3.10) = hQi; Qj i where in the last line we used the antisymmetry of the intersection pairing. 3.4 Local geometry for order n dualities The minimal local geometry that captures the order (n 1) duality of a (10 2n)dimensional theory (i.e. 4d duality, 2d triality and 0d quadrality) is local CPn. This is the Cn=Zn orbifold with action (1; : : : ; 1) on the di erent complex planes. Local CPn gives rise to n critical points and hence to n independent n-cycles. This number is precisely what is required for studying a generic eld content in every dimension. One of the cycles accounts for the gauge group and the remaining n 1 corresponds to nodes (which are also gauged) for all possible types of avors: two in 4d (fundamental and antifundamental chirals), three in 2d (fundamental and antifundamental chirals, and Fermi) and four in 0d (fundamental and antifundamental chirals, and fundamental and antifundamental Fermis). volume. ! Wi we can make the corresponding n-cycle, here C0, shrink to zero The Newton polynomial coming for the toric diagram for local CPn contains, in general, n + 1 terms. It is possible to rescale n of its coe cients to 1, leading to P (x1; : : : ; xn) = x1 + : : : + xn + (3.11) 1 x1 : : : xn + ; with 2 C. The parameter does not a ect the critical points i . However, it modi es the corresponding critical values Wi by an overall shift. Equivalently, we can think that the critical values Wi are xed and that the origin of the W -plane is shifted by . Thus, this geometry has enough freedom for studying the limit in which any of the n gauge groups go to in nite coupling. We can attain this by sending CP5, which makes the volume of the corresponding cycle vanish. ! Wi as shown in gure 7 for 4 Quadrality In the previous section we motivated quadrality using mirror symmetry for con gurations of D(-1)-branes probing toric CY 5-folds. It is however natural to conjecture that, as it occurs for Seiberg duality and triality, that quadrality applies to arbitrary N = 1 gauged matrix models. The elementary quadrality transformation, which we now explain in more detail, can be phrased in terms of the simple SQCD-like theory T that we introduced earlier. For quick reference, gure 8 reproduces the quiver diagrams for T and its quadrality dual. Let us discuss T rst. The four avor nodes and the gauge node in the quiver correspond to U(Ni) = SU(Ni) U(1)(i) groups. All matter elds transform in bifundamental representations, so the global diagonal combination of all of them, Pi4=0 Qi decouples.5 Without loss of generality, we can identify the global symmetry with SU(N1) U(1)(3).6 It is straightforward to read the transformation properties of the matter elds under the global symmetry group from the quiver. Cancellation of the abelian gauge anomaly constrains the ranks of the avor nodes to satisfy discussed earlier. N1 N2 + N3 N4 = 0 : (4.1) 5From now on Qi refers to the charge under U(1)(i). It should not be confused with the D-brane charges 6U(1)(4) is not independent, and the corresponding charge is simply Q4 = (Q0 + Q1 + Q2 + Q3). If more nodes are gauged, as in the D-branes examples considered in section section 7, there will be additional matter elds that ensure the cancellation of all gauge anomalies. 4.1 The quadrality dual We propose the quadrality dual T 0 is given by the quiver in gure 8, together with some J - and H-terms that we discuss below. Since N1 + N3 N2 N4 = 0, the abelian gauge anomaly still vanishes. The global symmetry of the dual is SU(N1) U(1)(3), in agreement with the original theory.7 Let us explain the arguments that lead to this proposal. Dual gauge group. The dual theory has a U(N00 ) gauge symmetry, with N00 = N1 N0: (4.2) This result was derived in section section 3 in the case of theories with a D-brane realization from conservation of the total brane charge. We postulate it holds in general. Dual avors. For theories arising on D-branes, we used mirror symmetry to derive the transformation of avors summarized in gure 5. Once again, we propose this transformation applies to general theories. Mesons. There are three types of mesons in T 0. They can be expressed as composites of the elds in the original theory. In all cases they must contain X10 which, for D-branes, is the chiral eld charged under the avor node whose brane charge changes under quadrality. 7When determining the global symmetry of a theory, it is necessary to take into account its J- and H-terms, which can in principle break the naive symmetries preserved by the quiver. We discuss these interactions below. (4.3) (4.4) The mesons are given by M14 = X10 X04 The types of elds obtained by taking these products nicely coincide with the ones established using the brane intersections in (3.9). Notice that in order to form the gauge invariant meson 31, it is necessary to conjugate X10. This is a novel feature of 0d, which does not arise in 4d or 2d. Dual avors-meson couplings. As it occurs in Seiberg duality and triality, there are new interaction terms coupling the mesons to the dual avors. These terms are the most general ones allowed by the gauge and global symmetries.8 In this case, the couplings are J 12 = X20X01 J 40 = X01M14 H 03; 31 = X01 ! ! ! Quiver loop ( 12X20X01) ( 40X01M14) To simplify visualization of the interactions, on the right column we give the corresponding loops in the quiver. Below we will explain how they are crucial for consistency of quadrality. Interestingly, the appearance of a novel type of meson 31 = 30 X10 is correlated with the existence of a new type of interactions in 0d, the H-terms. 4.2 The meaning of quadrality in 0d In this section we would like to examine what we can mean by duality in the context of a 0d QFT. Typically when we have a Seiberg-like duality we start with two distinct theories in the UV, and only in the IR the two theories will become identical, as they ow to the same conformal xed point. In such situations, some aspects, such as the chiral ring, are invariants of the ow and can be studied on either side of the duality even before the ow. Checking this match has been one of the key evidences for Seiberg-like dualities. But duality extends beyond the chiral sector and is expected that at the IR xed point, correlations function of any collection of elds, whether chiral or not, match on both sides, with a suitable dictionary of how operators from one side map to the other. In the case at hand, a robust check of our proposed quadrality is to verify that the chiral rings match for each dual version. However this is not enough to claim equivalence of two theories, as there are non-chiral operators in the theory. So one would like to have the analog notion of \IR" in such theories, so that one could say that the IR of all sides agree for all operators. However, there is di culty de ning the notion of IR xed point in the present context, because the dimension of space-time is 0 and so we have no such notion. Instead we propose the following alternative. As a supersymmetric theory ows to the IR in d > 0 cases, chiral elds do not get renormalized (which we will loosely call \F-terms") 8This principle also holds for more complicated theories. We should always include all interactions allowed by the gauge and global symmetries. but non-chiral elds do get renormalized (the \D-terms"). So it is natural to de ne the notion of IR in d = 0 theories by saying that there is a deformation of the \D-terms" in the Lagrangian (i.e. Q-trivial additions to the Lagrangian) which leads the theory to have the expected superconformal symmetry. So we need to address what is the superconformal group in 0d. It is natural to expect that the IR superconformal eld theory (SCFT) has a U(1) R-symmetry. Then, the superconformal group in 0d should have a bosonic conformal symmetry given by SO(1; 1) U(1)R. Let us call the generators of this group and R. Moreover we expect, as in higher dimensions, that the number of supersymmetries gets doubled at the conformal point. Since we started with one nilpotent supercharge, let us call it Q+, we should obtain another one, Q . We expect the following superconformal algebra: Q+2 =Q 2 = 0 ; Note that this symmetry algebra is at the level of elds in the theory and not the Hilbert space, because this theory has no time dimensions. So we conjecture that there is a distinguished xed point where the above algebra is a symmetry of eld space and correlation functions.9 Whether such a point is unique is not clear and requires further study. In this paper we check the quadrality only by checking the chiral rings match on all sides (as is the case with duality checks in higher dimension). This low dimension case allows us to possibly be able to check a more detailed statement by including all operators! It would be interesting to pursue this direction and see if one can precisely x the Lagrangian at the conformal xed point of these theories and prove the quadrality symmetry for correlation functions of all operators. 5 In this section we collect additional checks of the quadrality proposal. avor anomalies provides a non-trivial check of the proposal. As mentioned earlier, the non-abelian avor anomalies trivially vanish. The following table summarizes the anomalies. 1 2 T N0 N0 N0 N0. However, as explained above, it is not really independent and can be determined in terms of the other abelian anomalies. 9It is possible (and probably likely) that the symmetry is only a symmetry of the correlation functions and not the Lagrangian itself, because the path-integral measure may not be invariant. Interestingly, the matching of the U(1)(1) and U(1)(2) anomalies between the two theories tests the existence of the non-conventional 31 meson. However, it does not establish whether this eld is a chiral or a Fermi. If we did not have independent derivations of the transformations of the gauge group rank based on conservation of brane charge, matching of abelian avor anomalies would simultaneously test the combination of the rank, avor and mesons rules. 5.2 Periodicity As explained in section section 3, mirror symmetry implies that after four consecutive quadrality transformations we should return to the original theory. This sequence is shown in gure 9. At various steps we have integrated out chiral-Fermi and Fermi-Fermi pairs, due to mass terms of the form shown in gure 1. For this to be possible, the detailed form of the J - and H-terms coupling mesons to dual avors is crucial. The periodicity of the sequence of quadralities is hence a non-trivial check of these couplings. The rank of the gauge group evolves as follows N000 = N2 N0000 = N3 N00000 = N4 N0 N1 + N0 N2 + N1 N3 + N2 N0 We see that the gauge anomaly constraint on the ranks of the avors (4.1) is crucial for returning to the original rank of the gauge group. 5.3 Following Seiberg's seminal work on 4d N = 1 duality [1], deformations have become a standard tool for testing a wide range of equivalences between eld theories. Here we will show that similar arguments can be applied to quadrality. When testing the e ect of a deformation in Seiberg duality, one only needs to consider the magnetic dual. Remarkably, for quadrality we can, and actually must, study the e ect of any deformation on the full quadrality sequence. This is a general feature of order n dualities. The e ect of deformations on the entire collection of dual theories provides an (n-1)-fold increase in the number of constraints and consistency checks. N = 1 matrix models have a rich set of possible deformations with which to test quadrality. In particular, we can introduce X and mass terms, which correspond to J - and H-terms respectively. The original sequence. The original quadrality sequence was discussed in section section 5.2. It consists of four theories T , T 0, T 00 and T 000, corresponding to the four quivers in gure 9. Since the global symmetry is preserved, the ranks of the avor nodes remain equal to (N1; N2; N3; N4) in all theories. The rank of the gauge group evolves according to (5.1). Possible deformations and deformed sequences. Let us consider the original theory T , which is shown in gure 8. All bifundamental avors are, generically, rectangular matrices. We can use global and gauge symmetries to simplify them, such that all entries are zero except for, at most, those in N0 N0 diagonal submatrices. There are three possible mass deformations of this theory: a) X10 02 mass: T can be deformed by introducing an X10 02 mass term, i.e. J 02 = m 10 : For simplicity, here and in the deformations that we discuss below, we assume the rank of the mass matrix is 1. It is straightforward to extend our discussion to higher rank masses. We call T~ the resulting theory, which still has the same quiver diagram of T , but with reduced ranks for some of the global nodes, as we now explain. The mass term clearly breaks the global symmetry down to (N~1; N~2; N~3; N~4) = (N1 1; N2 1; N3; N4).10 The gauge group is una ected so its rank is N~0 = N0. We now take T~ as the new starting point. Acting with quadrality, we obtain a deformed sequence of theories T~, T~0, T~00 and T~000. Once again, the quivers for the deformed sequence are those in gure 9. The di erences with respect to the original sequence are the following. The ranks of the avor nodes are (N1 1; N2 1; N3; N4) for all the theories. In addition, using the initial N~i's in (5.1) we determine the evolution of the rank of the gauge group is ~ T N0 T~0 N00 1 T~00 N000 T~000 N0000 (5.2) (5.3) connected to a new, rank 1, global node. 30X04 mass: this deformation is very similar to the one we have just considered. It breaks the global symmetry down to (N1; N2; N3 1; N4 1). The rank of the gauge group is una ected in the initial theory and evolves as follows c) H 30; 02 = m. The global symmetry is reduced to (N1; N2 gauge group remains the same in the starting theory and then follows the sequence It is interesting to notice that a common feature of the three deformations considered above is that, for each of them, only one theory in the sequence gets higgsed.11 Connecting the original and the deformed sequences. Alternatively, if the quadrality proposal is correct, we should be able to obtain the deformed sequences by mapping the original deformations of T to deformations of T 0, T 00 and T 000. As we will now explain, it is possible to verify that this is indeed the case. Before doing so, it is convenient to discuss how general deformations are mapped under quadrality. In the context of N = 1 matrix models, we refer to any modi cation of J - or H-terms as a deformation. Deformations modify the global symmetry of a theory, while preserving its gauge symmetry and matter content. When moving to a dual theory we must now include all interactions that are consistent with the new global symmetry. The modi cation in the interactions of the dual theory is identi ed with the map of the original deformation. Typically, we are interested in introducing new interaction terms to the original theory, which reduces the global symmetry. As a result, the dual theory admits new interaction terms that are the translation of the deformation. This prescription is very general and is the one used when mapping deformations under dualities in other dimensions. For example, it is precisely the approach one uses in the well-known case of mapping superpotential deformations of 4d N = 1 theories under Seiberg duality.12 For concreteness, let us focus on case (a), namely on a rank 1 X10 02 mass term. The other deformations can be understood using similar ideas. The rst dual theory, T 0, 11In an abuse of language, what we mean by higgsing is that some of the gauginos become massive, which reduces the gauge symmetry. 12It is important to note that while it is relatively straightforward to map deformations under duality with our prescription, this cannot be done by simply rewriting the component expansions of some terms in the Lagrangian. Here, again, 4d Seiberg duality provides a familiar example. Deformations can be easily implemented at the level of the superpotential, just replacing combinations of chiral elds that are charged under the dualized gauge group by mesons. This process, however, cannot be implemented as a reorganization of the component expansion of the Lagrangian. the meson M21, which becomes massive by coupling to other elds and is integrated out. contains a Fermi meson 12 = X10 02. The deformation of T given by (5.2) maps to an additional constant contribution to J 12 as follows (5.6) (5.7) (5.8) (5.9) T : J 02 = m 10 T 0 : J 12 = X20X01 + m ; where the rst term in J 12 is the usual coupling between the Fermi meson 12 and the chiral avors in T 0. The constant term in J 12 , where m is the original rank 1 mass matrix, is precisely the new interaction that is allowed when the global symmetry is reduced to (N1 1; N2 1; N3; N4).13 The action now contains the term SJ = J 12 J 12 + : : : = j 20 01 + mj2 + : : : ; which xes at a non-zero value an entry in each of the diagonalized 20 and 01. This, in turn, reduces the rank of the gauge group to N0 1, in perfect agreement with (5.3). Having correctly reproduced T~0 by mapping the deformation, let us now consider the two remaining theories in the sequence. The xed non-zero values for 20 and 01 in T 0 map to mass terms in both T 00 and T 000 that break the global symmetry down to (N1 1; N2 1; N3; N4), as expected. The two mass terms are: T 00 ! T~00 T 000 ! T~000 J -term: J 10 = m 02 H-term: H 20; 01 = m We can understand these deformations as follows. When T 0 is dualized, a chiral meson M21 = X20X01 is generated. This meson is not shown in phases T 00 and T 000 of gure 9 because it becomes massive and can be integrated out. gure 10 shows where M21 would be in T 00 and T 000. This picture is just intended as a visual reference; to reintroduce M21 in these theories we should also integrate in the elds they paired with. These theories contain the following interactions: T 00 T 000 J -term: J 10 = H-term: H 20; 01 = 13This is the 0d analogue of the map, at the level of the superpotential, between a mass term for quarks in 4d SQCD and a linear term for mesons in its Seiberg dual. The non-zero values for 20 and 01 in T 0 that we discussed above, translate into a non-zero value for M21 . When plugged into (5.9), it produces the mass deformations in (5.8). In addition, no avor gets a non-zero value in either T 00 or T 000 so the gauge group is not higgsed. This is in agreement with the fact that the ranks of the gauge group in T~00 and T~000 are equal to those in the undeformed sequence. In summary, we have correctly reproduced the entire deformed sequence. This matching provides a rather non-trivial check of the quadrality proposal, which takes into account all types of mesons and interaction terms. As another check of quadrality, we compare the chiral ring of the four theories. For simplicity, we identify a few elements of the chiral ring, focusing on operators that can be expressed as super elds. We restrict to operators built out of a small number of matter elds, leaving a complete enumeration for future work. The chiral ring is given by cohomology classes of Q, i.e. it consists of gauge invariant operators that are Q-closed but not Q-exact. The top components of super elds are, by de nition, Q-exact. The only exception is X, which has a single component.14 We are thus interested in the case in which only the lowest component of a gauge invariant super eld, either elementary or a product, survives. Such a component then becomes an element of the chiral ring. The chiral ring is determined on-shell which, among other things, requires that Ga = Ja for all elementary Fermi elds. Let us rst consider gauge invariant elementary elds. For quick reference, below we summarize some of the super eld discussion of section section 2 and indicate whether the lowest components of the operators are in the chiral ring. Super eld lowest Q(lowest) Chiral ring? HJEP07(21)53 where we have separated the X and X contributions. we obtain We can repeat the exercise for products of two matter elds. Following (2.12)-(2.14), Super eld lowest Chiral ring? X X X1X2 X1X2 X1 2 X1 2 14As mentioned earlier, even though we sometimes consider X and X separately, we should keep in mind that they combine into a single super eld. The \ " category in (5.10) and (5.11) indicates operators that become elements of the chiral ring if the G-components of the corresponding Fermi elds vanish. This is automatically the case if such Fermis do not participate in any J -term oriented loop in the quiver. Below we list operators in the chiral ring for the quadrality sequence shown in gure 9. For convenience, we refer to the operators in terms of super elds, with the understanding that their lowest components are the objects of interest. O41 O31 O42 T X04X10 30X10 X04 02 T 0 M 14 31 40X20 M 14 12 T 00 M 14 31 42 42 X03 01 34M 14 (5.12) Matching T and T 0. In order to illustrate the main ideas that go into the matching, it is instructive to discuss the correspondence between T and T 0 in detail. The rest of the theories follow a similar logic. In particular the analysis of T 000 is identical to the one of T 0 up to a re ection with respect to a vertical axis. Let us discuss the rows in (5.12) that deserve special comments. The operator O31 is given by 30 10 in T and by 31 in T 0. It may appear that T 0 also contains the operator 01 03 between this pair of nodes. However, the theory has H 03; 31 = 01, which gives rise to the coupling 03 31 01. Since 31 does not participate in any other loop in the quiver, the equation of motion for 31 forces 01 03 to vanish on-shell. O42 corresponds to 04 02 in T . Interestingly, there are two operators in T 0 with the right properties: 40 02 and M14 12 . Which one should we use? First of all, the two operators are not related by an equation of motion, so we cannot restrict to just one of them. Remarkably, for Q-closedness it is necessary to consider a linear combination of them: 40 02 M14 12 . We have fQ; 40 02 M14 12 g = G40 20 M14 G 12 ' 14 10 20 14 10 20 = 0 : (5.13) Both 40 and 12 participate in J -term loops, so their respective G-components do not automatically vanish, but they compensate when combined. A similar explanation applies to O31 in T 000. Additional comments. Further things that work nicely for all the theories are: Elementary singlet Fermis. Notice that the three theories other than T have singlet Fermi elds: 12 in T 0, 23 in T 00 and 34 in T 000. They are not in the chiral ring, because they all participate in J -term loops and hence their G-components do not vanish. This is good, since there are no matching operators in the other theories, most notably in T . node, node 1, of a 2-gauge/6- avor quiver. X Fermis. There are two types of composite Fermi elds, which are of the general forms X and X . In (5.11) we argued that the X operators are not in the chiral ring. Indeed, such operators do not match between di erent theories. A simple example is X20 03 in T 0, which does not have a counterpart in T . Chiral rings can be fairly non-trivial in theories with multiple matter elds and gauge groups, even if restricted to special subsectors. This point is illustrated in section 7.3, where we compute all the chiral ring operators that only consist of chiral elds for two quadrality dual theories on D(-1)-branes probing local CP4. 6 Quadrality networks Theories connected by sequences of quadrality transformations can be nicely organized in terms of quadrality networks. Analogous constructions exist for Seiberg duality [4, 14, 15] and triality [2, 9]. These networks become particularly interesting for theories with multiple gauge groups. One simple generalization of the simple SQCD-like theory considered in previous sections consists of merging several copies of the basic 1-gauge/4- avor quiver to make up n-gauge/(2n + 2)- avor quivers. An n = 2 example with the quadrality action on gauge node 1 is shown in gure 11. The quadrality actions on the two gauge nodes do not commute, so all possible combinations of quadrality lead to a network of quivers. It turns out that there are 44 quivers at n = 2. The quivers can be divided into four types according to the type of eld connecting the two gauge nodes. Quivers of the same type may have di erent meson contents. gure 12 shows the complete quadrality network of n = 2 quivers. In the gure, the labels A, B, C, D denote the four types of quivers as in gure 11. The blue and pink arrows indicate quadrality transformations on node 1 and 2, respectively. The length-4 closed oriented loops consisting of arrows of a given color correspond to four consecutive quadrality transformations on the same node. As it occurs for Seiberg duality and triality, more general quivers can lead to in nite quadrality networks. 7 D-brane theories probing toric CY 5-folds. 7.1 C 5 In this section we study N = 1 matrix models arising on the worldvolume of D(-1)-branes Let us rst consider D(-1)-branes in at 10d spacetime, i.e. C5. This theory is often called the Type IIB matrix model and has been proposed as a nonperturbative formulation of type IIB string theory [16{18]. The models presented in this and the coming section illustrate how the general structures discussed in section section 2 arise in D-brane constructions. Bosons and fermions are decomposed in terms of the SU(5) SO(10) global symmetry as follows : 10v ! : 16s ! m(51) + (15=2) + m(5 1) mn(101=2) + m(5 3=2) Here m and n are indices in the fundamental representation of SU(5). We choose the convention where complex conjugation exchanges superscripts and subscripts. Contraction of conjugate indices (AmBm) implies summation. The 16 supercharges decompose as Q : 16c ! 1 5=2 + 10 1=2 + 53=2 : HJEP07(21)53 We will only use the singlet supercharge. The on-shell form of the action of the matrix model can be split into three parts: (7.1) (7.2) (7.3) (7.4) (7.5) (7.6) (7.7) (7.8) (7.9) The D-term is The J -term is The H-term is 1 2 1 2 S = SD + SJ + SH : SD = Tr where "mnpqr is the SU(5) invariant tensor. The supersymmetry variation is m = m ; m = 0 ; m = 0 ; mn = [ m; n] ; = [ m ; m] : It is easy to show that SD, SJ and SH are separately invariant under a supersymmetry transformation. We can introduce independent coupling constants for the three terms. Ratios among the three couplings are xed only if we turn on non-singlet supercharges from (7.2). Unlike SD or SJ , the supersymmetry of SH does not rely on a cancellation between a purely bosonic term and a fermion bilinear term. Explicitly, 8( SH ) = mnpqr Tr [ m; n] pq r mn[ p; q] r = mnpqr Tr mn[[ p; q]; r] = 0 : In the last step, we used the Jacobi identity. section section 2. We simply note that It is straightforward to rephrase this theory in the (mostly) o -shell formalism of J mn( ) = [ m ; n] ; Jmn( ) = [ m; n] ; H mn;pq( ) = "mnpqr r : Let us now consider D(-1)-branes probing local CP4, namely the C5=Z5 orbifold with action (1; 1; 1; 1; 1). This geometry is the simplest toric CY5 that can be studied using mirror symmetry and, as discussed in section section 3.4, it is the minimal local geometry realizing quadrality. The corresponding gauge theory is obtained from the one for C 5 presented in the previous section by standard orbifold techniques [ 19, 20 ]. Its quiver diagram is shown in gure 13. We have chiral multiplets Xim;i+1, Fermi multiplets i+2;i and gaugino multiplets i;i. mn The on-shell supersymmetry variation is a natural generalization of (7.7): m i;i+1 = m i;i+1 ; m i;i+1 = 0 ; i+1;i = 0 ; m im+n2;i = ( im+2;i+1 in+1;i i+2;i+1 i+1;i) ; n m i;i = ( im;i+1 m i+1;i m i;i 1 m i 1;i) : where the quiver node indices are de ned (mod 5). The absolute-value-square of a complex matrix is de ned as jAj2 = AAy. Once again, SD, SJ and SH are separately invariant under a supersymmetry transformation. The terms in the action are SD = SJ = SH = 5 X Tr i=1 5 X Tr i=1 1 2 1 8 i=1 jXim;i+1 m i;i 1Xim 1;ij2 + i;i( im;i 1 m m i 1;i m i;i+1 m i+1;i) ; m n 2 j i 1;i i;i+1 n m mn n i 1;i i;i+1j2 + i+1;i 1( im 1;i i;i+1 n m i 1;i i;i+1) ; i+2;i i;i 2 i 2;i+2 ; mn pq r (7.10) (7.11) For theories on D(-1)-branes, the chiral ring operators consisting exclusively of chiral elds should reproduce the coordinate ring of the probed CY5. More precisely, following the discussion in section section 5.4, the operators we are interested in are products of components of chiral elds. For simplicity, since we are focusing only on chiral elds, we can drop the conjugation in our discussion. Using local CP4 as an example, in this section we show that the CY5 can be recovered from two di erent quadrality phases. This fact simultaneously demonstrates the connection of the chiral ring to the probed CY5 and the invariance of this particular sector under quadrality. The original theory. Let us rst consider the theory presented in section section 7.2. For simplicity, let us focus on a single D(-1)-brane, namely we set N = 1 in the quiver of gure 13. For generic N , we can diagonalize all elds and the full answer is the N th symmetric product of the N = 1 result. We can construct the following 55 gauge invariant operators M m1m2m3m4m5 = 1m21 2m32 3m43 4m54 5m15 ; mi = 1; : : : ; 5 : (7.12) In terms of the SU(5) global symmetry, they decompose as follows 5 5 5 5 5 = 1 + 4(24) + 5(75) + 6(126) + 1260 + 5(1750) + 4(224) = [0; 0; 0; 0] + 4[1; 0; 0; 1] + 5[0; 1; 1; 0] + 6[2; 0; 1; 0] + [5; 0; 0; 0] (7.13) +5[1; 2; 0; 0] + 4[3; 1; 0; 0] ; where, in order to distinguish representations with equal dimension we also provided the corresponding Dynkin labels. Vanishing of the bosonic potential in (7.11) gives rise to the following relations i 1;i i;i+1 n m i 1;i i;i+1 = 0 ; (7.14) which fully symmetrize the indices in M m1m2m3m4m5 . Accordingly, only the 1260 = [5; 0; 0; 0], i.e. the totally symmetric 5-index representation of SU(5), survives from the gauge invariant operators in (7.13). We call them the generators of the chiral ring and label them Msm1m2m3m4m5 , with 1 m1 m2 m3 m4 m5 The generators satisfy rst order quadratic relations. First, note that 1260 1260 = 1004 + 2574 + 38500 + 4125 + 3150 + 11760 = [10; 0; 0; 0] + [8; 1; 0; 0] + [6; 2; 0; 0] + [4; 3; 0; 0] (7.15) +[2; 4; 0; 0] + [0; 5; 0; 0] : The J -term relations (7.14) imply that in a product Msm1m2m3m4m5 Msn1n2n3n4n5 any pair of indices (mi; ni) can be swapped leaving the product invariant. Using this, it is possible to show that the generators obey 7000 quadratic relations, which transform in the 38500 + 3150 representation of SU(5). They can be written explicitly as follows 38500 = [6; 2; 0; 0] : Rpijqkrlsmtun = Msijkv1v2 Mslmnw1w2 v1w1pqr v2w2stu = 0 ; 3150 = [2; 4; 0; 0] : Tpi1jp2p3q1q2q3r1r2r3s1s2s3 = Msim1m2m3m4 Msjn1n2n3n4 (7.16) { 28 { HJEP07(21)53 The variety formed by the generators subject to their rst order relations is not a complete intersection. The plethystic logarithm of the Hilbert series g(t; xi; C5=Z5) of the variety is thus not nite and takes the form PL[g(t; xi; C5=Z5)] = [5; 0; 0; 0]t ([6; 2; 0; 0] + [2; 4; 0; 0])t2 + : : : ; (7.17) where t is a fugacity counting the degree in terms of the generators Msm1m2m3m4m5 . The previous analysis is in precise agreement with the computation of the Hilbert series directly from the toric geometry of C5=Z5, as explained in appendix section A. We hence conclude the chiral ring reproduces the coordinate ring of the probed CY5. The SU(5) charges of the generators transforming in the 1260 representation form a convex polytope in Z4. This polytope | the lattice of generators | is the dual of the toric diagram of C5=Z5. Such relation between the generators and the polytope dual to the toric diagram is described in detail for the simpler example of C3=Z3 in appendix section B. The quadrality dual. Let us now consider the theory obtained by acting with quadrality on node 1. Quadrality generates the following mesons: M52 = X51 X12 = 5 5 = 15 + 10 54 = X51 14 = 5 35 = 31 X51 = 10 10 = 5 + 45 5 = 10 + 40 where we express them as composites of the elds in the original theory and indicate in blue the elds that become massive and are integrated out. The masses for elds in the rst two lines are J -terms, while the one for the last line is an H-term. The resulting quiver diagram is shown in gure 14. We do not write the J - and H-terms explicitly; they can be easily determined from the quiver and global symmetry. This theory is just one in an in nite web of quadrality dual theories, analogous to the web of 4d Seiberg dual theories on D3-branes probing local CP2 [4, 14]. Chiral elds can be labeled using fundamental and antifundamental SU(5) indices as follows i5j2 : 15 2k3 : 5 l34 : 5 4m1n : 10 ( 15)p : 5 (7.19) We restrict to i j and m < n to match the 15 (symmetric) and 10 (antisymmetric) representations, respectively. Once again, let us focus on the case of a single D(-1)-brane, namely N = 1. Proceeding as before, we construct gauge invariant operators using chiral elds, which transform under SU(5) as follows ij k l 52 23 34 4m1n( 15)p = 15 5 5 10 5 = 1 + 6(24) + 7(75) + 10(126) + 2(126) + 1260 + 7(1750)+1750 + 3(200) + 6(224) + 7000 + 4(1024) + 3(10500) + 2(1701) + 1750 : (7.20) where the U(4) color indices at node 1 are properly contracted. For brevity, we do not provide the Dynkin labels of the representations. HJEP07(21)53 The next step is to determine which of the representations in (7.20) survive once J terms are taken into account. This can be conveniently done in steps, as we explain below. The key idea is the following. In this theory, all J -terms correspond to cubic loops in the quiver, involving one Fermi and two chiral elds. This means that the J -terms are quadratic in chiral elds. For each Fermi eld R transforming in some representation R of SU(5), the condition J R = 0 sets the R representation in the corresponding product of scalar elds to zero. The J -terms for the ve types of Fermi elds in the theory end up eliminating several of the operators in (7.20). It is instructive to discuss in detail how to obtain the generators. First consider the product 34 j4k1 = 5 i 10 = 10 + 40 : These two chiral elds, appropriately contracted, form J 13 . Since 13 transforms in the 10, we know that the 10 is eliminated from the product so we are left with jk i 34 41jJ=0 = 40. Taking the product with ( 15)l, we then have 34 j4k1 ( 15)ljJ=0 i 40 5 = 10 + 15 + 175 ; where the inclusion sign indicates that we still have not considered all the relevant J -terms. J 54 = 0 removes the 45 in the product j4k1 ( 15)l which, in turn, leaves us with This can be conveniently written as 34 j4k1 ( 15)ljJ=0 = 15 = [2; 0; 0; 0] : i M(iAj) = i 34 j4m1 ( 15)m = i34Zj ; with i and j symmetrized and Zj = j4m1( 15)m. (7.21) (7.22) (7.23) (7.24) Next, let us consider the product J 35 = 0 eliminates the 40, resulting in which is symmetric in the three indices. i5j2 2k3 = 15 5 = 35 + 40 : M(iBjk) = i5j2 2k3jJ=0 = 35 = [3; 0; 0] ; Combining (7.24) and (7.26), we conclude the surviving gauge invariants are Mfsm1m2m3m4m5 = M(mA1)m2 M(mB3)m4m5 = m1m2 2m33 3m44 Zm5 ; 52 which transform in the 1260 = [5; 0; 0; 0] representation due to the symmetrization of all indices that follows from the remaining J -terms. This is in full agreement with the generators of the original theory. Furthermore, it is possible to verify explicitly that the generators satisfy the same 7000 quadratic relations of the original theory (7.16). In summary, with the matching of generators and their rst order relations, we conclude that the chiral rings of the two dual theories reproduce the coordinate ring of C5=Z5. The matrix models on D(-1)-branes probing toric CY 5-folds have, in general, an in nite number of dual phases connected by quadrality. Some of these phases are described by the brane hyperbrick models discussed in section section 3.1, equivalently by periodic quivers on T 4. We refer to them as toric phases. This is a straightforward generalization of the concept of toric phases in 4d and 2d. A toric node is a node in a toric phase whose dualization also results in a toric phase.15 It is natural to ask what is the structure of a minimal, i.e. with a minimum number of elds, toric node for N = 1 matrix models.16 In every dimension, minimal toric nodes involve two bifundamental elds of each possible type. In the corresponding periodic quivers, the two elds in each of these pairs emanate from the toric node in opposite directions. In 4d, such a node involves two incoming and two outgoing chirals. This con guration maps to a square face in the brane tiling [6]. Similarly, a minimal toric node in 2d has two incoming chirals, two outgoing chirals and two Fermis. This translates into a cube in the brane brick model [9]. Finally, a minimal toric node in 0d contains two incoming chirals, two outgoing chirals, two incoming Fermis and two outgoing Fermis. It maps to a hypercube in the brane hyperbrick model. Can we nd a relatively simple toric CY 5-fold that: a) contains at least a minimal toric node and b) has more than one toric phase? For 4d gauge theories, a standard example of a CY3 with these properties is local CP1 CP1, also known as F0 [6]. The analogous CY4 for 2d theories is local CP1 CP1 CP1, i.e. Q1;1;1=Z2 [9]. It is then natural to conjecture 15Notice that it is possible for a toric phase not to have any toric node. 16We emphasize minimality because we expect multiple possibilities for toric nodes. While toric nodes have a unique structure in 4d theories, this is no longer the case in 2d. It is reasonable to anticipate a similar behavior in 0d. (7.25) (7.26) (7.27) HJEP07(21)53 CP1 does the job for matrix models. Below we present some evidence supporting this claim. Consider the following choice of coe cients in the Newton polynomial of (CP1)4: 1 x 1 y P (x; y; z; w) = x + + i y + + 0:9(1 + i) z + + 0:9( 1 + i) w + w (7.28) Following the mirror symmetry analysis reviewed in section section 3.2, gure 15.a shows the resulting 16 critical values on the W -plane and the associated vanishing paths. Focusing on the y = +1 subset, shown in gure 15.b, we recognize the con guration of vanishing paths for phase B of the Q1;1;1=Z2 brane brick model [9].17 This is a phase that indeed contains minimal toric nodes, i.e. cubic brane bricks. By symmetry, the y = 1 and x = subsets give rise to almost identical con gurations. The z = +1 vanishing paths are shown in gure 15.c. They also correspond to phase B of Q1;1;1=Z2. Due to symmetry, the same is true for z = 1 and w = 1. These observations, combined, suggest that it is very plausible that the con guration in gure 15 corresponds to a simple phase with minimal toric nodes. A detailed study of the matrix model(s) associated to this geometry would be extremely interesting. It is however beyond the scope of this paper and we leave it for future work. 8 Towards quadrality from compacti cation Dualities in lower dimensional quantum eld theories can often be derived from the ones for higher dimensional theories via compacti cation. For IR dualities, the interplay between the low energy limit and the zero compacti cation size limit can be subtle and sometimes the two limits do not commute. This approach has been successfully exploited for deriving dualities in 3d and 2d SUSY theories starting from 4d Seiberg duality. In [21, 22], dualities of 3d N = 2 theories were 17Here we are making a comparison to a CY 4-fold. We say that the con guration of vanishing paths of the CY5 is equal to the one of the CY4, when they coincide on the W -plane. theories were studied in [23, 24]. The resulting 2d theories depend on how the theory is coupled to background elds, which maps to a choice of R-symmetry. It is possible to show that the reduction of Seiberg duality gives rise to triality by simultaneously picking an appropriate choice of R-charges and turning on an FI term. The index of the 4d theories plays a central role in the aforementioned studies. The superconformal S3 S1 index becomes the S3 partition function of the 3d theory [25{27] and the S2 T2 partition function becomes the elliptic genus in 2d. Agreement of the indices in the compacti ed theories provides substantial support for the lower dimensional dualities. These examples show that the relation between duality and compacti cation is rather nontrivial. Issues that need to be carefully addressed include the ratio between the UV cuto and the compacti cation radius, the di erence between compact and non-compact scalars in 3d, appearance of extra terms in the superpotential due to nonperturbative e ects and the need for summing over ux sectors that might correspond to defect operators. It is natural to ask whether 0d quadrality can be derived from Seiberg duality or triality along similar lines. The SUSY partition function of the higher dimensional theory on the compacti cation manifold should be compared with the matrix integral of the 0d theory. There are various alternatives for compactifying 4d N = 1 down to 0d N = 1, such as S 2 T2 and S 2 S2. If, instead, we try to connect 2d (0; 2) to 0d N = 1, we run into a puzzle. There is no known SUSY preserving S2 compacti cation of 2d (0; 2). On the other hand, T2 leads to too much SUSY in 0d. It might be possible to compactify on a more general Riemann surface, including punctures if necessary. Typically, higher genus surfaces give rise to relatively complicated theories in lower dimensions that re ect the geometric data. Before addressing quadrality, it may be instructive to consider theories with extended supersymmetry, relating 2d (2; 2) to 0d N = 2. We can investigate whether the S2 partition function of the 2d (2; 2) theory has a 0d N = 2 interpretation. Along similar lines, [23] studied compacti cations of 4d N = 2 to 2d (2; 2). It would be extremely interesting to provide a eld theoretic derivation of quadrality by obtaining it from a higher dimensional duality via compacti cation. This is however beyond the scope of this paper. We plan to revisit this question in the future. 9 We introduced quadrality, a new order 4 duality that applies to N = 1 supersymmetric gauged matrix models. Our proposal follows naturally from mirror symmetry, which provides a uni ed framework that puts 4d Seiberg duality, 2d GGP triality and quadrality on an equal footing. We expect that quadrality is not restricted to theories with a D-brane realization and holds for general N = 1 matrix models. We performed various checks of the proposal, including the matching of: global symmetries, abelian avor anomalies, deformations and the chiral ring. The chiral ring was computed in detail for a pair of quadrality dual theories on D(-1)-branes probing local CP4, for which we showed not only that it is the same in the two theories but that it reproduces the coordinate ring of the CY5 singularity. We also initiated the study of various aspects of the matrix models that arise on the worldvolume of D(-1)-branes probing toric CY 5-folds. There are various natural directions for future investigation. First, dualities are powerful tools for elucidating the dynamics of quantum eld theories in di erent dimensions. The application of Seiberg duality to map the phase space of 4d N = 1 SQCD is a prime example. It would be worth studying what quadrality can teach us about the dynamics of matrix models. It would also be interesting to establish whether it is possible to derive quadrality from a higher dimensional duality through compacti cation. Below we discuss two additional open questions. Evaluating the integral. Conceptually, the most explicit way to verify quadrality would be to compute the matrix integrals for the four dual theories and show that they are equal. This would be analogous to the computation of the elliptic genus to verify 2d triality [2, 28]. For triality the three dual theories ow to the same SCFT in the IR [29]. The elliptic genus is invariant under the RG ow, making it possible to probe the SCFT from the UV gauge theory. In 0d, the usual notion of RG ow does not exist, so determining the conditions under which the matrix integrals should agree becomes more subtle. The 2d elliptic genus can be re ned by turning on avor fugacities. In the case of non-compact target spaces, the fugacities regulate divergent contributions. In 0d, since there are no background gauge elds, it is not clear how to turn on fugacities. Due to these issues, evaluating the integrals and comparing between di erent quadrality phases is not straightforward. We hope to revisit these questions in future work. M-theory lift. Mirror symmetry relates D(-1)-branes probing a CY 5-fold to Euclidean D4-branes wrapping 5-cycles in the mirror CY5. It would be interesting to determine the M-theory lift of this con guration. The ED4-branes become EM5-branes wrapping the original 5-cycles times the M-theory circle. Wick-rotating and decompactifying the Mtheory circle, we arrive at an M-theory con guration with physical M5-branes wrapping 5-cycles. The result is a supersymmetric quantum mechanics. The original 0d matrix model can be reinterpreted as the dimensional reduction of the M-theory quantum mechanics. The situation is somewhat analogous, but not equivalent, to the relation between the Dinstanton matrix model and D0 quantum mechanics. In particular, the D(-1) matrix model contains information on the Witten index of D0 quantum mechanics [30]. In the D0/D(-1) connection, an explicit Lagrangian description is available on both sides. In contrast, in the relation between the IIA matrix model and the M-theory quantum mechanics we are considering, the Lagrangian is only known for the former but not for the latter. The precise nature of the quantum mechanics of wrapped M5-branes is an interesting open problem. More generally, it would also be interesting to determine whether some new duality for supersymmetric quantum mechanics can be inferred from quadrality. Acknowledgments We would like to thank P. Putrov, M. Romo, N. Seiberg, E. Witten and S.-T. Yau for useful and enjoyable discussions. We are also grateful to D. Ghim for collaboration on related topics. We gratefully acknowledge support from the Simons Center for Geometry and Physics, Stony Brook University, where some of the research for this paper was performed during the 2016 Simons Summer Workshop. The work of S. F. is supported by the U.S. National Science Foundation grant PHY-1518967 and by a PSC-CUNY award. The work of S. L. was supported by Samsung Science and Technology Foundation under Project Number SSTBA1402-08. The work of S. L. was also performed in part at the Institute for Advanced Study supported by the IBM Einstein Fellowship of the Institute for Advanced Study, and at the Aspen Center for Physics supported by National Science Foundation grant PHY-1066293. The work of R.-K. S. is supported by the ERC STG grant 639220 \Curved SUSY". The work of C.V. is supported in part by NSF grant PHY-1067976. A The Hilbert series for C3=Z3, C4=Z4 and C5=Z5 The Hilbert series [31, 32] is a powerful tool for enumerating operators in a chiral ring and for studying its geometric structure. It is formally de ned in algebraic geometry as the generating function g(t; R) = 1 n=0 X dim(Rn)tn ; (A.1) where R is an algebraic quotient ring and Rn is a component of R of degree n 2 N. The fugacity t counts the degree of the component Rn. For multi-graded rings with compoP1 n=0 dim(R~n)t1n1 : : : tknk , where t1; : : : ; tk are the fugacities of the grading. nents R~n and grading ~n = (n1; : : : ; nk), the Hilbert series takes the form g(t1; : : : ; tk; R) = Hilbert series from toric geometry. When the chiral ring is a toric variety, its Hilbert series can be derived directly from the toric diagram [31, 33]. For a toric CY n-fold, the toric diagram is an n 1 dimensional convex polytope that admits at least one triangulation in terms of (n 1)-simplices. From a triangulation, one can construct a dual web diagram. For Calabi-Yau 3-folds, these are the so-called (p; q)-webs [34{36]. gure 16 shows the toric diagrams and dual web diagrams for C3=Z3 and C4=Z4. The Hilbert series of the toric variety X then can be de ned from a triangulation of the toric diagram as follows g(t1; : : : ; tn; X) = ~t ~v(i;j)) ; (A.2) where ~t ~v(i;j) = Qan tvaa(i;j). The index i = 1; : : : ; r runs over the simplices making up the triangulation while j = 1; : : : ; n runs over the faces of each simplex. The vector ~v(i; j) is the n-dimensional outer normal to the face of the fan associated to face j of simplex i.18 18Notice that for a CY n-fold, these normal vectors are n-dimensional, while we said that the toric and web diagrams are (n 1)-dimensional. More precisely, the toric diagram of a CY n-fold lives on a hyperplane in n-dimensions at distance 1 from the origin. This fact allows a trivial projection to (n 1)-dimension. In the examples below, we reincorporate the nth coordinate in order to determine the normal vectors. r X n Y(1 i=1 j=1 Hilbert series for C3=Z3, C4=Z4, and C5=Z5. toric diagrams for these three geometries are The coordinates for the points in the C3=Z3 : (1; 0; 1); (0; 1; 1); ( 1; 1; 1); (0; 0; 1) C4=Z4 : (1; 0; 0; 1); (0; 1; 0; 1); (0; 0; 1; 1); ( 1; 1; 1; 1); (0; 0; 0; 1) C4=Z4 : (1; 0; 0; 0; 1); (0; 1; 0; 0; 1); (0; 0; 1; 0; 1); (0; 0; 0; 1; 1); ( 1; 1; 1; 1; 1); (0; 0; 0; 0; 1) Notice that we have included the nth coordinate, which will become important in the discussion that follows. Labeling the points in the toric diagrams from 1; : : : ; n + 1 in the order they are listed in (A.3), then the unique triangulations of these diagrams can be summarized in terms of the points in the toric diagram as follows C3=Z3 : ff1; 2; 4g; f1; 3; 4g f2; 3; 4gg C4=Z4 : ff1; 2; 3; 5g; f1; 2; 4; 5g; f1; 3; 4; 5g; f2; 3; 4; 5gg C4=Z4 : ff1; 2; 3; 4; 6g; f1; 2; 3; 5; 6g; f1; 2; 4; 5; 6g; f1; 3; 4; 5; 6g; f2; 3; 4; 5; 6gg (A.4) The corresponding Hilbert series are g(t; C3=Z3) = g(t; C4=Z4) = g(t; C5=Z5) = 1 + 7t + t2 t)3 ; 1 + 31t + 31t2 + t3 (1 1 + 121t + 381t2 + 121t3 + t4 (1 t)5 : (A.3) (A.5) The plethystic logarithms [31, 32] of the Hilbert series are PL[g(t; C3=Z3)] = 10t PL[g(t; C4=Z4)] = 35t PL[g(t; C5=Z5)] = 126t 27t2 + 105t3 465t2 + 8960t3 540t4 + : : : ; 201376t4 + : : : ; 7000t2 + 544500t3 48095250t4 + : : : : Note that none of these plethystic logarithms has a nite expansion, indicating that the three toric varieties are not complete intersections. Furthermore, the coe cients in (A.6) are sums of dimensions of irreducible representations of SU(3), SU(4) and SU(5), respectively. This computation con rms that the chiral ring discussed in section 7.3 indeed corresponds to the coordinate ring of C5=Z5. The rst term in the plethystic logarithm corresponds to the 126 generators of the toric variety transforming in the [5; 0; 0; 0] representation of SU(5). The second term in the expansion indicates that these 126 generators satisfy 7000 quadratic relations, which transform in the [6; 2; 0; 0]+[2; 4; 0; 0] representations of SU(5). B Review: C3=Z3 While the D-brane constructions of this paper focus on D(-1)-branes probing toric CY 5-folds, it is enlightening to review the case of D3-branes probing C3=Z3 in further detail. This example is useful because it exhibits many of the concepts that apply to the CY5 case in a considerably simpler context. For N D3-branes, the worldvolume theory is a 4d N = 1 supersymmetric gauge theory with the quiver diagram shown in gure 17 and superpotential The global symmetry of the theory is SU(3) U(1)R. The Hilbert series for a single D3-brane has been computed in [37]. It takes the form W = ijk i12 j23 3k1 : g(t; x1; x2; C3=Z3) = 1 X[3n; 0]tn ; n=0 (B.1) (B.2) generators form the convex polygon dual to the toric diagram. where [3n; 0] is the character of the SU(3) representation, with the entries being Dynkin labels of the representation. t is the fugacity for the U(1)R symmetry. The plethystic PL[g(t; x1; x2; C3=Z3)] = [3; 0]t [2; 2]t2 + ([1; 1] + [1; 4] + [2; 2] + [4; 1])t3 + : : : : (B.3) This matches the result obtained from toric geometry in (A.6). There are 10 generators which transform in the [3; 0] representation of SU(3) satisfying 27 relations transforming in the [2; 2] representation of SU(3). The vacuum moduli space is not a complete intersection and for the abelian theory it is precisely C3=Z3. The 10 generators can be written in terms of the chiral bifundamental elds as follows, where one sets i j k such that M ijk transform in the [3; 0] representation of SU(3). This follows from the fact that the F -terms associated to the superpotential in (B.1), i ab (B.4) (B.5) (B.6) (B.7) M ijk = i12 j23 3k1 ; = ijk jbc cka = 0 : following quotient variety, M ijk=hRmij n = 0i : fully symmetrize the indices of M ijk. The plethystic logarithm (B.3) indicates that there are quadratic relations between the generators M ijk transforming in the [2; 2] representation of SU(3). 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Sebastián Franco, Sangmin Lee, Rak-Kyeong Seong, Cumrun Vafa. Quadrality for supersymmetric matrix models, Journal of High Energy Physics, 2017, 53, DOI: 10.1007/JHEP07(2017)053