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Register a new userUnitarity, Crossing Symmetry and Duality in the scattering of ${cal N}=1$ Susy Matter Chern-Simons theories
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We study the most general renormalizable ${cal N}=1$ $U(N)$ Chern-Simons gauge theory coupled to a single (generically massive) fundamental matter multiplet. At leading order in the t Hooft large $N$ limit we present computations and conjectures for the $2 times 2$ $S$ matrix in these theories; our results apply at all orders in the t Hooft coupling and the matter self interaction. Our $S$ matrices are in perfect agreement with the recently conjectured strong weak coupling self duality of this class of theories. The consistency of our results with unitarity requires a modification of the usual rules of crossing symmetry in precisely the manner anticipated in arXiv:1404.6373, lending substantial support to the conjectures of that paper. In a certain range of coupling constants our $S$ matrices have a pole whose mass vanishes on a self dual codimension one surface in the space of couplings.
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We investigate phases of 3d ${cal N}=2$ Chern-Simons-matter theories, extending to three dimensions the celebrated correspondence between 2d gauged Wess-Zumino-Witten (GWZW) models and non-linear sigma models (NLSMs) with geometric targets. We find that although the correspondence in 3d and 2d are closely related by circle compactification, an important subtlety arises in this process, changing the phase structure of the 3d theory. Namely, the effective theory obtained from the circle compactification of a phase of a 3d ${cal N}=2$ gauge theory is, in general, different from the phase of the 3d ${cal N}=2$ theory on ${mathbb R}^2times S^{1}$, which means taking phases of a 3d gauge theory does not necessarily commute with compactification. We compute the Witten index of each effective theory to check this observation. Furthermore, when the matter fields have the same non-minimal charges, the 3d ${cal N}=2$ Chern-Simons-matter theory with a proper Chern-Simons level will decompose into several identical 2d gauged linear sigma models (GLSMs) for the same target upon reduction to 2d. To illustrate this phenomenon, we investigate how vacua of the 3d gauge theory for a weighted projective space $Wmathbb{P}_{[l,cdots,l]}$ move on the field space when we change the radius of $S^{1}$.
Aharony, Bergman, Jafferis and Maldacena have recently proposed a dual gravitational description for a family of superconformal Chern Simons theories in three spacetime dimensions. In this note we perform the one loop computation that determines the field theory superconformal index of this theory and compare with the index computed over the Fock space of dual supersymmetric gravitons. In the appropriate limit (large $N$ and large $k$) we find a perfect match.
We study the algebra of BPS Wilson loops in 3d gauge theories with N=2 supersymmetry and Chern-Simons terms. We argue that new relations appear on the quantum level, and that in many cases this makes the algebra finite-dimensional. We use our results to propose the mapping of Wilson loops under Seiberg-like dualities and verify that the proposed map agrees with the exact results for expectation values of circular Wilson loops. In some cases we also relate the algebra of Wilson loops to the equivariant quantum K-ring of certain quasi projective varieties. This generalizes the connection between the Verlinde algebra and the quantum cohomology of the Grassmannian found by Witten.
It has been conjectured that 3d fermions minimally coupled to Chern-Simons gauge fields are dual to 3d critical scalars, also minimally coupled to Chern-Simons gauge fields. The large $N$ arguments for this duality can formally be used to show that Chern-Simons-gauged {it critical} (Gross-Neveu) fermions are also dual to gauged `{it regular} scalars at every order in a $1/N$ expansion, provided both theories are well-defined (when one fine-tunes the two relevant parameters of each of these theories to zero). In the strict large $N$ limit these `quasi-bosonic theories appear as fixed lines parameterized by $x_6$, the coefficient of a sextic term in the potential. While $x_6$ is an exactly marginal deformation at leading order in large $N$, it develops a non-trivial $beta$ function at first subleading order in $1/N$. We demonstrate that the beta function is a cubic polynomial in $x_6$ at this order in $1/N$, and compute the coefficients of the cubic and quadratic terms as a function of the t Hooft coupling. We conjecture that flows governed by this leading large $N$ beta function have three fixed points for $x_6$ at every non-zero value of the t Hooft coupling, implying the existence of three distinct regular bosonic and three distinct dual critical fermionic conformal fixed points, at every value of the t Hooft coupling. We analyze the phase structure of these fixed point theories at zero temperature. We also construct dual pairs of large $N$ fine-tuned renormalization group flows from supersymmetric ${cal N}=2$ Chern-Simons-matter theories, such that one of the flows ends up in the IR at a regular boson theory while its dual partner flows to a critical fermion theory. This construction suggests that the duality between these theories persists at finite $N$, at least when $N$ is large.
We compute the two, three point function of the opearators in the spin zero multiplet of ${cal N}=2$ Supersymmetric vector matter Chern-Simons theory at large $N$ and at all orders of t Hooft coupling by solving the Schwinger-Dyson equation. Schwinger-Dyson method to compute four point function becomes extremely complicated and hence we use bootstrap method to solve for four point function of scaler operator $J_0^{f}=barpsi psi$ and $J_0^{b}=barphi phi$. Interestingly, due to the fact that $langle J_0^{f}J_0^{f}J_0^{b} rangle$ is a contact term, the four point function of $ J_0^{f}$ operator looks like that of free theory up to overall coupling constant dependent factors and up to some bulk AdS contact terms. On the other hand the $J_0^{b}$ four-point function receives an additional contribution compared to the free theory expression due to the $J_0^{f}$ exchange. Interestingly, double discontinuity of this single trace operator $J_0^{f}$ vanishes and hence it only contributes to AdS-contact term.
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