## Kavli IPMU Komaba Seminar

Seminar information archive ～12/11｜Next seminar｜Future seminars 12/12～

Date, time & place | Monday 16:30 - 18:00 002Room #002 (Graduate School of Math. Sci. Bldg.) |
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### 2009/11/09

16:30-18:00 Room #002 (Graduate School of Math. Sci. Bldg.)

Differential Graded Categories and heterotic string theory

**Makoto Sakurai**(東京大学大学院数理科学研究科)Differential Graded Categories and heterotic string theory

[ Abstract ]

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.

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.