Suppose that M is an A -progenerator.

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Then we have equivalences of categories,. In particular we obtain a description of the category associated to a closed unmarked surface S. We have an equivalence of categories,. Likewise we can identify global functions on the quantum character variety i. Let us briefly recall the classical construction here. Changing the choice of trivialization amounts to conjugating a given homomorphism by a group element. Hence, the G -character stack of S is the quotient stack.

The embedding of the annulus around the circle boundary of S induces a G -equivariant map,. The character stack of S , C is then the quotient stack. The main results of [ Ben-Zvi , Ben-Zvi ] , give identifications,. Therefore, we obtain quantizations of character stacks of closed surfaces. The natural mapping class group symmetry of the torus invariant then explains the well-known modularity of these characters.

In this section we describe the corresponding roles of the quantum character variety of the torus. Let us consider the oriented field theory defined by integrating a balanced tensor category A on oriented surfaces. We have identified codimension two defects for quantum A -character varieties with braided A -modules, i. In the language of extended topological field theory, this means we attach the 2-category U A - m o d to the circle. The excision axiom applied to a decomposition of T 2 into cylinders allows us to identify the torus integral as the monoidal Hochschild homology, or trace, of the cylinder hence annulus integral of A :.

Equivalently, we can describe the torus integral as the trace or Hochschild homology of the 2-category of braided A -modules. It follows from the general theory of characters in Hochschild homology see e. The 2-category of braided R e p q G -modules is a 4d gauge theory analog of the category of integrable level k representations of the loop group.

Note that since we are considering modules in R e p q G rather than V e c t , this category is a quantum analog not of the category of D -modules on G but of the category of D -modules on G which are weakly equivariant for the adjoint action from which the former can be obtained by de-equivariantization. It follows from the quantum Hamiltonian reduction formalism of the previous section that sealing up the puncture results in imposing the quantum moment map relations for the adjoint action — i.

Indeed the endomorphisms of the quantum structure sheaf are known in many cases see below for the t -analog to recover. We likewise expect a theory of quantum character sheaves to provide a natural q -analog of the Lusztig theory.

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## A topological origin of quantum symmetric pairs - INSPIRE-HEP

Interesting examples of such quantum character sheaves are provided by the quantum Springer sheaves — the characters of the braided module category tr R e p q B associated to the R e p q G -algebra defined by the quantum Borel or other parabolics , which can be expected via a quantum Hotta-Kashiwara theorem to be described by a q-analog of the Harish-Chandra system. In this section, we will explain how our theory of quantum character varieties naturally produces the spherical DAHA when fed the torus T 2 marked by the quantum mirabolic conjugacy class.

When q is a root of unity [ Varagnolo ] ,. More precisely, we have slightly reformulated each result here, for a more uniform presentation. Wen and Z. B 78 , Barkeshli and X. B 82 , Bonderson, California Institute of Technology, Ardonne, E. Kailasvuori, and E. Read and E. B 59 , Lu, X. Wen, Z. Wang, and Z.

### Mathematics Subject Classification

B 81 , Rezayi and F. B 50 , B 54 , Einarsson, Fractional Statistics on a Torus , Phys. Trugman and S. B 31 , Tao and D. B 28 , Jansen, E. Lieb, and R. Su, J.

Schrieffer, and A. Heeger, Soliton Excitations in Polyacetylene , Phys. B 22 , An h y -dependent phase has been dropped for simplicity. As a result, note that the original Laughlin state Eq. A , 45 Wilczek and A. Some care must be given to fermion negative signs at odd denominator filling factors, in equations such as 39 , 40 , and See Ref. Greiter, X. Wen, and F. Wilczek, Paired Hall States , Nucl. Note that a coordinate shift, in particular, changes both the magnetic vector potential and the quasiperiodic boundary condition in x on wave functions.

Here, the index n is always defined via properties under magnetic translations, Eq. Other choices for the configurations of the quasiholes will result in the same braiding matrices, as it should be. We leave it understood that this relation holds modulo 2. Price is set at 2, or best offer.

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## Quantum Field Theory of Particles with Braid Group Statistics in 2+1 Dimenions

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