Quiver Chern-Simons Research Articles

Dualities and flavored indices of M2-brane SCFTs

We study various conjectural dual descriptions of a stack of M2-branes in M-theory including ADHM, ABJ(M), BLG, discrete gauge theories and quiver Chern-Simons (CS) theories and propose several new dualities of the M2-brane SCFTs by analyzing flavored supersymmetric indices in detail. The mapping of local operators, Coulomb, Higgs and mixed branch operators as well as global symmetries under the dualities are obtained from the precise matching of the indices. Furthermore, we find closed form expressions for the Coulomb limit of the indices of the U(N) ADHM theory and the dual quiver CS theory for arbitrary N and propose a refined generating function for plane partitions with trace N. For the quiver CS theories we also find an infinite-sum expression for the Higgs limit of the indices which is more useful than the original expression.

3D N=2 ADE^ Chern-Simons quivers

We study 3d $\mathcal{N}=2$ Chern-Simons (CS) quiver theories on ${S}^{3}$ and ${\mathrm{\ensuremath{\Sigma}}}_{\mathfrak{g}}\ifmmode\times\else\texttimes\fi{}{S}^{1}$. Using localization results, we examine their partition functions in the large rank limit and requiring the resulting matrix models to be local, find a large class of quiver theories that include quivers in one-to-one correspondence with the $\stackrel{^}{ADE}$ Dynkin diagrams. We compute explicitly the partition function on ${S}^{3}$ for $\stackrel{^}{D}$ quivers and that on ${\mathrm{\ensuremath{\Sigma}}}_{\mathfrak{g}}\ifmmode\times\else\texttimes\fi{}{S}^{1}$ for $\stackrel{^}{AD}$ quivers, which lead to certain predictions for their holographic duals. We also provide a new and simple proof of the ``index theorem,'' extending its applicability to a larger class of theories than considered before in the literature.

Non-toric cones and Chern-Simons quivers

We obtain an integral formula for the volume of non-toric tri-Sasaki Einstein manifolds arising from nonabelian hyperkähler quotients. The derivation is based on equivariant localization and generalizes existing formulas for Abelian quotients, which lead to toric manifolds. The formula is particularly valuable in the context of AdS4 ×Y7 vacua of M-theory and their field theory duals. As an application, we consider 3d mathcal{N}=3 Chern-Simons theories with affine ADE quivers. While the  series corresponds to toric Y7, the widehat{D} and Ê series are non-toric. We compute the volumes of the corresponding seven-manifolds and compare to the prediction from supersymmetric localization in field theory, finding perfect agreement. This is the first test of an infinite number of non-toric AdS4/CFT3 dualities.

D2-brane Chern-Simons theories: F -maximization = a-maximization

We study a system of N D2-branes probing a generic Calabi-Yau three-fold singularity in the presence of a non-zero quantized Romans mass n. We argue that the low-energy effective N = 2 Chern-Simons quiver gauge theory flows to a superconformal fixed point in the IR, and construct the dual AdS_4 solution in massive IIA supergravity. We compute the free energy F of the gauge theory on S^3 using localization. In the large N limit we find F = c(nN)^{1/3}a^{2/3}, where c is a universal constant and a is the a-function of the "parent" four-dimensional N = 1 theory on N D3-branes probing the same Calabi-Yau singularity. It follows that maximizing F over the space of admissible R-symmetries is equivalent to maximizing a for this class of theories. Moreover, we show that the gauge theory result precisely matches the holographic free energy of the supergravity solution, and provide a similar matching of the VEV of a BPS Wilson loop operator.

Connecting mirror symmetry in 3D and 2D via localization

We explicitly apply localization results to study the interpolation between three- and two-dimensional mirror symmetries for Abelian gauge theories with four supercharges. We first use the ellipsoid [Formula: see text] partition functions to verify the mirror symmetry between a pair of general three-dimensional 𝒩 = 2 Abelian Chern–Simons quiver gauge theories. These expressions readily factorize into holomorphic blocks and their antiholomorphic copies, so we can also obtain the partition functions on S1×S2 via fusion procedure. We then demonstrate S1×S2 partition functions for the three-dimensional Abelian gauge theories can be dimensionally reduced to the S2 partition functions of 𝒩 = (2, 2) GLSM and Landau–Ginzburg model for the corresponding two-dimensional mirror pair, as anticipated previously in M. Aganagic et al., J. High Energy Phys.0107, 022 (2001). We also comment on the analogous interpolation for the non-Abelian gauge theories and compute the K-theory vortex partition function for a simple limit to verify the prediction from holomorphic block.

Reduced Chern-Simons quiver theories and cohomological 3-algebra models

We study the BPS spectrum and vacuum moduli spaces in dimensional reductions of Chern-Simons-matter theories with N>=2 supersymmetry to zero dimensions. Our main example is a matrix model version of the ABJM theory which we relate explicitly to certain reduced 3-algebra models. We find the explicit maps from Chern-Simons quiver matrix models to dual IKKT matrix models. We address the problem of topologically twisting the ABJM matrix model, and along the way construct a new twist of the IKKT model. We construct a cohomological matrix model whose partition function localizes onto a moduli space specified by 3-algebra relations which live in the double of the conifold quiver. It computes an equivariant index enumerating framed BPS states with specified R-charges which can be expressed as a combinatorial sum over certain filtered pyramid partitions.

The gravity dual of supersymmetric gauge theories on a biaxially squashed three-sphere

We present the gravity dual to a class of three-dimensional N=2 supersymmetric gauge theories on a biaxially squashed three-sphere, with a non-trivial background gauge field. This is described by a 1/2 BPS Euclidean solution of four-dimensional N=2 gauged supergravity, consisting of a Taub–NUT–AdS metric with a non-trivial instanton for the graviphoton field. The holographic free energy of this solution agrees precisely with the large N limit of the free energy obtained from the localized partition function of a class of Chern–Simons quiver gauge theories. We also discuss a different supersymmetric solution, whose boundary is a biaxially squashed Lens space S3/Z2 with a topologically non-trivial background gauge field. This metric is of Eguchi–Hanson–AdS type, although it is not Einstein, and has a single unit of gauge field flux through the S2 cycle.

The gravity dual of supersymmetric gauge theories on a squashed three-sphere

We present the gravity dual to a class of three-dimensional N=2 supersymmetric gauge theories on a U(1)×U(1)-invariant squashed three-sphere, with a non-trivial background gauge field. This is described by a supersymmetric solution of four-dimensional N=2 gauged supergravity with a non-trivial instanton for the graviphoton field. The particular gauge theory in turn determines the lift to a solution of eleven-dimensional supergravity. We compute the partition function for a class of Chern–Simons quiver gauge theories on both sides of the duality, in the large N limit, finding precise agreement for the functional dependence on the squashing parameter. This constitutes an exact check of the gauge/gravity correspondence in a non-conformally invariant setting.

Toric Fano varieties and Chern-Simons quivers

In favourable cases the low energy dynamics of a stack of M2-branes at a toric Calabi-Yau fourfold singularity can be described by an N=2 supersymmetric Chern-Simons quiver theory, but there still does not exists an "inverse algorithm" going from the toric data of the CY_4 to the CS quiver. We make progress in that direction by deriving CS quiver theories for M2-branes probing cones over a large class of geometries Ypq(B_4), which are S^3/Z_p bundles over toric Fano varieties B_4. We rely on the type IIA understanding of CS quivers, giving a firm string theory footing to our CS theories. In particular we give a derivation of some previously conjectured CS quivers in the case B_4= CP^1*CP^1, as field theories dual to M-theory backgrounds with nontrivial torsion G_4 fluxes.

Operator counting for $ \mathcal{N} = {2} $ Chern-Simons gauge theories with chiral-like matter fields

The localization formula of Chern-Simons quiver gauge theory on $S^3$ nicely reproduces the geometric data such as volume of Sasaki-Einstein manifolds in the large-$N$ limit, at least for vector-like models. The validity of chiral-like models is not established yet, due to technical problems in both analytic and numerical approaches. Recently Gulotta, Herzog and Pufu suggested that the counting of chiral operators can be used to find the eigenvalue distribution of quiver matrix models. In this paper we apply this method to some vector-like or chiral-like quiver theories, including the triangular quivers with generic Chern-Simons levels which are dual to in-homogeneous Sasaki-Einstein manifolds $Y^{p,k}(\mathbb{CP}^2)$. The result is consistent with AdS/CFT and the volume formula. We discuss the implication of our analysis.

On fractional quantum Hall solitons and Chern–Simons quiver gauge theories

We investigate a class of hierarchical multiple layers of fractional quantum Hall soliton (FQHS) systems from Chern–Simons quivers embedded in M-theory on the cotangent on a two-dimensional complex toric variety V2, which is dual to type IIA superstring on a three-dimensional complex manifold CP1 × V2 fibered over a real line . Based on M-theory/type IIA duality, FQHS systems can be derived from wrapped D4-branes on 2-cycles in CP1 × V2 type IIA geometry. In this realization, the magnetic source can be identified with gauge fields obtained from the decomposition of the R–R 3-form on a generic combination of 2-cycles. Using type IIA D-brane flux data, we compute the filling factors for models relying on CP2 and the zeroth Hirzebruch surface.

Seiberg duality for Chern-Simons quivers and D-brane mutations

Chern-Simons quivers for M2-branes at Calabi-Yau singularities are best understood as the low energy theory of D2-branes on a dual type IIA background. We show how the D2-brane point of view naturally leads to three dimensional Seiberg dualities for Chern-Simons quivers with chiral matter content: They arise from a change of brane basis (or mutation), in complete analogy with the better known Seiberg dualities for D3-brane quivers. This perspective reproduces the known rules for Seiberg dualities in Chern-Simons-Yang-Mills theories with unitary gauge groups. We provide explicit examples of dual theories for the quiver dual to the Y^{p,q}(CP^2) geometries. We also comment on the string theory derivation of CS quivers dual to massive type IIA geometries.

On fractional quantum Hall solitons in ABJM-like theory

Using D-brane physics, we study fractional quantum Hall solitons (FQHS) in ABJM-like theory in terms of type IIA dual geometries. In particular, we discuss a class of Chern–Simons (CS) quivers describing FQHS systems at low energy. These CS quivers come from R–R gauge fields interacting with D6-branes wrapped on 4-cycles, which reside within a blown up CP3 projective space. Based on the CS quiver method and mimicking the construction of del Pezzo surfaces in terms of CP2, we first give a model which corresponds to a single layer model of FQHS system, then we propose a multi-layer system generalizing the doubled CS field theory, which is used in the study of topological defect in graphene.

Comments on 3d Seiberg-like dualities

We study Seiberg-like dualities in three dimensional N=2 supersymmetric theories, emphasizing Chern-Simons terms for the global symmetry group, which affect contact terms in two-point functions of global currents and are essential to the duality map. We introduce new Seiberg-like dualities for Yang-Mills-Chern-Simons theories with unitary gauge groups with arbitrary numbers of matter fields in the fundamental and antifundamental representations. These dualities are derived from Aharony duality by real mass deformations. They allow to initiate the systematic study of Seiberg-like dualities in Chern-Simons quivers. We also comment on known Seiberg-like dualities for symplectic and orthogonal gauge groups and extend the latter to the Yang-Mills case. We check our proposals by showing that the localized partition functions on the squashed S^3 match between dual descriptions.

Quantum moduli space of Chern-Simons quivers, wrapped D6-branes and AdS 4 /CFT 3

We study the quantum moduli space of $ \mathcal{N} = 2 $ Chern-Simons quivers with generic ranks and CS levels, proving along the way exact formulas for the charges of bare monopole operators. We then derive $ \mathcal{N} = 2 $ Chern-Simons quiver theories dual to $ Ad{S_4} \times {Y^{p,q}}\left( {\mathbb{C}{\mathbb{P}^2}} \right) $ M-theory backgrounds, for the whole family of Sasaki-Einstein seven-manifolds and for any value of the torsion G 4 flux. The derivation of the gauge theories relies on the reduction to type IIA string theory, in which M2-branes become D2-branes while the conical geometry maps to RR flux and D6-branes wrapped on compact four-cycles. M5-branes on torsion cycles map to flux and wrapped D4-branes. The moduli space of the quiver is shown to contain the corresponding CY4 cone and all its crepant resolutions.

LargeNlimit of quiver matrix models and Sasaki-Einstein manifolds

We study the matrix models that result from the localization of the partition functions of $\mathcal{N}=2$ Chern-Simons--matter theories on the three-sphere. A large class of such theories are conjectured to be holographically dual to M-theory on Sasaki-Einstein seven-manifolds. We study the M-theory limit (large $N$ and fixed Chern-Simons levels) of these matrix models for various examples, and show that in this limit the free energy reproduces the expected AdS/CFT result of ${N}^{3/2}/\mathrm{Vol}(Y{)}^{1/2}$, where $\mathrm{Vol}(Y)$ is the volume of the corresponding Sasaki-Einstein metric. More generally we conjecture a relation between the large $N$ limit of the partition function, interpreted as a function of trial $R$ charges, and the volumes of Sasakian metrics on links of Calabi-Yau fourfold singularities. We verify this conjecture for a family of $U(N{)}^{2}$ Chern-Simons quivers based on M2 branes at hypersurface singularities, and for a $U(N{)}^{3}$ theory based on M2 branes at a toric singularity.

Moduli spaces of Chern-Simons quiver gauge theories andAdS4/CFT3

We analyze the classical moduli spaces of supersymmetric vacua of 3D $\mathcal{N}=2$ 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 $\mathcal{N}=1$ 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 existing 4D techniques, for constructing candidate 3D $\mathcal{N}=2$ superconformal Chern-Simons quivers with ${\mathrm{AdS}}_{4}$ gravity duals. As simple, but nontrivial, examples, we identify a family of Chern-Simons quiver gauge theories which are candidate ${\mathrm{AdS}}_{4}/{\mathrm{CFT}}_{3}$ duals to an infinite class of toric Sasaki-Einstein seven-manifolds with explicit metrics.

A simple class of 𝒩 = 3 gauge/gravity duals

We find the gravity duals to an infinite series of N=3 Chern-Simons quiver theories. They are AdS_4 x M_7 vacua of M-theory, with M_7 in a certain class of 3-Sasaki-Einstein manifolds obtained by a quotient construction. The field theories can be engineered from a brane configuration; their geometry is summarized by a hyperKaehler toric fan that can be read off easily from the relative angles of the branes. The singularity at the tip of the cone over M_7 is generically not an orbifold. The simplest new manifolds we consider can be written as the biquotient U(1)\U(3)/U(1). We also comment on the relation between our theories and four-dimensional N=1 theories with the same quiver.