## Kavli IPMU Komaba Seminar

開催情報 月曜日　16:30～18:00　数理科学研究科棟(駒場) 002号室 河野 俊丈

### 2009年11月09日(月)

16:30-18:00   数理科学研究科棟(駒場) 002号室
Makoto Sakurai 氏 (東京大学大学院数理科学研究科)
Differential Graded Categories and heterotic string theory
[ 講演概要 ]
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.