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Topological Dirac Sigma Models and the Classical Master Equation
High Energy Physics - Theory · Submitted 28 Jun 2022
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim We present the construction of the classical Batalin-Vilkovisky action for topological Dirac sigma models.
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The abstract, sentence by sentence
We present the construction of the classical Batalin-Vilkovisky action for topological Dirac sigma models. The latter are two-dimensional topological field theories that simultaneously generalise the completely gauged Wess-Zumino-Novikov-Witten model and the Poisson sigma model. Their underlying structure is that of Dirac manifolds associated to maximal isotropic and integrable subbundles of an exact Courant algebroid twisted by a 3-form. In contrast to the Poisson sigma model, the AKSZ construction is not applicable for the general Dirac sigma model. We therefore follow a direct approach for determining a suitable BV extension of the classical action functional with ghosts and antifields satisfying the classical master equation. Special attention is paid on target space covariance, which requires the introduction of two connections with torsion on the Dirac structure.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We present the construction of the classical Batalin-Vilkovisky action for topological Dirac sigma models. | contribution 1.00 | 0.96 | 0.41 | asserted 1.00 |
| s1 | The latter are two-dimensional topological field theories that simultaneously generalise the completely gauged Wess-Zumino-Novikov-Witten model and the Poisson sigma model. | background 0.98 | 0.13 | 0.06 | asserted 1.00 |
| s2 | Their underlying structure is that of Dirac manifolds associated to maximal isotropic and integrable subbundles of an exact Courant algebroid twisted by a 3-form. | background 0.91 | 0.23 | 0.07 | asserted 1.00 |
| s3 | In contrast to the Poisson sigma model, the AKSZ construction is not applicable for the general Dirac sigma model. | gap 0.32 | 0.49 | 0.12 | asserted 1.00 |
| s4 | We therefore follow a direct approach for determining a suitable BV extension of the classical action functional with ghosts and antifields satisfying the classical master equation. | method 0.92 | 0.93 | 0.16 | asserted 0.98 |
| s5 | Special attention is paid on target space covariance, which requires the introduction of two connections with torsion on the Dirac structure. | method 0.71 | 0.82 | 0.09 | asserted 1.00 |
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The paper
Original abstract
We present the construction of the classical Batalin-Vilkovisky action for topological Dirac sigma models. The latter are two-dimensional topological field theories that simultaneously generalise the completely gauged Wess-Zumino-Novikov-Witten model and the Poisson sigma model. Their underlying structure is that of Dirac manifolds associated to maximal isotropic and integrable subbundles of an exact Courant algebroid twisted by a 3-form. In contrast to the Poisson sigma model, the AKSZ construction is not applicable for the general Dirac sigma model. We therefore follow a direct approach for determining a suitable BV extension of the classical action functional with ghosts and antifields satisfying the classical master equation. Special attention is paid on target space covariance, which requires the introduction of two connections with torsion on the Dirac structure.
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How the literature uses it
6 arXiv papers cite this in 9 sentences
Papers citing it · 7 sentences without a confident label
Cites it as background 2 papers
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We are not going to repeat here the technical analysis of [ref] .
Basic curvature and the Atiyah cocycle in gauge theory · Dirac sigma models · 0.96
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Although the machinery of Dirac structures, Manin pairs, and Lie quasi-bialgebras, that we will employ, might be obscure to some physicists, it has previously made numerous appearances in string theory, including: in Poisson-Lie T-duality [ref] , in 2D integrable [ref] and topological [ref] sigma models, in the realisation of symmetries on string and brane worldvolumes [ref] , and in the geometry of flux compactifications [ref] , to give an incomplete list.
Localisation without supersymmetry: towards exact results from Dirac structures in 3D $N = 0$ gauge theory · Introduction · 0.98
What it cites · 38 sentences without a confident label
Builds on 1 paper
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Notably, they satisfy a number of useful identities, which we present in component form:[Note that these are the generalizations for arbitrary Dirac structure $E$ of equations (D.11)-(D.14) of Ref. [ref] for the Dirac structure on $T^∗M$, to which they reduce under the special choice [ref] and $g →0$.] T^±c_ab_̊c^ = -2ρ^_[a∇^±_^̊_b]+_̊a^_̊b^Θ^±^_ , ι_ρ_[aι_ρ_bR^±d_c] = ^̊_[a∇^±_T^±d_bc]+T^±d_e[aT^±e_bc] , [∇^±_,∇^±_]T^±a_bc = Θ^±^_∇^±_T^±a_bc+T^±d_bcR^±a_d-T^±a_dcR^±d_b-T^±a_bdR^±d_c , T^±^c_ab = G_±^-1^cρ (ρ^_[a∇^±_ρθ_b]- θ_[b ∇^±_ρρ^_a]-ρ^_[aθ_b]Θ^±^_ρ ) , where in the cases when the covariant derivative acts both on tangent and bundle indices, it is the covariant derivative combining the ones on $TM$ and $E$, $∇^♯±_v$ and $∇^±_v$, respectively.
BV and BFV for the H-twisted Poisson sigma model · Curvature and torsion vs. $E$-curvature and $E$-torsion · 0.85
Uses the method or tool of 2 papers
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For this purpose [ref] , one considers the subbundle $V_E := E ⊕(E^⊥∩V)$, where $V={ v + ι_v g, v ∈TM }$ and $E^⊥$ is the orthogonal to $E$ with respect to the canonical inner product on the generalized tangent bundle $TM ⊕T^*M$.
Transverse generalized metrics and 2d sigma models · Nonstandard gaugings and the gauge transformations · 0.91
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To achieve the goal of finding the BV action for Dirac sigma models we follow the usual steps of the field-antifield formalism [ref] .
Antibracket, Antifields and Gauge-Theory Quantization · BRST operator and field-antifield content · 0.92
Compares with 2 papers
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In that case, the transformation of$A$given in [ref] reduces to Ạ|_HPSM=ϵ-T(A,ϵ) , as found in [ref] .$]
BV and BFV for the H-twisted Poisson sigma model · Target space covariant formulation · 0.80
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To achieve this, one should consider $E=T^∗M$ and make the following identifications [ref] : ρ=Π^♯ and θ=id , or, in components, $ρ^_a→Π^$ and $θ_a→_̣^$.
Beyond the standard gauging: gauge symmetries of Dirac Sigma Models · Curvature and torsion vs. $E$-curvature and $E$-torsion · 0.94
Cites as support 1 paper
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Indeed, as proven in [ref] , the maps E_±:=θ^∗±ρ: E→TM , where $θ^∗=g^-1(θ,·)$, $g^-1$ being the inverse of the nondegenerate Riemannian metric $g$, are invertible.
Beyond the standard gauging: gauge symmetries of Dirac Sigma Models · Curvature and torsion vs. $E$-curvature and $E$-torsion · 0.99
Cites as background 3 papers
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A particular class of such theories consists of those whose moduli space of classical solutions modulo gauge transformations is finite-dimensional, in which case they are referred as topological or almost topological field theories.[A theory is called topological if, up to gauge transformations, the classical solutions do not depend on background data of the source manifold, but just on its topology—or likewise, on the quantum level, if expectation values of gauge invariant quantities have such a feature.] Well known examples of topological field theories in two dimensions include the A/B topological string models [ref] , the $G/G$ Wess-Zumino-Novikov-Witten (WZNW) model [ref] and the Poisson sigma model [ref] , the two-dimensional Yang-Mills theory is an example of an almost topological theory.
Mirror Manifolds And Topological Field Theory · Introduction · 0.99
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A particular class of such theories consists of those whose moduli space of classical solutions modulo gauge transformations is finite-dimensional, in which case they are referred as topological or almost topological field theories.[A theory is called topological if, up to gauge transformations, the classical solutions do not depend on background data of the source manifold, but just on its topology—or likewise, on the quantum level, if expectation values of gauge invariant quantities have such a feature.] Well known examples of topological field theories in two dimensions include the A/B topological string models [ref] , the $G/G$ Wess-Zumino-Novikov-Witten (WZNW) model [ref] and the Poisson sigma model [ref] , the two-dimensional Yang-Mills theory is an example of an almost topological theory.
The Topological G/G WZW Model in the Generalized Momentum Representation · Introduction · 0.99
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Thus, $ω$ is presymplectic rather than symplectic.[A discussion on presymplectic AKSZ constructions appears in Ref. [ref] .] It becomes a genuine symplectic structure when $θ$ is invertible, with primary example the $H$-twisted Poisson sigma model where $θ$ is the identity operator.
Presymplectic AKSZ formulation of Einstein gravity · The “Q versus QP problem” · 0.82
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