functorial quantum field theory
Reshetikhin?Turaev model? / Chern-Simons theory
FQFT and cohomology
geometry, complex numbers, complex line
$dim = 1$: Riemann surface, super Riemann surface
What is called the $\beta$-$\gamma$ system is a 2-dimensional quantum field theory defined on Riemann surfaces $X$ whose fields are pairs consisting of a $(0,0)$-form and a $(1,0)$-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
We state the definition of the $\beta$-$\gamma$-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
Let $X$ be a Riemann surface.
the field bundle $E \to X$ is
hence the (abelian) sheaf of local sections is
we write
$\mathcal{E}_c \hookrightarrow \Gamma_X(E)$
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 $(1,1)$-forms
hence the global pairing
is given by
is the Dolbeault differential $\bar \partial$
hence the elliptic complex of fields is
is the Dolbeault complex;
and hence the action functional
is
(…)
…holomorphic Chern-Simons theory…
The equations of motion are
Consider a sigma-model $X\hookrightarrow \mathbb{R}^{1,1}$ to the target space $\mathbb{R}^{1,1}$
which has an abelian right-moving Kac-Moody symmetry $q\mapsto q+\lambda$ with $\partial\lambda=0$. We can can consider a theory where this symmetry is promoted to a gauge symmetry, i.e.
where $\partial_\beta q := \partial q + \beta$ where $\beta\in\Omega^{1,\bullet}(X)$ is the connection. If we choose the gauge with $q=0$, we obtain the $\beta$-$\gamma$ system with action
Thus, a $\beta$-$\gamma$ system can be interpreted as a chiral (or Kac-Moody) quotient along a null killing vector of a sigma-model with target space $\mathbb{R}^{1,1}$ (LinRoc20).
Nikita Nekrasov, Lectures on curved beta-gamma systems, pure spinors, and anomalies, (hep-th/0511008)
Anton Zeitlin, Beta-gamma systems and the deformations of the BRST operator, (arxiv/0904.2234)
Discussion in the context of BV-quantization and factorization algebras is in chapter 6 of
A construction of chiral differential operators via quantization of $\beta\gamma$ system in BV formalism with an intermediate step using factorization algebras:
Vassily Gorbounov, Owen Gwilliam, Brian Williams, Chiral differential operators via Batalin-Vilkovisky quantization, pdf
Ulf Lindstrom, Martin Rocek, $\beta$-$\gamma$-systems interacting with sigma-models, (arXiv:2004.06544)