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homotopy BV-algebra

rough notes from a talk by Bruno Valette – raw material to be polished

see also framed little 2-disk operad

Homotopy theory for $A_{\mathrm{\infty }}$-algebras

homotopy theory for $A_{\mathrm{\infty }}$-algebras

for $V$ a complex with the structure of an $A_{\mathrm{\infty }}$-algebra and for $V\to W$ a morphism of chain cmoplexes, we get an induced $A_{\mathrm{\infty }}$-structure on $W$.

Application: for $V$ a differential graded algebra its chain cohomology inherits the structure of an $A_{\mathrm{\infty }}$-algebra. The product operations are the Massey products.

Theorem (Getzler–Jones, Hinich, Berger–Moerdijk, Spitzweck)

There is a cofibrantly generated model category structure on the category of differential graded operads (operads in the monoidal cateory of chain complexes or cochain complexes).

Proposition

If $P_{\mathrm{\infty }}$ is a cofibrant dg-operad, then under some assumptions $P_{\mathrm{\infty }}$-algebra for an operad structures are preserved by weak equivalences.

Definition

A quasi-free dg-operad is one that is free after forgetting the differential

Proposition

Cofibrant operads are the retracts of quasi-free operads.

In particular quasi-free operads are cofibrant.

So we look for quasi-free resolutions.

Definition

Gerstenhaber algebra: essentially a Poisson algebra in the dg context.

Question

How to define a Gerstenhaber algebra up to homotopy?

Extend the operations dfined by Gerstenhaber on $\mathrm{CH}(A,A)$ whjich induce the Gerstenhaber algebra on $\mathrm{HH}(A,A)$ to a Gerstenhaber algebra up to homotopy.

So we have a strict structure on Hochschild homology $\mathrm{HH}(A,A)$ and are asking for from which homotopy structure it may come on Hochschild chains in $\mathrm{CH}(A,A)$.

So we define the Gerstenhaber algebra operad? whose algebra for an operad are Gertsenhaber algebra?s.

It is generated of course from the product operation and the bracket operation modulo the associativity constraint for the product, the Jacobi identity for the bracket and the Leibnitz property for their interaction.

These relations are always encoded in quadratic expressions.

Koszul duality theory for operads

the dual notion of operad is that of cooperad (reverse all arrows)

Theorem (Ginzburg–Kapranov, Getzler–Jones)

There exist adjoint functors

$$B:\{\mathrm{dg}\mathrm{operads}\}\leftrightarrow \{\mathrm{dg}\mathrm{cooperads}\}:\Omega$$B : \{dg operads\} \leftrightarrow \{dg cooperads\} : \Omega

given by bar and cobar construction

under this equivalence

$$\mathrm{quadratic}\mathrm{operad}P\stackrel{\mathrm{Koszul}\mathrm{duality}}{\to }\mathrm{cooperad}P\prime$$quadratic operad P \stackrel{Koszul duality}{\to} cooperad P'

with $P_{\mathrm{\infty }}:=\Omega P\prime \to P$ morphism of dg-operads.

Definition (Koszul operad)

$P$ is Koszul if $P_{\mathrm{\infty }}=\Omega P\prime \stackrel{\simeq }{\to }P$ is a quasi-isomorphism, i.e. a cofibrant replacement.

Proposition (Ginzburg–Kapranov, Getzler–Jones)

If $P=F(V)/(R)$ be a quadratic operad with $\dim (V)<+\mathrm{\infty }$ The linear dual $P{\prime }^{*}$ is a quadratic operad aabd the suspension of $P{\prime }^{*}\simeq \prime :=F(V^{*}\otimes \mathrm{sgn})/(R^{\perp })$.

Method

Compute $P\prime$ and its operadic structure with this formula, then dualize everything to get this formula.

Proposition (Getzler–Jones, Markl)

The Gerstenhaber operad $G$ is Koszul.

Homotopy Gerstenhaber or $G_{\mathrm{\infty }}$-algebras

$$G=G\prime =\mathrm{Com}\circ {\mathrm{Lie}}^1$$G = G' = Com \circ Lie^1

Definition (Batalin–Vilkovisky algebra)

This is a Gerstenhaber algebra $A$ with a unary operator $\Delta :A\to A$ of degree +1 such that

• ${\Delta }^2=0$

• $[-,-]$ measures failure of $\Delta$ to be a derivation with respect to the product $\cdot$.

This also is the algebra over an operad. But this operad is no longer a quadratic operad?.

So we define:

Definition (quadratic BV-algebra)

As above but now demand $\Delta$ a derivation of both the product $\cdot$ and the bracket $[-,-]$.

This is a first step in the resolution process.

Now the corresponding operad is Koszul. So we get

$$\Rightarrow {\mathrm{qBV}}_{\mathrm{\infty }}:=\Omega q\mathrm{BV}\prime \stackrel{\simeq }{\to }$$\Rightarrow qBV_\infty := \Omega q BV' \stackrel{\simeq}{\to}

as a quasi-free resolution.

Definition-proposition** A homotopy BV-algebra is an algebra over an operad for that.

In terms of this we have a homotopy BV-algebra structure on $\mathrm{CH}(A,A)$.

Idea

Add a suitable differential $d_1:\mathrm{qBV}\prime \to \mathrm{qBV}\prime$.

Koszul duality theory

Let $P=F(V)/(R)$ be a quadratic and linear presentation of a dg-operad

let $q:F(V)\to F(V{)}^{(2)}$ the quadratic projection

and $\mathrm{qP}:=F(V)/(\mathrm{qR})$ the quadratic analogue of $P$

Definition (quadratic-linear Koszul operad)

$P=F(V)/(R)$ is a Koszul operad if

• $R\cap V=\{0\}\iff V\mathrm{is}\mathrm{minimal}$

• (more …)

• $qP$ is Koszul

This now yields a good machinery for cofibrant resolutions for Koszul operads.

Theorem

When $P$ is a quadratic linear Koszul operad then

$$P_{\mathrm{\infty }}:=\Omega P\prime \stackrel{\simeq }{\to }P$$P_\infty := \Omega P' \stackrel{\simeq}{\to} P

Theorem

This applies to the BV operad.

Definition

A ${\mathrm{BV}}_{\mathrm{\infty }}$-algebra is an algebra over this cofibrant replacement for the BV operad.

Applications

Theorem (Ginzburg, Tradler, Menichi)

• $A$ a Frobenius algebra.

• Its Hochschild cohomology $\mathrm{HH}(A,A)$ is a BV-algebra.

Theorem (Cyclic Deligne conjecture)

…

Comparison to other definitions

Another definition of homotopy BV-algebra by Kravchenko – this turns out to be a special case of the definition here by setting some operations to $0$ (her algebra is not an algebra over a cofibrant operad).

Another definition by Tamarkin–Tsygan: this is more general than the one here. TT have many more operations, namely operations with sevearl outputs.

The notion here also difers from that in Beilinson–Drinfeld.

PBW isomorphism

The free operad $F(V)$ is filtered $\Rightarrow P=F(V)/(R)$ is filtered. There is then a morphism of operads $qP\to \mathrm{gr}P$.

Theorem

For $P$ Koszul we have

$$qP\simeq \mathrm{gr}P$$q P \simeq gr P

For instance

$$q\mathrm{BV}\simeq \mathrm{gr}P$$q BV \simeq gr P

Relation with framed little disks

Let $\mathrm{dD}$ be the framed little disk operad; then:

Getzler: $H_{•}(\mathrm{fD})\simeq \mathrm{BV}$.

Theorem (Gianciracusa–Salvatore–Severa)

$\mathrm{fD}$ is formal, i.e. quasi-isomorphic by zig-zags to its homology

Relation with TCFT

A topological conformal field theory is an algebra over the PROP $C_{•}(ℛ)$ of Riemann surfaces.

Theorem

Any TCFT carries a homotopy BV-algebra structure which lifts the BV-algebra structure of Getzler in its homology.

Homotopy theory for $P_{\mathrm{\infty }}$ algebras

Deformation theory

Proposition

In some sense homotopy BV is a formal extension of homotopy Gerstenhaber.

Theorem (generalized Lian–Zuckermann conjecture)

For any topological vertex algebra? $A$ with $\mathbb{N}$-graded conformal weight there exists an expliciit ${\mathrm{BV}}_{\mathrm{\infty }}$-algebra structure on $A$ which extends Lian–Zuckermann operations on $A$ and which lifts the BV-algebra structure on $H(A)$.

References

The above material probably roughly follows the talk slides

• Bruno Valette, Homotopy Batalin-Vilkovisky algebra (pdf)

The corresponding article is

• Imma Galvez-Carrillo, Andy Tonks, Bruno Valette, Homotopy Batalin-Vilkovisky algebras (arXiv:0907.2246)

See also

• Imma Gálvez, Vassily Gorbounov, Andrew Tonks, Homotopy Gerstenhaber structures and vertex algebras, math/0611231.QA