arXiv subjects
Publications and source records attributed to Ezra Getzler.
The Batalin-Fradkin-Vilkovisky formalism studies a differential graded symplectic supermanifold whose cohomology vanishes in negative degree. This hypothesis is violated by the N=1 spinning particle arXiv:1605.04762; its cohomology is nontrivial in all negative degrees. Replacing the Poisson bracket by the Moyal bracket has the effect of eliminating these cohomology classes.
We study the Karoubi operator on the unnormalized chain complex of a paracyclic module; its restriction to the normalized chain complex has previously been considered by Dwyer and Kan, and in the cyclic case by Cuntz and Quillen. We obtain a direct proof of the Dold-Kan theorem for paracyclic modules of Dwyer and Kan, by directly relating the Karoubi operator to projection to the normalized subcomplex.
We construct a natural morphism $ρ$ from the nerve $\text{MC}_\bullet(L) = \text{MC}(Ω_\bullet \widehat{\otimes} L)$ of a pronilpotent curved L${}_\infty$-algebra $L$ to the simplicial subset $γ_\bullet(L) = \text{MC}(Ω_\bullet \widehat{\otimes} L,s_\bullet)$ of Maurer--Cartan element satisfying the Dupont gauge condition. This morphism equals the identity on the image of the inclusion $γ_\bullet(L) \hookrightarrow \text{MC}_\bullet(L)$. The proof uses the extension of Berglund's homotopical perturbation theory for L${}_\infty$-algebras to curved L${}_\infty$-algebras. The morphism $ρ$ equals the holonomy for nilpotent Lie algebras. In a sequel to this paper, we use a cubical analogue $ρ^\square$ of $ρ$ to identify $ρ$ with higher holonomy for semiabelian curved \Linf-algebras.
Using a homotopy introduced by de Wilde and Lecomte and homological perturbation theory for $A_\infty$-algebras, we give an explicit proof that the universal enveloping algebra $UL$ of a differential graded Lie algebra $L$ is Koszul, via an explicit contracting homotopy from the cobar construction $ΩCL$ of the Chevalley-Eilenberg chain coalgebra $CL$ of $L$ to $UL$.
We prove that the differential graded Lie algebra of functionals associated to the Chern-Simons theory of a semisimple Lie algebra is homotopy abelian. For a general field theory, we show that the variational complex in the Batalin-Vilkovisky formalism is a differential graded Lie algebra.
In the Batalin-Vilkovisky formalism, gauge conditions are expressed as Lagrangian submanifolds in the space of fields and antifields. We discuss a way of patching together gauge conditions over different parts of the space of fields, and apply this method to extend the light-cone gauge for the superparticle to a conic neighbourhood of the forward light-cone in momentum space.
We show that the classical Brink-Schwarz superparticle is a generalized AKSZ field theory. We work in the Batalin-Vilkovisky formalism: the main technical tool is the vanishing of Batalin--Vilkovisky cohomology below degree -1.
We express covariance of the Batalin-Vilkovisky formalism in classical mechanics by means of the Maurer-Cartan equation in a curved Lie superalgebra, defined using the formal variational calculus and Sullivan's Thom-Whitney construction. We use this framework to construct a Batalin-Vilkovisky canonical transformation identifying the Batalin-Vilkovisky formulation of the spinning particle with an AKSZ field theory.
Let L be a (pro-nilpotent) curved L-infinity algebra, and let h be a homotopy between L and a subcomplex M. Using homotopical perturbation theory, Fukaya constructed from this data a curved L-infinity structure on M. We prove that projection from L to M induces a bijection between the set of Maurer-Cartan elements x of L such that hx=0 and the set of Maurer-Cartan elements of M.
The mapping class group of a closed surface of genus $g$ is an extension of the Torelli group by the symplectic group. This leads to two natural problems: (a) compute (stably) the symplectic decomposition of the lower central series of the Torelli group and (b) compute (stably) the Poincaré polynomial of the cohomology of the mapping group with coefficients in a symplectic representation $V$. Using ideas from graph cohomology, we give an effective computation of the symplectic decomposition of the quadratic dual of the lower central series of the Torelli group, and assuming the later is Kozsul, it provides a solution to the first problem. This, together with Mumford's conjecture, proven by Madsen-Weiss, provides a solution to the second problem. Finally, we present samples of computations, up to degree 13.
We extend our previous calculation [arXiv:1511.02135] of the BV cohomology of the spinning particle with a flat target to the general case, in which the target carries a non-trivial pseudo-Riemannian metric and a magnetic field.
The derived Maurer-Cartan locus $\text{MC}^\bullet(L)$ is a functor from differential graded Lie algebras to cosimplicial schemes. If L is differential graded Lie algebra, let $L_+$ be the truncation of $L$ in positive degrees $i>0$. We prove that the differential graded algebra of functions on the cosimplicial scheme $\text{MC}^\bullet(L)$ is quasi-isomorphic to the Chevalley-Eilenberg complex of $L_+$.
We show that the axiom of Felder and Kazhdan on the vanishing of the cohomology groups in negative degree associated to solutions of the classical master equation in the Batalin-Vilkovisky formalism is violated by the spinning particle in a flat background coupled to D=1 supergravity. In this model, there are nontrivial cohomology groups in all negative degrees, regardless of the dimension of the spacetime in which the spinning particle is propagating.
In an enriched setting, we show that higher groupoids and higher categories form categories of fibrant objects. The nerve of a differential graded algebra is a higher category in the category of algebraic varieties, where covers are defined to be smooth epimorphisms.
Thick simplices are the nerves of the contractible groupoids obtained by inverting the arrows in the categories [n]. Using explicit expansions of simplicial subsets of the thick simplices, we present a new approach to results of Rezk and of Joyal and Tierney on quasi-categories and their associated Kan complexes of quasi-invertible morphisms.
We show that there is a sequence of operations on the positively graded part of a differential graded algebra making it into an L-infinity algebra. The formulas for the higher brackets involve Bernoulli numbers. The construction generalizes the derived bracket for Poisson manifolds, and the Lie 2-algebra associated to a Courant algebroid constructed by Roytenberg and Weinstein.
We show that the sum over planar trees formula of Kontsevich and Soibelman transfers C-infinity structures along a contraction. Applying this result to a cosimplicial commutative algebra A^* over a field of characteristic zero, we exhibit a canonical unital C-infinity structure on Tot(A^*), which is unital if A^* is; in particular, we obtain a canonical C-infinity structure on the cochain complex of a simplicial set.
The Deligne groupoid is a functor from nilpotent differential graded Lie algebras concentrated in positive degrees to groupoids; in the special case of Lie algebras over a field of characteristic zero, it gives the associated simply connected Lie group. We generalize the Deligne groupoid to a functor gamma from L-infinity algebras concentrated in degree >-n to n-groupoids. (We actually construct the nerve of the n-groupoid, which is an enriched Kan complex.) The construction of gamma is quite explicit (it is based on Dupont's proof of the de Rham theorem) and yields higher dimensional analogues of holonomy and of the Campbell-Hausdorff formula. In the case of abelian L-infinity algebras (i.e. chain complexes), the functor gamma is the Dold-Kan simplicial set.