The fundamental solution matrix and relative stable maps
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Abstract
Givental’s Lagrangian cone \({\mathscr {L}}_X\) is a Lagrangian submanifold of a symplectic vector space which encodes the genuszero Gromov–Witten invariants of X. Building on work of Braverman, Coates has obtained the Lagrangian cone as the pushforward of a certain class on the moduli space of stable maps to Open image in new window . This provides a conceptual description for an otherwise mysterious change of variables called the dilaton shift. We recast this construction in its natural context, namely the moduli space of stable maps to Open image in new window relative the divisor Open image in new window . We find that the resulting pushforward is another familiar object, namely the transform of the Lagrangian cone under the action of the fundamental solution matrix. This hints at a generalisation of Givental’s quantisation formalism to the setting of relative invariants. Finally, we use a hidden polynomiality property implied by our construction to obtain a sequence of universal relations for the Gromov–Witten invariants, as well as new proofs of several foundational results concerning both the Lagrangian cone and the fundamental solution matrix.
Keywords
Gromov–Witten invariants Quantisation formalism Relative Gromov–Witten invariantsMathematics Subject Classification
14N35 53D451 Introduction
The Gromov–Witten invariants of a smooth projective variety X are defined as certain intersection numbers on moduli spaces of stable maps to X. They can be thought of as counting curves of specified genus and degree passing through specified cycles in X. Their intrinsic interest aside, Gromov–Witten invariants have connections to numerous other areas of mathematics, from representation theory to symplectic topology. In algebraic geometry they have been used in the proofs of classification theorems, as a tool for distinguishing nondeformationequivalent varieties.
Many results in Gromov–Witten theory are expressed most cleanly via generating functions, that is, formal functions (usually polynomials or power series) whose coefficients are given by Gromov–Witten invariants. Oftentimes, a simple identity involving generating functions is all that is needed to express a relationship which, on the level of individual invariants, is extremely complicated. There is an underlying reason for this: Gromov–Witten theory has deep connections to theoretical physics, through which the aforementioned generating functions appear as the “partition functions” of physical theories. This circle of ideas has been extremely influential for the development of the subject, with the first major result in this direction being the celebrated Mirror Theorem [3, 17, 18].
The benefits of this quantisation formalism are twofold. From a theoretical viewpoint, it can be used to make rigorous sense of a number of deep predictions coming from physics. On the other hand, from a practical point of view, it has proven to be an extremely versatile framework in which to formulate and prove statements about Gromov–Witten invariants. Indeed, there are many results in Gromov–Witten theory which would be difficult to even state without the quantisation formalism: examples include the quantum Riemann–Roch formula [8], the crepant transformation conjecture [10], the Virasoro conjecture and various versions of the “genus zero implies higher genus” principle [20].
Building on work of Braverman [2], Coates shows in [7] that \({\mathscr {L}}_X\) can be obtained as a (\({\mathbb {C}}^*\)localised) pushforward from the moduli space of stable maps to Open image in new window (usually called the graph space). This is motivated by Givental’s heuristic description of \({\mathscr {H}}\) as the \(S^1\)equivariant cohomology of the loop space of X [16], and gives a natural geometric interpretation for a mysterious change of variables, called the “dilaton shift”, which is essential to the quantisation formalism.
Coates’ construction requires restricting to a certain open substack of the moduli space of stable maps to Open image in new window , before localising to a proper fixed locus (with respect to the natural \({\mathbb {C}}^*\)action on the moduli space) in order to push forward. With hindsight, this is really the pushforward from one of the \({\mathbb {C}}^*\)fixed loci in the moduli space of relative stable maps to the pair Open image in new window .
A natural question to ask is then: what happens if we sum over all the fixed loci? In this article we provide the answer (see Proposition 2.4): the result is the transform of the Lagrangian cone under the action of the fundamental solution matrix. The main tools used in the proof are the relative virtual localisation formula [23, Theorem 3.6], a virtual pushforward theorem for relative stable maps to the nonrigid target [15, Theorem 5.2.7] and a comparison lemma for psi classes, which we prove in Sect. 3.2.
Because we are now summing over all fixed loci, we know that the resulting class must actually belong to the nonlocalised equivariant cohomology. In practice, this means the following: we push forward and obtain a class which, a priori, looks like a rational function in z; however we know that, after performing suitable cancellations, we must end up with a polynomial (here z denotes the \({\mathbb {C}}^*\)equivariant parameter). We use this observation to give new and simple proofs of a number of foundational results belonging to the quantisation formalism theory.
Future directions
This construction provides a hint as to how one might obtain a quantisation formalism for relative (or logarithmic) Gromov–Witten invariants; see Remark 2.3. This was in fact the original motivation for this work.
User’s guide
Readers familiar with Gromov–Witten theory and the quantisation formalism may skip straight to Sect. 2.6 where we give the statement of the main result. For the uninitiated, we provide in Sects. 2.1–2.5 a brief introduction to Gromov–Witten invariants, the Lagrangian cone and relative Gromov–Witten theory. The proof of the main result is given in Sect. 3; this is mostly a computation, with the only geometric content being a lemma on psi classes which we prove in Sect. 3.2. Finally in Sect. 4 we provide examples of how the “hidden polynomiality” implied by our construction can be used to obtain universal relations for the Gromov–Witten invariants, as well as new proofs of a number of standard results concerning the Lagrangian cone and the fundamental solution matrix.
2 Background and statement of the main result
2.1 Gromov–Witten invariants
2.2 Givental space
The Lagrangian cone \({\mathscr {L}}_X\) is a geometric object which encodes all the genuszero Gromov–Witten invariants of X. It can be viewed as the graph of a certain generating function for these invariants. This generating function must keep track, through its formal variables, of both the cohomological insertions \(\gamma _i\) and the exponents \(k_i\) of the classes \(\psi _i\). We begin by defining a vector space \({\mathscr {H}}\) whose coordinates will give precisely these formal variables; the Lagrangian cone will then be a submanifold of \({\mathscr {H}}\).
Example 2.1
2.3 Lagrangian cone
As it has been presented, divorced from its origins in physics, \({\mathscr {L}}_X\) may come across as a mysterious object. Working with it takes some getting used to, but the eventual payoff is significant, and it is now recognised as a fundamental tool in Gromov–Witten theory. To give just a taste of this, we state a few basic facts about the Lagrangian cone.
Theorem 2.2

\({\mathscr {L}}_X\) is a cone (it is preserved under scalar multiplication by elements of \(\Lambda \));

for Open image in new window , we have Open image in new window ;

the set of all tangent spaces to \({\mathscr {L}}_X\) forms a finitedimensional family; thus \({\mathscr {L}}_X\) is ruled by a finitedimensional family of linear subspaces.
Thus we see that the geometry of \({\mathscr {L}}_X\) is very tightly constrained. The above theorem is actually equivalent [21, Theorem 1] to the following three fundamental results in Gromov–Witten theory: the string equation, the dilaton equation and the topological recursion relations. More generally, the Lagrangian cone can be used to conveniently express statements which would be exceedingly cumbersome to phrase otherwise. For more on this, see [9, 19].
Finally, we note that the dilaton shift \({\mathbf{q }}(z) = {\mathbf{t }}(z)  z\mathbb {1}_X\) is an essential part of the theory; for instance, \({\mathscr {L}}_X\) is not even a cone in the \({\mathbf{t }}(z)\) coordinates.
2.4 Fundamental solution matrix
2.5 Relative stable maps
We will assume that the reader is reasonably familiar with relative stable maps; all the facts which we will use can be found in [23, Sections 2–3], which also serves as a good introduction to relative Gromov–Witten theory.
Remark 2.3
More recently, the theory of logarithmic stable maps, as developed by Abramovich, Chen, Gross and Siebert, has provided an alternative (and significantly more general) approach to relative stable maps [1, 4, 24]. We expect that the computations we carry out here will carry over to the log setting, once a suitable localisation formula has been established for log stable maps. Indeed, log Gromov–Witten theory relative a simple normal crossings divisor seems to be the correct generality in which to apply the construction given in this article.
2.6 Statement of the main result
Proposition 2.4
An immediate corollary of the above result is that Open image in new window rather than just \({\mathscr {H}}\). For an application of this, as well as a deeper exploration of the “hidden polynomiality” arising from our construction, see Sect. 4.
Remark 2.5
The total transform \(S({\mathscr {L}}_X)\) has a geometric interpretation as a family of ancestor cones; see [8, Appendix 2].
Remark 2.6
Notice that for any choice of \(\beta \), the curve class \((\beta ,1)\) is nonzero. Hence the sum in Proposition 2.4 is over all\(\beta \) and n. This is in contrast to the sum which appears in the definition of the Lagrangian cone in Sect. 2.3, which is only over the stable range, i.e., excludes the cases \((\beta ,n) = (0,0)\) and (0, 1). This difference will become important during the proof of Proposition 2.4.
3 Proof of the main result
We will assume that the reader is familiar with the space of relative stable maps, and in particular with the torus localisation formula, established in [23] whenever the divisor is fixed pointwise by the action (as is the case for us). We will write \(X_0\) and \(X_\infty \) for Open image in new window and Open image in new window , viewing them either as divisors in Open image in new window or in Open image in new window , as appropriate.
3.1 Identifying the fixed loci
Note that for certain choices of \((\beta _0, A_0 \,  \, \beta _\infty , A_\infty )\) the moduli spaces which we have written down above do not exist, because the data defining them is not stable. In these degenerate cases, we still have fixed loci; it is simply that one (or both) of the factors becomes trivial. Hence we must deal with these separately. The possible situations are enumerated below.
Case 1: \((\beta ,n)=(0,0)\). This is the maximally degenerate case. The fixed locus is just X, which has virtual codimension 0; there is no virtual normal bundle.
Case 2: \((\beta ,n)=(0,1)\)and\(n_\infty =0\). In this case the fixed locus is again just X, with a single marked point \(x_1\) mapped to \(X_0\) and another marked point \(x_\infty \) mapped to \(X_\infty \) (there is no expansion of the target). The virtual codimension is 1, and the Euler class of the virtual normal bundle is Open image in new window .
Case 3: \(n \geqslant 1\)and\((\beta _0,n_0)=(0,0)\). In this case the fixed locus is a moduli space of relative maps to the nonrigid target, with \(n+2\) marked points. The virtual codimension is 1, and the virtual normal bundle contribution is \(z\psi _{q_\infty }\).
Case 4: \(n \geqslant 1\)and\((\beta _0,n_0)=(0,1)\). Here the fixed locus is the same as the one in the previous case, but it now has virtual codimension 2 because there is a marked point at the \(X_0\) end of the rational bridge; the Euler class of the virtual normal bundle is Open image in new window .
Case 5: \(n \geqslant 2\)and\((\beta _\infty ,n_\infty )=(0,0)\). In this case the fixed locus is just the moduli space of stable maps to X with \(n+1\) markings. The virtual codimension is 2, and the Euler class of the virtual normal bundle is Open image in new window .
3.2 Comparison lemma for psi classes
Lemma 3.1
The map \(\pi \) cannot contract any component of the source curve which contains a marking.
Proof
The components contracted by \(\pi \) are those with two or fewer special points which are mapped into a fibre of Open image in new window over X. Let \(C^\prime \) be such a component. Since it has two or fewer special points, the map f must be nonconstant on \(C^\prime \) (by stability), and hence there is at least one point of \(C^\prime \) which maps to \(X_\infty \) and at least one point which maps to \(X_0\). Thus, \(C^\prime \) contains exactly two special points, which must map to the special divisors of the nonrigid target.
Now suppose for a contradiction that some marking \(x_i\) belongs to \(C^\prime \). If \(x_i\) is a nonrelative marking then we immediately arrive at a contradiction, since such a marking cannot map into any special divisor. Otherwise, \(x_i = q_\infty \) or \(x_\infty \) and so is mapped into \(X_0\) or \(X_\infty \), respectively; without loss of generality we may suppose \(x_i = q_\infty \). By the stability condition for relative stable maps, there must exist some other component of the source curve which maps with positive degree into the same level of the nonrigid target as \(C^\prime \). But this would necessarily touch \(X_0\), which is a contradiction since \(q_\infty \) is the only point of the source curve which is allowed to map to \(X_0\) (here we are using the fact that Open image in new window is a global product; for nontrivial \({\mathbb {P}}^1\)bundles over X, it is no longer true that a component of the source curve which touches \(X_\infty \) must also touch \(X_0\)). \(\square \)
Corollary 3.2
\(\pi ^* \psi _i = \psi _i\) for any Open image in new window . Thus, we can identify any nonrigid invariant of Open image in new window with the corresponding invariant of X.
3.3 Calculating the contributions
3.4 Putting everything together
Remark 3.3
It is perhaps worth comparing our computation to the computation carried out in [7]. There, the moduli space under consideration is the space of ordinary stable maps to Open image in new window ; Coates restricts to an open substack of this space, consisting of stable maps such that only a single point of the curve is mapped to \(X_\infty \). He then applies torus localisation and pushes forward from the (proper) fixed loci. From our point of view, the loci from which he pushes forward are the degenerate loci which appear as Case 5 in Sect. 3.1 above. The special cases which he calls Case 2 and Case 3 are what we call Case 2 and Case 1, respectively. Our nonspecial case, which contributes a product of invariants from stable maps to X and stable maps to the nonrigid target, does not appear in his setting; nor do our special Cases 3 and 4.
4 Variants and applications
Since an equivariant pushforward must take values in Open image in new window , an immediate consequence of Proposition 2.4 is the following:
Theorem 4.1
Remark 4.2
Theorem 4.1 can be viewed as a generalisation of one of the fundamental results in the quantisation formalism, namely that the Jfunction is inverse to the fundamental solution matrix; see Remark 4.4 below.
In this section we will now extend the above line of argument, exploiting the “hidden polynomiality” implicit in our construction. We obtain new proofs and generalisations of several foundational results concerning both the fundamental solution matrix and the Lagrangian cone.
4.1 The fundamental solution matrix and its adjoint
Proposition 4.3
Proof
Remark 4.4
4.2 Properties of the Lagrangian cone
Here we reprove two fundamental facts concerning the Lagrangian cone. First, we modify the previous construction to give a concrete proof that \({\mathscr {L}}_X\) is Lagrangian (though it should be noted that this also follows from the general fact that the graph of any closed 1form is Lagrangian).
Proposition 4.5
\({\mathscr {L}}_X\) is Lagrangian.
Proof
Proposition 4.6
Proof
Remark 4.7
The idea of using torus localisation to prove that certain generating functions are polynomials is not new. It was used by Givental in the proof of the Mirror Theorem [17] and by CiocanFontanine and Kim in the proof of the wallcrossing formula for quasimap invariants [5]. The disussion above constitutes a small continuation of this story.
Notes
Acknowledgements
I owe a great deal of thanks to Tom Coates, for first suggesting this project, for patiently explaining the quantisation formalism to me and for pointing out some of the applications presented in Sect. 4. I would also like to thank Pierrick Bousseau, Elana Kalashnikov and Mark Shoemaker for useful discussions, and the referee for a number of helpful comments.
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