Maybe not so surprising.
The rather unobtrusive infingens $z^{n+1}\frac{d}{dz}$ 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 Johnathan Ethans (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.
This post imported from StackExchange MathOverflow at 2014-09-25 08:24 (UTC), posted by SE-user Tom Copeland
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