Janus configurations with SL(2, ℤ)-duality twists, strings on mapping tori and a tridiagonal determinant formula

• Published in: The journal of high energy physics Vol. 2014; no. 7
• Main Authors: Ganor, Ori J., Moore, Nathan P., Sun, Hao-Yu, Torres-Chicon, Nesty R.
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01-07-2014
• Subjects: Classical and Quantum Gravitation; Elementary Particles; Physics; Physics and Astronomy; Quantum Field Theories; Quantum Field Theory; Quantum Physics; Relativity Theory; String Theory; Topological Field Theories; Brane Dynamics in Gauge Theories; Duality in Gauge Field Theories; Chern-Simons Theories
• ISSN: 1029-8479; 1029-8479
• DOI: 10.1007/JHEP07(2014)010

A
bstract
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.