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Topological A-Type Models with Flux
High Energy Physics - Theory · Submitted 8 Jan 2008
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim Generically these are topological membrane models, and we show that the fluxes are related to deformations of the Courant bracket which generalize the twist by a closed 3-from $H$, in the sense that satisfying the AKSZ master equation implies the integrability conditions for an almost generalized complex structure with respect to the deformed Courant bracket.
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The abstract, sentence by sentence
We study deformations of the A-model in the presence of fluxes, by which we mean rank-three tensors with antisymmetrized upper/lower indices, using the AKSZ construction. Generically these are topological membrane models, and we show that the fluxes are related to deformations of the Courant bracket which generalize the twist by a closed 3-from $H$, in the sense that satisfying the AKSZ master equation implies the integrability conditions for an almost generalized complex structure with respect to the deformed Courant bracket. In addition, the master equation imposes conditions on the fluxes that generalize $dH=0$. The membrane model can be defined on a large class of $U(m)$- and $U(m) \times U(m)$-structure manifolds, including geometries inspired by $(1,1)$ supersymmetric $\sigma$-models with additional supersymmetries due to almost complex (but not necessarily complex) structures in the target space. Furthermore, we show that the model can be defined on three particular half-flat manifolds related to the Iwasawa manifold. When only $H$-flux is turned on it is possible to obtain a topological string model, which we do for the case of a Calabi-Yau with a closed 3-form turned on. The simplest deformation from the A-model is due to the $(2,0)+ (0,2)$ component of a non-trivial $b$-field. The model is generically no longer evaluated on holomorphic maps and defines new topological invariants. Deformations due to $H$-flux can be more radical, completely preventing auxiliary fields from being integrated out.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We study deformations of the A-model in the presence of fluxes, by which we mean rank-three tensors with antisymmetrized upper/lower indices, using the AKSZ construction. | contribution 0.97 | 0.91 | 0.08 | asserted 0.98 |
| s1 | Generically these are topological membrane models, and we show that the fluxes are related to deformations of the Courant bracket which generalize the twist by a closed 3-from $H$, in the sense that satisfying the AKSZ master equation implies the integrability conditions for an almost generalized complex structure with respect to the deformed Courant bracket. | result 0.89 | 0.91 | 0.49 | asserted 1.00 |
| s2 | In addition, the master equation imposes conditions on the fluxes that generalize $dH=0$. | result 0.99 | 0.86 | 0.12 | asserted 1.00 |
| s3 | The membrane model can be defined on a large class of $U(m)$- and $U(m) \times U(m)$-structure manifolds, including geometries inspired by $(1,1)$ supersymmetric $\sigma$-models with additional supersymmetries due to almost complex (but not necessarily complex) structures in the target space. | result 0.78 | 0.75 | 0.09 | asserted 0.90 |
| s4 | Furthermore, we show that the model can be defined on three particular half-flat manifolds related to the Iwasawa manifold. | result 0.78 | 0.90 | 0.07 | asserted 1.00 |
| s5 | When only $H$-flux is turned on it is possible to obtain a topological string model, which we do for the case of a Calabi-Yau with a closed 3-form turned on. | result 0.77 | 0.88 | 0.10 | hedged possibility 0.79 |
| s6 | The simplest deformation from the A-model is due to the $(2,0)+ (0,2)$ component of a non-trivial $b$-field. | result 0.83 | 0.60 | 0.05 | asserted 1.00 |
| s7 | The model is generically no longer evaluated on holomorphic maps and defines new topological invariants. | result 0.68 | 0.78 | 0.17 | asserted 1.00 |
| s8 | Deformations due to $H$-flux can be more radical, completely preventing auxiliary fields from being integrated out. | result 0.96 | 0.80 | 0.09 | hedged possibility 0.56 |
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The paper
Original abstract
We study deformations of the A-model in the presence of fluxes, by which we mean rank-three tensors with antisymmetrized upper/lower indices, using the AKSZ construction. Generically these are topological membrane models, and we show that the fluxes are related to deformations of the Courant bracket which generalize the twist by a closed 3-from $H$, in the sense that satisfying the AKSZ master equation implies the integrability conditions for an almost generalized complex structure with respect to the deformed Courant bracket. In addition, the master equation imposes conditions on the fluxes that generalize $dH=0$. The membrane model can be defined on a large class of $U(m)$- and $U(m) \times U(m)$-structure manifolds, including geometries inspired by $(1,1)$ supersymmetric $\sigma$-models with additional supersymmetries due to almost complex (but not necessarily complex) structures in the target space. Furthermore, we show that the model can be defined on three particular half-flat manifolds related to the Iwasawa manifold. When only $H$-flux is turned on it is possible to obtain a topological string model, which we do for the case of a Calabi-Yau with a closed 3-form turned on. The simplest deformation from the A-model is due to the $(2,0)+ (0,2)$ component of a non-trivial $b$-field. The model is generically no longer evaluated on holomorphic maps and defines new topological invariants. Deformations due to $H$-flux can be more radical, completely preventing auxiliary fields from being integrated out.
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References
How the literature uses it
6 arXiv papers cite this in 11 sentences
Papers citing it · 7 sentences without a confident label
Compares with it 1 paper
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The resolution might in fact be that since M-theory geometrizes $T$-duality, one should instead look for a membrane action which can be written in terms of the Roytenberg bracket[ [ref] contains an attempt in this direction however the author considers a somewhat different deformation of the Courant bracket than considered here and in [ref] .
Non-geometric String Backgrounds and Worldsheet Algebras · Why Are The Currents Not Conserved? · 0.86
Cites it as background 3 papers
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They have been extensively studied by introducing AKSZ membrane models with generalized complex structures arising from generalized geometry, which reduce to the A- and B-models [ref] .
AKSZ constructions for topological membranes on $G_2$-manifolds · Introduction · 0.95
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The reader is referred to [ref] and [ref] for a more complete discussion of this correspondence, the former from the physics and and the latter from the mathematics perspective.] The relation of the two-dimensional actions to the A- and B-models, and also to the topological string, is clearly established.
Doubled Formalism, Complexification and Topological Sigma-Models · The mathematical perspective · 0.94
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Three dimensional version of A model has been proposed in [ref]
Lectures on AKSZ Sigma Models for Physicists · Comments and Future Outlook · 0.89
What it cites · 91 sentences without a confident label
Uses the method or tool of 1 paper
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Our approach makes use of the AKSZ construction, originally introduced in [ref] by Alexandrov, Kontsevich, Schwarz, and Zoboronsky, which is a very general framework that enables a geometric construction of topological models by making use of the Batalin-Vilkovisky (BV) formalism [ref] .
The Geometry of the Master Equation and Topological Quantum Field Theory · Introduction · 0.96
Compares with 4 papers
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It would be interesting to gauge fix and compare this model with the topological model obtained by twisting the $(2,2)$ bi-Hermitian $σ$-model [ref] .
Topological sigma-models with H-flux and twisted generalized complex manifolds · Topological models on Calabi-Yau with three-form flux · 0.86
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It would be interesting to gauge fix and compare this model with the topological model obtained by twisting the $(2,2)$ bi-Hermitian $σ$-model [ref] .
Topological twisted sigma model with H-flux revisited · Topological models on Calabi-Yau with three-form flux · 0.86
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It would be interesting to gauge fix and compare this model with the topological model obtained by twisting the $(2,2)$ bi-Hermitian $σ$-model [ref] .
The biHermitian topological sigma model · Topological models on Calabi-Yau with three-form flux · 0.86
Cites as background 37 papers
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An almost generalized complex structure [ref] on a manifold $$ is an an endomorphism $J$ of the sum of the tangent and cotangent bundles $T ⊕T^*$ which satisfies $J^2 = - 1$, and is compatible with the canonical metric $G$ on $T ⊕T^*$: [equation] This generalizes the concept of an almost complex structure $I$, which is an endomorphism of $T$ that squares to $-1$.
Generalized complex geometry · Generalized Complex Geometry · 0.97
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We will call any supermanifold (finite or infinite dimensional) a P-manifold if it is equipped with an odd Poisson structure, and a Q-manifold if it is equipped with an odd vector field [equation] that obeys $Q^2=0$ [ref] .
Geometry of Batalin-Vilkovisky quantization · The AKSZ construction of the A-model · 0.90
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The correspondence between AKSZ membrane actions and Courant algebroids ([ref]) was pointed out in [ref] , and has subsequently been studied extensively [ref] .
On the structure of graded symplectic supermanifolds and Courant algebroids · The three dimensional A-type model · 0.86
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