Quadratic double ramification integrals and the noncommutative KdV hierarchy
Abstract.
In this paper we compute the intersection number of two double ramification cycles (with different ramification profiles) and the top Chern class of the Hodge bundle on the moduli space of stable curves of any genus. These quadratic double ramification integrals are the main ingredient for the computation of the double ramification hierarchy associated to the infinite dimensional partial cohomological field theory given by where is a parameter and is Hain’s theta class, appearing in Hain’s formula for the double ramification cycle on the moduli space of curves of compact type. This infinite rank double ramification hierarchy can be seen as a rank integrable system in two space and one time dimensions. We prove that it coincides with a natural analogue of the KdV hierarchy on a noncommutative Moyal torus.
Contents
1. Introduction
The main idea of this paper comes from the observation that the double ramification (DR) cycle, i.e. the class in the cohomology of the moduli space of stable curves representing the most natural compactification of the locus of smooth curves whose marked points support a principal divisor [Hai13], can be seen as a partial cohomological field theory (CohFT) [LRZ15, KM94] with an infinite dimensional phase space.
In [Bur15, BR16a, BDGR18] it was shown how to associate to any partial CohFT an integrable hierarchy of Hamiltonian systems of evolutionary PDEs in one space and one time dimensions. The number of dependent variables in these system of PDEs equals the dimension of the phase space of the partial CohFT. This integrable system is called the DR hierarchy and its properties and generalizations (including a quantization which exists in the case of actual CohFTs) where studied in [BR16b, BDGR19, BGR17, BR18].
This paper wants to answer the question: “what is the DR hierarchy associated to the infinite rank partial CohFT given by the DR cycle?”. As the general construction of the DR hierarchy already involves intersection numbers of a partial CohFT with the top Hodge class and a DR cycle, choosing as CohFT a second DR cycle leads us directly to having to compute intersection numbers of two different DR cycles and the top Hodge class. This is, of course, a question of its own geometric interest and, as we show in Chapter 2 of this paper, it has a very explicit answer.
In fact there is a natural deformation of the DR cycle which gives a oneparameter family of partial CohFTs. It comes from Hain’s formula [Hai13], expressing the DR cycle (restricted to the moduli space of stable curves of compact type) as the th power of (the pullback to the moduli space of stable curves of compact type) of the class of the theta divisor on the universal Jacobian. If we consider instead the exponential of such theta class, putting into play all of its powers, we get a more general but still very explicit infinite rank partial CohFT.
After showing in Chapter 3 how to trade the resulting infinite rank DR hierarchy in one space and one time dimensions as a rank hierarchy in two space and one time dimensions, we set out to compute it explicitly. The main result of this paper is in Chapter 4, where we show that the DR hierarchy of our infinite rank partial CohFT coincides with the noncommutative KdV hierarchy, the natural generalization of the ordinary KdV hierarchy on a circle to a torus with a noncommutative Moyal structure.
One of the applications is that, through our mixed intersection theory and integrable systems techniques, we are able to compute the intersection numbers of the top Hodge class, one DR cycle, any power of the theta class and any power of the psi class at one marked point.
Acknowledgements. This project has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie SkłodowskaCurie grant agreement No. 797635.
2. Quadratic double ramification integrals
For a pair of nonnegative integers in the stable range, i.e. satisfying , let be the moduli space of stable curves with genus and marked points labeled by the set . For integers , such that , the double ramification cycle is the Poincaré dual to the pushforward to of the virtual fundamental class of the moduli space of rubber stable maps from curves of genus with marked points to relative to and with ramification profile given by . Here “rubber” means that we consider maps up to the action in the target and a positive (negative) coefficient indicates a pole (zero) at the th marked point of order (), while just indicates that the th marked point is unconstrained. For future convenience we will also define the class to vanish in case .
Let us introduce the class defined by
(2.1) 
if and zero otherwise, where , , is the first Chern class of the th tautological line bundle, and, for and in the stable range and , and is the class of the irreducible boundary divisor of formed by stable curves with a separating node at which two stable components meet, one of genus and marked points labeled by and the other of genus and marked points labeled by the complement (naturally, if at least one of the stability conditions and is not satisfied).
By a result of Hain [Hai13], we know that
(2.2) 
where is the locus of stable curves with no nonseparating nodes. Moreover, always by [Hai13], the class represents the pullback to of the theta divisor on the universal Jacobian over , which implies the following relation in ,
(2.3) 
Let , , be the th Chern class of the Hodge bundle on . We have [FP00]
(2.4) 
The classes , and are algebraic, i.e. they belong in fact to the Chow ring . By the localization exact sequence, see e.g. [Ful98, Section 1.8], for all ,
where and are the inclusion maps of and into , and by equation (2.4), we deduce that, if is such that , then . This allows to deduce from identities (2.2) and (2.3) the following relations in :
(2.5)  
(2.6) 
For two pairs of nonnegative integers and in the stable range, let be the map that glues two stable curves of genus and with marked points labeled by the sets and , respectively, with , at their last marked points to form stable curve with a separating node with genus , and marked points labeled by . It is easy to see from the definitions that we have
(2.7) 
where and the classes on the righthand side are pulled back from each of the two factors in the product . By [BSSZ15], we also have
In the following we will denote by the pushforward .
We have the following result on the intersection number of two double ramification cycles and the top Chern class of the Hodge bundle.
Theorem 2.1.
Let and , then
(2.8) 
Proof.
By equation (2.5), Theorem 2.1 is equivalent to the formula
(2.9) 
Denote the lefthand side of equation (2.9) by , where , and , . Clearly, , so suppose that . In order to compute the integral, let us express one of the classes using formula (2.1). Using relation (2.3) and the formula for the intersection of a psi class with a double ramification cycle [BSSZ15], for any we compute
where . As a result,
Next we compute
Summarizing the above computations we get
that proves formula (2.9) and completes the proof of the theorem. ∎
3. An infinite rank partial CohFT and its DR hierarchy
Recall the following definition, which is a generalization first considered in [LRZ15] of the notion of cohomological field theory (CohFT) from [KM94].
Definition 3.1.
For a pair of nonnegative integers in the stable range , a partial CohFT is a system of linear maps , where is an arbitrary finite dimensional vector space, called the phase space, together with a special element , called the unit, and a symmetric nondegenerate bilinear form , called the metric, such that, chosen any basis of V, the following axioms are satisfied:

the maps are equivariant with respect to the action permuting the copies of in and the marked points in , respectively.

for , where is the map that forgets the last marked point.
Moreover for . 
for , and , where , , , and is the corresponding gluing map and where is defined by for .
Remark that a notion of infinite rank partial CohFT, i.e. a partial CohFT with an infinite dimensional phase space , requires some care. One needs to clarify what is meant by the matrix and to make sense of the, a priori infinite, sum over and , both appearing in Axiom (iii). One possibility is demanding that the image of the linear map induced by is contained in , where is the injective map induced by the bilinear form . Then in Axiom (iii), instead of using an undefined bilinear form on , one can use the bilinear form on induced by . This solves the problem with convergence.
A useful special case is the following. Let the basis of be countable and, for any in the stable range and each , let the set be finite. In particular this implies that the matrix is row and columnfinite (each row and each column have a finite number of nonzero entries), which is equivalent to , where is the dual “basis”. Let us further demand that the injective map is surjective too, i.e. that a unique twosided row and columnfinite matrix , inverse to , exists (it represents the inverse map ). Then the equation appearing in Axiom (iii) is well defined with the double sum only having a finite number of nonzero terms for each boundary divisor.
We will now construct an example of such infinite rank partial CohFT.
Proposition 3.2.
Let be a formal parameter. The classes form an infinite rank partial cohomological field theory with a phase space , where the unit is and the metric, written in the basis , is .
Proof.
Axioms (i) and (ii) follow directly from the definition of the classes . For Axiom (iii) notice that, for fixed , we have unless . Moreover we have and equation (2.7) implies Axiom (iii) where the double sum consists of just one term. ∎
In [Bur15] it was shown how to associate an integrable system of evolutionary Hamiltonian PDEs, called the double ramification (DR) hierarchy, to any CohFT and in [BDGR18] it was remarked how a partial CohFT is sufficient for the construction to work. Let us see how such construction generalizes to the infinite rank partial CohFT introduced in Proposition 3.2. Recall from [Bur15, BR16a] that the DR Hamiltonian densities are the generating series
(3.1) 
for and , seen as formal power series in the formal variables , .
Thanks to the fact that the intersection numbers appearing in equation (3.1) vanish unless and that, by formula (2.5), the class is a polynomial in homogeneous of degree , the above generating functions can be expressed uniquely (see e.g. [BR16a]) as a degree differential polynomial, i.e. a formal power series in and the new formal variables , , , of degree with respect to the grading , . The relation between the new variables and the old ones is given by the formula . The expression of the operator in the new variables is given by
Specifically, thanks to the fact that the intersection numbers appearing in equation (3.1) vanish unless , we obtain , where we put the superscript to denote the space of differential polynomials of degree .
The DR Hamiltonians are defined as the local functionals , which denote the equivalence classes of in the quotient vector space . Notice that, with respect to the formal variables , the symbol represents the coefficient of in the formal power series .
A result of [Bur15] says that the DR Hamiltonians mutually commute,
(3.2) 
with respect to the Poisson brackets of two local functionals , with , given by
(3.3) 
where and .
for
Now we make the following observation, specific for the infinite rank partial CohFT we are dealing with. For fixed , let us collect the DR Hamiltonian densities , for all , into a single generating function by use of the extra formal variable . Because the classes are polynomials in of top degree , where in particular the coefficient of is a polynomial of degree , and , we can consider the formal change of variables
and express uniquely as a differential polynomial in these new variables, specifically , where , , . Naturally, we have
We will denote simply by .
The DR Hamiltonian densities can be recovered from by the formula , which extracts the coefficient of from . Hence . This suggest to restrict our attention to the Hamiltonians , whose densities depend on through only. These are the simplest and most commonly considered kind of local functionals.
Let be the equivalence class of in the quotient vector space . Then, from equation (3.2), we deduce
(3.4) 
where the Poisson bracket of two local functionals , with is given by
(3.5) 
where and .
for
The evolutionary PDEs generated via the above Poisson structure by the DR Hamiltonians are all compatible and have the form
(3.6) 
4. The noncommutative KdV hierarchy and the main theorem
The classical construction of the KdV hierarchy as a system of Lax equations admits a generalization, where one doesn’t require the pairwise commutativity of the derivatives of the dependent variable. The formal algebraic construction is the following. Let be a triple, where is an associative, not necessarily commutative, algebra with a multiplication denoted by , , is a sequence of elements generating and is a linear operator satisfying the properties
Let us consider the algebra of pseudodifferential operators on of the form
Consider an operator . The noncommutative KdV hierarchy on is defined by (see e.g. [Ham05, DM00])
(4.1) 
where we put the subscript in the notation for a commutator in order to emphasize that it is taken with respect to the noncommutative product . The first equation of the hierarchy is
In what follows we will work with a specific example from the class of noncommutative KdV hierarchies.
The graded algebra of differential polynomials in two space dimensions introduced in Section 3, , where , , , can be endowed with the following graded associative Moyal starproduct. Let , with , , then define
(4.2) 