beta-gamma system



What is called the β\beta-γ\gamma system is a 2-dimensional quantum field theory defined on Riemann surfaces XX whose fields are pairs consisting of a (0,0)(0,0)-form and a (1,0)(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

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


We state the definition of the β\beta-γ\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

Abelian massless theory

Let XX be a Riemann surface.



Abelian massive theory


Holomorphic Chern-Simons theory

holomorphic Chern-Simons theory


Euler-Lagrange equations of motion

The equations of motion are

¯γ=0,¯β=0. \bar \partial \gamma = 0 \;\;, \;\; \bar \partial\beta = 0 \,.

Relation with σ\sigma-models

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

S[q,γ]= Xq¯γ, S[q,\gamma] = \int_X \partial q \wedge \bar{\partial}\gamma,

which has an abelian right-moving Kac-Moody symmetry qq+λq\mapsto q+\lambda with λ=0\partial\lambda=0. We can can consider a theory where this symmetry is promoted to a gauge symmetry, i.e.

S gauged[q,γ]= X βq¯γ, S_{\mathrm{gauged}}[q,\gamma] = \int_X \partial_\beta q \wedge \bar{\partial}\gamma,

where βq:=q+β\partial_\beta q := \partial q + \beta where βΩ 1,(X)\beta\in\Omega^{1,\bullet}(X) is the connection. If we choose the gauge with q=0q=0, we obtain the β\beta-γ\gamma system with action

S βγ[β,γ]= Xβ¯γ. S_{\beta\gamma}[\beta,\gamma] = \int_X \beta \wedge \bar{\partial}\gamma.

Thus, a β\beta-γ\gamma system can be interpreted as a chiral (or Kac-Moody) quotient along a null killing vector of a sigma-model with target space 1,1\mathbb{R}^{1,1} (LinRoc20).


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

A construction of chiral differential operators via quantization of βγ\beta\gamma system in BV formalism with an intermediate step using factorization algebras: