Abstract
We give an interpretation of the holographic correspondence between two-dimensional BF theory on the punctured disk with gauge group PSL(2, ℝ) and Schwarzian quantum mechanics in terms of a Drinfeld-Sokolov reduction. The latter, in turn, is equivalent to the presence of certain edge states imposing a first class constraint on the model. The constrained path integral localizes over exceptional Virasoro coadjoint orbits. The reduced theory is governed by the Schwarzian action functional generating a Hamiltonian S1-action on the orbits. The partition function is given by a sum over topological sectors (corresponding to the exceptional orbits), each of which is computed by a formal Duistermaat-Heckman integral.
Highlights
We give an interpretation of the holographic correspondence between twodimensional BF theory on the punctured disk with gauge group PSL(2, R) and Schwarzian quantum mechanics in terms of a Drinfeld-Sokolov reduction
The theory reduces to Schwarzian quantum mechanics whose action functional generates an S1-action on the orbits
We show that in the presence of a fixed edge state the constrained partition function reduces to a sum of integrals over non-exceptional Virasoro coadjoint orbits
Summary
Let us start with a non-exhaustive recollection of finite-dimensional Duistermaat-Heckman integration. Let (M, ω) be a compact symplectic 2n-dimensional manifold endowed with an action of S1. Suppose that this circle action is Hamiltonian, that is we assume that the action is generated by a vector field ξ and the existence of a smooth function H on M which satisfy the relation ιξω + dH = 0. Suppose that H has only isolated critical points. Duistermaat and Heckman showed in [11] that the integral. Where the sum runs over all (isolated) critical points m of H and the wj(m) are the weights of the S1-action on the tangent space TmM of M at m. The integration measure is taken to be the Liouville measure ωn/n! defined by the symplectic form ω
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