Computation of cohomology of vertex algebras
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Computation of cohomology of vertex algebras Bojko Bakalov · Alberto De Sole · Victor G. Kac Received: 19 June 2020 / Revised: 17 July 2020 / Accepted: 24 July 2020 Published online: 16 November 2020 © The Mathematical Society of Japan and Springer Japan KK, part of Springer Nature 2020 Communicated by: Yasuyuki Kawahigashi Abstract. We review cohomology theories corresponding to the chiral and classical operads. The first one is the cohomology theory of vertex algebras, while the second one is the classical cohomology of Poisson vertex algebras (PVA), and we construct a spectral sequence relating them. Since in “good” cases the classical PVA cohomology coincides with the variational PVA cohomology and there are well-developed methods to compute the latter, this enables us to compute the cohomology of vertex algebras in many interesting cases. Finally, we describe a unified approach to integrability through vanishing of the first cohomology, which is applicable to both classical and quantum systems of Hamiltonian PDEs. Keywords and phrases: chiral and classical operads, vertex algebra cohomology, classical and variational Poisson cohomology, Harrison cohomology, spectral sequences Mathematics Subject Classification (2020): Primary 17B69; Secondary 17B63, 17B65, 17B80, 18D50
B. Bakalov Department of Mathematics, North Carolina State University, Raleigh, NC 27695, USA (e-mail: [email protected]) A. De Sole Dipartimento di Matematica, Sapienza Università di Roma, P.le Aldo Moro 2, 00185 Rome, Italy (e-mail: [email protected]) V.G. Kac Department of Mathematics, MIT, 77 Massachusetts Ave., Cambridge, MA 02139, USA (e-mail: [email protected])
B. Bakalov, A. De Sole and V.G. Kac
Contents 1. 2. 3. 4. 5. 6. 7. 8. A.
Introduction ........................................................................................... Basic definitions ...................................................................................... The chiral and classical operads ................................................................ LCA, VA and PVA cohomology ................................................................ Wick-type formula for VA cocycles and bounded VA cohomology .................. A spectral sequence for VA cohomology ..................................................... Examples of computations of VA cohomology ............................................. Application to integrability of evolution PDE.............................................. The spectral sequence of a filtered complex.................................................
1. Introduction In the series of papers [BDSHK18], [BDSHK19], [BDSK19], [BDSKV19], [BDSHKV20] we developed, with our collaborators, the foundations of cohomology theory of vertex algebras. This theory is the last in the series of cohomology theories beyond the Lie (super)algebra cohomology, which are intimately related to each other. All these theories are based on a Z-graded Lie superalgebra k WP = WP , (1.1) k≥−1
associated to a linear symmetric operad P, governing the
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