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Register a new userSuperconformal duality-invariant models and $mathcal{N} = 4$ SYM effective action
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Authors
Sergei M. Kuzenko
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We present $mathcal{N}=2$ superconformal $mathsf{U}(1)$ duality-invariant models for an Abelian vector multiplet coupled to conformal supergravity. In a Minkowski background, such a nonlinear theory is expected to describe (the planar part of) the low-energy effective action for the $mathcal{N}=4$ $mathsf{SU}(N)$ super-Yang-Mills (SYM) theory on its Coulomb branch where (i) the gauge group $mathsf{SU}(N)$ is spontaneously broken to $mathsf{SU}(N-1) times mathsf{U}(1)$; and (ii) the dynamics is captured by a single $mathcal{N}=2$ vector multiplet associated with the $mathsf{U}(1)$ factor of the unbroken group. Additionally, a local $mathsf{U}(1)$ duality-invariant action generating the $mathcal{N}=2$ super-Weyl anomaly is proposed. By providing a new derivation of the recently constructed $mathsf{U}(1)$ duality-invariant $mathcal{N}=1$ superconformal electrodynamics, we introduce its $mathsf{SL}(2,{mathbb R})$ duality-invariant coupling to the dilaton-axion multiplet.
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We construct a manifestly N=3 supersymmetric low-energy effective action of N=3 super Yang-Mills theory. The effective action is written in the N=3 harmonic superspace and respects the full N=3 superconformal symmetry. On mass shell this action is responsible for the four-derivative terms in the N=4 SYM effective action, such as F^4/X^4 and its supersymmetric completions, while off shell it involves also higher-derivative terms. For constant Maxwell and scalar fields its bosonic part coincides, up to the F^6/X^8 order, with the bosonic part of the D3 brane action in the AdS_5 x S^5 background. We also argue that in the sector of scalar fields it involves the correctly normalized Wess-Zumino term with the implicit SU(3) symmetry.
Superconformal indices (SCIs) of 4d ${mathcal N}=4$ SYM theories with simple gauge groups are described in terms of elliptic hypergeometric integrals. For $F_4, E_6, E_7, E_8$ gauge groups this yields first examples of integrals of such type. S-duality transformation for G_2 and F_4 SCIs is equivalent to a change of integration variables. Equality of SCIs for SP(2N) and SO(2N+1) group theories is proved in several important special cases. Reduction of SCIs to partition functions of 3d $mathcal{N}=2$ SYM theories with one matter field in the adjoint representation is investigated, corresponding 3d dual partners are found, and some new related hyperbolic beta integrals are conjectured.
In this note, I describe the space of vacua $mathcal{V}$ of four dimensional $mathcal{N}=4$ SYM on $mathbb{R}^4$ with gauge group a compact simple Lie Group $G$ as a stratified space. On each stratum, the low energy effective field theory is different. This language allows one to make precise the idea of moving in the space of vacua $mathcal{V}$. A particular subset of the strata of $mathcal{N}=4$ SYM can be efficiently described using the theory of sheets in a Lie algebra. For these strata, I study the conjectural action of S-duality. I also indicate some benefits of using such a language for the study of the available space of vacua $overline{mathcal{V}}$ on the boundary of GL twisted $mathcal{N}=4$ SYM on a half-space $mathbb{R}^3 times mathbb{R}^+$. As an application of boundary symmetry breaking, I indicate how a) the Local Nilpotent Cone arises as part of the available symmetry breaking choices on the boundary of the four dimensional theory and b) the Global Nilpotent Cone arises in the theory reduced down to two dimensions on a Riemann Surface $C$. These geometries play a critical role in the Local and Global Geometric Langlands Program(s).
We review the basic results concerning the structure of effective action in N=4 supersymmetric Yang-Mills theory in Coulomb phase. Various classical formulations of this theory are considered. We show that the low-energy effective action depending on all fileds of N=4 vector multiplet can be exactly found. This result is discussed on the base of algebraic analysis exploring the general harmonic superspace techniques and on the base of straightforward quantum field theory calculations using the N=2 supersymmetric background field method. We study the one-loop effective action beyond leading low-energy approximation and construct supersymmetric generalization of Heisenberg-Euler-Schwinger effective action depending on all fields of N=4 vector multiplet. We also consider the derivation of leading low-enrgy effective action at two loops.
We continue the research initiated in hep-th/0607215 and apply our method of conformal automorphisms to generate various N=4 superconformal quantum many-body systems on the real line from a set of decoupled particles extended by fermionic degrees of freedom. The su(1,1|2) invariant models are governed by two scalar potentials obeying a system of nonlinear partial differential equations which generalizes the Witten-Dijkgraaf-Verlinde-Verlinde equations. As an application, the N=4 superconformal extension of the three-particle (A-type) Calogero model generates a unique G_2-type Hamiltonian featuring three-body interactions. We fully analyze the N=4 superconformal three- and four-particle models based on the root systems of A_1 + G_2 and F_4, respectively. Beyond Wyllards solutions we find a list of new models, whose translational non-invariance of the center-of-mass motion fails to decouple and extends even to the relative particle motion.
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