Abstract
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.
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Ganor, O.J., Moore, N.P., Sun, HY. et al. Janus configurations with SL(2, ℤ)-duality twists, strings on mapping tori and a tridiagonal determinant formula. J. High Energ. Phys. 2014, 10 (2014). https://doi.org/10.1007/JHEP07(2014)010
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DOI: https://doi.org/10.1007/JHEP07(2014)010