Combinatorial Quantisation of $GL(1|1)$ Chern-Simons Theory I: The Torus

Abstract : Chern-Simons Theories with gauge super-groups appear naturally in string theory and they possess interesting applications in mathematics, e.g. for the construction of knot and link invariants. This paper is the first in a series where we propose a new quantisation scheme for such super-group Chern-Simons theories on 3-manifolds of the form $\Sigma \times \mathbb{R}$. It is based on a simplicial decomposition of an n-punctured Riemann surface $\Sigma=\Sigma_{g,n}$ of genus g and allows to construct observables of the quantum theory for any g and n from basic building blocks, most importantly the so-called monodromy algebra. In this paper we restrict to the torus case, i.e. we assume that $\Sigma = T^2$, and to the gauge super-group G=GL(1|1). We construct the corresponding space of quantum states for the integer level k Chern-Simons theory along with an explicit representation of the modular group SL(2,Z) on these states. The latter is shown to be equivalent to the Lyubachenko-Majid action on the centre of a restricted version of the quantised universal enveloping algebra of the Lie super-algebra gl(1|1) at the primitive k-th root of unity.
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Soumis le : mercredi 12 décembre 2018 - 12:54:13
Dernière modification le : vendredi 18 janvier 2019 - 01:01:32

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Nezhla Aghaei, Azat M. Gainutdinov, Michal Pawelkiewicz, Volker Schomerus. Combinatorial Quantisation of $GL(1|1)$ Chern-Simons Theory I: The Torus. 2018. 〈hal-01952715〉



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