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

eq1

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

eq2

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 provedthe 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

eq3

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

 

eq4

such that

 

eq5

 

It follows that

eq6

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

eq8

on \not {\rm H}, one gets

 

eq9

 

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 …