Chern-Simons Matter Theories Research Articles

The supersymmetric NUTs and bolts of holography

We show that a given conformal boundary can have a rich and intricate space of supersymmetric supergravity solutions filling it, focusing on the case where this conformal boundary is a biaxially squashed Lens space. Generically we find that the biaxially squashed Lens space S3/Zp admits Taub-NUT-AdS fillings, with topology R4/Zp, as well as smooth Taub-Bolt-AdS fillings with non-trivial topology. We show that the Taub-NUT-AdS solutions always lift to solutions of M-theory, and correspondingly that the gravitational free energy then agrees with the large N limit of the dual field theory free energy, obtained from the localized partition function of a class of N=2 Chern–Simons-matter theories. However, the solutions of Taub-Bolt-AdS type only lift to M-theory for appropriate classes of internal manifold, meaning that these solutions exist only for corresponding classes of three-dimensional N=2 field theories.

Phases of large N vector Chern-Simons theories on S 2 × S 1

We study the thermal partition function of level $k$ U(N) Chern-Simons theories on $S^2$ interacting with matter in the fundamental representation. We work in the 't Hooft limit, $N,k\to\infty$, with $\lambda = N/k$ and $\frac{T^2 V_{2}}{N}$ held fixed where $T$ is the temperature and $V_{2}$ the volume of the sphere. An effective action proposed in arXiv:1211.4843 relates the partition function to the expectation value of a `potential' function of the $S^1$ holonomy in pure Chern-Simons theory; in several examples we compute the holonomy potential as a function of $\lambda$. We use level rank duality of pure Chern-Simons theory to demonstrate the equality of thermal partition functions of previously conjectured dual pairs of theories as a function of the temperature. We reduce the partition function to a matrix integral over holonomies. The summation over flux sectors quantizes the eigenvalues of this matrix in units of ${2\pi \over k}$ and the eigenvalue density of the holonomy matrix is bounded from above by $\frac{1}{2 \pi \lambda}$. The corresponding matrix integrals generically undergo two phase transitions as a function of temperature. For several Chern-Simons matter theories we are able to exactly solve the relevant matrix models in the low temperature phase, and determine the phase transition temperature as a function of $\lambda$. At low temperatures our partition function smoothly matches onto the $N$ and $\lambda$ independent free energy of a gas of non renormalized multi trace operators. We also find an exact solution to a simple toy matrix model; the large $N$ Gross-Witten-Wadia matrix integral subject to an upper bound on eigenvalue density.

Duality and higher temperature phases of large N Chern-Simons matter theories on S 2 × S 1

It has been recently demonstrated that the thermal partition function of any large $N$ Chern-Simons gauge theories on $S^2$, coupled to fundamental matter, reduces to a capped unitary matrix model. The matrix models corresponding to several specific matter Chern-Simons theories at temperature $T$ were determined in arXiv:1301.6169. The large $N$ saddle point equations for these theories were determined in the same paper, and were solved in the low temperature phase. In this paper we find exact solutions for these saddle point equations in three other phases of these theories and thereby explicitly determine the free energy of the corresponding theories at all values of $T^2/N$. As anticipated on general grounds in arXiv:1301.6169, our results are in perfect agreement with conjectured level rank type bosonization dualities between pairs of such theories.

A systematic study on matrix models for Chern–Simons-matter theories

We investigate the planar solution of matrix models derived from various Chern–Simons-matter theories compatible with the planar limit. The saddle-point equations for most of such theories can be solved in a systematic way. A relation to Fuchsian systems play an important role in obtaining the planar resolvents. For those theories, the eigenvalue distribution is found to be confined in a bounded region even when the ʼt Hooft couplings become large. As a result, the vevs of Wilson loops are bounded in the large ʼt Hooft coupling limit. This implies that many of Chern–Simons-matter theories have quite different properties from ABJM theory. If the gauge group is of the form U(N1)k1×U(N2)k2, then the resolvents can be obtained in a more explicit form than in the general cases.

NONPLANAR INTEGRABILITY AND PARITY IN ABJ THEORY

In this paper, we study the action of the nonplanar two-loop dilatation operator in an SU(2)×SU(2) subsector of the ABJ Chern–Simons-matter theory. The gauge invariant operators we consider are the restricted Schur polynomials. As in ABJM theory, there is a limit in which the spectrum reduces to a set of decoupled harmonic oscillators, indicating integrability in the large M and N double limit of the theory. We then consider parity transformations on the gauge invariant operators. In this case the nonplanar anomalous dimensions break parity invariance. Our analysis shows that (M-N) is related to the holonomy in the string theory, confirming one of the main features of the theory and its string dual. Furthermore, in the limit where ABJ theory reduces to ABJM theory, parity invariance is restored.

Formula omitted] superconformal symmetry of [formula omitted] Chern–Simons theory

We demonstrate that the general D=3, N=5 Chern–Simons matter theory possesses a full OSp(5|4) superconformal symmetry, and construct the corresponding superconformal currents. The closure of the superconformal algebra is verified in detail. We also show that the conserved OSp(6|4) superconformal currents in the general N=6 theory can be obtained as special cases of the OSp(5|4) currents by enhancing the R-symmetry of the N=5 theory from USp(4) to SU(4).

The partition function of ABJ theory

We study the partition function of the N=6 supersymmetric U(N_1)_k x U(N_2)_{-k} Chern-Simons-matter (CSM) theory, also known as the ABJ theory. For this purpose, we first compute the partition function of the U(N_1) x U(N_2) lens space matrix model exactly. The result can be expressed as a product of q-deformed Barnes G-function and a generalization of multiple q-hypergeometric function. The ABJ partition function is then obtained from the lens space partition function by analytically continuing N_2 to -N_2. The answer is given by min(N_1,N_2)-dimensional integrals and generalizes the "mirror description" of the partition function of the ABJM theory, i.e. the N=6 supersymmetric U(N)_k x U(N)_{-k} CSM theory. Our expression correctly reproduces perturbative expansions and vanishes for |N_1-N_2|>k in line with the conjectured supersymmetry breaking, and the Seiberg duality is explicitly checked for a class of nontrivial examples.

EquivalentD=3Supergravity Amplitudes from Double Copies of Three-Algebra and Two-Algebra Gauge Theories

We show that three-dimensional supergravity amplitudes can be obtained as double copies of either three-algebra super-Chern-Simons matter theory or two-algebra super-Yang-Mills theory when either theory is organized to display the color-kinematics duality. We prove that only helicity-conserving four-dimensional gravity amplitudes have nonvanishing descendants when reduced to three dimensions, implying the vanishing of odd-multiplicity S-matrix elements, in agreement with Chern-Simons matter theory. We explicitly verify the double-copy correspondence at four and six points for N = 12,10,8 supergravity theories and discuss its validity for all multiplicity.

Free energy of $ {{\widehat{D}}_n} $ quiver Chern-Simons theories

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 addition of massive fundamental flavor fields, verifying that integrating these out decreases the free energy in accordance with the F-theorem.

Thermodynamics of the brane in Chern-Simons matter theories with flavor

We study the holographic dual of flavors in a Chern-Simons matter theory at non-zero temperature, realized as D6-branes in the type IIA black hole dual in the ABJM background geometry. We consider both massive and massless flavors. The former are treated in the quenched approximation, whereas the massless ones are considered as dynamical objects and their backreaction on the geometry is included in the black hole background. We compute the holographically renormalized action of the probe by imposing several physical conditions. In the limit of massless flavors the free energy and entropy of the probe match non-trivially the first variation of these quantities for the backreacted background when the number of flavors is increased by one unit. We compute several thermodynamical functions for the system and analyze the meson melting phase transition between Minkowski and black hole embeddings.

BPS vortices, Q-balls, and Q-vortices in $ \mathcal{N} $ = 6 Chern-Simons matter theory

We investigate the vortex-type BPS equations in the ABJM theory without and with mass-deformation. We systematically classify the BPS equations in terms of the number of supersymmetry and the R-symmetries of the undeformed and mass-deformed ABJM theories. For the undeformed case, we analyze the ${\cal N}=2$ BPS equations for U(2)$\times$U(2) gauge symmetry and obtain a coupled differential equation which can be reduced to either Liouville- or Sinh-Gordon-type vortex equations according to the choice of scalar functions. For the mass-deformed case with U($N$)$\times$U($N$) gauge symmetry, we obtain some number of pairs of coupled differential equations from the ${\cal N}=1,2$ BPS equations, which can be reduced to the vortex equations in Maxwell-Higgs theory or Chern-Simons matter theories as special cases. We discuss the solutions. In ${\cal N}=3$ vortex equations Chern-Simons-type vortex equation is not allowed. We also show that ${\cal N}=\frac52, \frac32, \frac12$ BPS equations are equivalent to those with higher integer supersymmetries.

Construction of new D = 3, $ \mathcal{N}=4 $ quiver gauge theories

In this paper we propose a special class of 3-algebras, called double-symplectic 3-algebras. We further show that a consistent contraction of the double-symplectic 3-algebra gives a new 3-algebra, called an N=4 three-algebra, which is then identified as the exact gauged three-algebra in the N=4 quiver gauge theories. A systematic construction is proposed for the 3-brackets and fundamental identities used in building up the N=4 theories, by starting with two superalgebras whose bosonic parts share at least one simple factor or U(1) factor. This leads to a systematic way of constructing D=3, N=4 quiver theories, of which several examples with new gauge groups are presented in detail. The general N=4 superconformal Chern-Simons matter theories in terms of ordinary Lie algebras can be also re-derived in our new 3-algebra approach.

NUMERICAL STUDIES OF THE ABJM THEORY FOR ARBITRARY N AT ARBITRARY COUPLING CONSTANT

We show that the ABJM theory, which is an [Formula: see text] superconformal U (N) × U (N) Chern-Simons matter theory, can be studied for arbitrary N at arbitrary coupling constant by applying a simple Monte Carlo method to the matrix model derived by using the localization method. Here we calculate the free energy, and show that some results obtained by the Fermi gas approach can be clearly understood from the constant map contribution obtained by the genus expansion.

Topologically gauged superconformal Chern-Simons matter theories

By coupling $ \mathcal{N}=8 $ superconformal matter to $ \mathcal{N}=8 $ superconformal Chern-Simons gravity in three dimensions we obtain theories with novel terms in the scalar potential leading to AdS 3 solutions and superconformal symmetry breaking. If we start from the theory derived by Bagger, Lambert and Gustavsson, our coupled theory either inherits the SO(4) gauge group or reduces it to SO(3). If the construction is instead based on a free matter theory we find that the gravitational topological gauging also requires the introduction of a Chern-Simons gauge sector resulting in a consistent theory for any SO(N ) gauge group.

Brane resolution through fibration

We consider p-branes with one or more circular directions fibered over the transverse space. The fibration, in conjunction with the transverse space having a blown-up cycle, enables these p-brane solutions to be completely regular. Some such circularly-wrapped D3-brane solutions describe flows from SU(N)^3 N=2 theory, F_0 theory, as well as an infinite family of superconformal quiver gauge theories, down to three-dimensional field theories. We discuss the operators that are turned on away from the UV fixed points. Similarly, there are wrapped M2-brane solutions which describe smooth flows from known three-dimensional supersymmetric Chern-Simons matter theories, such as ABJM theory. We also consider p-brane solutions on gravitational instantons, and discuss various ways in which U-duality can be applied to yield other non-singular solutions.

Duality between $ \mathcal{N}=5 $ and $ \mathcal{N}=6 $ Chern-Simons matter theory

We provide evidences for the duality between ${\cal N}=6$ $U(M)_{4} \times U(N)_{-4}$ Chern-Simons matter theory and ${\cal N}=5$ $O(\hat{M})_{2} \times USp(2\hat{N})_{-1}$ theory for a suitable $\hat{M},\hat{N}$ by working out the superconformal index, which shows perfect matching. For ${\cal N}=5$ theories, we show that supersymmetry is enhanced to ${\cal N}=6$ by explicitly constructing monopole operators filling in $SO(6)_R$ $R$-currents. Finally we work out the large $N$ index of $O(2N)_{2k} \times USp(2N)_{-k}$ and show that it exactly matches with the gravity index on $AdS_4 \times S^7/D_k$, which further provides additional evidence for the duality between the ${\cal N}=5$ and ${\cal N}=6$ theory for $k=1$

The large N limit of toric Chern-Simons matter theories and their duals

We compute the large N limit of the localized three dimensional free energy of various field theories with known proposed AdS duals. We show that vector-like theories agree with the expected supergravity results, and with the conjectured F-theorem. We also check that the large N free energy is preserved by the three dimensional Seiberg duality for general classes of vector like theories. Then we analyze the behavior of the free energy of chiral-like theories by applying a new proposal. The proposal is based on the restoration of a discrete symmetry on the free energy before the extremization. We apply this procedure at strong coupling in some examples and we discuss the results. We conclude the paper by proposing an alternative geometrical expression for the free energy.

One-loop corrections to a holographic Wilson loop in AdS 4 × ℂℙ3

The evaluation of BPS Wilson loops in N=6, D=3 Chern-Simons matter theory is reduced to ordinary matrix integrals via localization technique. It is easy to check that the vacuum expectation value of 1/2 BPS Wilson loops at leading order in planar limit agrees with the regularized classical string action, via AdS/CFT. Then the subleading terms in principle can be calculated by treating the string theory semi-classically. In this article we calculate the one-loop determinant for fluctuation modes of holographic Wilson loop in the dual geometry AdS4xCP3. The fermionic normal mode frequencies are expressed in terms of the hypergeometric function, and we compute the one-loop effective action numerically. The discrepancy with localization formula is due to the zero mode normalization constant, which is yet to be determined.

New Relations for Three-Dimensional Supersymmetric Scattering Amplitudes

We provide evidence for a duality between color and kinematics in three-dimensional supersymmetric Chern-Simons matter theories. We show that the six-point amplitude in the maximally supersymmetric N=8 theory can be arranged so that the kinematic factors satisfy the fundamental identity of three-algebras. We further show that the four- and six-point N=8 amplitudes can be squared into the amplitudes of N=16 three-dimensional supergravity, thus providing evidence for a hidden three-algebra structure in the dynamics of the supergravity.

$ \mathcal{Z} $ extremization in chiral-like Chern-Simons theories

We study the localized free energy on S^3 of three-dimensional N=2 Chern-Simons matter theories at weak coupling. We compute the two loop R charge in three different ways, namely by the standard perturbative approach, by extremizing the localized partition function at finite N and by applying the standard saddle point approximation for large N. We show that the latter approach does not reproduce the expected result when chiral theories are considered. We circumvent these problems by restoring a reflection symmetry on the eigenvalues in the free energy. Thanks to this symmetrization we find that the three methods employed agree. In particular we match the computation for a model whose four dimensional parent is the quiver gauge theory describing D3 branes probing the Hirzebruch surface. We conclude by commenting on the application of our results and to the strong coupling regime.