# IPMU Komaba Seminar

Speaker: Makoto Sakurai (The Univ. of Tokyo) Differential Graded Categories and heterotic string theory Mon, Nov 09, 2009, 16:30 - 18:00 Room 002, Mathematical Sciences Building, Komaba Campus The saying "category theory is an abstract nonsense" is even physically not true. The schematic language of triangulated category presents a new stage of string theory. To illuminate this idea, I will draw your attention to the blow-up minimal model of complex algebraic surfaces. This is done under the hypothetical assumptions of "generalized complex structure" of cotangent bundle due to Hitchin school. The coordinate transformation Jacobian matrices of the measure of sigma model with spin structures cause one part of the gravitational "anomaly cancellation" of smooth Kahler manifold $X$ and Weyl anomaly of compact Riemann surface $\Sigma$. $Anom = c_1 (X) c_1 (\Sigma) \oplus ch_2 (X)$, in terms of 1st and 2nd Chern characters. Note that when $\Sigma$ is a puctured disk with flat metric, the chiral algebra is nothing but the ordinary vertex algebra. Note that I do not explain the complex differential geometry, but essentially more recent works with the category of DGA (Diffenreial Graded Algebra), which is behind the super conformal field theory of chiral algebras. My result of "vanishing tachyon" (nil-radical part of vertex algebras) and "causality resortation" in compactified non-critical heterotic sigma model is physically a promising idea of new solution to unitary representation of operator algebras. This idea is realized in the formalism of BRST cohomology and its generalization in $\mathcal{N} = (0,2)$ supersymmetry, that is, non-commutative geometry with non-linear constraint condition of pure spinors for covariant quantization.