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Recently I encountered in a class the fact that there is a generating function of Gromov--Witten invariants that satisfies the Korteweg--de Vries hierarchy. Let me state the fact more precisely. Define $$\langle \tau_{k_1}\cdots \tau_{k_n}\rangle := \int_{\overline{\mathcal{M}}_{g,n}} \psi_1^{k_1}\cdots \psi_n^{k_n},$$ where $\overline{\mathcal{M}}_{g,n}$ is Deligne--Mumford space and $\psi_i$ is the first Chern class of the line bundle over $\mathcal{M}_{g,n}$ whose fiber at a given curve is the cotangent line at the $i$-th marked point of that curve. Next, define $$F(t,\lambda) := \sum_{g=0}^\infty \lambda^{2g-2} \sum_{n = (n_1,\ldots,n_k)} \frac {t^n} {n!} \langle \tau^n\rangle_g,$$ where $t = (t_0, t_1, \ldots)$ and $\lambda$ are formal variables. Then for all $n \geq 1$, $F$ satisfies the following PDE: $$(2n+1)\lambda^{-2}\partial_{t_n}\partial_{t_0}^2F = \partial_{t_{n-1}}\partial_{t_0}F\partial_{t_0}^3F + 2\partial_{t_{n-1}}\partial_{t_0}^2F\partial_{t_0}^2F + \frac 1 4\partial_{t_{n-1}}\partial_{t_0}^4 F.$$ Note that when $n=1$, it follows that (up to coefficients and $\lambda$) $\partial_{t_0}^2F$ satisfies the KdV equation: $$q_t = qq_x + \frac 1 2q_{xxx}.$$

I was very surprised that a generating function whose coefficients come from the geometry of Deligne--Mumford space should satisfy a nonlinear PDE for waves in shallow water. My question is:

Is there any "moral" reason for why a water wave PDE should have any connection with Gromov--Witten invariants?

My understanding (correct me if I'm wrong) is that Kontsevich's proof of this (which went by forming a cell decomposition of $\overline{M}_{g,n}$ using ribbon graphs) doesn't shed light on my question.

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    $\begingroup$ I'm not an expert, so take this with a grain of salt. I don't think you should search for a direct link between water waves and GW theory. Instead the answer is that water waves are examples of solitons, and that the theory of solitons can be interpreted in terms of integrable systems. Also, there are "moral" reasons for why GW theory should have a link to integrable hierarchies, in this case the KdV hierarchy. $\endgroup$ – Dan Petersen Oct 22 '13 at 18:20
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    $\begingroup$ Moreover, Kontsevich's proof does provide a link to integrable systems -- he rewrites $F$ in terms of a particular matrix integral, and it's known more generally that certain types of matrix integrals give rise to $\tau$-functions of integrable hierarchies. $\endgroup$ – Dan Petersen Oct 22 '13 at 18:20
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    $\begingroup$ Some other references are the five arXiv papers by Yuji Kodama and Lauren Williams. $\endgroup$ – Richard Stanley Sep 25 '14 at 0:50
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    $\begingroup$ Don't think about KdV as a fluid equation. It is just a normal form in the realm of nonlinear dispersive PDEs. You encounter it at every intersection. $\endgroup$ – Denis Serre Jun 5 '15 at 20:45
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    $\begingroup$ @DenisSerre... Exactly. This reminds me also of the "mysteries" that are corollaries of the Strong Law of Large Numbers: a particular example satisfies it... because it is universal. $\endgroup$ – paul garrett Jun 5 '15 at 21:48
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Maybe not so surprising.

The rather unobtrusive infinigens $z^{n+1}\frac{d}{dz}$ weave a web of connections among hydrodynamical equations, moduli spaces of Riemann surfaces, and the combinatorics of associahedra.

$$Summary$$

The most salient link between hydrodynamical equations and moduli spaces seems to be the infinite dimensional Witt Lie algebra/group and its central extension, the Virasoro-Bott Lie algebra/group. Hydrodynamical Euler equations give the geodesics of these groups, which govern the topology of the moduli space of the punctured Riemann surfaces of string theory. Somewhere in between lurk the Stasheff polytopes, the associahedra, whose combinatorics can be related to flow fields and the collisions of particles on a line that are related to the topology of punctured Riemann surfaces.

Example 1) Flows, the geometry of associahedra, and moduli spaces for marked Riemann sufaces of genus 0

A) Flows, streamlines, integral curves, and compositional inversion:

Let the inverse of the formal power series $\omega=h(z)=a_1\:z+ a_2 \: z^2+ \cdots$ be $z=h^{-1}(\omega)=b_1 \: \omega + b_2 {\omega}^2 + \cdots$ ; then, with $g(z)=1/[dh(z)/dz]$, a flow field is generated by

$$\exp \left[ {t \cdot g(z)\frac{d}{{dz}}} \right]z = \exp \left[ {t\frac{d}{{d\omega }}} \right]{h^{ - 1}}(\omega ) = {h^{ - 1}}[t + \omega] = {h^{ - 1}}[t + h(z)]=W(t,z),$$

and it is easy to show that the flow map has the following features;

$$<Identity>\:\:\: W(0,z)= z $$

$$<Orbit>\:\:\: W(t,0)= h^{(-1)}(t)$$ $$$$

$$<Velocity/generator>\:\:\: \frac{dW(0,z)}{dt} = g(z) = [h^{(-1)}]^{'}(h(z))$$

$$<Autonomous\:\: ODE>\:\:\: g(h^{(-1)}(\omega)) = [h^{(-1)}]^{'}(\omega)$$

$$<Group\:\:property>\:\:\: W[s,W(t,z)] = W(s+t,z)$$

$$<Tangency>\:\:\left [\frac{d}{dt}-g(z)\frac{d}{dz} \right ]\:W(t,z) = 0,$$

so $(1,-g(z))$ are the components of a vector orthogonal to the gradient of $W$ and, therefore, tangent to the contour of $W$ at $(t,z)$.

B) Compositional (Lagrange) inversion and associahedra (cf. Loday):

The iterated derivatives acting on $z$ and evaluated a $z=0$ generate the coefficients of the inverse power series. E.g.,

$$b_5=\frac{1}{5!}[g(z)\frac{d}{{dz}}]^{5}z|_{z=0} = \frac{1}{a_1^{9}} [14\: a_2^{4} - 21\: a_1 a_2^2 a_3 + a_1^2[6 \:a_2 a_4+ 3\: a_3^2] - 1\: a_1^3 a_5].$$

This is related to a refined f-vector (face-vector) for the 3-D Stasheff polytope, or 3-D associahedron, with 14 vertices (0-D faces), 21 edges (1-D faces), 6 pentagons (2-D faces), 3 rectangles (2-D faces), 1 3-D polytope (3-D faces). Subtracting two from the index of $a_n$, and ignoring the resulting indeterminates with indices with values less than one, allows one to read off the geometry of the associahedron from cartesian products of the lower dimensional associahedra (Loday), e.g., $3\: a^2_3$ becomes $3\: a^2_1$, the cartesian product of the 1-D associahedron with itself, which is a tetragon, or square in some reps.

This correspondence between the refined f-vectors of the $n$-D associahedron and $b_{n+2}$ holds in general, (see OEIS-A133437).

C) Associahedra and marked Riemann surfaces of genus 0:

Brown and Bergstrom in "Inversion of series and the cohomology of the moduli spaces of $M_{0,n}^\delta$" state:

For $n \geq 3$, let $M_{0,n}$ denote the moduli space of genus $0$ curves with $n$ marked points, and $\overline{M}_{0,n}$ its smooth compactification. ... In this paper, we prove that the inverse of the ordinary generating series for the Poincare polynomial of $H^\bullet(M_{0,n})$ is given by the corresponding series for $H^\bullet(M^{\delta}_{0,n})$, where $M_{0,n}\subset M^{\delta}_{0,n} \subset \overline{M}_{0,n}$ is a certain smooth affine scheme.

And on page 3, they give the abbreviated formula

$$M^{\delta}_{0,6}=14\; M_{0,3} \cup 21\: M_{0,4} \cup [6\: M_{0,5} \cup 3\: M^2_{0,4}] \cup M_{0,6}.$$

So, we have a connection between flows determined by the combinatorics of the associahedra and moduli spaces.

Example II) The inviscid Burgers-Hopf equation and associahedra

Define $$U(x,t)=\frac{x-A(x,t)}{t}$$ and $$A^{-1}(x,t)=x+t\;F(x).$$ Then it is easy to show that with $A(0,t)=0$ that $U$ satisfies the inviscid Burgers equation

$$U_t(x,t)+U(x,t)U_x(x,t)=0 , \:\:\:\: U(x,0)=F(x).$$

For details, see my sketch "Compositional inverse pairs, the Burgers-Hopf equation, and associahedra" at my mini-arxiv.

With $F(x)=c_2\:x^2+c_3\:x^3+ \cdots\;$, we have as asserted in Example I that

$$A(x,t)=x+(-c_2t)x^2+(-c_3t+2c_2^2t^2)x^3+(-c_4t+5c_2c_3t^2-5c_2^3t^3)x^4+(-c_5t+(6c_2c_4+3c_3^2)t^2+21c_2^2c_3t^3+14c_2^4t^4)x^5+\cdots\:,$$

the associahedra again. For $F(x)=x^n$, with $n>2$, $A(x,t)$ is the o.g.f. for the Fuss-Catalan numbers, which are related to dissections of polygons (cf. OEIS-A001764, particularly the Schuetz/Whieldon link). For $n=2$, we obtain the celebrated Catalan numbers and relations to Brownian motion, Lax pairs, random matrix theory, and Wigner's semicircle law/distribution, as discussed by Govind Meno in "Burgers turbulence: kinetic theory and complete integrability" and a similarly titled paper by Ravi Srinivasan. Victor Buchstaber in "Toric Topology of Stasheff Polytopes" even derives the Catalan numbers from an infinite set of conservation laws reminiscent of those for the KdV equation.

$$General\:\:\: Discussion$$

The Lie algebra of the diffeomorphism group of a manifold, Diff(M), consists of all vector fields on M, i.e., the infinitesimal generators $g(z)\frac{d}{dz}$ in Example I above (1-D or 2-D case, real or complex $z$), which induce an infinitesimal change in the coordinates $z \rightarrow z+t\:g(z)$. A basis for this algebra is the infinite dimensional Witt Lie algebra with elements $l_n=-z^{n+1}\frac{d}{dz}$. The geodesics for this group are given by the particular Euler eqn. the inviscid Burgers-Hopf equation (Example II). Already, with the subgroup $(l_{-1},l_0,l_1)$, related to linear fractional transformations, we can see connections to the moduli space of the Riemann sphere through the Riemann-Roch theorem as discussed by Gleb Arutyunov on page 87 of "Lectures on String Theory".

Making a central extension of the Witt algebra (on a circle) leads to the Virasoro algebra and group, whose geodesics are related to the KdV equation

$$\partial_t U+U\:\partial_xU=-c\partial^3_xU,$$

which is essentially a perturbed inviscid Burgers-Hopf with the constant parameter $c$ being the "depth" of the fluid. For more on this, see "Hydrodynamics and infinite dimensional Riemannian geometry" by Jonathan Evans (a review of The Geometry of Infinite Dimensional Groups by Boris Khesin and Robert Wendt), "Groups and topology in the Euler hydrodynamics and KdV" by Khesin, or " Euler equations on homogeneous spaces and Virasoro orbits" by Khesin and Gerard Misiolek.

The Virasoro algebra in conformal field theory governs the topology of the string world-sheet interactions generating the moduli spaces of Riemann surfaces with punctures corresponding to particles interacting on a line segment (Zwiebach, A First Course in String Theory, pg. 310). The Stasheff associahedra make another cameo appearance being intimately related to the moduli spaces of colliding particles (Devadoss, Devadoss/Heath/Vipismakul, Devadoss/Fehrman/Heath/Vashist, and the beautifully illustrated book Discrete and Computational Geometry by Satayan Devadoss and Joseph O'Rourke).

Alexander Givental in "Gromov–Witten invariants and quantization of quadratic Hamiltonians" relates a Virasoro algebra to the Witten–Kontsevich tau-function/potential and Euler fields. (The corresponding Witt algebra rep is rife with enumerative combinatorics. See my sketch of the algebra in "Infinitesimal generators, the Pascal Triangle, and the Witt and Virasoro algebras".)

So, the connecting element that these hydrodynamical and topological characters seem to share are the simple infinigens--the ghosts of Lie.

Another example (update Oct 12, 2015)

A Ricatti equation related to quadratic infinigens in sl(2) is linked to a soliton solution $1-tanh^2(x-ct)=d[tanh(x-ct)]/dx$ of a Kdv equation in The Elliptic Lie Triad (following up on my comment below on Rzadowski's paper). The hyperbolic tangent can be regarded as an exponential generating function for the number of connected components in the space of M-polynomials in hyperbolic functions (ref. in OEIS A000111) or for a proportionality factor in the Kervaire-Milnor formula in homotopy theory for hyper-spheres involving normalized Bernoulli numbers.

More generally, the bivariate e.g.f. for the Eulerian numbers (A008292/A123125) with its associated quadratic (sl2) infinigen provides a soliton solution of the 1-D KdV equation, and the Eulerians are rife with ($A_n$ and $B_n$) connnections to enumerative algebraic geometry, as discussed by Hirzebruch, Losev and Manin, Batryev and Blume, Cohen, et al.

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  • $\begingroup$ Also see "Bernoulli numbers and solitons" by G Rzadkowski tandfonline.com/doi/pdf/10.1142/S1402925110000635 The connections of the Bernoulli numbers to topology are well-known. $\endgroup$ – Tom Copeland Nov 27 '14 at 11:28
  • $\begingroup$ See also Friedrich and McKay, "Formal groups, Witt vectors and free probability" (page 5). $\endgroup$ – Tom Copeland Dec 26 '14 at 23:22
  • $\begingroup$ A similar situation occurs with the hypercubes (oeis.org/A038207) and a Fokker-Planck equation. $\endgroup$ – Tom Copeland Oct 17 '15 at 16:25
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    $\begingroup$ Tom, please stop constantly editing this answer; you bump it to the top every time you do so. If you have so much to say about this topic then write a blog post and constantly edit that instead. $\endgroup$ – Qiaochu Yuan Oct 22 '15 at 17:22
  • $\begingroup$ From the article on Mirzakhani in the Nov AMS Notices (pg. 1239): Witten’s posited relations are expressed in terms of the Virasoro Lie algebra generated by differential operators $ L_n= -x^{n+1} \frac{d}{dx}$ for $n > −2$, with commutators $[L_n,L_m] = (n-m) L_{n+m}$. ... Mirzakhani was able to decipher the mystery ... . Theorem 6 (The Witten-Kontsevich conjecture). The moduli volume recursion gives the Virasoro constraints. $\endgroup$ – Tom Copeland Oct 27 '18 at 18:29

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