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String Theory, Strongly Correlated Systems, and Duality Twists
 Torres Chicon, Nesty Ray
 Advisor(s): Ganor, Ori J
Abstract
In the first part of this dissertation (Chapter 1), I present a construction of a six dimensional (2,0)theory model that describes the dynamics of the Fractional Quantum Hall Effect (FQHE). The FQHE appears as part of the low energy description of the Coulomb branch of the A_1 (2,0)theory formulated on a geometry (S^1 x R^2)/Z_k. At lowenergy, the configuration is described in terms of a 4+1D supersymmetric YangMills (SYM) theory on a cone (R^2/Z_k) with additional 2+1D degrees of freedom at the tip of the cone that include fractionally charged particles. These fractionally charged "quasiparticles" are BPS strings of the (2,0)theory wrapped on short cycles. In this framework, a Wboson can be modeled as a bound state of k quasiparticles, which can be used to understand the dynamics of the FQHE.
In the second part of this dissertation (Chapters 23), I investigate the N=4 SYM theory compactified on a circle, with a varying coupling constant (Janus configuration) and an Sduality twist. I relate this setup to a three dimensional topological theory and to a dual string theory. The equality of these descriptions is exhibited by matching the operator algebra, and the dimensions of the Hilbert space. Additionally, this dissertation addresses a classic result in number theory, called quadratic reciprocity, using string theory language. I present a proof that quadratic reciprocity is a direct consequence of Tduality of TypeII string theory. This is demonstrated by analyzing a partition function of abelian N=4 SYM theory on a certain supersymmetrypreserving fourmanifold with variable coupling constant and a SL(2,Z)duality twist.
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