MAT 649

Topics in algebraic Yang-Mills theory

Spring 2009

Instructor: Michael Movshev

N=1 D=10 supersymmetric Yang-Mills theory has the richest structure among the gauge theories. This is due of course to the large group of symmetries. This,however, becomes an obstacle for the standard methods of analysis , like super-space technique, that work well for theories with less super-symmetries. This calls for an explanation. The language of homological algebra turns to be appropriate for problems of this kind. The result is that we not only locate the obstruction for super-space formulation, we explain how to resolve it. As a byproduct we classify all supersymmetric deformations of the standard N=1 D=10 Lagrangian and explain how to construct the global deformations.

No knowledge of physics is required. Some familiarity with homological algebra and algebraic geometry will be helpful.

I am planning to assign several problem sets, which will be posted here.

Schedule of Topics

(Note: This schedule is tentative and may be revised at a later date.)

Week of


Jan. 26

Yang-Mills theory and Yang-Mills algebra.

Hochschild homology and cohomology of algebras.

Feb. 2

Yang-Mills algebra and its homological properties.

Geometric interpretation of Hochschild homology. Chen iterated integrals.

Feb. 9

BV formulation of Yang-Mills theory. Its homological explanation.

Deformation theory for pure Yang-Mills algebra

Feb. 16

Supersymmetric Yang-Mills algebra in dimensions 3,4,6,10

Superspace formulation in dimensions 3 and 4.

Feb. 23

Pure spinors in dimension 10.

Homological nature of pure spinors.

March 2

    Resolution of Corti and Reid. Analogs of pure spinors in lower dimensions.

March 9

    Koszul cohomology of graded rings of Fano varieties. Fano varieties of co-index 3. Mukai classification.

March 16

    Methods to put supersymmetry in dimension 10 off-shell.

March 23

    Mathematical formulation of a supersymmetric deformation.

    The equivariant complex.

March 30

Kontsevich deformation quantization

April 6

Spring Break

April 13

Application of Kontsevich quantization to the principal deformation.

April 20

Construction of the global deformations.

April 27

Complexes related to nonintegrable distributions.

May 4

Applications to Yang-Mills theory and 11-dimensional supergravity.

Suggested reading

Yang-Mills algebra:

Homological Algebra:

BV formalism:

Deformation Quantization:

Pure spinors in dimension 10 and Fano: