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.
Highlights
(ii) via a dual type-IIA string theory system
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 on T 2; and (ii) the space of ground states of strings on an associated mapping torus with T 2 fiber
We have argued that a duality between U(1)n Chern-Simons theory on T 2 with coupling constant matrix (3.1) and string configurations on a mapping torus provide a geometrical realization to the algebra of Wilson loop operators in the Chern-Simons theory
Summary
The low-energy system is described by a 2+1D Chern-Simons action with gauge group U(1)n and action. [(K−1)ij is the i, j element of the matrix K−1.] In particular, for any j = 1, . The Ui’s generate a finite abelian group, which we denote by Gα. We denote by Gβ the finite abelian group generated by the Vi’s. Both groups are isomorphic and can be described as follows. Zn/Λ is a finite abelian group and Gα ∼= Gβ ∼= Zn/Λ, since an element of Zn represents the powers of a monomial in the Ui’s (or Vi’s), and an element in Λ corresponds to a monomial that is a central element.
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