The deformation quantization mapping of Poisson- to associative structures in field theory
Arthemy V. Kiselev
TL;DR
This paper constructs a deformation quantization map for Poisson structures in field theory, transforming the commutative algebra of local functionals into a noncommutative star-product algebra using geometric techniques.
Contribution
It introduces a well-defined deformation quantization procedure for variational Poisson brackets in field theories, extending the geometric approach to iterated variations.
Findings
Defined a star-product in the algebra of local functionals
Established a deformation quantization map for Poisson structures
Provided a geometric framework for quantization in field theory
Abstract
Let be a variational Poisson bracket in a field model on an affine bundle over an affine base manifold . Denote by the commutative associative multiplication in the Poisson algebra of local functionals that take field configurations to numbers. By applying the techniques from geometry of iterated variations, we make well defined the deformation quantization map that produces a noncommutative -linear star-product in .
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The deformation quantization mapping
of Poisson- to associative structures
in field theory
Arthemy V. Kiselev{}^{\text{*\S}}
(Date: April 30, 2017)
Abstract.
Let be a variational Poisson bracket in a field model on an affine bundle over an affine base manifold . Denote by the commutative associative multiplication in the Poisson algebra of local functionals that take field configurations to numbers. By applying the techniques from geometry of iterated variations, we make well defined the deformation quantization map that produces a noncommutative -linear star-product in .
Key words and phrases:
Deformation quantization, star-product, field theory, Poisson bracket, associativity
2010 Mathematics Subject Classification:
53D55, 58E30, 81S10; secondary 53D17, 58Z05, 70S20.
Address: Johann Bernoulli Institute for Mathematics and Computer Science, University of Groningen, P.O. Box 407, 9700 AK Groningen, The Netherlands. E-mail: [email protected]
∗ Max Planck Institute for Mathematics, Vivatsgasse 7, D-53111 Bonn, Germany
§ Present address: IHÉS, Le Bois–Marie, 35 route de Chartres, Bures-sur-Yvette, F-91440 France
Introduction
Starting from a Poisson bi-vector on a given finite-dimensional affine Poisson manifold , the Kontsevich graph summation formula [25] yields an explicit deformation of the commutative product in the algebra of smooth functions. The new operation on the space of power series is specified by the Poisson structure on : namely, such that all the bi-differential terms at higher powers of the formal parameter are completely determined by the Poisson bracket in the leading deformation term. The deformed product is no longer commutative if but it stays associative,
[TABLE]
by virtue of bi-vector’s property to be Poisson, cf. [5].
In this paper we extend the Poisson set-up and graph summation technique in the deformation to the jet-space (super)geometry of -valued fields over another, -dimensional affine base manifold in a given bundle and secondly, of variational Poisson bi-vectors that encode the Poisson brackets on the space of local functionals taking . We explain why an extension of the Kontsevich graph technique [25] is possible and how it is done by using the geometry of iterated variations [17, 20]. For instance, we derive a variational analogue of the Moyal associative -product, f\mathop{\star}g=(f)\,\exp\bigl{(}\overleftarrow{\partial_{i}}\cdot\hbar\mathcal{P}^{ij}\cdot\overrightarrow{\partial_{j}}\bigr{)}\,(g), in the case where the coefficients of bi-vector are constant (hence the Jacobi identity holds trivially). To process variational Poisson structures with nonconstant coefficients, we analyse (see [5, 6]) the factorization mechanism \operatorname{Assoc}_{\star_{\hbar}}(f,g,h)=\Diamond\bigl{(}\mathcal{P},\operatorname{Jac}_{\mathcal{P}}(\cdot,\cdot\cdot)\bigr{)}(f,g,h) using the Kontsevich graphs at higher powers of the deformation parameter . We explain why, holding up to , the associativity of can start leaking at orders in the variational Poisson geometry of field-theoretic models.
Its concept going back to Weyl–Grönewold [14] and Moyal [34], the problem of associativity-preserving deformation quantisation of commutative product in the algebras of functions on smooth finite-dimensional symplectic manifolds was considered by Bayen–Flato–Frønsdal–Lichnerowicz–Sternheimer [2]. Further progress within the symplectic picture was made by De Wilde–Lecomte [10] and independently, Fedosov [12]. To tackle the deformation quantisation problem in the case of finite-dimensional affine Poisson geometries \bigl{(}N^{n},\{\cdot,\cdot\}_{\mathcal{P}}\bigr{)} – that is, in absence of the Darboux lemma which guarantees the existence of canonical coordinates on a chart in the symplectic case – Kontsevich developed the graph complex technique [28, 27]; it yields an explicit construction of each term in the series , see [25, 29]. We recall this approach and analyse some of its features in section 1 below. Specifically, the sum over a suitable set of weighted oriented graphs determines on a star-product which (i) contains a given Poisson bracket in the leading deformation term at and which (ii) is associative modulo the Jacobi identity for ,
[TABLE]
where
[TABLE]
The construction of polydifferential operator has been analysed up to order in [5, 6].
A key distinction between associativity mechanisms for the Darboux-symplectic and Poisson cases is a possibility of the star-product self-action on non-constant coefficients of the bracket . It is readily seen that whenever those coefficients are constant, the graph summation formula for then yields the Moyal star-product [34],
[TABLE]
This formula’s geometric extension to the infinite-dimensional space of -valued fields over a given -dimensional affine manifold will be obtained in §2.3.1, see Eq. (14) on p. 14 below.
Valid in finite-dimensional set-up, the result of [25] was known to be not working in the infinite dimension. It could not be applied to field-theoretic models, should one attempt to assemble such geometries via a limiting procedure by first taking infinitely many “zero-dimensional field theories” over the discrete topological space . In fact, not only is the geometry of -valued fields (here, internal degrees of freedom attached at every base point ) infinite-dimensional if but also does the mathematical apparatus to encode it become substantially more complex, cf. [17, 20]. Many elements of differential calculus are known to be fragile in the course of transition from finite-dimensional geometry of to the infinite jet spaces for the bundles of -valued fields over , or to the infinite jet spaces of maps , cf. [24] vs [40] and [21] vs [1] or contrast [17] vs [37], [20] vs [30], and [25] vs this paper.
The aim of this paper is to develop a tool for regular deformation quantisation ({\boldsymbol{\mathcal{A}}},{\times})\mapsto\bigl{(}{\boldsymbol{\mathcal{A}}}[[\hbar]],{\star}_{\hbar}\bigr{)} of field theory models. The commutative associative unital algebras of local functionals equipped with variational Poisson structures are the input data of quantisation algorithm; in the output one obtains the noncommutative products , associative up to , in the unital algebras .
This paper is structured as follows. In the next section we review the concept of deformation quantisation [25] for finite-dimensional affine Poisson manifolds \bigl{(}N^{n}, \{\cdot,\cdot\}_{\mathcal{P}}\bigr{)}: the deformation is approached via summation over a class of weighted oriented graphs. In section 2 we proceed with finite-dimensional affine bundles over affine manifolds ; the infinitesimal parts of deformations are now specified by the variational Poisson brackets for algebras of local functionals taking . To extend the deformation quantisation technique to such set-up, we let elements of the Gel’fand calculus of singular linear integral operators enter the game in §2.2. Each graph in the Kontsevich summation formula now encodes a local variational (poly)differential operator. We then inspect in §2.3 the geometric mechanism through which the new star-products can stay associative. Taking star-product (2) by Weyl–Grönewold–Moyal as prototype, we illustrate the algorithm of variational deformation quantisation by presenting this structure’s associative analogue for the class of Hamiltonian total differential operators \bigl{\|}P^{ij}_{\tau}({\boldsymbol{x}})\,{\mathrm{d}}^{|\tau|}/{\mathrm{d}}{\boldsymbol{x}}^{\tau}\bigr{\|}_{i=1\ldots n}^{j=1\ldots n} whose coefficients do not depend on sections . On the other hand, for field-dependent Hamiltonian operators we indicate a channel for the associativity to leak at orders . This effect was altogether suppressed in the seminal picture; originally invisible, it can appear only in the framework of fibre bundles over the base manifold of positive dimension .
1. The Kontsevich -product on finite-dimensional Poisson manifolds
In this section we recall the graph technique [25] for deformation quantisation on finite-dimensional affine Poisson manifolds \bigl{(}N^{n}, \{\cdot,\cdot\}_{\mathcal{P}}\bigr{)}.
1.1.
Let us first consider the direct problem of producing Lie algebra structures from a given associative product in the algebra of functions on .
Lemma 1**.**
Let be an associative algebra. Denote by the associative multiplication in . Then the bi-linear skew-symmetric operation
[TABLE]
is a Lie bracket satisfying the Jacobi identity.111For example, let be the algebra of smooth functions or the algebra of formal power series on a given manifold . Then Lie bracket (3) is not necessarily a bi-derivation and its differential order with respect to either of the arguments is not necessarily bounded.
Proof.
Indeed, the Jacobiator is assembled by using the sum of associators:
[TABLE]
This tells us that the Jacobi identity for the bracket is an obstruction to the associativity of the product . ∎
Corollary 2**.**
Let be a Poisson bracket on and be the multiplication in the algebra . Suppose that a deformation of the product in to a multiplication in is such that is associative at all orders of the deformation parameter . Then this deformation yields a transformation of the Poisson bracket to a Lie, but not necessarily Poisson bracket (3).
Lemma 3** ([25]).**
Denote by the associative multiplication in the algebra . Suppose that a deformation is such that is a bi-derivation and let be associative up to , that is,
[TABLE]
Then the bi-linear skew-symmetric bi-derivation222By assumption, the leading deformation term in is a bi-derivation, hence same are its symmetric and skew-symmetric parts, and , respectively.
[TABLE]
is a Poisson bracket.
Proof.
In the leading order, . At we have that \operatorname{Assoc}_{\star\ \text{mod}\>\bar{o}(\hbar^{2})}(f,g,h)=\hbar\,\bigl{[}B_{1}(f,g)\times h+B_{1}(f\times g,h)-f\times B_{1}(g,h)-B_{1}(f,g\times h)\bigr{]}+\bar{o}(\hbar)=\bar{o}(\hbar) because is a derivation with respect to each argument. Next, at we obtain that
[TABLE]
We are given that this expression’s leading term vanishes. By taking an alternating sum over the group of permutations of three arguments and recalling that the product is commutative, we deduce that
[TABLE]
that is, all the four terms containing cancel out; in the sum over permutations they are grouped by (1st 2nd) (3rd) (4th). Finally, let us split and obtain that in fact, its symmetric part also cancels out in the alternating sum:
[TABLE]
whence the assertion readily follows. ∎
Example 1**.**
Let be functions in the Cartesian coordinates and on . Consider the associative star-product
[TABLE]
We have that
[TABLE]
By construction, we obtain
[TABLE]
which is the two functions’ Poisson bracket referred to the canonical Darboux coordinates and .
Remark 1*.*
From now on we shall always assume that the leading deformation term at in is skew-symmetric. In the Kontsevich star-product, the symmetric part of a given deformation term might not be vanishing identically ab initio but it can then be trivialised – at the expense of using suitable gauge transformations , of its arguments (see [25]).
1.2.
Now suppose that a Poisson bracket is given on in advance. Can the commutative associative multiplication in the algebra be deformed to an associative star-product such that the formal power series is well defined ? More specifically, the bi-linear, not necessarily commutative star-product must satisfy the four axioms:
- (1)
it is associative,
[TABLE]
i.e. modulo the property of bracket on to be Poisson; 2. (2)
the unit function remains the neutral element for ; whatever , one has that ; 3. (3)
each term , including the skew-symmetric Poisson bracket to start with at , is a bi-linear differential operator of bounded order; 4. (4)
the product is (let to be) -linear over .
Theorem 4** ([25]).**
For every affine -dimensional Poisson manifold there exists a star-product in satisfying the above four axioms.
The proof is constructive (cf. [7] and [26, 39]); the graph technique [25, 29, 28, 27] is a convenient way to encode the bi-differential terms in perturbation series . Every term in at , is encoded by an oriented graph with vertices, of which two sinks contain the respective arguments and and each of the remaining internal vertices is a source for two oriented edges. (In total, there are such wedges with arrows in every such graph .) Neither tadpoles nor multiple edges are permitted (cf. [7]). Next, install a copy of the given Poisson bi-vector at each of the tops of the wedges and decorate every edge in the graph at hand with a summation index running from to The two edges issued from each internal vertex are ordered, so that the precedent and antecedent edges correspond to the first and second indexes in a copy of the Poisson bi-vector .
1.3.
To encode multi-vectors \Xi\in\Gamma\bigl{(}\bigwedge^{*}TN^{n}\bigr{)} in a standard way, consider the parity-odd neighbour of cotangent bundle to the manifold and denote by the -tuple of -parity odd fibre coordinates over a chart with an -tuple of local coordinates. Whenever the values are given, construct the bi-vector \mathcal{P}=\tfrac{1}{2}\langle\xi_{i}P^{ij}\bigr{|}_{{\boldsymbol{u}}}\xi_{j}\rangle\in\Gamma(\bigwedge^{2}TU_{\alpha}); bi-vectors are Poisson if they satisfy the classical master-equation , see footnote 5 on p. 5.
Convention 1**.**
The correspondence between every decorated oriented edge and analytic expressions occurring in is established in Fig. 1;
at every set of index values, the respective content of vertices in a connected graph component is multiplied using (cf. footnote 4 on p. 4). The expressions determined by different connected components of one graph in their formal sum are also multiplied by using the original product .
Remark 2*.*
Because other arrows may stick into the vertices connected by an edge in Fig. 1, the objects and contained there can be derivatives (with respect to ’s) of the bi-vector or, specifically to but never possible to , arguments and of the star-product. On the same grounds, because there is another arrow issued from the tail vertex with , the formula encoded by a graph does in fact not depend on any of the auxiliary, parity-odd variables .
In the Kontsevich star-product, every graph is accompained with its weight ; these numbers are obtained by calculating certain explicitly given integrals over the configuration spaces of distinct points – in fact, the graphs’ vertices containing – on the Lobachevsky plane (in its Poincaré model in the upper half-plane), see [25]. The full set of rational values of weights for all graphs in an expansion of the Kontsevich star-product has been obtained in [6].
Example 2**.**
For any functions , the expansion reads as follows:333The precedence-antecedence of edges is given by the ordering of indexes , , , and in the analytic formula, see (5) below.
[TABLE]
Referred to any system of affine local coordinates on , for a given Poisson bi-vector \mathcal{P}{\bigr{|}}_{{\boldsymbol{u}}}=\tfrac{1}{2}\langle\xi_{i}P^{ij}({\boldsymbol{u}})\xi_{j}\rangle this sum of weighted graphs is realised by the formula444Note that a graph itself suggests the easiest-to-read way to write down the respective differential operator’s formula; this will be particularly convenient in the variational setting of section 2, see p. 9.
[TABLE]
The values of (derivatives of) both arguments and coefficients of the Poisson bi-vector are calculated at in the right-hand side of the above formula.
Example 3** (Moyal–Weyl–Grönewold).**
Suppose that all coefficients of the Poisson bi-vector are constant, which is a well defined property with respect to all local coordinate systems on the affine manifold at hand. In effect, the graphs with at least one arrow ariving at a vertex containing make no contribution to the star-product . The only contributing graphs are portrayed in this figure,
[TABLE]
These graphs are such that their weights in the power series combine it to the Moyal exponent,
[TABLE]
Here we accept that the use of every next copy of the bi-vector creates a new pair of summation indexes.
Remark 3*.*
The introduction of two identical copies, and , of the geometry where the objects and are defined reveals an idea that will be used heavily in what follows.
Proposition 5**.**
The associativity of Moyal star-product (2) is established by the a posteriori congruence mechanism.
Proof (see [8]).
From the identity
[TABLE]
we infer that
[TABLE]
which is due to the Baker–Campbell–Hausdorff formula for the exponent of sums of commuting derivatives, and by having indexes relabelled. ∎
1.4.
Whenever the coefficients are not constant on the domain , the classical master-equation555The Jacobi identity for Poisson bracket is equivalent to the zero-value condition for all Hamiltonians ; the tri-vector is viewed here as a tri-linear totally antisymmetric mapping and we denote by the Schouten bracket (i.e., parity-odd Poisson bracket); in coordinates, one proves that is a nontrivial constraint for the bi-vector . Where is the Jacobi identity for the Poisson bracket hidden in the associator for the full star-product ?
Example 4**.**
It is easy to see that .
By definition, we put
[TABLE]
In formulae, by ascribing the index to the unlabeled edge, the identity reads
[TABLE]
The coefficient of is the familiar form of the Jacobi identity.
To understand how sums of graphs can vanish by virtue of differential consequences of Jacobi identity (6), let us note that for a given Poisson bi-vector and for every derivation falling on the Jacobiator , the Leibniz rule yields that
[TABLE]
The last three terms in the right-hand side of the above formula amount to a redefinition of Jacobiator’s arguments; hence every such term vanishes. Consequently, the first term in which the derivation acts on the two internal vertices of the Jacobiator itself is equal to zero: \bigl{(}\partial_{i}\bigl{(}\operatorname{Jac}(\mathcal{P})\bigr{)}\bigr{)}(\cdot,\cdot,\cdot)=0. One now proceeds recursively over arbitrarily large finite set of derivations that sumltaneously fall on the Jacobiator, then acting independently from each other according to the Leibniz rule.
Definition 1**.**
A Leibniz graph is a graph whose vertices are either sinks, or the sources for two arrows, or the Jacobiator (which is a source for three arrows); there must be at least one Jacobiator vertex. The three arrows originating from a Jacobiator vertex must land on three distinct vertices (and not on the Jacobiator itself).666Each edge falling on a Jacobiator works by the Leibniz rule on the two internal vertices in it. Combined with expansion (6) of the Jacobiator using graphs, this tells us that every Leibniz graph expands to a sum of Kontsevich graphs which were introduced before.
Example 5**.**
An example of a Leibniz graph is given in Fig. 2.
Leibniz graphs encode (poly)differential operators whose arguments are at least one copy of the tri-vector and possibly, the bi-vector itself.777In Example 4 the Leibniz graph amounts to just one tri-vector vertex and no extra copies of the Poisson bi-vector in other internal vertices, of which there are none.
Proposition 6**.**
For every Poisson bi-vector the value – at the Jacobiator – of every (poly)differential operator encoded by the Leibniz graph(s) is zero.
Corollary 7**.**
To prove that the associator for the Kontsevich star-product vanishes for every Poisson structure contained in each internal vertex within a graph expansion of , it suffices to realize the associator as a sum of Leibniz graphs:888The same technique, showing the vanishing of a sum of Kontsevich graphs by writing it as a sum of Leibniz graphs, has been used in [4] to solve another problem in the graph calculus.
[TABLE]
From Example 4 we already know that the factorizing polydifferential operator in (1) is .
Example 6**.**
The assemply of factorizing operator , i.e. at order in the expansion , is explained in [5]; linear in its argument at , the operator has differential order one with respect to the Jacobiator.
The next step in factorization (1), now at order with respect to in the associator, is achieved in [6].
Proposition 8** ([6]).**
No solution of factorization problem (1) can have differential order less than two with respect ot the Jacobiator ; conversely, there always exists a Leibniz graph at in the polydifferential operator such that at least two arrows fall on the Jacobiator.
2. Deformation quantisation in the algebras of local functionals for field models
In this section we lift the Kontsevich graph technique from a quantisation of the product in the algebra of smooth functions on a finite-dimensional affine Poisson manifold \bigl{(}N^{n}, {\mathcal{P}}\bigr{)} to the deformation of the product of local functionals in the geometry of -valued fields over an affine base manifold . We shall analyse the construction of local variational polydifferential operators which are encoded by the Kontsevich graphs (in particular, by the Leibniz graphs), now containing at each internal vertex a copy of the variational Poisson structure . It is the Gel’fand formalism of singular linear integral operators supported on the diagonal [13] that becomes our working language.
2.1. Field model geometry
To extend the affine geometry of section 1, let us list the ingredients of the affine bundle set-up.999In retrospect, the construction in section 1 can be viewed as a special case of such “bundles” over a point .
Let \bigl{(}M^{m}, \operatorname{dvol}(\cdot)\bigr{)} be an -dimensional oriented affine real manifold equipped with a volume element.101010Not excluding the case where the volume element can nontrivially depend on the jets of sections over points , for the sake of brevity let us not write such admissible second argument in \operatorname{dvol}\bigl{(}\cdot,j_{\infty}(\phi)(\cdot)\bigr{)}. Let be an affine bundle with -dimensional fibres over the base . Denote by , , an -tuple of local coordinates in the fibre .
Denote by the total space of the bundle of infinite jets for sections of the bundle over ; the infinite jet space is the projective limit of the sequence of finite jet spaces ,
[TABLE]
It is clear that affine reparametrisations of local coordinates on the base induce linear transformations of smooth sections’ derivatives up to positive order for all . By definition, we put for an object’s dependence on sections and their derivatives up to arbitrarily large but still finite order, which is well defined by the above.
Denote by the vector space of integral functionals of form F=\int f\bigl{(}{\boldsymbol{x}}_{1},[{\boldsymbol{u}}]\bigr{)}\cdot\operatorname{dvol}({\boldsymbol{x}}_{1}) such that F({\boldsymbol{s}})=\int_{M^{m}}f\bigl{(}{\boldsymbol{x}}_{1},j^{\infty}_{{\boldsymbol{x}}_{1}}({\boldsymbol{s}})\bigr{)}\cdot\operatorname{dvol}({\boldsymbol{x}}_{1}). Viewed as functionals that take sections over to numbers, the integral objects can be shifted by using the null functionals . Those can be of topological nature,111111For instance, set , let , take the usual angle variables on the two circles, and consider the null Lagrangian that takes every section of an affine bundle over such to . Obviously, the cohomology class in is nonzero for the top-degree form ; for it is only locally but not globally exact. . We always quotient them out in this paper by taking the factorgroup . Secondly, null integral functionals can mark the zero class in the top-degree horizontal cohomology group121212The integrations by parts over are nominally present in the construction of horizontal cohomolohgy groups for the jet space over the bundle . By default, let us technically assume that no boundary terms would ever appear from in any formulae. For that, either let the base be a closed manifold (hence ; for instance, take ) or choose the class of admissible sections for the bundle is such a way that they decay so rapidly towards the boundary that all the integrands under study also vanish at . For instance, suppose that and all the field profiles are Schwarz. for over .
By brute force, introduce the multiplication {\times}\colon F\otimes G\mapsto F\times G=\int f\bigl{(}{\boldsymbol{x}}_{1},[{\boldsymbol{u}}]\bigr{)}\cdot\operatorname{dvol}({\boldsymbol{x}}_{1})\times\int g\bigl{(}{\boldsymbol{x}}_{2},[{\boldsymbol{u}}]\bigr{)}\cdot\operatorname{dvol}({\boldsymbol{x}}_{2})=\iint f\bigl{(}{\boldsymbol{x}}_{1},[{\boldsymbol{u}}]\bigr{)}\times g\bigl{(}{\boldsymbol{x}}_{2},[{\boldsymbol{u}}]\bigr{)}\cdot\operatorname{dvol}({\boldsymbol{x}}_{1})\,\operatorname{dvol}({\boldsymbol{x}}_{2})\colon\Gamma(\pi)\to\Bbbk for G=\int g\bigl{(}{\boldsymbol{x}}_{2},[{\boldsymbol{u}}]\bigr{)}\cdot\operatorname{dvol}({\boldsymbol{x}}_{2}). This yields the algebra of local functionals,131313We recall that in the (graded-)commutative set-up one has that \smash{\bigl{(}F\mathbin{\stackrel{{\scriptstyle\overline{\mathfrak{M}}^{m}(\pi)}}{{\times}}}G\bigr{)}({\boldsymbol{s}})=F({\boldsymbol{s}})\mathbin{\stackrel{{\scriptstyle\Bbbk}}{{\times}}}G({\boldsymbol{s}})} but a known mechanism destroys this algebra homomorphism in a larger setting of formal noncommutative variational symplectic geometry and its calculus of cyclic words ([20], cf. [30, 36]). also denoted by in [17, 20]
Referring only to the fibre’s local portrait but not to its global organisation, we introduce the -parity odd coordinates , , in the reversed-parity cotangent spaces to the fibres of the bundle , see [20, §2.1] and [17, §2.1]. Let us note that for vector spaces , the vector space isomorphism reduces this construction of Kupershmidt’s variational cotangent bundle [31] over to the Whitney sum J^{\infty}\bigl{(}\pi\mathbin{{\times}_{M}}\Pi\widehat{\pi}\bigr{)}.
Convention 2**.**
The notation will be used in what follows to avoid an agglomeration of formulae. Indeed, the case of affine bundle already impels the construction of horizontal jet bundle over the space .
The variational bi-vectors {\boldsymbol{\mathcal{P}}}\in\bar{H}^{m}\bigl{(}\pi\mathbin{{\times}_{M}}\Pi\widehat{\pi}\bigr{)} are integral functionals of the form
[TABLE]
where the linear total differential operators A=\bigl{\|}P^{ij}_{\tau}\cdot\bigl{(}\tfrac{{\mathrm{d}}}{{\mathrm{d}}{\boldsymbol{x}}}\bigr{)}^{\tau}\bigr{\|}_{i=1,\ldots,n}^{j=1,\ldots,n} are skew-adjoint (to make the object well defined); for all multi-indexes , the parity-odd symbols , , , , , are the respective jet fibre coordinates.
The construction of variational -vectors with is alike, see [20]. Due to the introduction of parity-odd variables as canonical conjugates of the -tuples , the vector space of all variational multivectors is naturally endowed with the parity-odd variational Poisson bracket, or variational Schouten bracket . Its construction – as descendent structure with respect to the Batalin–Vilkovisky Laplacian – was recalled in [17, 20].
Definition 2**.**
A variational bi-vector is called Poisson if it satisfies the classical master-equation . The horizontal cohomology class equivalence means, in particular, that the variational tri-vector , viewed as an integral functional, takes .
Every variational Poisson bi-vector induces the respective variational Poisson bracket on the space of integral functionals . An axiomatic construction of is explained in Definition 3 on p. 3; it is the derived bracket of two Hamiltonians (see [20, §3]).141414Note that an attempt to modify the volume element on can affect the output of .
The bracket is extended, via the Leibniz rule, from the vector space of integral functionals , , to the Poisson structure on the algebra of (sums of) such functionals’ formal products .
Remark 4*.*
The value of in at two integral functionals does not depend on a choice of representatives for the two arguments and for the variational Poisson bi-vector {\boldsymbol{\mathcal{P}}}\in\bar{H}^{m}\bigl{(}\pi\mathbin{{\times}_{M}}\Pi\widehat{\pi}\bigr{)}, taken modulo those integral functionals which map all sections of the respective (super)bundle to . This is no longer necessarily so for the higher-order terms, beyond the variational Poisson bracket at , in expansions (4).
Remark 5*.*
Let us remember that every integral functional – e.g., taken as a building block in a local functional – does carry its own integration variable which runs through that integral functional’s own copy of the base for the respective (super)bundle. For a given model over \bigl{(}M^{m}, \operatorname{dvol}(\cdot)\bigr{)}, the variational Poisson bi-vector {\boldsymbol{\mathcal{P}}}=\tfrac{1}{2}\int\xi_{i}\,P^{ij}_{\tau}({\boldsymbol{x}},[{\boldsymbol{u}}])\,\bigl{(}\tfrac{{\mathrm{d}}}{{\mathrm{d}}{\boldsymbol{x}}}\bigr{)}^{\tau}(\xi_{j})\cdot\operatorname{dvol}({\boldsymbol{x}}) and two Hamiltonians, and , are integral functionals defined at sections of the bundles and , respectively. In total, these three objects carry three copies of the given volume element on .
On the other hand, the variational Poisson bracket of and with respect to is an integral functional that carries one copy of the volume element. Why and where to have the two copies of gone ? We now recall an answer to this question.
2.2. Elements of the geometry of iterated variations
2.2.1.
Let be a two-component section of the Whitney sum of bundles. Suppose that this section undergoes an infinitesimal shift along the direction
[TABLE]
which we decompose with respect to the adapted basis in the tangent space . At their attachment point, the vectors and are – by definition – tangent to the respective coordinate lines for variables and . By construction, these vectors and are dual; at every running from to , the two ordered couplings of (co)vectors attached over at the fibres’ points – with values and of the respective coordinates – are
[TABLE]
Likewise, the coefficients and of the virtual shifts along the coordinate lines and are normalised via
[TABLE]
over all internal points .
Convention 3**.**
The differentials of functionals’ densities are expanded with respect to the bases , in the fibres tangent spaces; the plus or minus signs in the sections’ shifts are chosen in such a way that the couplings always evaluate to .
The directed variations and , as well as and , are singular linear integral operators supported, due to (7), on the diagonal. Each variation contains copies of Dirac’s -distribution weighted by the respective coefficients and . We have that
[TABLE]
see [17, §2.2–3] for details; for brevity, the indication of fibre points for given and is omitted in such formulae. Whenever acting on local functionals, these linear operators yield those functionals’ responses to infinitesimal shifts of their arguments. i. e. of the sections at which the functionals are evaluated.
Let us now recall the mechanism of integration by parts (see [17] and [20, §2.5]).
Lemma 9**.**
In absence of boundary terms, the on-the-diagonal integration by parts converts derivatives along one copy of the integration domain into derivatives with respect to the same variables, now referred to another copy of the base.
Proof.
Consider a point of the affine manifold and denote by a near-by point with coordinates , here and immediately below ; the notation makes obvious sense. For the sake of brevity, put . We have that, due to the absence of boundary terms and then by definition,151515The definition of total derivative , which is
\Bigl{(}j^{\infty}(s)^{*}\Bigl{(}\Bigl{(}\frac{{\mathrm{d}}}{{\mathrm{d}}x}f\Bigr{)}(x,[u])\Bigr{)}\Bigr{)}(x_{0})\stackrel{{\scriptstyle\text{def}}}{{=}}\Bigl{(}\frac{\partial}{\partial x}\Bigl{(}j^{\infty}(s)^{*}\Bigl{(}f(x,[u])\Bigr{)}\Bigr{)}\Bigr{)}(x_{0}),
explains why the partial derivatives reshape into as soon as they arrive to the graph’s vertices and there, they act on the objects which are defined over jet bundles and which are evaluated at the infinite jets of sections .
[TABLE]
For multi-indexes longer than the powers are processed by repeated integrations by parts; this yields . In the course of derivation of densities with respect to not but and so, in the course of using the other of two (co)vectors’ couplings, all reasonings are still performed in the exactly same way. ∎
Corollary 10**.**
The derivatives w.r.t. base variables are transported along an edge to the arrowhead according to the scenarios drawn in Fig. 3;
each derivative is referred to the copy of base manifold over which the object or structure it acts on is defined.
2.2.2.
Let us now explain how the edges in Kontsevich’s graphs get oriented; this mechanism is unseparable from the integration by parts. Every edge is realised by the linking of variations – with respect to the canonical conjugate variables and – of objects that are contained in the two adjacent vertices. The orientation of such edge is the ordering of singular linear integral operators; initially, they act as shown in Fig. 4.
Every edge in an oriented graph contributes to the summand (which encodes in the star-product ) by the linking of variations and by the linking of differentials of objects contained in the vertices. Novel with respect to the classical set-up of section 1, the variations and absorb the derivatives – previously, non-existent – along the base manifold .
The singular linear integral operators and now act not only on the spaces of local functionals but also on elements of their native space of singular distributions. At the same time, regular integral functionals – like and or from , which are contained in vertices of a graph at hand, – themselves can discard the copy of volume elements which they are equipped with and by this, reshape into singular integral operators. The linking of normalised variations yields singular linear integral operators that act via multiplication by . At the end of the day, the linking of every two neighbouring objects converts one of them into a singular linear integral operator such that the (co)vectors contained in it act on their duals, resulting in the multiples or .
Example 7**.**
The edge encodes the formula
[TABLE]
The singular distributions wright the diagonal ; both couplings evaluate to , and normalisation (8) makes the edge’s cargo invisible (indeed, it contributes via multiplication by ).
Remark 6*.*
The three singular operators in (9) can be directed in the opposite way, i.e. against the edge orientation along which the derivatives are transported in any case. This would keep the volume element at the arrow tail.
Example 8**.**
Consider the edge encoding the formula
[TABLE]
In this case, both couplings evaluate to by (7) still their values’ product is ; the diagonal-making and normalisation mechanism remain the same as before.
Summarising, the ordering in (7) is the only mechanism that creates sign factors; the direction in which the operators act along the edge does not necessarily coincide with that edge’s orientation in a graph . The arrow specifies the direction to transport the derivatives by using the integrations by parts.
Definition 3**.**
The variational Poisson bracket of two integral functionals and with respect to a given variational Poisson bi-vector is the graph
[TABLE]
By using two pairs of normalised variations and by letting the volume element stay in the vertex containing , we realise the geometry of singular distributions encoded by this picture via the formula161616The notation , not yet essential in the one-time variations here, will be explained in Convention 4 on p. 4.
[TABLE]
The two pairs of couplings evaluate to . The algorithm’s output is (cf. [31, 35])
[TABLE]
where is the Hamiltonian operator built into the variational Poisson bi-vector .
Remark 7*.*
Because the operator is skew-adjoint, one could now integrate by parts, obtaining an even shorter expression,
[TABLE]
Still let it be remembered that it is the wedge graph in (10) that does define the variational Poisson bracket, whereas this formula is its remote consequence.
Remark 8*.*
When the variational Poisson bracket of two given functionals is assembled by Definition 3 – to be evaluated at a section , – the total derivatives immediately follow the partial derivative in the construction of variational derivatives . Such inseparability of the horizontal and vertical derivations referring to their own geometries and , respectively, is specific only to one-step reasonings (for instance, derivation of the Euler–Lagrange equations of motion from a given action functional). However, a necessity to iterate the virtual shifts of a section reveals a difficulty of the classical jet bundle geometry (e. g., this was acknowledged in [15, §15.1]).
We shall now explain how the graded permutability of iterated variations is achieved in the course of both all the couplings evaluation and transporting the derivatives along to their final positions.
2.2.3.
Consider a vertex where two or more arrows arrive — or a vertex that contains (so that two partial derivatives, and , retro-act on its content) and that serves as the head for another arrow (hence, bringing the partial derivative followed by and possibly, by the total derivative(s) specified by that arrow’s tail), see (12):
[TABLE]
In which consecutive order are those partial and total derivatives, related to different edges, applied to the content of a vertex ?
Furthermore, the associativity of Kontsevich star-product is achieved in particular due to many cancellations of similar terms in the associator . Consider those three-sink graphs which can be built using at least two pairs of weighted graphs in the inner and outer products, respectively. Their cancellation prescribes that the resulting analytic expressions, encoded by every such non-contributing graph with three sinks, must not depend on a scenario to compose that graph. We conclude that the action of total derivatives in the inner star-products in the associator is delayed until all the partial derivatives would have finished acting in the outer star-products.171717This allowed an intrinsic regularisation of the Laplacian and variational Schouten bracket in the Batalin–Vilkovisky formalism (see [17, 18, 19] and [7, 15]), the concept was furthered to the formal noncommutative symplectic supergeometry and calculus of cyclic words (see [20] and [30]).
Proposition 11** (see [17, 18, 19, 20]).**
The vertical derivations and (the lifts of) horizontal derivations are performed at different stages. First, the vertical derivations along , together with their counterparts from the parity-odd symplectic dual, frame the edges of entire graph . In the meantime, the derivatives along the base are stored inside the variations by using . Lastly, all the horizontal derivatives are channelled from to , at the end of the day acting on the objects which are targets of .
Convention 4**.**
To indicate the delayed arrival of total derivatives to their final places (where we write them at once), let us embrace these operators by using in all formulas (e. g., see Definition 3 on p. 3 above).
Remark 9*.*
We emphasize that by the definition of total derivative (see footnote 15 on p. 15), the derivatives of virtual shifts for sections reshape, under integration by parts, into the derivatives of those sections but they do not affect the parity-odd variables which parametrise the fibres of another bundle. Consequently, the total derivatives refer only to the jet space where they act on the respective fibre variables in a vertex’ content.
Example 9**.**
The first graph in (12) corresponds to the formula
[TABLE]
where the multi-indexes and arrive from the respective arrow tails.
The second graph in Fig. 12 contributes with the expression
[TABLE]
where the multi-index arrives from the tail of arrow decorated with and where the copies of base for objects at the heads of arrows that carry and are marked using and , respectively.
Summarising, the local portrait of oriented edges around every vertex in a given graph determines the vertex-incoming partial derivatives with respect to variables , in-coming graded partial derivatives with respect to the parity-odd variables , and (powers of) delayed total derivatives. All these derivations act on the object contained in a vertex at hand, that is, on either a Hamiltonian density or structure constants P^{\alpha\beta}_{\zeta}\bigl{(}{\boldsymbol{x}},[{\boldsymbol{u}}]\bigr{)} of the variational Poisson bracket. Note that in both cases, the arguments are referred to the geometry of , hence those objects are expressed in terms of sections of the parity-even jet bundle .
Globally, each oriented graph in the Kontsevich summation formula (4) encodes a singular linear integral operator that acts on a local functional contained in one of the sinks.
2.3. On the associativity of star-product
2.3.1.
Let be a variational Poisson bi-vector such that its coefficients do not depend explicitly on the jet fibre variables in the bundle over . Let and be integral functionals referred to two identical copies of the jet space . A variational generalisation of the Moyal–Grönewold–Weyl star-product for and is the local functional which is constructed from (2) for the variational Poisson bracket by using the techniques from §2.2. The variational Moyal product of two integral functionals is expressed by the formula
[TABLE]
The angular brackets in (14) embrace the total derivatives whose action – in every term of the towered wedge graph expansion of – antecedes181818We recall that the action of total derivatives contained, e. g., in itself constituting a part of the object is also delayed until all the partial derivatives would have acted on the densities , , or . the action of partial derivatives with respect to and . For instance, the expansion starts as follows:
[TABLE]
Let , be an affine change of variables in such that the Jacobian matrix is locally constant on the intersection of two charts , . Then formula (14) is invariant with respect to such coordinate reparametrisation. The associativity of (14) is proved in a standard way (see the proof of Proposition 5 on p. 5). The associator of three given integral functionals over itself is an integral functional whose density is identically zero at all points of over .
Proposition 12**.**
Formula (14) provides the deformation quantisation of first and, via factorisation by using the junior Poisson bracket for the modified system, of second variational Poisson structures for the Drinfel’d–Sokolov hierarchies.
Example 10** (root system ).**
Consider the Korteweg–de Vries equation
[TABLE]
realised by using its second, field-dependent variational Poisson structure.191919Through the Fourier transform, the Hamiltonian operator encodes the Virasoro algebra, cf. [3]. Consider the Miura substitution [33] ; let us explain in advance that the conserved current stems – via the First Noether theorem – from the Noether symmetry of the action for the Liouville equation .
The mapping w=w\bigl{(}x,[u]\bigr{)} is determined by the integral ; we recall that the coefficient “2” in the right-hand side of is the only entry of the Cartan matrix for Lie algebra .
By definition, put so that is the first Hamiltonian operator of modified KdV hierarchy and so that . Denote by the adjoint of linearisation . By using the chain rules and , we cast the (potential) modified KdV equations,
[TABLE]
This factorisation pattern involving the Fréchet derivatives (or linearisations),
[TABLE]
is common to all the root systems of ranks , that is, for the (modified) Drinfel’d–Sokolov hierarchies [11]. It is seen that the hierarchy for respective analogue of potential modified KdV equation for constitutes the maximal commutative subalgebra in the Lie algebra of Noether symmetries for Leznov–Saveliev’s nonperiodic 2D Toda chains [32] u_{xy}^{i}=\exp\Bigl{(}\sum\limits_{j=1}^{r}\frac{2\langle\alpha_{i},\alpha_{j}\rangle}{\langle\alpha_{j},\alpha_{j}\rangle}\cdot u^{j}\Bigr{)}. The algorithm for construction of integrals [41] , , is known from [38], see [22] for an illustration. The De Donder–Weyl formalism [9] furthers the approach: the variables , , are the canonical conjugate momenta, , for the genuine coordinates , , satisfying the 2D Toda equations. The Lagrangian density is , where each row of the Cartan matrix is symmetrised to by using the root lengths, at every . Consequently, the junior variational Poisson structure for the modified Drinfel’d–Sokolov hierarchy is
[TABLE]
for every root system . Its coefficients are constants .
Having thus factorised a higher, field-dependent variational Poisson structure through the junior variational Poisson structures whose coefficients do not depend explicitly on the new fields , we reduce the large deformation quantisation problem for functionals , , to a much smaller Moyal–Grönewold–Weyl case (14) of the same functionals F\bigl{[}{\boldsymbol{w}}[{\boldsymbol{u}}]\bigr{]}, G\bigl{[}{\boldsymbol{w}}[{\boldsymbol{u}}]\bigr{]}, H\bigl{[}{\boldsymbol{w}}[{\boldsymbol{u}}]\bigr{]}\colon\Gamma(\pi)\to\Bbbk, now referred to the jet bundle of (potential) modified hierarchies.
Indeed, let and be two affine bundles over the base . Consider a jet space morphism specified by a Miura-type substitution of positive differential order. The Hamiltonian differential operators factorise via202020Integrating by parts,
\tfrac{1}{2}\int\Biggl{\langle}({\boldsymbol{\xi}})\Bigl{(}\overleftarrow{\ell}_{{\boldsymbol{w}}}^{({\boldsymbol{u}})}\Bigr{)}^{\dagger}\cdot\Bigl{(}P_{\tau}{\bigr{|}}_{({\boldsymbol{x}},[{\boldsymbol{u}}])}\Bigl{(}\frac{{\mathrm{d}}}{{\mathrm{d}}{\boldsymbol{x}}}\Bigr{)}^{\tau}\circ\overrightarrow{\ell}_{{\boldsymbol{w}}}^{({\boldsymbol{u}})\,\dagger}\Bigr{)}({\boldsymbol{\xi}})\Biggr{\rangle}\cong{}\\ {}\cong\tfrac{1}{2}\int\Biggl{\langle}{\boldsymbol{\xi}}\cdot\Bigl{(}\overrightarrow{\ell}_{{\boldsymbol{w}}}^{({\boldsymbol{u}})}\circ P_{\tau}{\bigr{|}}_{({\boldsymbol{x}},[{\boldsymbol{u}}])}\Bigl{(}\frac{{\mathrm{d}}}{{\mathrm{d}}{\boldsymbol{x}}}\Bigr{)}^{\tau}\circ\overrightarrow{\ell}_{{\boldsymbol{w}}}^{({\boldsymbol{u}})\,\dagger}\Bigr{)}({\boldsymbol{\xi}})\Biggr{\rangle}=\tfrac{1}{2}\int\langle{\boldsymbol{\xi}},\overrightarrow{A}({\boldsymbol{\xi}})\rangle\ ,
we construct the Hamiltonian differential operator in total derivatives that takes variational covectors to (the generating sections of) evolutionary vector fields.
[TABLE]
where is the variational Poisson bracket induced for functionals from a given variational Poisson structure for the pull-backs {H\bigl{[}{\boldsymbol{w}}[{\boldsymbol{u}}]\bigr{]}\colon\Gamma(\pi)\to\Bbbk}.
Formula (15) correlates senior Poisson structures for multi-Hamiltonian hierarchies with the junior Hamiltonian operators for the respective modified hierarchies of completely integrable PDE systems (see [16, 23] and references therein). Solutions \bigl{(}\pi, {\boldsymbol{w}}\bigl{(}{\boldsymbol{x}},[{\boldsymbol{u}}]\bigr{)}, B\bigr{)} are “good” if the coefficients of differential operator , referred to , do not contain the jet variables explicitly (that is, the star-product for the Hamiltonian operator amounts to formula (14)). In this case the output of deformation quantisation procedure is associative at all orders of the deformation parameter . However, for a given Hamiltonian differential operator over , its factorisation problem can be very hard.
2.3.2.
Finally, let us take a generic variational Poisson brackets with field-dependent coefficients . For instance, suppose that factorisation (15), reducing a given Hamiltonian differential operator on to the Moyal case for on , is not yet known.
It is readily seen that the splitting of differential consequences from the Jacobi identity into the separately vanishing homogeneous components, see Proposition 6 in §1.4, no longer takes place without reservations in the variational setting. This is because not only the vertical derivatives along the fibre of work by the Leibniz rule over the five vertices in every Jacobiator (6) but also do the total derivatives in their trail, as in (13), convert the Jacobiator on its three arguments into an indivisible object. Therefore, in the variational picture only those Kontsevich graph expansions of Leibniz graphs can vanish in which the Jacobiator is not split. Yet we remember from [5] that this is already not the case at in the associator for , see §1.4 above.
Secondly, for every triple of arguments, the Jacobiator is the map which, in terms of [17, 19, 20], can be a synonym of zero. Namely, if the density of this cohomologically trivial integral functional is not vanishing over all points of , the local variational polydifferential operator in
[TABLE]
can produce a nonzero integral functional from its zero-value argument . Indeed, whenever two or more arrows arrive at a vertex in the argument of , see (12), the order in which partial and then total derivatives act is (13). Therefore, the mechanism that guarantees the vanishing of the first variation for a cohomologically trivial argument is stepped over.
Corollary 13**.**
In the field-theoretic setting, the associativity of star-product can start leaking at order for a variational Poisson structure with field-dependent coefficients in the leading deformation term, so that .
Acknowledgements
The author is grateful to M. Kontsevich for advice and R. Buring for discussion. The author also thanks the organizers of 50th Sophus Lie Seminar (26–30 September 2016 in Bedlewo, Poland) for hospitality. This research was supported in part by JBI RUG project 106552 (Groningen) and by the IHÉS and MPIM (Bonn), to which the author is grateful for their warm hospitality.
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