18.276 Mathematical Methods in Physics
Topics: Operads and 2d Quantum Field Theory
Graduate course
Mon Wed 2:30 - 4:00 pm; Room 2-151
Office hours: Mon 1:30-2:30 pm; 2-246
Lecturer: Alexander A. Voronov
We plan to give a mathematical introduction to 2d quantum field theories
(QFT's), aiming at recent developments enhanced by applying methods of
operad theory. 2d QFT's are models of elementary particle physics, which
include conformal field theory (CFT), string theory, and quantum
gravity. The theory of operads is a tool of algebraic topology, which
proved to be useful in the study of loop spaces in the seventies and is
now undergoing a period of renaissance, mainly because of recent
applications to homotopy algebra structures and physics.
Prerequisites: familiarity with Riemann surfaces (complex algebraic
curves) and basic homology theory will be helpful. No knowledge of
physics or operads is required - all necessary notions will be
introduced along the way. During the course, some problems, including
open ones, will be given.
Outline of the course: introduction to CFT; operads of Riemann surfaces
as underlying geometric structures of CFT's and other 2d QFT's, such as
topological QFT's and CFT's and quantum gravity; introduction to
operads; operads related to moduli spaces of Riemann surfaces;
$L_\infty$, $A_\infty$, and $G_\infty$ algebras; the homotopy algebraic
structure of TCFT; cohomology of CFT's and vertex algebras; cyclic and
modular operads. Time permitting and depending on the interests of the
audience, we may make the following digressions: operads and
Fulton-MacPherson's resolution of diagonals; Beilinson-Ginzburg's work
on the local structure of moduli spaces; Deligne's question about
homotopy algebraic structures on the Hochschild complex; using operads
to study the cohomology of moduli spaces of Riemann surfaces
(Grothendieck's ideas).