UC Berkeley

In this thesis we explore two interesting relationships between string theory and quantum field theory.

Firstly, we develop an equivalence between two Hilbert spaces: (i) the space of states of U(1)^n Chern-Simons theory with a certain class of tridiagonal matrices of coupling constants (with corners) on T^2; and (ii) the space of ground states of strings on an associated mapping torus with T^2 fiber. The equivalence is deduced by studying the space of ground states of SL(2,Z)-twisted circle compactifications of U(1) gauge theory, connected with a Janus configuration, and further compactified on T^2. The equality of dimensions of the two Hilbert spaces (i) and (ii) is equivalent to a known identity on determinants of tridiagonal matrices with corners. The equivalence of operator algebras acting on the two Hilbert spaces follows from a relation between the Smith normal form of the Chern-Simons coupling constant matrix and the isometry group of the mapping torus, as well as the torsion part of its first homology group.

Secondly, the Fractional Quantum Hall Effect appears as part of the low-energy description of the Coulomb branch of the A_1 (2,0)-theory formulated on (S^1xR^2)/Z_k, where the generator of Z_k acts as a combination of translation on S^1 and rotation by 2&pi/k on R^2. At low-energy the configuration is described in terms of a 4+1D Super-Yang-Mills theory on a cone (R^2/Z_k) with additional 2+1D degrees of freedom at the tip of the cone. Fractionally charged quasi-particles have a natural description in terms of BPS strings of the (2,0)-theory. We analyze the large k limit, where a smooth cigar-geometry provides an alternative description. In this framework a W-boson can be modeled as a bound state of k quasi-particles. The W-boson becomes a Q-ball, and it can be described by a soliton solution of BPS monopole equations on a certain auxiliary curved space. We show that axisymmetric solutions of these equations correspond to singular maps from AdS_3 to AdS_2, and we present some numerical results.