Reflexive polygons have been extensively studied in a variety of contexts in mathematics and physics. We generalize this programme by looking at the 45 different lattice polygons with two interior points up to SL(2,$\mathbb{Z}$) equivalence. Each corresponds to some affine toric 3-fold as a cone over a Sasaki-Einstein 5-fold. We study the quiver gauge theories of D3-branes probing these cones, which coincide with the mesonic moduli space. The minimum of the volume function of the Sasaki-Einstein base manifold plays an important role in computing the R-charges. We analyze these minimized volumes with respect to the topological quantities of the compact surfaces constructed from the polygons. Unlike reflexive polytopes, one can have two fans from the two interior points, and hence give rise to two smooth varieties after complete resolutions, leading to an interesting pair of closely related geometries and gauge theories.
We construct N=1 supersymmetric theories on worldvolumes of D5 branes wrapped around 2-cycles of threefolds which are A-D-E fibrations over a plane. We propose large N duals as geometric transitions involving blowdowns of two cycles and blowups of three-cycles. This yields exact predictions for a large class of N=1 supersymmetric gauge systems including U(N) gauge theories with two adjoint matter fields deformed by superpotential terms, which arise in A-D-E fibered geometries with non-trivial monodromies.
We study effective quiver gauge theories arising from a stack of D3-branes on certain Calabi-Yau singularities. Our point of view is a first principle approach via open topological string theory. This means that we construct the natural A-infinity-structure of open string amplitudes in the associated D-brane category. Then we show that it precisely reproduces the results of the method of brane tilings, without having to resort to any effective field theory computations. In particular, we prove a general and simple formula for effective superpotentials.
We study Grothendieck's dessins d'enfants in the context of the $\mathcal{N}=2$ supersymmetric gauge theories in $\left(3+1\right)$ dimensions with product $SU\left(2\right)$ gauge groups which have recently been considered by Gaiotto et al. We identify the precise context in which dessins arise in these theories: they are the so-called ribbon graphs of such theories at certain isolated points in the Coulomb branch of the moduli space. With this point in mind, we highlight connections to other work on trivalent dessins, gauge theories, and the modular group.
We identify Moore-Read wavefunctions, describing non-abelian statistics in fractional quantum Hall systems, with the instanton partition of N=2 superconformal quiver gauge theories at suitable values of masses and Ω-background parameters. This is obtained by extending to rational conformal field theories the SU(2) gauge quiver/Liouville field theory duality recently found by Alday-Gaiotto-Tachikawa. A direct link between the Moore-Read Hall $n$-body wavefunctions and Z_n-equivariant Donaldson polynomials is pointed out.
Magnetic quivers and Hasse diagrams for Higgs branches of rank 1 $4d$ $\mathcal{N}=2$ SCFTs are provided. These rank 1 theories fit naturally into families of higher rank theories, originating from higher dimensions, which are addressed.
We study metastable dynamical breaking of supersymmetry in A_n quiver gauge theories. We present a general analysis and criteria for the perturbative existence of metastable vacua in quivers of any length. Different mechanisms of gauge mediation can be realized.
We investigate orthosymplectic quivers that take the shape of D-type and B-type Dynkin diagrams. The D-type orthosymplectic quivers explored here contain a balanced "fork", i.e., a balanced subquiver with a D-type bifurcation, whereas the B-type orthosymplectic quivers are obtained by folding the D-type quivers. The Coulomb branches of these quivers are products of two moduli spaces. In the second part, the relevant orthosymplectic quivers are shown to emerge as magnetic quivers for brane configurations involving ON$^0$ planes. Notably, the appearance of ON$^0$ plane clarifies the product natu…
We construct a new infinite family of quiver gauge theories which blow down to the X^{p,q} quiver gauge theories found by Hanany, Kazakopoulos and Wecht. This family includes a quiver gauge theory for the third del Pezzo surface. We show, using Z-minimaization, that these theories generically have irrational R-charges. The AdS/CFT correspondence implies that the dual geometries are irregular toric Sasaki-Einstein manifolds, although we do not know the explicit metrics.
In this note we construct magnetic quivers for the known rank-2 four dimensional $\mathcal{N}=2$ superconformal field theories. For every rank-1 theory one can find a unitary magnetic quiver; we observe that this is no longer possible at rank 2. Our list of magnetic quivers necessarily includes orthosymplectic quivers, in addition to unitary ones, of both the simply and non-simply laced variety. Using quiver subtraction, one can compute Higgs branch Hasse diagrams and compare with the results obtained via other methods finding nearly perfect agreement.
We compute the index for orbifold quiver gauge theories. We compare it with the results obtained from the type IIB supergravity (superstring) on AdS_5 \times S^5/Γ.
We initiate a systematic investigation of the space of 2+1 dimensional quiver gauge theories, emphasising a succinct "forward algorithm". Few "order parametres" are introduced such as the number of terms in the superpotential and the number of gauge groups. Starting with two terms in the superpotential, we find a generating function, with interesting geometric interpretation, which counts the number of inequivalent theories for a given number of gauge groups and fields. We demonstratively list these theories for some low numbers thereof. Furthermore, we show how these theories arise from M2-br…
We construct a new infinite family of N=1 quiver gauge theories which can be Higgsed to the Y^{p,q} quiver gauge theories. The dual geometries are toric Calabi-Yau cones for which we give the toric data. We also discuss the action of Seiberg duality on these quivers, and explore the different Seiberg dual theories. We describe the relationship of these theories to five dimensional gauge theories on (p,q) 5-branes. Using the toric data, we specify some of the properties of the corresponding dual Sasaki-Einstein manifolds. These theories generically have algebraic R-charges which are not quadrat…
We argue that quiver gauge theories with $SU(N)$ gauge groups give rise to lattice gauge theories with matter possessing fractonic properties, where the lattice is the quiver itself. This idea extends a recent proposal by Razamat. This class of theories exhibit a $\mathbb{Z}_N$ 1-form global symmetry that can be used to classify their phases. The order parameter of this transition is the expectation value of Wilson loops, which correspond to mesonic operators in the underlying quiver gauge theory. We discuss how this perspective naturally fits with the deconstruction of a higher dimensional th…
We consider N=2 quiver Chern-Simons theories described by brane tilings, whose moduli spaces are toric Calabi-Yau 4-folds. Simple prescriptions to obtain toric data of the moduli space and a corresponding brane crystal from a brane tiling are proposed.
We study Seiberg duality of quiver gauge theories associated to the complex cone over the second del Pezzo surface. Homomorphisms in the path algebra of the quivers in each of these cases satisfy relations which follow from a superpotential of the corresponding gauge theory as F-flatness conditions. We verify that Seiberg duality between each pair of these theories can be understood as a derived equivalence between the categories of modules of representation of the path algebras of the quivers. Starting from the projective modules of one quiver we construct tilting complexes whose endomorphism…
We discuss a general procedure to obtain 1/2 BPS partition functions for generic N=1 quiver gauge theories. These functions count the gauge invariant operators (bosonic and fermionic), charged under all the global symmetries (mesonic and baryonic), in the chiral ring of a given quiver gauge theory. In particular we discuss the inclusion of the spinor degrees of freedom in the partition functions.
We propose a relation between correlation functions in the 2d A_{N-1} conformal Toda theories and the Nekrasov instanton partition functions in certain conformal N=2 SU(N) 4d quiver gauge theories. Our proposal generalises the recently uncovered relation between the Liouville theory and SU(2) quivers. New features appear in the analysis that have no counterparts in the Liouville case.
The possibility that non-supersymmetric quiver theories may have a renormalization-group fixed point at which there is conformal invariance requires non-perturbative information.
We demonstrate a practical and efficient method for generating toric Calabi-Yau quiver theories, applicable to both D3 and M2 brane world-volume physics. A new analytic method is presented at low order parametres and an algorithm for the general case is developed which has polynomial complexity in the number of edges in the quiver. Using this algorithm, carefully implemented, we classify the quiver diagram and assign possible superpotentials for various small values of the number of edges and nodes. We examine some preliminary statistics on this space of toric quiver theories.
We show by direct construction that a large class of quiver gauge theories admits actions of finite Heisenberg groups. We consider various quiver gauge theories that arise as AdS/CFT duals of orbifolds of C^3, the conifold and its orbifolds and some orbifolds of the cone over Y(p,q). Matching the gauge theory analysis with string theory on the corresponding spaces implies that the operators counting wrapped branes do not commute in the presence of flux.
Quiver theories arising on D3-branes at orbifold and del Pezzo singularities are studied using mirror symmetry. We show that the quivers for the orbifold theories are given by the soliton spectrum of massive 2d N=2 theory with weighted projective spaces as target. For the theories obtained from the del Pezzo singularities we show that the geometry of the mirror manifold gives quiver theories related to each other by Picard-Lefschetz transformations, a subset of which are simple Seiberg duals. We also address how one indeed derives Seiberg duality on the matter content from such geometrical tra…
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 show that the Higgs branch of a four-dimensional Yang-Mills theory, with gauge and matter content summarised by an ADE quiver diagram, is identical to the generalised Coulomb branch of a four-dimensional superconformal strongly coupled gauge theory with ADE global symmetry. This equivalence suggests the existence of a mirror symmetry between the quiver theories and the strongly coupled theories.
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.
In this paper we detail the lattice constructions of several classes of supersymmetric quiver gauge theories in two and three Euclidean spacetime dimensions possessing exact supersymmetry at finite lattice spacing. Such constructions are obtained through the methods of topological twisting and geometric discretization of Euclidean Yang--Mills theories with eight and sixteen supercharges in two and three dimensions. We detail the lattice constructions of two-dimensional quiver gauge theories possessing four and eight supercharges and three-dimensional quiver gauge theories possessing eight supe…
We describe an infinite family of quiver gauge theories that are AdS/CFT dual to a corresponding class of explicit horizon Sasaki-Einstein manifolds. The quivers may be obtained from a family of orbifold theories by a simple iterative procedure. A key aspect in their construction relies on the global symmetry which is dual to the isometry of the manifolds. For an arbitrary such quiver we compute the exact R-charges of the fields in the IR by applying a-maximization. The values we obtain are generically quadratic irrational numbers and agree perfectly with the central charges and baryon charges…
We consider non-supersymmetric quiver theories obtained by orbifolding the N=4 supersymmetric U(K) gauge theory by a discrete Z_Γgroup embedded in the SU(4) R-symmetry group. We explicitly find that in such theories there are no one-loop quadratic divergences in the effective potential. Moreover, when the gauge group U(n)^Γof the quiver theory is spontaneously broken down to the diagonal U(n), we identify a custodial supersymmetry which is responsible for the fermion-boson degeneracy of the mass spectrum.
We study N=1 four dimensional quiver theories arising on the worldvolume of D3-branes at del Pezzo singularities of Calabi-Yau threefolds. We argue that under local mirror symmetry D3-branes become D6-branes wrapped on a three torus in the mirror manifold. The type IIB (p,q) 5-brane web description of the local del Pezzo, being closely related to the geometry of its mirror manifold, encodes the geometry of 3-cycles and is used to obtain gauge groups, quiver diagrams and the charges of the fractional branes.
We compute R-charges of the BPS-monopole operators in $\mathcal{N} = 3$ $\widehat{ADE}$ Chern-Simons quiver gauge theories, along the lines of the work of Benna, Klebanov and Klose in \cite{bkk}. These theories have a weakly coupled UV completion in terms of $\mathcal{N}=3$ supersymmetric Chern-Simons Yang-Mills theories. In the UV limit the monopole operators are well approximated by classical solutions. We construct classical BPS and anti-BPS monopole solutions to these theories which preserve $\frac{1}{3}$ supersymmetry all along the RG flow. We compute the $SU(2)_R$ charges in these backgr…
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 compute superpotentials for quiver gauge theories arising from marginal D-Brane decay on collapsed del Pezzo cycles S in a Calabi-Yau X. This is done using the machinery of A-infinity products in the derived category of coherent sheaves of X, which in turn is related to the derived category of S and quiver path algebras. We confirm that the superpotential is what one might have guessed from analyzing the moduli space, i.e., it is linear in the fields corresponding to the Ext2's of the quiver and that each such Ext2 multiplies a polynomial in Ext1's equal to precisely the relation represente…
The open string sector of the topological B-model on CY $(m+2)$-folds is described by $m$-graded quivers with superpotentials. This correspondence generalizes the connection between CY $(m+2)$-folds and gauge theories on the worldvolume of D$(5-2m)$-branes for $m=0,\ldots,3$ to arbitrary $m$. In this paper we introduce the Calabi-Yau product, a new algorithm that starting from the known quiver theories for a pair of toric CY$_{m+2}$ and CY$_{n+2}$ produces the quiver theory for a related CY$_{m+n+3}$. This method significantly supersedes existing ones, enabling the simple determination of quiv…
We compute superconformal indices for $\mathcal{N} = 3$ $\hat{ADE}$ Chern-Simons quiver gauge theories with a product gauge group $\prod_i U(N)_i$, using the method of supersymmetric localization. We also perform a large $N$ analysis of the index. This index includes contribution from non zero magnetic flux sector. The fact that these theories have a weakly coupled UV completion in terms of $\mathcal{N}= 3$ supersymmetric Chern-Simons Yang-Mills theories enables us to apply the localization technique. Such theories have dual M-theory description on $\mathrm{AdS}_4\times M_7$, where $M_7$ is a …
For a special family of 4d $\mathcal{N}=2$ superconformal quiver theories, the worldsheet dual corresponding to the free theory is identified. This result is obtained from the recently proposed worldsheet dual of free $\mathcal{N}=4$ SYM by applying a $\mathbb{Z}_k$ orbifold. In support of our proposal we show that the spectrum of both the untwisted as well as the twisted sectors coincides between the two descriptions.
We consider a general class of symmetries of hyper-Kahler quotients which can be interpreted as classical analogs of Seiberg duality for N=2 supersymmetric quiver gauge theories in the baryonic Higgs branch. Along the way we find that a limit application of this duality yields new exotic realizations of ALE spaces as hyper-Kahler quotients. We also discuss how these results admit in some particular cases a natural interpretation in string theory. We finally comment on a possible relationship with Fourier-Mukai tranforms.
We study Coulomb branch moduli spaces of a class of three dimensional $\mathcal{N}=4$ gauge theories whose quiver satisfies the balance condition. The Coulomb branch is described by dressed monopole operators which can be counted using the Monopole formula. We mainly focus on A-type quivers in this paper, using Hilbert Series to study their moduli spaces, and present the interesting pattern which emerges. All of these balanced A-type quiver gauge theories can be realized on brane intervals in Type IIB string theory, where mirror symmetry acts by exchanging the five branes and induces an equiva…
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 …
We investigate the N=2 superconformal index for supersymmetric quiver Chern-Simons theories with large N gauge groups. After general arguments about the large N limit, we compute the first few terms in the series expansion of the index for theories proposed as dual theories to homogeneous spaces V^{5,2}, Q^{1,1,1}, Q^{2,2,2}, M^{1,1,1}, and N^{0,1,0}. We confirm that the indices have symmetries expected from the isometries of dual manifolds.
We discuss quiver gauge models with matter fields based on Dynkin diagrams of Lie superalgebra structures. We focus on A(1,0) case and we find first that it can be related to intersecting complex cycles with genus $g$. Using toric geometry, A(1,0) quivers are analyzed in some details and it is shown that A(1,0) can be used to incorporate fundamental fields to a product of two unitary factor groups. We expect that this approach can be applied to other kinds of Lie superalgebras;
Using the Eguchi-Kawai equivalence, we provide regularizations of supersymmetric quiver and Chern-Simons gauge theories which leave the supersymmetries unbroken. This allow us to study many interesting theories on a computer. As examples we construct large-$N$ reduced models of supersymmetric QCD with flavor and the ABJM model of multiple M2 branes.
We construct new integrable systems describing particles with internal spin from four-dimensional $\mathcal{N}=2$ quiver gauge theories. The models can be quantized and solved exactly using the quantum inverse scattering method and also using the Bethe/Gauge correspondence.
We utilise SUSY quiver gauge theories to compute properties of Slodowy slices; these are spaces transverse to the nilpotent orbits of a Lie algebra $\mathfrak g$. We analyse classes of quiver theories, with Classical gauge and flavour groups, whose Higgs branch Hilbert series are the intersections between Slodowy slices and the nilpotent cone $\mathcal S\cap \mathcal N$ of $\mathfrak{g}$. We calculate refined Hilbert series for Classical algebras up to rank $4$ (and $A_5$), and find descriptions of their representation matrix generators as algebraic varieties encoding the relations of the chir…
Effective field theories in type I and II superstring theories for D-branes located at points in the orbifold C^2/Z_n are supersymmetric gauge theories whose field content is conveniently summarized by a `quiver diagram,' and whose Lagrangian includes non-metric couplings to the orbifold moduli: in particular, twisted sector moduli couple as Fayet-Iliopoulos terms in the gauge theory. These theories describe D-branes on resolved ALE spaces. Their spaces of vacua are moduli spaces of smooth ALE metrics and Yang-Mills instantons, whose metrics are explicitly computable. For U(N) instantons, the …
We analyse the classical moduli spaces of supersymmetric vacua of 3d N=2 Chern-Simons quiver gauge theories. We show quite generally that the moduli space of the 3d theory always contains a baryonic branch of a parent 4d N=1 quiver gauge theory, where the 4d baryonic branch is determined by the vector of 3d Chern-Simons levels. In particular, starting with a 4d quiver theory dual to a 3-fold singularity, for certain general choices of Chern-Simons levels this branch of the moduli space of the corresponding 3d theory is a 4-fold singularity. Our results lead to a simple general method, using ex…
A large class of quiver gauge theories admits the action of finite Heisenberg groups of the form Heis(Z_q x Z_q). This Heisenberg group is generated by a manifest Z_q shift symmetry acting on the quiver along with a second Z_q rephasing (clock) generator acting on the links of the quiver. Under Seiberg duality, however, the action of the shift generator is no longer manifest, as the dualized node has a different structure from before. Nevertheless, we demonstrate that the Z_q shift generator acts naturally on the space of all Seiberg dual phases of a given quiver. We then prove that the space …
We explore the relationship between four-dimensional N=2 quantum field theories and their associated BPS quivers. For a wide class of theories including super-Yang-Mills theories, Argyres-Douglas models, and theories defined by M5-branes on punctured Riemann surfaces, there exists a quiver which implicitly characterizes the field theory. We study various aspects of this correspondence including the quiver interpretation of flavor symmetries, gauging, decoupling limits, and field theory dualities. In general a given quiver describes only a patch of the moduli space of the field theory, and a ke…
We attempt to derive quiver Chern-Simons-matter theories from the Bagger-Lambert theory with Nambu bracket, through an orbifold prescription which effectively induces a dimensional reduction of the internal space for 3-algebra. We consider M2-branes on an N=4 orbifold $C^2/Z_k\times C^2$, and compare the result with the so-called dual Aharony-Bergman-Jafferis-Maldacena model, proposed recently by Hanany, Vegh, and Zaffaroni. Unlike the N=6 example $C^4/Z_k$, we find ambiguities in the matrix regularization.
We study topological and integrable aspects of $\hat{c}=1$ strings. We consider the circle line theories 0A and 0B at particular radii, and the super affine theories at their self-dual radii. We construct their ground rings, identify them with certain quotients of the conifold, and suggest topological B-model descriptions. We consider the partition functions, correlators and Ward identities, and construct a Kontsevich-like matrix model. We then study all these aspects via the topological B-model description. Finally, we analyse the corresponding Dijkgraaf-Vafa type matrix models and quiver gau…
The web of dual gauge theories engineered from a class of toric Calabi-Yau threefolds is explored. In previous work, we have argued for a triality structure by compiling evidence for the fact that every such manifold $X_{N,M}$ (for given $(N,M)$) engineers three a priori different, weakly coupled quiver gauge theories in five dimensions. The strong coupling regime of the latter is in general described by Little String Theories. Furthermore, we also conjectured that the manifold $X_{N,M}$ is dual to $X_{N',M'}$ if $NM=N'M'$ and $\text{gcd}(N,M)=\text{gcd}(N',M')$. Combining this result with the…
Quiver gauge theories with a large number of nodes host a wealth of Wilson loop operators. Expectation values are obtained, using supersymmetric localization, for Wilson loops in the antisymmetric representations associated with each individual gauge node, for a sample of 5d long quiver gauge theories whose UV fixed points have holographic duals in Type IIB. The sample includes the $T_N$ theories and the results are uniformly given in terms of Bloch-Wigner functions. The holographic representation of the Wilson loops is identified. It comprises, for each supergravity solution, a two-parameter …
We build on previous studies of the Higgs and Coulomb branches of SUSY quiver theories having 8 supercharges, including $3d~{\cal N}=4$, and Classical gauge groups. The vacuum moduli spaces of many such theories can be parameterised by pairs of nilpotent orbits of Classical Lie algebras; they are transverse to one orbit and intersect the closure of the second. We refer to these transverse spaces as Slodowy intersections. They embrace reduced single instanton moduli spaces, nilpotent orbits, Kraft-Procesi transitions and Slodowy slices, as well as other types. We show how quiver subtractions, b…
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.
We apply the matrix model of Kapustin, Willett and Yaakov to compute the free energy of N=3 Chern-Simons matter theories with D_n quivers in the large N limit. We conjecture a general expression for the free energy that is explicitly invariant under Seiberg duality and show that it can be interpreted as a sum over certain graphs known as signed graphs. Through the AdS/CFT correspondence, this leads to a prediction for the volume of certain tri-Sasaki Einstein manifolds. We also study the unfolding procedure, which relates these D_n quivers to A_{2n-5} quivers. Furthermore, we consider the addi…
We study two types of discrete operations on Coulomb branches of $3d$ $\mathcal{N}=4$ quiver gauge theories using both abelianisation and the monopole formula. We generalise previous work on discrete quotients of Coulomb branches and introduce novel wreathed quiver theories. We further study quiver folding which produces Coulomb branches of non-simply laced quivers.
We show that the Bagger-Lambert-Gustavsson (BLG) theory with two pairs of negative norm generators is derived from the scaling limit of an orbifolded Aharony-Bergman-Jafferis-Maldacena (ABJM) theory. The BLG theory with many Lorentzian pairs is known to be reduced to the Dp-brane theory via Higgs mechanism, so our scaling procedure can be used to derive Dp-branes directly from M2-branes in the field theory language. In this paper, we focus on the D3-brane case and investigate the scaling limits of various quiver Chern-Simons theories obtained from different orbifolding actions. Remarkably, in …
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}^…
