Introduction to Supergeometry

Annotation

The course gives an extended introduction into supergeometry and its applications. Ideas and methods of super- and graded geometry play an important role in modern theoretical physics and mathematics. Supergeometry gives a concise and powerful language to describe a collection  of compatible structures, leading to applications in homological algebra, index theory, quantization of gauge systems and supersymmetry. In addition to thorough introduction into the foundations of the subject, the course will also consider various recent developments and applications.

Course plan

1. Functional-algebraic duality.  Example of a space of a non-set-theoretic nature: the double point. Idea of affine superspace

2. Elements of superalgebra. Graded vector spaces and modules.  Sign rule. Tensor, exterior and symmetric algebras of a superspace. Parity reversion functor. (Super)commutator.  Commutative superalgebras and modules over them. Dimension of a free module.  First examples of Lie superalgebras. Supertrace. Berezinian: axiomatic approach, explicit formula, relation with supertrace, multiplicativity. First examples of Lie supergroups.

3. Affine superspace and open superdomains. Algebra of smooth functions of even and odd variables. Homomorphisms and automorphisms of such algebras. The category of open superdomains. Differential calculus. Berezin integral and change of variables theorem. 

4. Supermanifolds.  Definition of smooth and complex-analytic supermanifolds as local ringed spaces with given "local models". The language of charts and atlases. The language of points. Graded (super)manifolds.  First examples: submanifolds in affine superspace. Superspheres. Projective superspaces. Grassmann supermanifolds.  Lie supergroups. Classification theorem for smooth supermanifolds.

5. Supermanifold language for "ordinary" differential geometry. Vector bundles $\Pi TM$ and $\Pi T^*M$. Interpretation of differential forms and multivector fields as functions. 

6. Various superanalogs of differential forms. Bernstein-Leites integral forms, pseudodifferential forms and their integration. Superforms as Lagrangians. 

7. Graded manifolds as generalization of vector bundles. Homological vector fields. Their application to describing structures. Examples: Lie algebroids, L-infinity algebras..

8. Examples of applications. Index of operator as supertrace. “Super proof” of the Atiyah-Singer Index theorem.  Volumes of classical supermanifolds. Stanford-Witten formula for symplectic supervolume. Odd symplectic geometry and Batalin-Vilkovisky formalism. “Microformal geometry”.  Universal recurrence relations for exterior powers. Generalization of Buchstaber-Rees theory.  “Super Pluecker” map and examples of "supercluster structure".

Literature

1. F. A. Berezin. Introduction to superanalysis. Reidel, Dordrecht,1987.
2. D. A. Leites. Introduction to the theory of supermanifolds, Russian Math. Surveys 35 (1) (1980),13-64.
3. Yu. I. Manin. Gauge field theory and complex geometry, Springer-Verlag, Berlin,1997.
4. Th. Voronov. Geometric Integration Theory on Supermanifolds. Harwood Academic Publ., 1992. (2nd ed.: Cambridge Scientific Publ., 2014)
5. P. Deligne and J. Morgan. Notes on supersymmetry (following Joseph Bernstein). In book: Quantum fields and strings: a course for mathematicians, Vol.1, 41-97, Amer. Math. Soc., Providence, RI, 1999.
6. A. Rogers. Supermanifolds: Theory and applications. World Scientific Publishing. Ltd., Hackensack, NJ, 2007.
7. Additional literature:
8. Th. Voronov.  Quantization on supermanifolds and the analytic proof of the Atiyah-Singer index theorem. J. Soviet Math. 64 (4) (1993),  993--1069.
9. Th. Voronov. On volumes of  classical supermanifolds. Sbornik: Mathematics 207 (11) (2016), 1512-1536.
10. Th. Voronov.  Graded geometry, Q-manifolds, and microformal geometry, Fortschritte der Physik 67 (2019), 1910023, DOI 10.1002/prop.201910023.