Frobenius Structures, the Total Descendent Potential and Gromov – Witten Theory

By George Shiber Saturday, May 14, 2016Saturday, August 17, 2019 In Cosmology, Grand Unification Physics, M Theory, Mathematical Physics, Mathematics, Philosophy Of Physics, Philosophy Of Science, Physics, Super String Theory Frobenius Structures, Gromov–Witten Theory

In this post, continuing from part one on Gromov-Witten invariants, I shall derive two deep properties, the second being central about the genus 1 Gromov-Witten potential, and along the way, discuss some propositions regarding the total ancestor potential. Let me introduce the total ancestor potential

where the genus g ancestor potential $\not \tilde F_t^g$ is defined by

with

${\bar \psi _i}: = {\pi ^ * }\left( {{\psi _i}} \right)$

referring to the pull-backs of the classes ${\psi _i}$ , i = 1, …, m, from ${\bar M_{g,m}}$ relative to the composition

$\pi :{X_{g,\,m + l,\,d}} \to {\bar M_{g,\,m + l,\,d}} \to {\bar M_{g,\,m}}$

Since the sum does not contain the terms with (g, m) = (0, 0),(0, 1),(0, 2) and (1, 0), one can treat ${\tilde {\rm A}_t}$ subject to the dilaton shift

$\not q(z) = \not t(z) - z \in {\tilde H_t}$

as an element in the Fock space depending on the parameter $t \in {\rm H}$ .

Let us implicitly define the operator ${S_t}$ on the Laurent 1/z-series completion of the space ${\rm H}$ defined by

with the 2-point gravitational descendent in genus 0 being

Now, it is one of the basic facts of quantum cohomology theory that

$S_t^ * \left( { - 1/z} \right){S_t}\left( {1/z} \right) = 1$

that is, given a metaplectic completion of $\left( {{{\tilde H}_t},\Omega } \right)$ , ${\hat S_t}$ defines a symplectic transformation depending on the parameter $t \in {\rm H}$ . Put

${\hat S_t}: = \exp {\left( {{\rm{In}}\,{S_t}} \right)^ \wedge }$

Here is a deep theorem with an excellent proof ‘here’

${\tilde D_{wk}} = {e^{{F^1}\left( t \right)}}{\hat S_t}^{ - 1}{\tilde {\rm A}_t}$

For the definition of ${\tilde D_{wk}}$ , see my last post. Getzler proved the 3g − 2-jet conjecture of Eguchi–Xiong and Dubrovin about genus g descendent potential

$\frac{{{{\not \partial }^m}}}{{\not \partial t_{{k_1} + 1}^{{\alpha _1}}...\not \partial t_{{k_m} + 1}^{{\alpha _m}}}}\left( {\tilde F_t^g} \right)\left| {_{t = 0}} \right.$

if k1 + … + km > 3g − 3.

Proposition 1

Any quantized symplectic operator of the form

$S\left( {1/z} \right) = \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\frown$}} \over 1} + {S_1}/z + {S_2}/{z^2} + ...$

acts on elements $G$ of the Fock space as

with ${\left[ {{S_{\bar q}}} \right]_ + }$ the power series truncation of $S\left( {{z^{ - 1}}} \right)\bar q\left( z \right)$ with the quadratic form

$W = \sum {\left( {{W_{kl{q_k},q}}} \right)}$

defined via

Proposition 2

The genus 0 descendent potential equals

which is a deep and famous reconstruction for genus 0 gravitational descendents due to Dubrovin and Dijkgraaf–Witten.

Let me charter into Frobenius structures. The operator-valued 1-form

$A\left( t \right): = z\left( {{d_t}{S_t}\left( {1/z} \right)} \right)S_t^{ - 1}\left( {1/z} \right)$

is independent of z and thus defines a linear pencil of connections

${\nabla _z}: = d - {z^{ - 1}}A\left( t \right) \wedge$

on the tangent bundle $T{\rm H}$ flat for all values of the parameter ${z^{ - 1}}$

hence

$\left\{ {\begin{array}{*{20}{c}}{A \wedge A = 0}\\{dA = 0}\end{array}} \right.$

and in coordinate form, $t = \sum {{t^\alpha }} {\phi _\alpha }$ the first condition is equivalent to commutativity

$\left\{ {\begin{array}{*{20}{c}}{{A_\alpha }{A_\beta } = {A_\beta }{A_\alpha }}\\{A = \sum {{A_\alpha }\left( t \right)d{t^\alpha }} }\end{array}} \right.$

The correspondence

${\not \partial _\alpha } \Rightarrow {A_\alpha }\left( t \right) = {i_{{{\not \partial }_\alpha }}}A$

defines commutative associative multiplications ${ \bullet _t}$ on the tangent spaces ${T_t}{\rm H}$ , and those are the quantum cup-product, and deeply, we have a coincidence of

$\left( {{\phi _\alpha }{ \bullet _t}{\phi _\beta },{\phi _\gamma }} \right)$

with the third directional derivatives

${\not \partial _\alpha }{\not \partial _\beta }{\not \partial _\gamma }\bar F_X^0\left( t \right)$

of the genus zero Gromov-Witten potential

$\bar F_X^0$

which explains why $dA = 0$ and $A_\alpha ^ * = {A_\alpha }$ yielding hence that the quantum cup-product is Frobenius: $\left( {a \bullet b,c} \right) = \left( {a,b \bullet c} \right)$ .

A Frobenius structure on a manifold H consists of:

(i) a flat pseudo-Riemannian metric (·, ·),

(ii) a function $F$ whose 3-rd covariant derivatives ${F_{abc}}$ are structure constants $\left( {a{ \bullet _t}b,c} \right)$ of a Frobenius algebra structure: that is, an associative commutative multiplication ${ \bullet _t}$ satisfying

$\left( {a{ \bullet _t}b,c} \right) = \left( {a,b{ \bullet _t}c} \right)$

on the tangent spaces ${T_t}{\rm H}$ which depends smoothly on t;

(iii) the vector field of unities 1 of the ${ \bullet _t}$ -product which has to be covariantly constant and preserve the multiplication and the metric.

Now, in Gromov–Witten theory, genus 0 Gromov-Witten invariants of a compact almost Kähler manifold $X$ define on

$H = {H^ * }\left( {X;\hat Q\left[ {\left\{ Q \right\}} \right]} \right)$

a formal structure of a calibrated conformal Frobenius manifold of conformal dimension $D = {\rm{di}}{{\rm{m}}_\mathbb{C}}X$ , and in the coordinate system

$t = \sum {{t^\alpha }} {\phi _\alpha }$

corresponding to a graded basis $\left\{ {{\phi _\alpha }} \right\}$ in ${H^ * }\left( {X,\hat Q} \right)$ , the Euler field takes on the form

$E = \sum {\left( {1 - {\rm{deg}}\,{\phi _\alpha }/2} \right)} \,{t_\alpha }{\not \partial _\alpha } + \rho$

where the constant part $\rho \in {H^ * }\left( X \right)$ is the 1-st Chern class of the tangent bundle $TX$ .

So, proposition 3: The equation ${\nabla _z}S = 0$ in a neighborhood of a semisimple point $u$ has a fundamental solution in the form

${\Psi _u}{R_u}\left( z \right){e^{U/z}}$

with

${R_u}\left( z \right) = \not 1 + {R_1}\,z + {R_2}\,{z^2} + ...$

a formal matrix power series satisfying: $R_u^ * \left( { - \,z} \right){R_u}\left( z \right) = \not 1$

and the series ${R_u}\left( z \right)$ satisfying the content of proposition 3 is unique up to right multiplication by diagonal matrices $\exp \left( {{a_1}z + {a_2}{z^3} + {a_3}{z^5} + ...} \right)$ with ${a_k} = {\rm{diag}}\left( {a_k^1,...,a_k^N} \right)$ constants. And, in the case of conformal Frobenius structures the series is uniquely determined by the homogeneity condition

$\left( {z{{\not \partial }_z} + \sum {{u^i}{{\not \partial }_i}} } \right){R_u}\left( z \right) = 0$

Now, let ${\tilde H^L}$ be the space of Laurent polynomials in z with coefficients in the tangent space ${T_u}H$ to the Frobenius manifold at a semisimple point $u$ . For $\bar q\left( z \right) \in {\bar H^L}_ +$ , let

$\Psi _u^{ - 1}\bar q\left( z \right) \equiv \left( {{{\bar q}^1}\left( z \right),...,{{\bar q}^N}\left( z \right)} \right)$

and let us introduce the direct product

${\Gamma ^ \otimes } = \tau \left( {\hbar ;{{\bar q}^1}} \right)...\tau \left( {\hbar ;{{\bar q}^N}} \right)$

of N copies of the Witten-Kontsevich tau-function as an element of the Fock space of functions on $\bar H_ + ^L$ . The series ${R_u}\left( z \right)$ defines a symplectic transformation on ${\bar H^L}$ and set

${\hat R_u} = \exp {\left( {{\rm{In}}\,{R_u}} \right)^ \wedge }$

and let

${\hat \Psi _u} \equiv G\left( {{\Psi ^{ - 1}}\bar q} \right) \Rightarrow G\left( {\bar q} \right)$

identify the Fock space with its coordinate version, and lastly,

$\sum\nolimits_{i = 1}^N {R_1^{ii}} d{u^i}$

with $R_1^{ii}$ the diagonal entries of the matrix ${R_1}$ in the series

${R_u}\left( z \right) = \not 1 + {R_1}/z + ...$

which is the 1-form on the Frobenius manifold. Define also the function

$C\left( u \right) = \frac{1}{2}\int_L^u {\sum {R_1^{ii}} } d{u^i}$

of $u$ defined up to an additive constant.

Definition: let the total descendent potential of a semisimple Frobenius manifold be defined by the formula

and introduce the total ancestor potential of a semisimple Frobenius manifold

Property 1 The total descendent potential ${\tilde D_{wk}}$ of a semisimple Frobenius manifold defined does not depend on the choice of a semisimple point $u$ since both

$\left\{ {\begin{array}{*{20}{c}}{{S_u}\left( {1/z} \right)}\\{{\Psi _u}{R_u}\left( z \right){e^{\left( {U/z} \right)}}}\end{array}} \right.$

satisfy the same equation

${\nabla _z}S = 0$

with coefficients rational in $z$ . Hence, derivatives of ${\hat D_{wk}}$ in the directions of the parameter $u$ vanish as if

$\left\{ {\begin{array}{*{20}{c}}{S_u^{ - 1}}\\{{\Psi _u}{R_u}{e^{\left( {U/z} \right)}}}\end{array}} \right.$

were inverse to each other.

Consequentially, the main property of this post:

Property 2

The genus 1 Gromov-Witten potential ${F^1}\left( t \right)$ of a semisimple Frobenius manifold is given by the formula

with

$\Delta _t^{ - 1}\left( u \right) = \left( {{{\not \partial }_{{u^i}}},{{\not \partial }_{{u^j}}}} \right)$

being the inner squares of the canonical idempotents ${\not \partial _{{u^i}}}$ of the semisimple Frobenius multiplication ${ \bullet _u}$ and the genus 1 descendent potential equals