nLab 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.


  • the field bundle EXE \to X is

    E 0,Γ(TX) 1,Γ(TX) E \coloneqq \wedge^{0,\bullet}\Gamma(T X) \oplus \wedge^{1,\bullet} \Gamma(T X)
  • hence the (abelian) sheaf of local sections is

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

    we write

    cΓ X(E)\mathcal{E}_c \hookrightarrow \Gamma_X(E)

    for the sections of compact support

  • the local pairing

    , loc:EEDens X \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)(1,1)-forms

    γ 1+β 1,γ 2,β 2 loc(γ 1β 2+γ 2β 1) |(1,1) \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

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

    is given by

    γ 1+β 1,γ 2,β 2 loc X(γ 1β 2+γ 2β 1) \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)


  • the differential operator

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

    is the Dolbeault differential ¯\bar \partial

  • hence the elliptic complex of fields is

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

    is the Dolbeault complex;

  • and hence the action functional

    S: c𝒞 S \colon \mathcal{E}_c \to \mathcal{C}


    (γ+β) 12 Xγ+β,¯(γ+β) = Xβ¯γ \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


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).

  • pure spinor formalism?



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 βγ\beta\gamma system in BV formalism with an intermediate step using factorization algebras:

As a fractional level or logarithmic CFT

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

and its lift to a logarithmic CFT:

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

Last revised on June 16, 2023 at 23:10:03. See the history of this page for a list of all contributions to it.