Mirror Maps in Chern-Simons Gauge Theory

TL;DR

This paper explores mirror symmetry in N=2 superconformal field theories through topological changes in gauge theory, revealing new insights into dualities and their applications to quantum Hall states.

Contribution

It introduces a novel topological perspective on mirror symmetry in gauge theories and connects it to quantum Hall hierarchies, expanding the understanding of dualities in physics.

Findings

Mirror map corresponds to interchange of matter and gauge degrees of freedom.

Magnetic monopole-instantons realize discrete symmetries in three dimensions.

Quantum Hall hierarchy is transformed into a new hierarchy with identical filling fractions.

Abstract

We describe mirror symmetry in N=2 superconformal field theories in terms of a dynamical topology changing process of the principal fiber bundle associated with a topological membrane. We show that the topological symmetries of Calabi-Yau sigma-models can be obtained from discrete geometric transformations of compact Chern-Simons gauge theory coupled to charged matter fields. We demonstrate that the appearence of magnetic monopole-instantons, which interpolate between topologically inequivalent vacua of the gauge theory, implies that the discrete symmetry group of the worldsheet theory is realized kinematically in three dimensions as the magnetic flux symmetry group. From this we construct the mirror map and show that it corresponds to the interchange of topologically non-trivial matter field and gauge degrees of freedom. We also apply the mirror transformation to the mean field theory of the quantum Hall effect. We show that it maps the Jain hierarchy into a new hierarchy of states in which the lowest composite fermions have the same filling fractions.