Combinatorial Quantisation of $GL(1|1)$ Chern-Simons Theory I: The Torus
| Autor: | Aghaei, Nezhla, Gainutdinov, Azat M., Pawelkiewicz, Michal, Schomerus, Volker |
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| Přispěvatelé: | Fédération de recherche Denis Poisson (FDP), Université de Tours-Centre National de la Recherche Scientifique (CNRS)-Université d'Orléans (UO), Institut de Physique Théorique - UMR CNRS 3681 (IPHT), Centre National de la Recherche Scientifique (CNRS)-Université Paris-Saclay-Commissariat à l'énergie atomique et aux énergies alternatives (CEA), Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Université Paris-Saclay-Centre National de la Recherche Scientifique (CNRS), Université d'Orléans (UO)-Université de Tours-Centre National de la Recherche Scientifique (CNRS) |
| Jazyk: | angličtina |
| Rok vydání: | 2018 |
| Předmět: |
High Energy Physics - Theory
algebra: Hopf Chern-Simons term [PHYS.HTHE]Physics [physics]/High Energy Physics - Theory [hep-th] supersymmetry: algebra monodromy [PHYS.MPHY]Physics [physics]/Mathematical Physics [math-ph] FOS: Physical sciences torus Mathematical Physics (math-ph) group: modular High Energy Physics::Theory High Energy Physics - Theory (hep-th) Riemann surface GL(1|1) Mathematics - Quantum Algebra FOS: Mathematics Quantum Algebra (math.QA) quantization supersymmetry Z) Mathematical Physics group: SL(2 |
| DOI: | 10.3204/PUBDB-2018-04852 |
| Popis: | 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. Comment: 46 pages |
| Databáze: | OpenAIRE |
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