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homotopy BV-algebra

rough notes from a talk by Bruno Valetteraw material to be polished

see also framed little 2-disk operad

Contents

Homotopy theory for A A_\infty-algebras

homotopy theory for A A_\infty-algebras

for VV a complex with the structure of an A A_\infty-algebra and for VWV \to W a morphism of chain cmoplexes, we get an induced A A_\infty-structure on WW.

Application: for VV a differential graded algebra its chain cohomology inherits the structure of an A A_\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 P_\infty is a cofibrant dg-operad, then under some assumptions P P_\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 CH(A,A)CH(A,A) whjich induce the Gerstenhaber algebra on HH(A,A)HH(A,A) to a Gerstenhaber algebra up to homotopy.

So we have a strict structure on Hochschild homology HH(A,A)HH(A,A) and are asking for from which homotopy structure it may come on Hochschild chains in CH(A,A)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:{dgoperads}{dgcooperads}:Ω B : \{dg operads\} \leftrightarrow \{dg cooperads\} : \Omega

given by bar and cobar construction

under this equivalence

quadraticoperadPKoszuldualitycooperadP quadratic operad P \stackrel{Koszul duality}{\to} cooperad P'

with P :=ΩPPP_\infty := \Omega P' \to P morphism of dg-operads.

Definition (Koszul operad)

PP is Koszul if P =ΩPPP_\infty = \Omega P' \stackrel{\simeq}{\to} P is a quasi-isomorphism, i.e. a cofibrant replacement.

Proposition (Ginzburg–Kapranov, Getzler–Jones)

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

Method

Compute PP' and its operadic structure with this formula, then dualize everything to get this formula.

Proposition (Getzler–Jones, Markl)

The Gerstenhaber operad GG is Koszul.

Homotopy Gerstenhaber or G G_\infty-algebras

G=G=ComLie 1 G = G' = Com \circ Lie^1
Definition (Batalin–Vilkovisky algebra)

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

  • Δ 2=0\Delta^2 = 0

  • [,][-,-] measures failure of Δ\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 Δ\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

qBV :=ΩqBV \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 CH(A,A)CH(A,A).

Idea

Add a suitable differential d 1:qBVqBVd_1 : qBV' \to qBV'.

Koszul duality theory

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

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

and qP:=F(V)/(qR)qP := F(V)/(qR) the quadratic analogue of PP

Definition (quadratic-linear Koszul operad)

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

  • RV={0}VisminimalR \cap V = \{0\} \Leftrightarrow V is minimal

  • (more …)

  • qPq P is Koszul

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

Theorem

When PP is a quadratic linear Koszul operad then

P :=ΩPP P_\infty := \Omega P' \stackrel{\simeq}{\to} P
Theorem

This applies to the BV operad.

Definition

A BV BV_\infty-algebra is an algebra over this cofibrant replacement for the BV operad.

Applications

Theorem (Ginzburg, Tradler, Menichi)
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 00 (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)F(V) is filtered P=F(V)/(R)\Rightarrow P = F(V)/(R) is filtered. There is then a morphism of operads qPgrPq P \to gr P.

Theorem

For PP Koszul we have

qPgrP q P \simeq gr P

For instance

qBVgrP q BV \simeq gr P

Relation with framed little disks

Let dDdD be the framed little disk operad; then:

Getzler: H (fD)BVH_\bullet(fD) \simeq BV.

Theorem (Gianciracusa–Salvatore–Severa)

fDfD 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 ()C_\bullet(\mathcal{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 P_\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? AA with \mathbb{N}-graded conformal weight there exists an expliciit BV BV_\infty-algebra structure on AA which extends Lian–Zuckermann operations on AA and which lifts the BV-algebra structure on H(A)H(A).

References

The above material probably roughly follows the talk slides

The corresponding article is

See also

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

Revised on April 9, 2011 11:28:24 by Urs Schreiber (82.113.99.12)