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An introduction to quantum sheaf cohomology

Eric Sharpe (2011)

Annales de l’institut Fourier

In this note we review “quantum sheaf cohomology,” a deformation of sheaf cohomology that arises in a fashion closely akin to (and sometimes generalizing) ordinary quantum cohomology. Quantum sheaf cohomology arises in the study of (0,2) mirror symmetry, which we review. We then review standard topological field theories and the A/2, B/2 models, in which quantum sheaf cohomology arises, and outline basic definitions and computations. We then discuss (2,2) and (0,2) supersymmetric Landau-Ginzburg...

Electronic properties of disclinated nanostructured cylinders

R. Pincak, J. Smotlacha, M. Pudlak (2013)

Nanoscale Systems: Mathematical Modeling, Theory and Applications

The electronic structure of the nanocylinder is investigated. Two cases of this kind of the nanostructure are explored: the defect-free nanocylinder and the nanocylinder whose geometry is perturbed by 2 heptagonal defects lying on the opposite sides. The characteristic quantity which is of our interest is the local density of states. To calculate it, the continuum gauge field-theory model will be used. In this model, the Dirac-like equation is solved on a curved surface. This procedure was used...

Factorization constraints and boundary conditions in rational CFT

Carl Stigner (2011)

Banach Center Publications

Among (conformal) quantum field theories, the rational conformal field theories are singled out by the fact that their correlators can be constructed from a modular tensor category 𝓒 with a distinguished object, a symmetric special Frobenius algebra A in 𝓒, via the so-called TFT-construction. These correlators satisfy in particular all factorization constraints, which involve gluing homomorphisms relating correlators of world sheets of different topology. We review the action...

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...

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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