We study the interplay between mass deformations and unoriented projections of super-conformal quiver gauge theories resulting from D3-branes at (toric) Calabi-Yau singularities. We focus on simple orbifold cases ($\mathbb{C}^3/\mathbb{Z}_3$ and $\mathbb{C}^3/\mathbb{Z}_4$) and their non-orbifold descendants. This allows us to generalize the construction rules and clarify points that have been previously overlooked. In particular we spell out the conditions of anomaly cancellations as well as super-conformal invariance that typically require the introduction of flavour branes, which in turn may spoil toric symmetry. Finally, we discuss duality cascades in this context and the interplay between Seiberg/toric duality and unoriented projection with (or without) mass deformations.
We discuss unoriented quivers with flavour that arise from D3-branes at local orbifold singularities, in the presence of Omega-planes and non-compact D7-branes. We produce a wide class of unoriented quiver gauge theories, including new instances of N=1 superconformal theories. We then consider unoriented D-brane instanton corrections of both `gauge' and `exotic' kinds. In particular, we show that conformal symmetry can be dynamically broken via the generation of exotic superpotentials. Finally we discuss aspects of the recently proposed N=1 remnant of N=4 S-duality. We identify new candidate d…
We study various orientifold projections of 4d $\mathcal{N}=1$ toric gauge theories, associated with CY singularities known as $L^{a,b,a}/\mathbb{Z}_2$, with $a+b$ even. We obtain superconformal chiral theories that have the same central charge, anomalies and superconformal index, whereas they were different before the orientifold. Some of these projections are implemented by a novel type of orientifold without fixed loci, known as glide orientifold. We claim that these theories flow to the same conformal manifold, and they are connected by quadratic exactly marginal deformations. The latter c…
We study the non-perturbative dynamics of an unoriented Z_5-quiver theory of GUT kind with gauge group U(5) and chiral matter. At strong coupling the non-perturbative dynamics is described in terms of set of baryon/meson variables satisfying a quantum deformed constraint. We compute the effective superpotential of the theory and show that it admits a line of supersymmetric vacua and a phase where supersymmetry is dynamically broken via gaugino condensation.
We compute the exact path integral of $\mathcal{N}=2$ supersymmetric gauge theories with general gauge group on $\mathbb{RP}^4$ and a $\mathbb{Z}_2$-quotient of the hemi-$S^4$. By specializing to $SU(2)$ superconformal quivers, we show that these, together with hemi-$S^4$ partition functions, compute Liouville correlators on unoriented/open Riemann surfaces. We perform explicit checks for Riemann surfaces obtained as $\mathbb{Z}_2$ quotients of the sphere and the torus. We also discuss the coupled $3d\!-\!4d$ systems associated to Liouville amplitudes with boundary punctures.
Starting from the $\mathcal{N}=2$ SYM$_{4}$ quiver theory living on wrapped $% N_{i}D5$ branes around $S_{i}^{2}$ spheres of deformed ADE fibered Calabi-Yau threefolds (CY3) and considering deformations using \textit{% massive} vector multiplets, we explicitly build a new class of $\mathcal{N}% =1 $ quiver gauge theories. In these models, the quiver gauge group $% \prod_{i}U(N_{i}) $ is spontaneously broken down to $% \prod_{i}SU(N_{i}) $ and Kahler deformations are shown to be given by the real part of the integral $(2,1) $ form of CY3. We also give the superfield correspondence between the $…
We study the relation between exactly marginal deformations in a large class of N=1 superconformal quiver gauge theories described by brane tilings and the degrees of freedom in the corresponding 5-brane systems. We show, with the help of NSVZ exact beta functions, that there are generically d-1 complex exactly marginal deformations of a gauge theory, where d is the perimeter of the corresponding n it, and the other two, the diagonal gauge coupling and a beta-like deformation, as background supergravity fields.toric diagram. We identify d-3 complex marginal deformations as deformations of the …
We discuss several examples of non-toric quiver gauge theories dual to Sasaki-Einstein manifolds with U(1)^2 or U(1) isometry. We give a general method for constructing non-toric examples by adding relevant deformations to the toric case. For all examples, we are able to make a complete comparison between the prediction for R-charges based on geometry and on quantum field theory. We also give a general discussion of the spectrum of conformal dimensions for mesonic and baryonic operators for a generic quiver theory; in the toric case we make an explicit comparison between R-charges of mesons an…
We study 4d SCFTs obtained by orientifold projections on necklace quivers with fractional branes. The models obtained by this procedure are $\mathcal{N}=1$ linear quivers with unitary, symplectic and orthogonal gauge groups, bifundamental and tensorial matter. Remarkably, models that are not dual in the unoriented case can have the same central charges and superconformal index after the projection. The reason for this behavior rests upon the ubiquitous presence of adjoint fields with R-charge one. We claim that the presence of such fields is at the origin of the notion of inherited S-duality o…
We study superconformal deformations of the $T_ρ^{\hatρ}[SU(N)]$ theories of Gaiotto-Hanany-Witten, paying special attention to mixed-branch operators with both electrically- and magnetically-charged fields. We explain why all marginal ${\cal N}=2$ operators of an ${\cal N}=4$ CFT$_3$ can be extracted unambiguously from the superconformal index. Computing the index at the appropriate order we show that the mixed moduli in $T_ρ^{\hatρ}[SU(N)]$ theories are double-string operators transforming in the (Adjoint, Adjoint) representation of the electric and magnetic flavour groups, up to some overco…
We describe how the geometry of the Higgs branch of 5d superconformal field theories is transformed under movement along the extended Coulomb branch. Working directly with the (unitary) magnetic quiver, we demonstrate a correspondence between Fayet-Iliopoulos deformations in 3d and 5d mass deformations. When the Higgs branch has multiple cones, characterised by a collection of magnetic quivers, the mirror map is not globally well-defined, however we are able to utilize the correspondence to establish a local version of mirror symmetry. We give several detailed examples of deformations, includi…
In this note we explore the stringy interpretation of non-perturbative effects in N=1^* deformations of the A_{k-1} quiver models. For certain types of deformations we argue that the massive vacua are described by Nk fractional D3-branes at the orbifold polarizing into k concentric 5-brane spheres each carrying fractional brane charge. The polarization of the D3-branes induces a polarization of D-instantons into string world-sheets wrapped on the Myers spheres. We show that the superpotentials in these models are indeed generated by these world-sheet instantons. We point out that for certain p…
Placing a set of branes at a Calabi-Yau singularity leads to an N=1 quiver gauge theory. We analyze F-term deformations of such gauge theories. A generic deformation can be obtained by making the Calabi-Yau non-commutative. We discuss non-commutative generalisations of well-known singularities such as the Del Pezzo singularities and the conifold. We also introduce new techniques for deriving superpotentials, based on quivers with ghosts and a notion of generalised Seiberg duality. The curious gauge structure of quivers with ghosts is most naturally described using the BV formalism. Finally we …
Magnetic quivers and Hasse diagrams for Higgs branches of rank $r$ 4d $\mathcal{N}=2$ SCFTs arising from $\mathbb{Z}_{\ell}$ $\mathcal{S}$-fold constructions are discussed. The magnetic quivers are derived using three different methods: 1) Using clues like dimension, global symmetry, and the folding parameter $\ell$ to guess the magnetic quiver. 2) From 6d $\mathcal{N}=(1,0)$ SCFTs as UV completions of 5d marginal theories, and specific FI deformations on their magnetic quiver, which is further folded by $\mathbb{Z}_{\ell}$. 3) From T-duality of Type IIA brane systems of 6d $\mathcal{N}=(1,0)$…
We study supersymmetry breaking metastable vacua arising from beta deformed quiver gauge theories. The relation between the bounds on metastability and the deformation are discussed. Metastable supersymmetry breaking vacua are found in the IR of beta deformed cascading quivers with vector-like field content. Furthermore the limiting case of massive Nf=Nc SQCD appears in the IR of gauge theories with chiral-like field content. We comment on the field theory origin of the deformation and on possible applications in AdS/CFT.
Renormalization group flows of quiver gauge theories play a central role in determining the low-energy properties of string vacua. We demonstrate that useful predictions about the RG dynamics of a quiver gauge theory may be extracted from the global structure of its quiver diagram. For quiver theories of a certain type, we develop an efficient and practical method for determining which superpotential deformations generate a flow to an interacting conformal fixed point.
We introduce bi-fermion fishnet theories, a class of models describing integrable sectors of four-dimensional gauge theories with non-maximal supersymmetry. Bi-fermion theories are characterized by a single complex scalar field and two Weyl fermions interacting only via chiral Yukawa couplings. The latter generate oriented Feynman diagrams forming hexagonal lattices, whose fishnet structure signals an underlying integrability that we exploit to compute anomalous dimensions of BMN-vacuum operators. Furthermore, we investigate Lunin-Maldacena deformations of $\mathcal{N}=2$ superconformal field …
We compute the prepotential for gauge theories descending from ${\cal N}=4$ SYM via quiver projections and mass deformations. This accounts for gauge theories with product gauge groups and bifundamental matter. The case of massive orientifold gauge theories with gauge group SO/Sp is also described. In the case with no gravitational corrections the results are shown to be in agreement with Seiberg-Witten analysis and previous results in the literature.
Dijkgraaf and Vafa have conjectured the correspondences between topological string theories, ${\cal N}=1$ gauge theories and matrix models. By the use of this conjecture, we calculate the quantum deformations of Calabi-Yau threefolds with ADE singularities from ADE multi-matrix models. We obtain the effective superpotentials of the dual quiver gauge theories in terms of the geometric engineering for the deformed geometries. We find the Veneziano-Yankielowicz terms in the effective superpotentials.
We construct a first order deformation of the complex structure of the cone over Sasaki-Einstein spaces Y^{p,q} and check supersymmetry explicitly. This space is a central element in the holographic dual of chiral symmetry breaking for a large class of cascading quiver theories. We discuss a solution describing a stack of N D3 branes and M fractional D3 branes at the tip of the deformed spaces.
Exact results are derived for 5d SCFTs with holographic duals in Type IIB supergravity. These theories have relevant deformations that flow to linear quiver gauge theories, with the number of nodes large in the large-$N$ limits described by supergravity. Starting from a suitable formulation of the matrix models resulting from supersymmetric localization of the squashed $S^5$ partition functions, the saddle point equations are solved for generic quivers with $N_f=2N$ at all interior nodes, which includes the $T_N$ theories, and for a sample of theories with $N_f\neq 2N$ nodes including theories…
We consider a four dimensional N=1 gauge theory with bifundamental matter and a superpotential, defined on stacks of fractional branes. By turning on a flux for the R-R graviphoton field strength and computing open string amplitudes with insertions of R-R closed string vertices, we introduce a non-anticommutative deformation and obtain the N=1/2 version of the theory. We also comment on the appearance of a new structure in the effective Lagrangian.
We develop a new method for constructing $3d$ $\mathcal{N}=4$ Coulomb branch chiral rings in terms of gauge-invariant generators and relations while making the global symmetry manifest. Our examples generalise to all balanced quivers of type $A$ and $D$ whose Coulomb branches are closures of nilpotent orbits. This new approach is a synthesis of operator counting using Hilbert series and explicit algebraic construction introduced by Bullimore, Dimofte and Gaiotto with significant potential for further generalisation to other quivers, including non-simply laced. The method also identifies comple…
We consider SU(3)-equivariant dimensional reduction of Yang-Mills theory on Kaehler manifolds of the form M x SU(3)/H, with H = SU(2) x U(1) or H = U(1) x U(1). The induced rank two quiver gauge theories on M are worked out in detail for representations of H which descend from a generic irreducible SU(3)-representation. The reduction of the Donaldson-Uhlenbeck-Yau equations on these spaces induces nonabelian quiver vortex equations on M, which we write down explicitly. When M is a noncommutative deformation of the space C^d, we construct explicit BPS and non-BPS solutions of finite energy for …
We review the role of integrability in the planar spectral problem of four-dimensional superconformal gauge theories besides N=4 SYM. The cases considered include the Leigh-Strassler marginal deformations of N=4 SYM, quiver theories which arise as orbifolds of AdS5xS5 on the dual gravity side, as well as various theories involving open spin chains.
The AdS/CFT correspondence between Sasaki-Einstein spaces and quiver gauge theories is studied from the perspective of massless BPS geodesics. The recently constructed toric Lpqr geometries are considered: we determine the dual superconformal quivers and the spectrum of BPS mesons. The conformal anomaly is compared with the volumes of the manifolds. The U(1)^2_F x U(1)_R global symmetry quantum numbers of the mesonic operators are successfully matched with the conserved momenta of the geodesics, providing a test of AdS/CFT duality. The correspondence between BPS mesons and geodesics allows to …
The complex dimension of the space of exactly marginal deformations for quiver CFTs dual to IIB theory compactified on $Y^{p,q}$ is known to be generically three. Simple general formulas already exist for two of the exactly marginal directions in the space of couplings, one of which corresponds to the sum of the (inverse squared of) gauge couplings, and the other to the $β$-deformation. Here we identify the third exactly marginal direction, which is dual to the modulus $\int B_{2}$ on the gravity side. This identification leads to a relation between the field theory gauge couplings and the vac…
We review the quiver matrix model (the ITEP model) in the light of the recent progress on 2d-4d connection of conformal field theories, in particular, on the relation between Toda field theories and a class of quiver superconformal gauge theories. On the basis of the CFT representation of the beta deformation of the model, a quantum spectral curve is introduced as << det (x- i g_s \partial ϕ(z)) >>=0 at finite N and for beta \neq 1. The planar loop equation in the large N limit follows with the aid of W_n constraints. Residue analysis is provided both for the curve of the matrix model with the…
Mirror symmetry has proven to be a powerful tool to study several properties of higher dimensional superconformal field theories upon compactification to three dimensions. We propose a quiver description for the mirror theories of the circle reduction of twisted $A_{2N}$ theories of class $\mathsf{S}$ in four dimensions. Although these quivers bear a resemblance to the star-shaped quivers previously studied in the literature, they contain unitary, symplectic and special orthogonal gauge groups, along with hypermultiplets in the fundamental representation. The vacuum moduli spaces of these quiv…
We study the Coulomb branches of 3d N=4 `star-shaped' quiver gauge theories and their deformation quantizations, by applying algebraic techniques that have been developed in the mathematics and physics literature over the last few years. The algebraic techniques supply an abelianization map, which embeds the Coulomb-branch chiral ring into a vastly simpler abelian algebra A. Relations among chiral-ring operators, and their deformation quantization, are canonically induced from the embedding into A. In the case of star-shaped quivers -- whose Coulomb branches are related to Higgs branches of 4d…
D-branes can end on orbifold planes if the action of the orbifold group includes (-1)^{F_L}. We consider configurations of D-branes ending on such orbifolds and study the low-energy theory on their worldvolume. We apply our results to gauge theories with eight supercharges in three and four dimensions. We explain how mirror symmetry for N=4 d=3 gauge theories with gauge group Sp(k) and matter in the antisymmetric tensor and fundamental representations follows from S-duality of IIB string theory. We argue that some of these theories have hidden Fayet-Iliopoulos deformations, not visible classic…
We study the duality group of $\hat{A}_{n-1}$ quiver gauge theories, primarily using their M5-brane construction. For $\mathcal{N}=2$ supersymmetry, this duality group was first noted by Witten to be the mapping class group of a torus with $n$ punctures. We find that it is a certain quotient of this group that acts faithfully on gauge couplings. This quotient group contains the affine Weyl group of $\hat{A}_{n-1}$, $\mathbb{Z}_n$ and $SL(2,\mathbb{Z})$. In fact there are $n$ non-commuting $SL(2,\mathbb{Z})$ subgroups, related to each other by conjugation using the $\mathbb{Z}_n$. When supersym…
We will demonstrate how calculations in toric geometry can be used to compute quantum corrections to the relations in the chiral ring for certain gauge theories. We focus on the gauge theory of the del Pezzo 2, and derive the chiral ring relations and quantum deformations to the vacuum moduli space using Affleck-Dine-Seiberg superpotential arguments. Then we calculate the versal deformation to the corresponding toric geometry using a method due to Altmann, and show that the result is equivalent to the deformation calculated using gauge theory. In an appendix we will apply this technique to a f…
Fractional branes added to a large stack of D3-branes at the singularity of a Calabi-Yau cone modify the quiver gauge theory breaking conformal invariance and leading to different kinds of IR behaviors. For toric singularities admitting complex deformations we propose a simple method that allows to compute the anomaly free rank distributions in the gauge theory corresponding to the fractional deformation branes. This algorithm fits Altmann's rule of decomposition of the toric diagram into a Minkowski sum of polytopes. More generally we suggest how different IR behaviors triggered by fractional…
We present a local computation of deformations of the tangent bundle for a resolved orbifold singularity C^d/G. These correspond to (0,2)-deformations of (2,2)-theories. A McKay-like correspondence is found predicting the dimension of the space of first-order deformations from simple calculations involving the group. This is confirmed in two dimensions using the Kronheimer-Nakajima quiver construction. In higher dimensions such a computation is subject to nontrivial worldsheet instanton corrections and some examples are given where this happens. However, we conjecture that the special crepant …
We construct an $ε$-deformation of W algebras, corresponding to the additive version of quiver $\text{W}_{q,t^{-1}}$ algebras which feature prominently in the 5d version of the BPS/CFT correspondence and refined topological strings on toric Calabi-Yau's. This new type of algebras fill in the missing intermediate level between $q$-deformed and ordinary W algebras. We show that $ε$-deformed W algebras are spectral duals of conventional W algebras, in particular the $ε$-deformed conformal blocks manifestly reproduce instanton partition functions of 4d $\mathcal{N}=2$ quiver gauge theories in the …
We study half-BPS surface operators in N=2supersymmetric asymptotically conformal gauge theories in four dimensions with SU(N) gauge group and 2N fundamental flavours using localization methods and coupled 2d/4d quiver gauge theories. We show that contours specified by a particular Jeffrey-Kirwan residue prescription in the localization analysis map to particular realizations of the surface operator as flavour defects. Seiberg duality of the 2d/4d quivers is mapped to contour deformations of the localization integral which in this case involves a residue at infinity. This is reflected as a mod…
A new dynamic SU(3)-structure solution in type-IIA is found by T-dualising a deformation of the Maldacena-Nastase solution along an SU(2) isometry. It is argued that this is dual to a quiver gauge theory with multiple Chern-Simons levels. A clear way of defining Chern-Simons levels in terms of Page charges is presented, which is also used to define a Chern-Simons term for the G2-structure analogue of Klebanov-Strassler, providing evidence of a cascade in both the ranks and levels of the dual quiver.
Deformation of N=2 quiver gauge theories by adjoint masses leads to fixed manifolds of N=1 superconformal field theories. We elaborate on the role of the complex three-form flux in the IIB duals to these fixed point theories, primarily using field theory techniques. We study the moduli space at a fixed point and find that it is either the two (complex) dimensional ALE space or three-dimensional generalized conifold, depending on the type of three-form flux that is present. We describe the exactly marginal operators that parameterize the fixed manifolds and find the operators which preserve the…
We study low-energy dynamics of $[SU(N)]^K$ chiral quiver gauge theories in connection with $\mathcal{N}=1$ super Yang-Mills (SYM) theory, and quantum chromodynamics with bi-fundamental fermions (QCD(BF)). These theories can be obtained by $\mathbb{Z}_K$ orbifold projections of $\mathcal{N}=1$ $SU(NK)$ SYM theory, but the perturbative planar equivalence does not extend nonperturbatively for $K\ge 3$. In order to study low-energy behaviors, we analyze these systems using 't~Hooft anomaly matching and reliable semiclassics on $\mathbb{R}^3\times S^1$. Thanks to 't~Hooft anomaly that involves $1$…
5d superconformal field theories (SCFTs) can be obtained from 6d SCFTs by circle compactification and mass deformation. Successive decoupling of hypermultiplet matter and RG-flow generates a decoupling tree of descendant 5d SCFTs. In this paper we determine the magnetic quivers and Hasse diagrams, that encode the Higgs branches of 5d SCFTs, for entire decoupling trees. Central to this undertaking is the approach in arXiv:2008.05577, which, starting from the generalized toric polygons (GTPs) dual to 5-brane webs/tropical curves, provides a systematic and succinct derivation of magnetic quivers …
We investigate a squashing deformation of 3d N=2 supersymmetric theories on three-sphere, which have four supercharges. The deformation preserves SU(2)_L x U(1)_r isometry and all four supersymmetries. We compute the partition function and find non-trivial dependence on the squashing parameter. We also consider the large N limit of a certain class of quiver gauge theories which have free energy of order N^{3/2}, and show that the free energy on the squashed sphere differs from that on round sphere by a certain factor depending only on the squashing parameter.
We introduce and explore the relation between quivers and 3-manifolds with the topology of the knot complement. This idea can be viewed as an adaptation of the knots-quivers correspondence to Gukov-Manolescu invariants of knot complements (also known as $F_K$ or $\hat{Z}$). Apart from assigning quivers to complements of $T^{(2,2p+1)}$ torus knots, we study the physical interpretation in terms of the BPS spectrum and general structure of 3d $\mathcal{N}=2$ theories associated to both sides of the correspondence. We also make a step towards categorification by proposing a $t$-deformation of all …
In the so-called (2,2) theory, which is the U(N)^4 circular quiver superconformal Chern-Simons theory with levels (k,0,-k,0), it was known that the instanton effects are described by the free energy of topological strings whose Gopakumar-Vafa invariants coincide with those of the local D_5 del Pezzo geometry. By considering two types of one-parameter rank deformations U(N) x U(N+M) x U(N+2M) x U(N+M) and U(N+M) x U(N) x U(N+M) x U(N), we classify the known diagonal BPS indices by degrees. Together with other two types of one-parameter deformations, we further propose the topological string exp…
The prepotentials for the quiver supersymmetric gauge theories are defined as quasiclassical tau-functions, depending on two different sets of variables: the parameters of the UV gauge theory or the bare compexified couplings, and the vacuum condensates of the theory in IR. The bare couplings are introduced as periods on the UV base curve, and the consistency of corresponding gradient formulas for the tau-functions is proven using the Riemann bilinear relations. It is shown, that dependence of generalised prepotentials for the quiver gauge theories upon the bare couplings turns to coincide wit…
We construct three-dimensional N=2 Chern-Simons-quiver theories which are holographically dual to the M-theory Freund-Rubin solutions AdS_4 x V_{5,2}/Z_k (with or without torsion G-flux), where V_{5,2} is a homogeneous Sasaki-Einstein seven-manifold. The global symmetry group of these theories is generically SU(2) x U(1) x U(1)_R, and they are hence non-toric. The field theories may be thought of as the n=2 member of a family of models, labelled by a positive integer n, arising on multiple M2-branes at certain hypersurface singularities. We describe how these models can be engineered via gener…
The Coulomb branches of certain 3-dimensional N=4 quiver gauge theories are closures of nilpotent orbits of classical or exceptional algebras. The monopole formula, as Hilbert series of the associated Coulomb branch chiral ring, has been successful in describing the singular hyper-Kähler structure. By means of the monopole formula with background charges for flavour symmetries, which realises real mass deformations, we study the resolution properties of all (characteristic) height two nilpotent orbits. As a result, the monopole formula correctly reproduces (i) the existence of a symplectic res…
We derive the structure of the Higgs branch of 5d superconformal field theories or gauge theories from their realization as a generalized toric polygon (or dot diagram). This approach is motivated by a dual, tropical curve decomposition of the $(p,q)$ 5-brane-web system. We define an edge coloring, which provides a decomposition of the generalized toric polygon into a refined Minkowski sum of sub-polygons, from which we compute the magnetic quiver. The Coulomb branch of the magnetic quiver is then conjecturally identified with the 5d Higgs branch. Furthermore, from partial resolutions, we iden…
We study macroscopically two dimensional $\mathcal{N}=(2,2)$ supersymmetric gauge theories constructed by compactifying the quiver gauge theories with eight supercharges on a product $\mathbb{T}^{d} \times \mathbb{R}^{2}_ε$ of a $d$-dimensional torus and a two dimensional cigar with $Ω$-deformation. We compute the universal part of the effective twisted superpotential. In doing so we establish the correspondence between the gauge theories, quantization of the moduli spaces of instantons on $\mathbb{R}^{2-d} \times \mathbb{T}^{2+d}$ and singular monopoles on $\mathbb{R}^{2-d} \times \mathbb{T}^…
