Low energy string theory is governed by type II supergravity. A solution to the supersymmetry equations combined with the Bianchi identities form a solution to the equations of motions, i.e., a vacuum. On the one hand, these can be solved with the aid of G-structures, principal subbundles of the frame bundle of the manifold. On the other, it turns out that there is a certain equivalence between the supersymmetry equations and twisted integrability conditions of generalized almost complex structures.
In this talk, I will give a brief overview of these geometrical structures and their relation to the flux vacua.
(Homotopy) algebras and bimodules over them can be viewed as factorization algebras on R which are locally constant (with respect to a certain stratification). Moreover, Lurie proved that E_n-algebras are equivalent to locally constant factorization algebras on R^n. Starting from this I will explain how to model the Morita category of E_n-algebras as an (\infty, n)-category. Every object in this category, i.e. any E_n-algebra A, is "fully dualizable" and thus gives rise to a (fully extended, in the sense of Lurie) TFT by the cobordism hypothesis of Baez-Dolan-Lurie. It turns out that this TFT can be explicitly constructed by (essentially) taking factorization homology with coefficients in the E_n-algebra A.
We introduce a notion of a graded Poisson manifold up to homotopy, namely a Poisson$[n,k]$-manifold, motivated by studying the dual of a Lie 2-algebra. We further study Maurer-Cartan elements on Poisson$[n,k]$-manifolds and symplectic$[n,n]$-manifolds. There are many interesting examples such as $n$-term $L_\infty$-algebras, twisted Poisson manifolds, quasi-Poisson $\g$-manifolds and twisted Courant algebroids. As a byproduct, we justify that the symplectic$[n,n]$-manifold is a homotopy version of the symplectic NQ-manifold and the Maurer-Cartan equation is a homotopy version of the master equation. The dual of an $n$-term $L_\infty$-algebra is a Poisson$[n,n]$-manifold. We prove that the cotangent bundle of a Poisson$[n,n]$-manifold is a symplectic NQ-manifold of degree $n+1$. In particular, we construct a Courant algebroid from a $2$-term $L_\infty$-algebra. By analyzing these structures, we obtain a Lie-quasi-Poisson groupoid from a Lie 2-algebra, which we propose to be the geometric structure on the dual of a Lie $2$-algebra. At last, we obtain an Ikeda-Uchino algebroid from a $3$-term $L_\infty$-algebra. See arXiv: 1312.4609 for more details.
In this talk we study the cohomology graph complexes, introduced by M. Kontsevich. We sketch the proof, using spectral sequences, that graph cohomology is not changed by allowing multiple adges. Furthermore, we sketch the derivation of formulas for the dimension of these complexes, which can be used to compute the Euler characteristic, and present results obtained by the computer.
Atiyah's theorem states that there is a diffeomorphism between the moduli space of based instantons on S^4 with a gauge group G and the parameter space of based holomorphic maps from S^2 to the loop group \Omega G. I'm working on the conjecture, which asserts, that there is a bijection between moduli space of based Yang-Mills fields with a gauge group G and the parameter space of based harmonic maps from S^2 into the loop group. I will talk about what is done on this way -- the twistor lift for the loop group \Omega G (via Birkhoff-Grothendieck decomposition).
Given a smooth complex algebraic curve, one can consider the moduli space of stable vector bundles over it. There is a well known formula for the Euler characteristic of line bundles over this moduli space, called the Verlinde formula. In this talk we outline a method to prove this formula through Atiyah-Bott localization over the Quot scheme of quotient sheaves.