Supersymmetric Chern-Simons Theory Research Articles

40 bilinear relations of q-Painlevé VI from mathcal{N} = 4 super Chern-Simons theory

We investigate partition functions of the circular-quiver supersymmetric Chern-Simons theory which corresponds to the q-deformed Painlevé VI equation. From the partition functions with the lowest rank vanishing, where the circular quiver reduces to a linear one, we find 40 bilinear relations. The bilinear relations extend naturally to higher ranks if we regard these partition functions as those in the lowest order of the grand canonical partition functions in the fugacity. Furthermore, we show that these bilinear relations are a powerful tool to determine some unknown partition functions. We also elaborate the relation with some previous works on q-Painlevé equations.

Non-perturbative tests of duality cascades in three dimensional supersymmetric gauge theories

It has been conjectured that duality cascade occurs in the mathcal{N} = 3 supersymmetric Yang-Mills Chern-Simons theory with the gauge group U(N)k × U(N + M)−k coupled to two bi-fundamental hypermultiplets. The brane picture suggests that this duality cascade can be generalized to a class of 3d mathcal{N} = 3 supersymmetric quiver gauge theories coming from so-called Hanany-Witten type brane configurations. In this paper we perform non-perturbative tests of the duality cascades using supersymmetry localization. We focus on S3 partition functions and prove predictions from the duality cascades. We also discuss that our result can be applied to generate new dualities for more general theories which include less supersymmetric theories and theories without brane constructions.

Nambu brackets, Chern–Simons theories, quantum curves and M2-branes

Abstract A non-technical review on recent progress in M2-branes is given. We focus especially on (1) historical progress in searching for the worldvolume theory of M2-branes using Nambu brackets and supersymmetric Chern–Simons theories, (2) studies of the ABJM matrix model obtained from localization using various techniques and (3) new insights of quantum curves derived from the studies of the ABJM matrix model.

The [formula omitted] enhancement of super Chern–Simons theories in [formula omitted], Calabi HyperKähler metrics and M2-branes on the [formula omitted] conifold

Considering matter coupled supersymmetric Chern–Simons theories in three dimensions we extend the Gaiotto–Witten mechanism of supersymmetry enhancement N3=3→N3=4 from the case where the hypermultiplets span a flat HyperKähler manifold to that where they live on a curved one. We derive the precise conditions of this enhancement in terms of generalized Gaiotto–Witten identities to be satisfied by the tri-holomorphic moment maps. An infinite class of HyperKähler metrics compatible with the enhancement condition is provided by the Calabi metrics on T⋆Pn. In this list we find, for n=2 the resolution of the metric cone on N0,1,0 which is the unique homogeneous Sasaki–Einstein 7-manifold leading to an N4=3 compactification of M-theory. This leads to challenging perspectives for the discovery of new relations between the enhancement mechanism in D=3, the geometry of M2-brane solutions and also for the dual description of super Chern–Simons theories on curved HyperKähler manifolds in terms of gauged fixed supergroup Chern–Simons theories.

Mass-deformed N=3 supersymmetric Chern-Simons-matter theory

The maximal extension of supersymmetric Chern-Simons theory coupled to fundamental matter has $\mathcal{N} = 3$ supersymmetry. In this short note, we provide the explicit form of the action for the mass-deformed $\mathcal{N} = 3$ supersymmetric $U(N)$ Chern-Simons-Matter theory. The theory admits a unique triplet mass deformation term consistent with supersymmetry. We explicitly construct the mass-deformed $\mathcal{N} = 3$ theory in $\mathcal{N} = 1$ superspace using a fundamental and an anti-fundamental superfield.

Supersymmetric Solutions and Borel Singularities for N=2 Supersymmetric Chern-Simons Theories.

In supersymmetric (SUSY) field theory, there exist configurations that formally satisfy SUSY conditions but are not on the original path integral contour. We refer to such configurations as complexified supersymmetric solutions (CSS). In this Letter we discuss that CSS provide important information on the large order behavior of the weak coupling perturbative series in SUSY field theories. We conjecture that CSS with a bosonic (fermionic) free parameter give poles (zeros) of the Borel transformation of perturbative series whose locations are uniquely determined by actions of the solutions. We demonstrate this for various SUSY observables in 3D N=2 SUSY Chern-Simons matter theories on a sphere. First we construct an infinite number of CSS in the general 3D N=2 SUSY theory with a Lagrangian where an adjoint scalar in the vector multiplet takes a complex value and matter fields are nontrivial. Then we compare their actions with Borel transformations of perturbative expansions by inverse Chern-Simons levels for the observables and see agreement with our conjecture. It turns out that the CSS explain all the Borel singularities for this case.

Exact Holography and Black Hole Entropy in mathcal{N}=8 and mathcal{N}=4 String Theory

We compute the exact entropy of one-eighth and one-quarter BPS black holes in mathcal{N}=8 and mathcal{N}=4 string theory respectively. This includes all the mathcal{N}=4 CHL models in both K3 and T4 compactifications. The main result is a measure for the finite dimensional integral that one obtains after localization of supergravity on AdS2× S2. This measure is determined entirely by an anomaly in supersymmetric Chern-Simons theory on local AdS3 and takes into account the contribution from all the supergravity multiplets. In Chern-Simons theory on compact manifolds, this is the anomaly that computes a certain one-loop dependence on the volume of the manifold. For one-eighth BPS black holes, our results are a first principles derivation of a measure proposed in arXiv:1111.1161, while in the case of one-quarter BPS black holes our result computes exactly all the perturbative or area corrections. Moreover, we argue that instantonic contributions can be incorporated and give evidence by computing the measure, which matches precisely the microscopics. Along with this, we find a unitary condition that truncates the answer to a finite sum of instantons in perfect agreement with a microscopic formula. Our results therefore solve a number of puzzles related to localization in supergravity and constitute a larger number of examples where holography can be shown to hold exactly.

Localization on AdS2 \xd7 S1

Conformal symmetry relates the metric on AdS2 × S1 to that of S3. This implies that under a suitable choice of boundary conditions for fields on AdS2 the partition function of conformal field theories on these spaces must agree which makes AdS2 × S1 a good testing ground to study localization on non-compact spaces. We study supersymmetry on AdS2 × S1 and determine the localizing Lagrangian for mathcal{N} = 2 supersymmetric Chern-Simons theory on AdS2 × S1. We evaluate the partition function of mathcal{N} = 2 supersymmetric Chern-Simons theory on AdS2 × S1 using localization, where the radius of S1 is q times that of AdS2. With boundary conditions on AdS2 × S1 which ensure that all the physical fields are normalizable and lie in the space of square integrable wave functions in AdS2, the result for the partition function precisely agrees with that of the theory on the q-fold covering of S3.

A -function for N=2 supersymmetric gauge theories in three dimensions

Recently, the existence of a candidate $a$-function for renormalizable theories in three dimensions was demonstrated for a general theory at leading order and for a scalar-fermion theory at next-to-leading order. Here we extend this work by constructing the $a$-function at next-to-leading order for an $\mathcal{N}=2$ supersymmetric Chern-Simons theory. This increase in precision for the $a$-function necessitated the evaluation of the underlying renormalization-group functions at four loops.

Low dimensional supersymmetries in SUSY Chern–Simons systems and geometrical implications

We study in detail the underlying graded geometric structure of abelian [Formula: see text] supersymmetric Chern–Simons theory in (2 + 1)-dimensions. This structure is an attribute of the hidden unbroken one-dimensional [Formula: see text] supersymmetries that the system also possesses. We establish the result that the geometric structures corresponding to the bosonic and to the fermionic sectors are equivalent fiber bundles over the (2 + 1)-dimensional manifold. Moreover, we find a geometrical answer to the question why some and not all of the fermionic sections are related to a [Formula: see text] supersymmetric algebra. Our findings are useful for the quantum theory of Chern–Simons vortices.

Instanton effects in orientifold ABJM theory

We investigate another supersymmetric Chern-Simons theory called the orientifold ABJM theory, which replaces the unitary supergroup structure of the ABJM theory with an orthosymplectic one. Its non-perturbative structure is completely clarified by considering the duplication of the quiver.

Supersymmetric Chern–Simons theory in presence of a boundary in the light-like direction

In this paper, we will analyze a three dimensional supersymmetric Chern–Simons theory on a manifold with a boundary. The boundary we will consider in this paper will be defined by n⋅x=0, where n is a light-like vector. It will be demonstrated that this boundary is preserved under the action of the SIM(1) subgroup of the Lorentz group. Furthermore, the presence of this boundary will break half of the supersymmetry of the original theory. As the original Chern–Simons theory had N=1 supersymmetry in absence of a boundary, it will only have N=1/2 supersymmetry in presence of this boundary. We will also observe that the Chern–Simons theory can be made gauge invariant by introducing new degrees of freedom on the boundary. The gauge transformation of these new degrees of freedom will exactly cancel the boundary term obtained from the gauge transformation of the Chern–Simons theory.

Quantum Hall effect in supersymmetric Chern-Simons theories

We introduce a supersymmetric Chern-Simons theory whose low energy physics is that of the fractional quantum Hall effect. The supersymmetry allows us to solve the theory analytically. We quantise the vortices and, by relating their dynamics to a matrix model, show that their ground state wavefunction is in the same universality class as the Laughlin state. We further construct coherent state representations of the excitations of a finite number of vortices. These are quasi-holes. By an explicit computation of the Berry phase, without resorting to a plasma analogy, we show that these excitations have fractional charge and spin.

Chern–Simons theory in SIM(1) superspace

In this paper, we will analyse a three dimensional supersymmetric Chern-Simons theory in SIM(1) superspace formalism. The breaking of the Lorentz symmetry down to the SIM(1) symmetry, breaks half the supersymmetry of the Lorentz invariant theory. So, the supersymmetry of the Lorentz invariant Chern-Simons theory with N=1 supersymmetry will break down to N=1/2 supersymmetry, when the Lorentz symmetry is broken down to the SIM(1) symmetry. First, we will write the Chern-Simons action using SIM(1) projections of N=1 superfields. However, as the SIM(1) transformations of these projections are very complicated, we will define SIM(1) superfields which transform simply under SIM(1) transformations. We will then express the Chern-Simons action using these SIM(1) superfields. Furthermore, we will analyse the gauge symmetry of this Chern-Simons theory. This is the first time that a Chern-Simons theory with N=1/2 supersymmetry will be constructed on a manifold without a boundary.

Phases of planar 5-dimensional supersymmetric Chern-Simons theory

In this paper we investigate the large-$N$ behavior of 5-dimensional $\mathcal{N}=1$ super Yang-Mills with a level $k$ Chern-Simons term and an adjoint hypermultiplet. As in three-dimensional Chern-Simons theories, one must choose an integration contour to completely define the theory. Using localization, we reduce the path integral to a matrix model with a cubic action and compute its free energy in various scenarios. In the limit of infinite Yang-Mills coupling and for particular choices of the contours, we find that the free-energy scales as $N^{5/2}$ for $U(N)$ gauge groups with large values of the Chern-Simons 't\,Hooft coupling, $\tilde\lambda\equiv N/k$. If we also set the hypermultiplet mass to zero, then this limit is a superconformal fixed point and the $N^{5/2}$ behavior parallels other fixed points which have known supergravity duals. We also demonstrate that $SU(N)$ gauge groups cannot have this $N^{5/2}$ scaling for their free-energy. At finite Yang-Mills coupling we establish the existence of a third order phase transition where the theory crosses over from the Yang-Mills phase to the Chern-Simons phase. The phase transition exists for any value of $\tilde\lambda$, although the details differ between small and large values of $\tilde\lambda$. For pure Chern-Simons theories we present evidence for a chain of phase transitions as $\tilde\lambda$ is increased. We also find the expectation values for supersymmetric circular Wilson loops in these various scenarios and show that the Chern-Simons term leads to different physical properties for fundamental and anti-fundamental Wilson loops. Different choices of the integration contours also lead to different properties for the loops.

Large N phase transitions in supersymmetric Chern-Simons theory with massive matter

We study three-dimensional {\cal N}=2 U(N) Chern-Simons theory on S^3 coupled to 2N_f chiral multiplets deformed by mass terms. The partition function localizes to a matrix integral, which can be exactly computed in the large N limit. In a specific decompactification limit, the theory exhibits quantum (third-order) phase transitions at finite critical values of the coupling. The theory presents three phases when 0 < N_f < N and two phases when N_f\geq N. The vacuum expectation value of the supersymmetric circular Wilson loop has a discontinuity in the first derivative.

Superconformal Chern-Simons theories beyond leading order

We discuss higher-order corrections to superconformal invariance for a class of N=2 supersymmetric Chern-Simons theories including the ABJM model. We argue that corrections are inevitable for general theories in this class; but that it is probable that any corrections are of a particular "maximally transcendental" form.

Effective superpotential in the supersymmetric Chern-Simons theory with matter

We develop a superfield approach to the effective potential in the supersymmetric Chern-Simons theory coupled to matter and, in the Landau gauge, calculate the one-loop K\"ahlerian effective potential and give some qualitative prescriptions for the two-loop one.

Integral Formula for Supersymmetric Scattering Amplitudes in Three Dimensions

We propose a new integral formula for all tree-level scattering amplitudes of N = 6 supersymmetric Chern-Simons theory. It resembles the Roiban-Spradlin-Volovich-Witten formula for N = 4 supersymmetric Yang-Mills theory based on a twistor string theory formulation. Our formula implies that the (2k)-point tree-level amplitude is closely related to degree (k-1) curves in CP(k-1).

New supersymmetric Wilson loops in ABJ(M) theories

We present two new families of Wilson loop operators in N=6 supersymmetric Chern–Simons theory. The first one is defined for an arbitrary contour on the three dimensional space and it resembles the Zarembo construction in N=4 SYM. The second one involves arbitrary curves on the two dimensional sphere. In both cases one can add certain scalar and fermionic couplings to the Wilson loop so it preserves at least two supercharges. Some previously known loops, notably the 1/2 BPS circle, belong to this class, but we point out more special cases which were not known before. They could provide further tests of the gauge/gravity correspondence in the ABJ(M) case and interesting observables, exactly computable by localization techniques.