12/9/16: The supermoduli space: the Mumford and super Mumford
isomorphisms. The Grothendieck-Riemann-Roch Theorem. The
Belavin-Polyakov theorem on the relation of the Polyakov measure to
the Mumford form. Thank you very much for being dedicated auditors!
[Class notes; Donagi and Witten, Supermoduli space is not
projected: Section 3.2 (pages 25-26)]
12/7/16: The supermoduli space: the local structure, the tangent
space. [Class notes; Donagi and Witten, Supermoduli space is not
projected: Section 3.2 (pages 25-26)]
12/5/16: The supermoduli space: density of singular points. The moduli
space of spin curves. The Kodaira-Spencer isomorphism for the ordinary
moduli space. [Class notes; Donagi and Witten, Supermoduli space is
not projected: Section 3.2 (pages 24-25)]
12/2/16: A primer on moduli spaces: automorphisms obstruct the
existence of a fine moduli space; the idea of an algebraic (Artin)
stack. [Class notes; Donagi and Witten, Supermoduli space is not
projected: Section 3.2 (page 24)]
11/30/16: The uniqueness of an SRS structure
over P1. A primer on moduli spaces: the moduli space
as an object representing the functor of families of geometric
objects; the universal family; the fine and coarse moduli space of
algebraic curves. [Class notes; Manin, Topics in Noncommutative
Geometry: Sections 2.2.1-4; Donagi and Witten, Supermoduli
space is not projected: Section 3.2 (page 24)]
11/28/16: Extended Thanksgiving break: no class.
11/25/16: Thanksgiving Break: University closed.
11/23/16: Extended Thanksgiving break: no class.
11/21/16: Classification of SRS structures over a Riemann
surface. [Class notes; Manin, Topics in Noncommutative
Geometry: Sections 2.2.1-4]
11/18/16: A family of SRS structures on P1|1. Super
Riemann surfaces and projective superspaces are split. [Class notes;
Donagi and Witten, Supermoduli space is not projected: Section
3.1; Manin, Gauge Field Theory and Complex Geometry: Section
4.3.5]
11/16/16: General projective superspaces: the tautological line bundle
and the tangent bundle. [Class notes; Manin, Gauge Field Theory and
Complex Geometry: Sections 4.3.6 and 4.3.11]
11/14/16: Example of a super Riemann surface: projective superspace of
dimenison 1|1. General projective superspaces: construction. [Class
notes; Manin, Topics in Noncommutative Geometry: Section 2.1;
Manin, Gauge Field Theory and Complex Geometry: Section
4.3.4]
11/11/16: Relation between points and divisors on a super Riemann
surface (SRS). [Class notes; Donagi and Witten, Supermoduli space
is not projected: Section 3.1]
11/9/16: The local structure of a maximally nonitegrable, odd
distribution. Superconformal coordinates. [Class notes; Donagi and
Witten, Supermoduli space is not projected: Section 3.1]
11/7/16: Super Riemann surfaces. Frobenius' theorem. Shander's
theorem. [Class notes; Manin: Section 4.4; Donagi and Witten, Supermoduli space
is not projected: Section 3.1]
11/4/16: A finite-dimensional version of the Faddeev-Popov procedure
(proof). Further integration of the string theory Feynman integral
along the group of diffeomorphisms of the Riemann surface: getting the
Faddeev-Popov determinant. Integration along conformal transformations
of the metric: the Liouville action and the critical dimension in
string theory. [Class notes; E. D'Hoker, String Theory
in Quantum Fields and Strings: A Course for Mathematicians,
volume 2: Lecture 3]
11/2/16: Integration of the string theory Feynman integral along the
space of maps from the surface to the Euclidean space-time: getting
the determinant of the negative Laplace operator on the surface. A
finite-dimensional version of the Faddeev-Popov procedure
(formulation). [Class notes; E. D'Hoker, String Theory
in Quantum Fields and Strings: A Course for Mathematicians,
volume 2: Lecture 3]
10/31/16: Introduction to string theory: Gaussian
integration. Relation to Riemann surfaces (complex curves). Plan for
reducing the Feynman integral to the sum of integrals over the moduli
spaces of complex curves. [Class notes]
10/28/16: Introduction to string theory: the Polyakov action and the
Feynman integral. [Class notes; E. D'Hoker, String Theory
in Quantum Fields and Strings: A Course for Mathematicians,
volume 2: Lecture 1]
10/26/16: Left and right connections. Stokes' theorem on a
supermanifold. [Class notes; Manin: 4.5.1-6; Deligne-Morgan: 3.12]
10/24/16: The complex of integral forms on a supermanifold. Stokes'
theorem on a supermanifold. [Class notes; Deligne-Morgan: 3.12]
10/21/16: Integration on a general paracompact
supermanifold. Differential forms on a manifold as Clifford
modules. [Class notes; Varadarajan: 4.6, Deligne-Morgan: 3.12.1]
10/19/16: Change of variables in the integral: the general
case. [Class notes; Varadarajan: 4.6]
10/17/16: The Berezin integral in a superdomain U
⊆ Rm|n. Change of variables: the purely odd
case. [Class notes; Varadarajan: 4.6]
10/14/16: Internal Hom (R0|1, M) =
ΠTM. Motivation for the Berezin integral: integration
over R0|1. [Class notes; Deligne-Morgan: Section
2.8]
10/12/16: The functor of points. Yoneda's lemma. Representable
functors. The moduli space of maps and the internal Hom. [Class notes;
Deligne-Morgan: Section 2.8]
10/10/16: The sign rule for ZxZ2-graded
objects and rectification of the homological interpretation of Ber(M)
for a free R-module M. Differential forms. De Rham cohomology. [Class
notes; Deligne-Morgan: Section 3.3; Manin: Sections 3.4.4-5]
10/7/16: Homological interpretation of the Berezinian of a module: the
right Koszul complex. Superderivations and the tangent bundle. [Class
notes; Deligne-Morgan: Sections 1.10-11, 3.2-3; Manin: Sections
3.4.6-7; Varadarajan: Section 4.4]
10/5/16: Homological interpretation of the Berezinian of a module: the
left Koszul complex. Showing it gives a free resolution of R as a
module over SR(M) for a free R-module M of a finite
rank. [Class notes; Deligne-Morgan: Sections 1.10-11; Manin: Section
3.4.6]
10/3/16: The universal enveloping algebra. Superdeterminant
(Berezinian). The Berezinian of a free module. [Class notes;
Deligne-Morgan: Sections 1.3, 1.10(A,C), 1.11; Varadarajan: Sections
3.5-6; Manin: Section 3.3.7]
9/30/16: Superderivations. Super multilinear
algebra. Supertrace. [Class notes; Varadarajan: Sections 3.5 and 3.1;
Deligne-Morgan: Sections 1.3 and 1.6]
9/28/16: Correction: I have
mistreated G1/G2
= Aut(V*). Super linear algebra. Lie
superalgebras. [Class notes; Donagi and Witten, Supermoduli space
is not projected: Section 2.2.1; Varadarajan: Sections 3.1, 3.7]
9/26/16: Obstructions to splitting: obstruction classes in
cohomology. [Class notes; Donagi and Witten, Supermoduli space is
not projected: Section 2.2.1; Manin: Section 4.2]
9/23/16: The vector bundle ("odd tangent bundle") corresponding to a
supermanifold. Obstructions to splitting: description of geometric
objects as nonabelian cohomology of degree one. [Class notes; Donagi
and Witten, Supermoduli space is not projected: page 11; Manin:
Section 4.2]
9/21/16: Complex projective superspace. Obstructions to splitting:
complex and smooth supermanifolds are locally split. [Class notes;
Varadarajan: Section 4.2; Manin: Section 4.3.4]
9/19/16: Examples: the de Rham algebra of a manifold; the split
supermanifold S(M,V) associated with a vector bundle over a
manifold. Applications to classical, nonsuper mathematics, such as Lie
algebroids as integrable odd vector fields on supermanifolds
S(M,V). [Class notes; Varadarajan: Section 4.2; Donagi and
Witten, Supermoduli space is not projected: Section 2.1]
9/16/16: Common constructions for a superspace M: the sheaf
JM of ideals, Mrd, Gr M. Split and projected
supermanifolds. [Class notes; Manin: Section 4.1.3; Varadarajan:
Section 4.2]
9/14/16: Local description of morphisms (proof). [Class notes; Manin:
Proposition 4.1.8; Varadarajan: Theorem 4.3.1]
9/12/16: Three categories of superspaces: algebraic, analytic, and
smooth. Local description of morphisms. [Class notes; Manin: Sections
4.1.1-2, 4.1.5, 4.1.7, Proposition 4.1.8; Varadarajan: Sections 4.2-3]
9/9/16: The categories of supercommutative rings and affine
superschemes. The structure sheaf on Spec(R). The notions of a
superspace and a superscheme. [Class notes; Manin: Sections 3.1.3,
4.1.1-2, 4.1.6]
9/7/16: Introduction. Superalgebra: supercommutative rings and
algebras, examples, the symmetric monoidal category of super vector
spaces, a supercommutative algebra as a commutative monoid therein.
[Class notes; Manin: Sections 4.1.1-2, 4.1.5, 4.1.7, Proposition
4.1.8; Varadarajan: Sections 2.6, 3.1, 3.7]
Last modified: (2016-12-09 16:20:33 CST)