Higher-Dimensional BF Theories¶in the Batalin-Vilkovisky Formalism:¶The BV Action and Generalized Wilson Loops

Cattaneo, Alberto S.; Rossi, Carlo A.

doi:10.1007/s002200100484

Cited by 43 publications

(55 citation statements)

References 35 publications

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“…For a definition of generalized holonomy see Section 6 (and Cattaneo and Rossi [4]). For a loop 2 LM , representing an element of H…”

Section: ‰W H Smentioning

confidence: 99%

“…We will later, in equations (3)(4)(5) and (3)(4)(5)(6), apply the above to fibers of appropriate bundles.…”

Section: Invariant Functions and Principal Bundlesmentioning

confidence: 99%

“…M endowed with a flat connection r , the trace of the holonomy yields a well-defined map on the set of the free homotopy classes of loops in the base manifold M . It was discussed in Cattaneo and Rossi [4] how this may generalize to families of loops, more precisely, to the homology classes in the free loop space of the underlying manifold. We will give a mathematical account of this in this section.…”

Section: Generalized Holonomymentioning

confidence: 99%

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String bracket and flat connections

Abbaspour

Zeinalian

2007

Algebr. Geom. Topol.

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Let G → P → M be a flat principal bundle over a compact and oriented manifold M of dimension m = 2d. We construct a map of Lie algebras Ψ :, where H S 1 2 * (LM ) is the even dimensional part of the equivariant homology of LM , the free loop space of M , and MC is the Maurer-Cartan moduli space of the graded differential Lie algebra Ω * (M, adP ), the differential forms with values in the associated adjoint bundle of P . For a 2-dimensional manifold M , our Lie algebra map reduces to that constructed by Goldman in [Go2]. We treat different Lie algebra structures on H S 1 2 * (LM ) depending on the choice of the linear reductive Lie group G in our discussion. This paper provides a mathematician-friendly formulation and proof of the main result of [CFP] for G = GL(n, C) and GL(n, R) together with its natural generalization to other reductive Lie groups.Contents 24 Appendix B. Parallel transport and iterated integrals 25 References 27

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“…For a definition of generalized holonomy see Section 6 (and Cattaneo and Rossi [4]). For a loop 2 LM , representing an element of H…”

Section: ‰W H Smentioning

confidence: 99%

“…We will later, in equations (3)(4)(5) and (3)(4)(5)(6), apply the above to fibers of appropriate bundles.…”

Section: Invariant Functions and Principal Bundlesmentioning

confidence: 99%

Section: Generalized Holonomymentioning

confidence: 99%

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String bracket and flat connections

Abbaspour

Zeinalian

2007

Algebr. Geom. Topol.

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“…Those may be viewed as the A ∞ -category structure on D b (Coh(X), whose compositions are given by correlation functions. 18 . Being an A ∞ -category (as the collection of all A ∞ modules over the A ∞ -algebra (O(ΠT X), ∂)) it satisfies a certain MC equation.…”

Section: Towards Physical Descriptions Ofmentioning

confidence: 99%

“…More fundamentally we showed that topological open membrane theory can describes and control A, B and C models of topological closed string as different boundary theories. The relation between A and B models are much like different twistings of the same superconformal 18 It is rather correlation maps than correlation functions, though we will stick to the original terminology field theory. Furthermore the topological open membrane theory leads to further (bulk) deformations of the mathematical or physical structure of the boundary string theory.…”

Section: Topological Open P-branes and Generalized Homological Mirrormentioning

confidence: 99%

TOPOLOGICAL OPEN P-BRANES

Park

2001

Symplectic Geometry and Mirror Symmetry

100

155

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By exploiting the BV quantization of topological bosonic open membrane, we argue that flat 3-form C-field leads to deformations of the algebras of multi-vectors on the Dirichlet-brane world-volume as 2-algebras. This would shed some new light on geometry of M-theory 5-brane and associated decoupled theories. We show that, in general, topological open p-brane has a structure of (p + 1)-algebra in the bulk, while a structure of p-algebra in the boundary. The bulk/boundary correspondences are exactly as of the generalized Deligne conjecture (a theorem of Kontsevich) in the algebraic world of p-algebras. It also imply that the algebras of quantum observables of (p − 1)-brane are "close to" the algebras of its classical observables as p-algebras. We interpret above as deformation quantization of (p − 1)-brane, generalizing the p = 1 case. We argue that there is such quantization based on the direct relation between BV master equation and Ward identity of the bulk topological theory. The path integral of the theory will lead to the explicit formula. We also discuss some applications to topological strings and conjecture that the homological mirror symmetry has further generalizations to the categories of p-algebras.

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L_∞‐Algebras of Classical Field Theories and the Batalin–Vilkovisky Formalism

Jurčo

Raspollini

Sämann

et al. 2019

Fortschritte der Physik

102

221

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We review in detail the Batalin–Vilkovisky formalism for Lagrangian field theories and its mathematical foundations with an emphasis on higher algebraic structures and classical field theories. In particular, we show how a field theory gives rise to an L∞‐algebra and how quasi‐isomorphisms between L∞‐algebras correspond to classical equivalences of field theories. A few experts may be familiar with parts of our discussion, however, the material is presented from the perspective of a very general notion of a gauge theory. We also make a number of new observations and present some new results. Most importantly, we discuss in great detail higher (categorified) Chern–Simons theories and give some useful shortcuts in usually rather involved computations.

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Higher-Dimensional BF Theories¶in the Batalin-Vilkovisky Formalism:¶The BV Action and Generalized Wilson Loops

Cited by 43 publications

References 35 publications

String bracket and flat connections

String bracket and flat connections

TOPOLOGICAL OPEN P-BRANES

L_∞‐Algebras of Classical Field Theories and the Batalin–Vilkovisky Formalism

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Higher-Dimensional BF Theories¶in the Batalin-Vilkovisky Formalism:¶The BV Action and Generalized Wilson Loops

Cited by 43 publications

References 35 publications

String bracket and flat connections

String bracket and flat connections

TOPOLOGICAL OPEN P-BRANES

L∞‐Algebras of Classical Field Theories and the Batalin–Vilkovisky Formalism

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Product

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L_∞‐Algebras of Classical Field Theories and the Batalin–Vilkovisky Formalism