nLab
beta-gamma system
Context
Quantum field theory
Complex geometry
Contents
Idea
What is called the - system is a 2-dimensional quantum field theory defined on Riemann surfaces whose fields are pairs consisting of a -form and a -form and whose equations of motion demand these fields to be holomorphic differential forms.
The name results from the traditional symbols for these fields, which are
Definition
We state the definition of the --system as a free field theory (see there) in BV-BRST formalism, following (Gwilliam, section 6.1).
We first give the standard variant of the theory, the
Then we consider the
Abelian massless theory
Let be a Riemann surface.
kinematics
-
the field bundle is
-
hence the (abelian) sheaf of local sections is
we write
for the sections of compact support
-
the local pairing
with values in the density bundle is given by wedge product followed by projection on the -forms
-
hence the global pairing
is given by
dynamics
-
the differential operator
is the Dolbeault differential
-
hence the elliptic complex of fields is
is the Dolbeault complex;
-
and hence the action functional
is
Abelian massive theory
(…)
Holomorphic Chern-Simons theory
…holomorphic Chern-Simons theory…
Properties
Euler-Lagrange equations of motion
The equations of motion are
References
Discussion in the context of BV-quantization and factorization algebras is in chapter 6 of
A construction of chiral differential operators via quantization of system in BV formalism with an intermediate step using factorization algebras:
- Vassily Gorbounov, Owen Gwilliami?, Brian Williams, Chiral differential operators via Batalin-Vilkovisky quantization, pdf
Last revised on December 31, 2017 at 08:57:32.
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