On higher Poisson and Koszul--Schouten brackets

TL;DR

This paper demonstrates how to construct a homotopy BV-algebra on differential forms over a higher Poisson manifold using the Lie derivative as the generating operator.

Contribution

It introduces a method to build a homotopy BV-algebra structure on differential forms in the context of higher Poisson geometry, utilizing the Lie derivative.

Findings

Homotopy BV-algebra structure constructed on differential forms

Lie derivative acts as the generating operator

Extension of Poisson geometry to higher structures

Abstract

In this note we show how to construct a homotopy BV-algebra on the algebra of differential forms over a higher Poisson manifold. The Lie derivative along the higher Poisson structure provides the generating operator.

Equations

$$\{f_1,f_2,\cdots ,f_r{\}}_P={[[\cdots [[[[P,f_1]],f_2]]\cdots ,f_r]]|}_M,$$

$$X⇝L_X=[d,i_X],$$

$$L_{[[X,Y]]}=[L_X,L_Y].$$

$$(L_P{)}^2=\frac{1}{2}[L_P,L_P]=\frac{1}{2}{L}_{[[P,P]]}=0.$$

$$[d,L_X]=0.$$

$$[{\alpha }_1,{\alpha }_2,\cdots ,{\alpha }_r{]}_P:=[\cdots [[L_P,{\alpha }_1],{\alpha }_2]\cdots ,{\alpha }_r]\text{mathds}1,$$

$$d[{\alpha }_1,\cdots ,{\alpha }_r{]}_P+\sum _{i=1}^r(-1{)}^{{ϵ}_i}[{\alpha }_1,\cdots ,d{\alpha }_i,\cdots {\alpha }_r{]}_P=0,$$

$$\epsilon_{j}=\left\{\begin{array}[]{cc}0&j=1\\ \sum^{j-1}_{k=1}\widetilde{\alpha}_{k}&j>1.\end{array}\right.$$

$$\{f_1,f_2,\cdots f_r{\}}_P=-L_{\stackrel{r}{P}}(f_{1}d{f}_2\dots d{f}_r)=i_{\stackrel{r}{P}}(d{f}_{1}d{f}_2\cdots d{f}_r).$$

Full text

**Keywords: Strong homotopy Lie algebra, derived bracket, higher Poisson bracket, higher Koszul–Schouten bracket, Batalin–Vilkoviski algebra, odd symplectic geometry.

**

1 Introduction and Background

Recall that a higher Poisson manifold is the pair $(M,P)$, where $M$ is a supermanifold and the higher Poisson structure $P\in\mathfrak{X}^{*}(M)(=C^{\infty}(\Pi T^{*}M))$ is an even parity (but otherwise arbitrary) multivector field that satisfies the “classical master equation”; $[[P,P]]=0$. Here the bracket is the canonical Schouten–Nijenhuis bracket on $\Pi T^{*}M$. The original formulation of higher Poisson manifolds was laid down by Voronov [voronov-2004, voronov-2005]. A similar notion was put forward by de Azc$\acute{\textnormal{a}}$rraga et.al [deAzcarraga:1996zk, deAzcarraga:1996jk].

Associated with the higher Poisson structure is a homotopy Poisson algebra on $C^{\infty}(M)$. That is an $L_{\infty}$-algebra in the sense of Lada & Stasheff [Lada:1992wc] (suitably “superised”) such that the series of brackets are multi-derivations over the (supercommutative) product of functions. The series of higher Poisson brackets are given by;

[TABLE]

where $f_{I}\in C^{\infty}(M)$. The “classical master equation” is directly equivalent to the higher Poisson brackets satisfying the so-called Jacobiators. This is fundamental as without this condition the series of higher brackets would not form an $L_{\infty}$-algebra.

In this note we show how to construct a series of higher brackets on the algebra of differential forms222we define the algebra of differential forms over a supermanifold to be $\Omega^{*}(M):=C^{\infty}(\Pi TM)$. As we will not delve into the theory of integration over supermanifolds, such a definition is perfectly adequate for our purposes. on a higher Poisson manifold. We do this by following Koszul [Koszul;1985] (also see [Akman:1995tm, Bering:1996kw, voronov-2004]), as higher antibrackets generated by the Lie derivative along the higher Poisson structure. The $n+1$-th bracket in essence measures the failure of $n$-th bracket to be a multi-derivation. As the Poisson structure is an even multivector field, the Lie derivative is an odd operator acting on differential forms and thus the series of brackets are odd. The “classical master equation” guarantees that the Lie derivative along the higher Poisson structure is nilpotent. Such a series of brackets forms an $L_{\infty}$-algebra commonly referred to as a homotopy BV-algebra.

We will refer to this series of brackets as higher Koszul–Schouten brackets. These brackets are considered as the natural analogues of the classical Koszul–Schouten bracket [Kosmann-Schwarzbach:1995, Kosmann-Schwarzbach:1996, Koszul;1985]. We show that the exterior derivative acts as a differential over the higher Koszul–Schouten brackets, just as it does in the classical case. Thus we have what we will call a differential homotopy BV-algebra.

It must be noted that Khudaverdian & Voronov [khudaverdian-2008] constructed a homotopy Schouten algebra on the algebra of differential forms over a higher Poisson manifold. That is they have an odd analogue of a homotopy Poisson algebra constructed from the higher Poisson structure. Importantly, the higher Schouten brackets are multi-derivations over the algebra of differential forms. This is in stark contrast to the higher Koszul–Schouten brackets.

Thus, there is at least two ways to equip the algebra of differential forms over a higher Poisson manifold with the structure of an $L_{\infty}$-algebra. However, these two constructions are not completely independent. Due to ideas present in [voronov-2004], the higher Schouten brackets can be thought of as the “classical limit” of the Koszul–Schouten brackets. It is not surprising that some relation exists between the two classes of higher bracket as both are build from the initial geometric data of a higher Poisson structure.

The distinction between Koszul–Schouten and Schouten brackets in the case of classical (binary) Poisson manifolds is non-existent. Simply put, there is no tri-bracket or higher and the distinction is not present.

Although we will not present details here, it is clear that the constructions presented in this note carry over to higher Poisson structures on Lie algebroids.

However, it is clear that a direct analogue for higher Schouten manifolds does not exist. Associated to such a structure is a homotopy Poisson algebra on differential forms over the supermanifold. One can construct the higher Poisson structure using arguments almost identical to that found in [khudaverdian-2008]. As such brackets are not all odd, no operator can generate something resembling a homotopy Poisson algebra in some “classical limit”.

As is customary in super-mathematics, the prefix “super” will generally be omitted. For example, by manifold we explicitly mean supermanifold. We will denote the parity of an object $A$, by tilde; $\widetilde{A}\in\mathds{Z}_{2}$. By even or odd we will be explicitly referring to parity. By a Q-manifold will mean the pair $(M,Q)$ where $M$ is a supermanifold and $Q\in\operatorname{Vect}(M)$ is an odd vector field known as a homological vector field satisfying $[Q,Q]=0$.

The notion of homotopy Poisson/Schouten/BV algebra as used here is much more restrictive that that found elsewhere in the literature [galvezcarrillo-2009, Ginzburg-1994]. We make no use of the theory of operads. However, the notion used throughout this work seems very well suited to geometric considerations and suits the purposes explored here.

2 Higher Koszul–Schouten brackets

Differential forms on a manifold $M$, are understood as functions on antitangent bundle $\Pi TM$. In natural local coordinates $\{x^{A},dx^{A}\}$ on $\Pi TM$ an arbitrary differential form is locally given by $\alpha(x,dx)\in\Omega^{*}(M)=C^{\infty}(\Pi TM)$. The parities of the local coordinates are $\widetilde{x}^{A}=\widetilde{A}$ and $d\widetilde{x}^{A}=(\widetilde{A}+1)$. Under transformations induced by coordinate changes on the base the fibre coordinates transform as $d\overline{x}^{A}=dx^{B}\left(\frac{\partial\overline{x}^{A}}{\partial x^{B}}\right)$.

Similarly, multivector fields on $M$, are identified with functions on the anticotangent bundle $\Pi T^{*}M$. We pick natural local coordinates $\{x^{A},x^{*}_{A}\}$, such that $\widetilde{x}^{*}_{A}=(\widetilde{A}+1)$. Multivector fields on $M$ are locally described by $X(x,x^{*})\in\mathfrak{X}^{*}(M)=C^{\infty}(\Pi T^{*}M)$. The fibre coordinates transform as $\overline{x}^{*}_{A}=\left(\frac{\partial x^{B}}{\partial\overline{x}^{A}}\right)x^{*}_{B}$ under transformations induced by coordinate changes on the base. A filtration can be defined on multivectors (and differential forms) that are polynomial in fibre coordinates. We will refer to this as multivector degree. Where no confusion will arise we will simply use degree for multivector degree. We will call a multivector field an $r$-vector if it is a monomial of degree $r$ in the fibre coordinates.

Warning: The parity of an object is generally independent of the degree. By odd and even we will be explicitly referring to parity and not weight.

Importantly, the anticotangent bundle comes equipped with a canonical odd symplectic structure. That is we have an odd Poisson bracket on $\mathfrak{X}^{*}(M)$. This bracket is the Schouten–Nijenhuist bracket.

From an arbitary multivector field $X\in\mathfrak{X}^{*}(M)$ we can associate a differential operator333Generically it will in fact be a pseudo-differential operator if we do not restrict attention to multivectors that are polynomial in $x^{*}$. acting on differential forms on $M$ known as the Lie derivative;

[TABLE]

where $d=dx^{A}\frac{\partial}{\partial x^{A}}$ is the exterior derivative and $i_{X}=(-1)^{\tilde{X}}X(x,\partial/\partial dx)$ is the interior product. As a differential operator, the Lie derivative is of order equal to that of the degree of the multivector field (assuming it is defined).

The Lie derivative is clearly not a derivation on the algebra of differential forms, apart form the isolated case of vector fields. Instead of the derivation property we have;

[TABLE]

Thus we see directly that for a higher Poisson structure $P$, the Lie derivative is nilpotent;

[TABLE]

A further property needed later on is that the exterior derivative is natural with respect to the Lie derivative, that is;

[TABLE]

• Remark

The Lie derivative of a differential form along a multivector field should be understood as the infinitesimal action of $\operatorname{Can}(\Pi T^{*}M)$ on differential forms on $M$ (functions on $\Pi TM$). See [khudaverdian-2006, Khudaverdian:2000zt, Khudaverdian:1999mz, Schwarz:1992nx] for more details.

Definition 2.1

Let $(M,P)$ be a higher Poisson manifold. The higher Koszul–Schouten brackets on $\Omega^{*}(M)$ are the higher derived brackets generated by the operator $L_{P}=[d,i_{P}]$,

[TABLE]

with $\alpha_{i}\in\Omega^{*}(M)$.

Here $\mathds{1}$ is the constant value $1$ zero form. That is the identity element in $\Omega^{*}(M)$. As the Lie derivative along the higher Poisson structure is odd and nilpotent the series on higher Koszul–Schouten brackets form a homotopy BV-algebra.

We know the Lie derivative $L_{P}$ is a differential operator of order equal to the degree (should it be defined) of $P$. The order of $L_{P}$ is at most $k$ if $\Phi_{P}^{r}$ vanishes (for all $\alpha_{i}$) for $r\geq k+1$. Thus for any multivector of (finite) degree $k$, the above series of brackets terminates after the $k$ place. The top non-zero bracket is a multi-derivation. The $(r+1)$ bracket is the obstruction to the Leibnitz rule for the $r$ bracket. However, it should be noted that we do not require polynomial multivector fields in order to define the higher Koszul–Schouten brackets.

Proposition 2.2

The exterior derivative satisfies a derivation rule over the Koszul brackets. Specifically,

[TABLE]

where

[TABLE]

• Proof

The above proposition follows directly from the naturality of the exterior derivative with respect to the generalised Lie derivative; $[d,L_{P}]=0$.

We take the attitude that the higher Koszul–Schouten brackets give a clear geometric example of “higher antibrackets”, [Akman:1995tm, Alfaro:1995vw, Bering:2006eb, Bering:1996kw]. Slightly more than this, we have a “differential homotopy BV-algebra” with the differential being provided by the exterior derivative.

3 Relations between higher Poisson and higher Koszul–Schouten brackets

In section we will generalise the known relations between the classical (binary) Poisson and Koszul–Schouten brackets. In order to achieve this we need to restrict our attention to finite degree Poisson structures. This is clear as although the higher Poisson brackets rely only on the Taylor expansion of the higher Poisson structure near $M\subset\Pi T^{*}M$ the higher Koszul–Schouten brackets require the full Poisson structure. First we need a lemma.

Lemma 3.1

Let $P\in\mathfrak{X}^{*}(M)$ be a higher Poisson structure of degree $k$. Then the $r$-th higher Poisson bracket is given by

[TABLE]

• Proof

The above can be proved via direct computation.