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Noncommutative Hodge-to-de Rham spectral sequence and the Heegaard Floer homology of double covers

Robert Lipshitz, David Treumann (2016)

Journal of the European Mathematical Society

Let A be a dg algebra over 𝔽 2 and let M be a dg A -bimodule. We show that under certain technical hypotheses on A , a noncommutative analog of the Hodge-to-de Rham spectral sequence starts at the Hochschild homology of the derived tensor product M A L M and converges to the Hochschild homology of M . We apply this result to bordered Heegaard Floer theory, giving spectral sequences associated to Heegaard Floer homology groups of certain branched and unbranched double covers.

Quantum invariants of periodic links and periodic 3-manifolds

Qi Chen, Thang Le (2004)

Fundamenta Mathematicae

We give criteria for framed links and 3-manifolds to be periodic of prime order. As applications we show that the Poincaré sphere is of periodicity 2, 3, 5 only and the Brieskorn sphere Σ(2,3,7) is of periodicity 2, 3, 7 only.

Quantum mechanics and nonabelian theta functions for the gauge group SU(2)

Răzvan Gelca, Alejandro Uribe (2015)

Fundamenta Mathematicae

We propose a direction of study of nonabelian theta functions by establishing an analogy between the Weyl quantization of a one-dimensional particle and the metaplectic representation on the one hand, and the quantization of the moduli space of flat connections on a surface and the representation of the mapping class group on the space of nonabelian theta functions on the other. We exemplify this with the cases of classical theta functions and of the nonabelian theta functions for the gauge group...

Quantum Singularity Theory for A ( r - 1 ) and r -Spin Theory

Huijun Fan, Tyler Jarvis, Yongbin Ruan (2011)

Annales de l’institut Fourier

We give a review of our construction of a cohomological field theory for quasi-homogeneous singularities and the r -spin theory of Jarvis-Kimura-Vaintrob. We further prove that for a singularity W of type A our construction of the stack of W -curves is canonically isomorphic to the stack of r -spin curves described by Abramovich and Jarvis. We further prove that our theory satisfies all the Jarvis-Kimura-Vaintrob axioms for an r -spin virtual class. Therefore, the Faber-Shadrin-Zvonkine proof of the...

Racks and orbits of dressing transformations

A. A. Balinsky (2000)

Commentationes Mathematicae Universitatis Carolinae

A new algebraic structure on the orbits of dressing transformations of the quasitriangular Poisson Lie groups is provided. This gives the topological interpretation of the link invariants associated with the Weinstein-Xu classical solutions of the quantum Yang-Baxter equation. Some applications to the three-dimensional topological quantum field theories are discussed.

Representations of the Kauffman bracket skein algebra of the punctured torus

Jea-Pil Cho, Răzvan Gelca (2014)

Fundamenta Mathematicae

We describe the action of the Kauffman bracket skein algebra on some vector spaces that arise as relative Kauffman bracket skein modules of tangles in the punctured torus. We show how this action determines the Reshetikhin-Turaev representation of the punctured torus. We rephrase our results to describe the quantum group quantization of the moduli space of flat SU(2)-connections on the punctured torus with fixed trace of the holonomy around the boundary.

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