nLab beta-gamma system

Idea

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

$$(\gamma,\beta) \in \Omega^{0,\bullet}(X) \oplus \Omega^{1, \bullet}(X) \,.$$

Definition

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

• Abelian massless theory.

Then we consider the

• Abelian massive theory

• Holomorphic Chern-Simons theory

Abelian massless theory

Let $X$ be a Riemann surface.

kinematics

• the field bundle $E \to X$ is

  $$E \coloneqq \wedge^{0,\bullet}\Gamma(T X) \oplus \wedge^{1,\bullet} \Gamma(T X)$$

• hence the (abelian) sheaf of local sections is

  $$\mathcal{E} = \Omega_X^{0,\bullet} \oplus \Omega_X^{1, \bullet} \,,$$

  we write

  $\mathcal{E}_c \hookrightarrow \Gamma_X(E)$

  for the sections of compact support

• the local pairing

  $$\langle -,-\rangle_{loc} \colon E \otimes E \to Dens_X$$

  with values in the density bundle is given by wedge product followed by projection on the $(1,1)$-forms

  $$\langle \gamma_1 + \beta_1, \gamma_2, \beta_2\rangle_{loc} \coloneqq (\gamma_1 \wedge \beta_2 + \gamma_2 \wedge \beta_1)_{|(1,1)}$$

• hence the global pairing

  $$\langle -,-\rangle \colon \mathcal{E}_c \otimes \mathcal{E}_c \to \mathbb{C}$$

  is given by

  $$\langle \gamma_1 + \beta_1, \gamma_2, \beta_2\rangle_{loc} \coloneqq \int_{X}\left(\gamma_1 \wedge \beta_2 + \gamma_2 \wedge \beta_1\right)$$

dynamics

• the differential operator

  $$Q \colon \mathcal{E} \to \mathcal{E}$$

  is the Dolbeault differential $\bar \partial$

• hence the elliptic complex of fields is

  $$(\mathcal{E}, Q) = (\Omega_X^{0,\bullet}\oplus \Omega_X^{1,\bullet}, \bar \partial)$$

  is the Dolbeault complex;

• and hence the action functional

  $$S \colon \mathcal{E}_c \to \mathcal{C}$$

  is

  $$\begin{aligned} (\gamma + \beta) & \mapsto \frac{1}{2}\int_X \langle \gamma+ \beta, \; \bar \partial (\gamma + \beta)\rangle \\ & = \int_X \beta \wedge \bar \partial \gamma \end{aligned}$$

Abelian massive theory

(…)

Holomorphic Chern-Simons theory

…holomorphic Chern-Simons theory…

Properties

Euler-Lagrange equations of motion

The equations of motion are

$$\bar \partial \gamma = 0 \;\;, \;\; \bar \partial\beta = 0 \,.$$

Relation with $\sigma$-models

Consider a sigma-model $X\hookrightarrow \mathbb{R}^{1,1}$ to the target space $\mathbb{R}^{1,1}$

$$S[q,\gamma] = \int_X \partial q \wedge \bar{\partial}\gamma,$$

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.

$$S_{\mathrm{gauged}}[q,\gamma] = \int_X \partial_\beta q \wedge \bar{\partial}\gamma,$$

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

$$S_{\beta\gamma}[\beta,\gamma] = \int_X \beta \wedge \bar{\partial}\gamma.$$

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).

Related concepts

• pure spinor formalism?

References

General

• 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, J.Phys. A42:355401 (2009) (doi arXiv/0904.2234)

Discussion in the context of BV-quantization and factorization algebras is in chapter 6 of

• Owen Gwilliam, Factorization algebras and free field theories PhD thesis (pdf)

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)

As a fractional level or logarithmic CFT

On the beta-gamma system as the $\mathfrak{su}(2)$ WZW model at fractional level $-1/2$:

• F. Lesage, P. Mathieu, J. Rasmussen, H. Saleur, The $\mathfrak{su}(2)_{-1/2}$ WZW model and the beta-gamma system, Nucl. Phys. B 647 (2002) 363-403 $[$arXiv:hep-th/0207201, doi:10.1016/S0550-3213(02)00905-7a$]$

and its lift to a logarithmic CFT:

• F. Lesage, P. Mathieu, J. Rasmussen, H. Saleur, Logarithmic lift of the $\mathfrak{su}(2)_{-1/2}$ model, Nuclear Physics B 686 3 (2004) 313-346 [doi:10.1016/j.nuclphysb.2004.02.039]