Reviews in Mathematical Physics

We review the AKSZ construction as applied to the topological open membranes and Poisson sigma models. We describe a generalization to open topological p-branes. Also, we propose a related (not necessarily BV) Nambu–Poisson sigma model.

We describe the quantization of 2-plectic manifolds as they arise in the context of the quantum geometry of M-branes and non-geometric flux compactifications of closed string theory. We review the groupoid approach to quantizing Poisson manifolds in detail, and then extend it to the loop spaces of 2-plectic manifolds, which are naturally symplectic manifolds. In particular, we discuss the groupoid quantization of the loop spaces of ℝ³, 𝕋³ and S³, and derive some interesting implications which match physical expectations from string theory and M-theory.

We generalize several results on Chern–Simons models on Σ × S¹ in the so-called "torus gauge" which were obtained in [A. Hahn, An analytic approach to Turaev's shadow invariant, J. Knot Theory Ramifications17(11) (2008) 1327–1385] (= arXiv:math-ph/0507040) to the case of general (simply-connected simple compact) structure groups and general link colorings. In particular, we give a non-perturbative evaluation of the Wilson loop observables corresponding to a special class of simple but non-trivial links and show that their values are given by Turaev's shadow invariant. As a byproduct, we obtain a heuristic path integral derivation of the quantum Racah formula.

In this paper a list of R-matrices on a certain coupled Lie algebra is obtained. With one of these R-matrices, we construct infinitely many bi-Hamiltonian structures for each of the two-component BKP and the Toda lattice hierarchies. We also show that, when such two hierarchies are reduced to their subhierarchies, these bi-Hamiltonian structures are reduced correspondingly.