Abstract Hodge Decomposition and Minimal Models for Cyclic Algebras

Chuang, Joseph; Lazarev, Andrey

doi:10.1007/s11005-009-0314-7

Cited by 11 publications

(14 citation statements)

References 17 publications

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“…This 'cyclic' SDR from a space onto its homology is equivalent to that of a Hodge decomposition, cf. [15,16]. A Hodge decomposition always exists for a finite-dimensional (super) vector space.…”

Section: Formal Odd Symplectic Geometrymentioning

confidence: 99%

Minimal models of quantum homotopy Lie algebras via the BV-formalism

Braun

Maunder

2018

Journal of Mathematical Physics

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Using the BV-formalism of mathematical physics an explicit construction for the minimal model of a quantum L∞-algebra is given as a formal super integral. The approach taken herein to these formal integrals is axiomatic, and they can be approached using perturbation theory to obtain combinatorial formulae as shown in the appendix. Additionally, there exists a canonical differential graded Lie algebra morphism mapping formal functions on homology to formal functions on the whole space. An inverse L∞-algebra morphism to this differential graded Lie algebra morphism is constructed as a formal super integral.

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Section: Formal Odd Symplectic Geometrymentioning

confidence: 99%

Minimal models of quantum homotopy Lie algebras via the BV-formalism

Braun

Maunder

2018

Journal of Mathematical Physics

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“…One can show that the notion a strong deformation retract is equivalent to that of an abstract Hodge decomposition, cf. [7,6].…”

Section: Application To Chern-simons Theorymentioning

confidence: 99%

“…Theorem B.11 (1) is by now rather well known and we give a proof here mainly for comparison with part (2). The decomposition theorem for cyclic algebras was proved in [6] from which the existence of cyclic minimal models was deduced, however it does not immediately follow from this that cyclic minimal models are unique up to homotopy equivalence, as P-algebra structures on the homology.…”

Section: Application To Chern-simons Theorymentioning

confidence: 99%

Unimodular homotopy algebras and Chern–Simons theory

Braun

Lazarev

2015

Journal of Pure and Applied Algebra

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Abstract. Quantum Chern-Simons invariants of differentiable manifolds are analyzed from the point of view of homological algebra. Given a manifold M and a Lie (or, more generally, an L∞) algebra g, the vector space H * (M ) ⊗ g has the structure of an L∞ algebra whose homotopy type is a homotopy invariant of M . We formulate necessary and sufficient conditions for this L∞ algebra to have a quantum lift. We also obtain structural results on unimodular L∞ algebras and introduce a doubling construction which links unimodular and cyclic L∞ algebras.

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“…The Ward identities of quantum closed SFT can be interpreted as the loop homotopy algebra axioms [5,37]. In chapter II, we pointed out that loop homotopy algebras are indeed algebras over the Feynman transform of a modular operad [22], and the minimal model theorem corresponding to such algebras has been established in [38,39]. The explicit construction of such minimal models resembles that in the case of A ∞ -algebras, but where one has to consider graphs (allowing loops) instead of trees.…”

Section: Decomposition Theorem For Closed String Loop Algebramentioning

confidence: 99%

“…Furthermore we have A P ⊥ A U ⊕ A T , A U ⊥ A U and A T ⊥ A T . These definitions are borrowed from [39].…”

Section: Definition 2 a Hodge Decomposition Of A Is A Pre Hodge Decomentioning

confidence: 99%

Homotopy Classification of Bosonic String Field Theory

Münster¹,

Sachs²

2014

Commun. Math. Phys.

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We prove the decomposition theorem for the loop homotopy algebra of quantum closed string field theory and use it to show that closed string field theory is unique up to gauge transformations on a given string background and given S-matrix. For the theory of open and closed strings we use results in open-closed homotopy algebra to show that the space of inequivalent open string field theories is isomorphic to the space of classical closed string backgrounds. As a further application of the openclosed homotopy algebra we show that string field theory is background independent and locally unique in a very precise sense. Finally we discuss topological string theory in the framework of homotopy algebras and find a generalized correspondence between closed strings and open string field theories.

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Abstract Hodge Decomposition and Minimal Models for Cyclic Algebras

Abstract: Abstract. We show that an algebra over a cyclic operad supplied with an additional linear algebra datum called Hodge decomposition admits a minimal model whose structure maps are given in terms of summation over trees. This minimal model is unique up to homotopy.

Cited by 11 publications

References 17 publications

Minimal models of quantum homotopy Lie algebras via the BV-formalism

Minimal models of quantum homotopy Lie algebras via the BV-formalism

Unimodular homotopy algebras and Chern–Simons theory

Homotopy Classification of Bosonic String Field Theory

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