Gromov – Witten Invariants, The W/K – Tau – function and Hodge Integrals

By George Shiber Thursday, May 5, 2016Saturday, August 17, 2019 In Cosmology, Grand Unification Physics, M Theory, Mathematical Physics, Mathematics, Philosophy Of Physics, Physics, Super String Theory Gromov–Witten invariants, Hodge Integrals

In this post, part one, I will discuss the importance of Gromov-Witten invariants, and for an excellent read, Bernd Siebert’s SGWI is a must. For X a compact almost Kähler manifold of complex dimension D, let $X_{g,m,d}^{Orb}$ be the moduli orbifold of degree d stable holomorphic maps to X of genus g where d takes values in the lattice ${H_2}\left( {{X^{Orb}}} \right)$ . By compactness, $X_{g,m,d}^{Orb}$ is equipped with a rational coefficient virtual fundamental cycle $\left[ {X_{g,m,d}^{Orb}} \right]$ of complex dimension

and the total descendent potential of $X_{g,m,d}^{Orb}$ is

${\not D_{X_{g,m,d}^{Orb}}}: = \exp \sum {{\hbar ^{g - 1}}} \not F_{X_{g,\,m,\,d}^{Orb}}^g$

with $\not F_{X_{g,\,m,\,d}^{Orb}}^g$ the genus g descendent potential

with $\psi _i^k$ being the powers of the 1-st Chern class of the universal cotangent line bundle over $X_{g,m,d}^{Orb}$ corresponding to the i-th marked point, and $ev_i^ *$ are pull-backs by the evaluation map $e{v_i}:X_{g,m,d}^{Orb} \to X$ at the i-th symplectic-marked point of the cohomology classes ${t_0},{t_1}... \in {H^ * }\left( {X,Q} \right)$ and ${Q^d}$ is the representative of d in the semigroup ring of the semigroup of degrees of holomorphic curves in X. The genus g Gromov-Witten potential of X is defined as the restriction

$F_X^g(t): = \not F_X^g\left| {_{{t_0} = 1,{t_1} = 2 = ...0}} \right.$

and the genus g descendent potentials are functions on the super-space of vector Laurent polynomials $\not t(z) = {t_0} + {t_1}z\,{t_2}{z^2} + ...$ with coefficients in $H: = {H^ * }\left( {X;\not Q\left[ {\left[ Q \right]} \right]} \right)$ , the cohomology space of X over the Novikov ring.

According to Witten’s conjecture, which Kontsevich proved, the total descendent potential coincides with the tau-function of the KdV-hierarchy satisfying the string equation

Let the Witten-Kontsevich tau-function be denoted by $\tau _s^{WK}\left( {\hbar ,\not t} \right)$ and let $H,(.,.)$ be an N-dimensional vector space equipped with a non-degenerate symmetric bilinear form. With $\not {\rm H}$ the space of Laurent polynomials in one indeterminate z with vector coefficients from $\not {\rm H}$ , introduce a symplectic bilinear form in $\not {\rm H}$ by

and let $\not {\rm H} = {\not {\rm H}_ + } \oplus {\not {\rm H}_ - }$ correspond to the decomposition

$f\left( {z,{z^{ - 1}}} \right) = {f_ + }\left( z \right) + {f_ - }\left( {1/z} \right)/z$

of the Laurent polynomials into polynomial and polar parts, and deeply, note that the subspaces ${\not {\rm H}_ \pm }$ are Lagrangian. One then quantizes infinitesimal symplectic transformations $\not L$ on $\not {\rm H}$ to order ≤ 2 linear differential operators $\not \tilde L$ . In a Darboux coordinate system $\left\{ {{p_\alpha },{p_\beta }} \right\}$ compatible with our decomposition $\not {\rm H} = {\not {\rm H}_ + } \oplus {\not {\rm H}_ - }$ one has

$\left\{ {\begin{array}{*{20}{c}}{{{\left( {{p_\alpha },{p_\beta }} \right)}^ \wedge } = \hbar {{\not \partial }_{{q_\alpha }}}{{\not \partial }_{{q_\beta }}}}\\{{{\left( {{p_\alpha },{p_\beta }} \right)}^ \wedge } = {q_\beta }{{\not \partial }_{{q_\alpha }}}}\\{{{\left( {{p_\alpha },{p_\beta }} \right)}^ \wedge }{q_\alpha }{q_\beta }/\hbar }\end{array}} \right.$

with

$\left[ {\not \tilde F,\not \tilde G} \right] = {\left\{ {\not F,\not G} \right\}^ \wedge } + \not C\left( {\not F,\not G} \right)$

holding and $\not C$ satisfying

$\left\{ {\begin{array}{*{20}{c}}{\not C\left( {p_\alpha ^2,q_\alpha ^2} \right) = 2}\\{\not C\left( {{p_\alpha }{p_\beta },{q_\alpha }{q_\beta }} \right) = 1\quad ,\quad \alpha = 1}\end{array}} \right.$

and $\not C = 0$ for any other pairs of quadratic Darboux monomials.

Let us move now to the Witten-Kontsevich Tau-function: let

such that

It follows that

which entails

$\Omega \left( {{L_m}f,g} \right) = - \,\Omega \left( {f,{L_m}g} \right)$

and thus shows that the operators ${L_m}$ are infinitesimal symplectic transformations on $\not {\rm H}$ . However, ${\tilde D_{wk}}$ is conjugate to ${z^2}d/dz = - \,d/dw$ , with $w = 1/z$ , therefore ${L_m}$ commute as $- w{d^{m + 1}}/d{w^{m + 1}}$ and by the Fourier transform, as the vector fields $- {x^{m + 1}}d/dx$ on the line. So, the Poisson brackets satisfy $\left\{ {{L_m},{L_n}} \right\} = \left( {m,n} \right){L_{m + n}}$ and hence we have a representation of the Lie algebra of vector fields on the line to the Lie algebra of quadratic hamiltonians on $\not {\rm H}$ .

In the 1-dimensional $H$ with the standard inner product, using the Darboux coordinate system

on $\not {\rm H}$ , one gets

We then get

$\left[ {{{\hat L}_m},{{\hat L}_n}} \right] = \left( {m - n} \right){\hat L_{m + n}}$

unless m, n = ±1, in which case:

$\left[ {{{\hat L}_1},{{\hat L}_{ - 1}}} \right] = 2\left[ {{{\hat L}_0} + 1/6} \right]$

Thus the operators ${\hat L_m} + {\delta _{m,0}}/16$ form a representation of the Lie algebra of vector fields on the line.

So, we get a nice formulation of Kontsevich theorem confirming Witten strikingly bold conjecture. Hence:

Proposition The Witten-Kontsevich tau-function is annihilated by the operators ${\hat L_m} + {\delta _{m,0}}/16$ , m = −1, 0, 1, 2, …; and thus is completely characterized by this property – up to a scalar factor

Hodge integrals: Let ${\rm E}$ denote the Hodge bundle over the moduli space $X_{g,m,d}^{Orb}$ . Then analytically, the fiber of ${\rm E}$ over the point represented by a stable map $\Sigma \to X$ is the complex space of dimension $g$ dual to

${H^1}\left( {\Sigma ,{\vartheta _\Sigma }} \right)$

and so ${\rm E}$ is the pull-back of by the contraction map

$ct:X_{g,m,d}^{Orb} \to {\bar {\rm M}_{g,m}}$

of the Hodge bundle over the Deligne-Mumford space. It is a beautiful theorem that even components $c{h_{2k}}\left( {\rm E} \right)$ of the Chern character vanish. Hence, let us define the total Hodge potential of $X$ as an extension of the total descendent potential depending on the sequence $\not s = \left( {{s_1},{s_2},...} \right)$ of new variables and incorporating intersection indices with characteristic classes of the Hodge bundles