Batalin–Vilkovisky algebras
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| Batalin–Vilkovisky algebras | |
|---|---|
| Name | Batalin–Vilkovisky algebras |
| Field | Algebraic topology; Mathematical physics |
| Introduced | 1981 |
| Founders | Evgeny Batalin; Grigori Vilkovisky |
Batalin–Vilkovisky algebras are graded commutative algebras equipped with a second-order differential operator that encodes a compatible Lie bracket, arising in quantization and topology. They connect structures studied by Evgeny Batalin and Grigori Vilkovisky with developments in algebraic topology, homological algebra, and mathematical physics, and have influenced work associated with institutions such as the Steklov Institute and universities linked to Michael Atiyah and Maxim Kontsevich. Their formalism appears alongside notions from Henri Cartan, Jean Leray, and Ilya Prigogine in contexts that bridge techniques from René Thom and Vladimir Arnold.
Definition
A Batalin–Vilkovisky algebra is a Z-graded commutative algebra A endowed with an odd linear operator Δ of degree −1 satisfying Δ^2 = 0 and a compatibility condition that makes the deviation of Δ from being a derivation produce a graded Lie bracket; this bracket often relates to constructions by John Milnor and Raoul Bott. The defining identity expresses that for homogeneous x, y in A the bracket {x,y} = (−1)^{|x|}(Δ(xy) − (Δx)y − (−1)^{|x|} x(Δy)) yields a Gerstenhaber algebra structure, linking to work by Murray Gerstenhaber and Daniel Quillen. The axioms are formulated in ways reminiscent of methods used by Alexander Grothendieck and Jean-Pierre Serre in sheaf cohomology and influenced later expositions by Pierre Deligne and Maxim Kontsevich.
Examples and basic constructions
Standard examples include the algebra of multivector fields on a Calabi–Yau manifold with the divergence operator determined by a volume form, connecting to concepts studied by Simon Donaldson and Shing-Tung Yau. The Hochschild cohomology of associative algebras studied by Maxim Kontsevich and Boris Feigin admits BV structures under conditions explored by Mikhail Gerstenhaber and Vladimir Hinich. The de Rham complex on an oriented manifold with the contraction by a volume form gives classical examples linked to the work of Henri Poincaré and Élie Cartan, while examples from symplectic geometry tie to Arnold's conjectures as addressed by Andreas Floer and Dietmar Salamon. Constructions via odd symplectic supermanifolds reflect approaches of Felix Berezin and Alexander Schwarz, and algebraic constructions using Koszul duality relate to ideas of Alexandre Beilinson and Vladimir Drinfeld.
Properties and algebraic structure
Algebraically, a BV algebra refines a Gerstenhaber algebra, providing an operator Δ that generates the bracket similarly to operators studied by Jean Leray and John Milnor; this interplay has been analyzed using spectral sequence techniques associated with Jean Leray and Lars Hörmander. The second-order property of Δ is critical and parallels second-order operators in the study of elliptic operators by Peter Lax and Lars Hörmander, while the graded Lie bracket satisfies Jacobi identities studied by Sophus Lie and Wilhelm Killing. Modules over BV algebras and morphisms between them are explored using categorical methods influenced by Saunders Mac Lane and Alexander Grothendieck, and deformation theory of BV structures connects to the work of Gerhard Hochschild and Daniel Quillen.
Homological and cohomological aspects
BV algebras appear naturally in homological algebra contexts such as cyclic cohomology developed by Alain Connes and noncommutative geometry themes associated with Connes and Henri Moscovici. The cohomology of a BV algebra inherits a Gerstenhaber algebra structure, a phenomenon paralleling results by John Milnor and Jean-Pierre Serre in cohomology operations. Homological perturbation theory and formality theorems used by Maxim Kontsevich and Ezra Getzler clarify obstructions and transfer of BV structures, while techniques from Dennis Sullivan and Daniel Quillen illuminate rational homotopy implications. Long exact sequences and spectral sequences for BV algebras mirror tools used in the work of Jean Leray and Friedrich Hirzebruch.
Operadic and homotopical formulations
Operadic descriptions employ the BV operad and its homotopy versions, building on operad theory developed by Jean-Louis Loday and Martin Markl and influenced by Boardman–Vogt style resolutions used by J. Peter May. Homotopy BV algebras (BV∞) relate to infinity-structures studied by Jim Stasheff and Murray Gerstenhaber, and Koszul duality for operads formulated by Victor Ginzburg and Mikhail Kapranov provides conceptual foundations. Model category techniques of Daniel Quillen and Alexander Heller are used to handle homotopical properties, while formality and deformation arguments appeal to techniques from Maxim Kontsevich and Pavel Etingof.
Applications in mathematical physics
BV algebras form the algebraic core of the Batalin–Vilkovisky formalism for gauge theory quantization developed by Evgeny Batalin and Grigori Vilkovisky, and are integral to path integral approaches influenced by Richard Feynman and Freeman Dyson. They appear in topological field theories studied by Edward Witten and Graeme Segal, in string theory contexts explored by Michael Green and John Schwarz, and in perturbative renormalization treated by Kenneth Wilson and Gerard 't Hooft. The algebraic apparatus interfaces with BRST cohomology from Becchi–Rouet–Stora and Carlo Becchi, and with factorization algebras studied by Kevin Costello and Owen Gwilliam, connecting to work by Edward Witten on topological sigma models.
Historical development and key references
The BV construction originated in papers by Evgeny Batalin and Grigori Vilkovisky in the early 1980s, building on BRST methods of Carlo Becchi and Raymond Stora and Alain Rouet. Subsequent mathematical formulations and expansions were influenced by Maxim Kontsevich, Ezra Getzler, and Thomas Tradler, while operadic and homotopical treatments were advanced by Martin Markl and Victor Ginzburg. Foundational texts and surveys by Kevin Costello, Owen Gwilliam, and Alberto S. Cattaneo synthesize developments linked to institutions such as the Steklov Institute and the Institut des Hautes Études Scientifiques. Important expository and research contributions come from Michael Atiyah, Edward Witten, and Jean-Michel Bismut.