Quantum Cohomology Algebra, the Dubrovin Connection and Holomorphy

By George Shiber Monday, February 1, 2016Sunday, August 4, 2019 In Grand Unification Physics, M Theory, Mathematical Physics, Mathematics, Philosophy Of Physics, Physics, Super String Theory Dubrovin Connection, Gromov–Witten Theory

The art of doing mathematics consists in finding that special case which contains all the germs of generality ~ David Hilbert

Continuing my analysis of quantum cohomology here, where I derived, for a Witten-section

$\sum\nolimits_{j = 0}^s {{y_j}} {\left[ {{P_j}} \right]^ * }$

that it is covariant constant if the following expression is zero for all $k$

$\begin{array}{c}\left( {\nabla _{{{\not \partial }_i}}^*{{\sum\limits_{j = 0}^s {{y_i}\left[ {{P_j}} \right]} }^*}} \right)\left[ {{P_k}} \right] = \\\begin{array}{*{20}{c}}{\left( {\sum\limits_{j = 0}^s {{{\not \partial }_i}{{\left[ {{P_j}} \right]}^*} + \sum\limits_{j = 0}^s {{y_i}\nabla _{{{\not \partial }_i}}^*{{\left[ {{P_j}} \right]}^*}} } } \right)}\\{ \cdot \left[ {{P_k}} \right]}\end{array} = \\{{\not \partial }_i}{y_k} - \sum\limits_{j = 0}^s {{y_j}} \left( {{{\left[ {{P_j}} \right]}^*}\sum\limits_{l = 0}^s {\Omega _{lk}^i\left[ {{P_k}} \right]} } \right)\\ \cdot {{\not \partial }_i}{y_k} - \sum\limits_{j = 0}^s {{y_j}{\mkern 1mu} } \Omega _{lk}^i\end{array}$

in this post, I will derive the second-order holomorphic Dubrovin equivalence relation of the associated quantum algebra. Let us recall that the Dubrovin connection is a meromorphic flat connection $\nabla$ on

${\widetilde \pi ^ * }\overline T M \cong {H_X} \times \left( {M \times {{\widetilde A}^1}} \right)$

defined by

${\nabla _{\frac{{\not \partial }}{{\not \partial {t_i}}}}} = \frac{{\not \partial }}{{\not \partial {t_i}}} - \frac{1}{z}\left( {{\phi _i} * } \right)$

${\nabla _{z\frac{{\not \partial }}{{{{\not \partial }_z}}}}} = z\frac{{\not \partial }}{{{{\not \partial }_z}}} + \frac{1}{z}\left( {E * } \right) + \mu$

$0 \le i \le N$ and $z$ the coordinate on ${\widetilde A^1}$ , and by the Poincaré pairing property, the Dubrovin connection equips $M$ with a Frobenius manifold with extended structure connection, and thus the genus-zero Gromov–Witten potential $F\,_X^0$ converges to an analytic function, allowing a definition of a Fredholm-Calabi-Yau measure for the Kähler–Witten integral, due to the quantum product $*$

$\int_{{{\left[ {{X_{g,n.d}}} \right]}^{{\rm{vir}}}}} {\prod\limits_{k = 1}^{k = n} {{\rm{ev}}\,_k^ * } } \left( {{a_k}} \right) \cup \psi \,_k^{ik}$

and allows the Gromov–Witten invariants

$\left\langle {{a_1}\psi _1^{{i_1}},...,{a_n}\psi _n^{{i_n}}} \right\rangle \,_{g,n,d}^X$

to be metaplectic invariants on the homotopy group-manifold of $X$ again due to, and since it is, equipped with the quantum product $*$ . Also recalling that $M$ is a compact Kähler manifold of complex dimension $n$ , whose cohomology algebra, with complex coefficients, is of the form

${H^ * }M = \mathbb{C}\left( {{b_1},...,{b_r}} \right)/\left( {{R_{1,}}...,{R_u}} \right)$

with ${b_1},...,{b_r}$ are additive generators of ${H^ * }M$ and ${R_1},...,{R_u}$ and are polynomials relations in ${b_1},...,{b_r}$ . It follows from the Bernstein–Sato polynomial function, that the quantum cohomology algebra is of the form

$Q{H^ * }M = \mathbb{C}\left( {{b_1},...,{b_r}} \right)/K\left[ {\widetilde {{{\not R}_1}},...,\widetilde {{{\not R}_u}}} \right]$

where $K = \mathbb{C}\left( {{q_1},...,{q_r}} \right)$ and each $\widetilde {{{\not R}_i}}$ is a $q$ -deformation of ${R_i}$ and ${q_i}$ are

functions

${q_i}:t = \sum\nolimits_{j = 1}^r {{t_i}} {b_i}{ \to ^\dagger }{e^{{t_i}}}$

on ${H^ * }M$ . Now, quantum cohomology theory gives, in addition to $Q{H^ * }M$ , a quantum product operation on ${H^ * }M$ , and so, the Dubrovin connection above is expandable as

$\nabla \equiv d + \frac{1}{h}\omega$

on the bundle ${\mathbb{C}^r} \times {\mathbb{C}^{s + 1}} \to {\mathbb{C}^r}$ with $\omega$ is the complex ${\rm{End}}{\mathbb{C}^{s + r}}$ -valued $1$ -form on ${\mathbb{C}^r}$ defined by

${\omega _t}(x)(y) = x{ \circ _t}y$

with h is a non-zero complex parameter, so actually we have a family of connections.

Hence,

Theorem: for any $h$ the connection

$\nabla \equiv d + \frac{1}{h}\omega$

is flat, thus

$d\omega = \omega \wedge \omega = 0$

The proof is given here.

Now introduce the ring ${D^h}$ of differential operators generated by $h{\not \partial _1},...,h{\not \partial _r}$ with coefficients in $K\left[ h \right]$ , and let me define a quantization of the cohomology algebra ${A^{QC}}$ and $D$ -module ${M^h} = {D^h}/\left( {D_1^h,...,D_u^h} \right)$ such that ${M^h}$ is free over $K\left[ h \right]$ of rank $s + 1$ , and

$\mathop {\lim }\limits_{h \to 0} S\left( {D_i^h} \right) = \widetilde {{{\not R}_i}}$

where $S\left( {D_i^h} \right)$ is the result of replacing $h{\not \partial _1},...,h{\not \partial _r}$ by ${b_1},...,{b_r}$ in $D_i^h$ for $i = 1,...,u$ .

Now define the quantum algebraic connection form

${\Omega ^h} = \sum\nolimits_{i = 1}^r {\Omega _i^h} d{t_i}$

via

$\left\{ {\begin{array}{*{20}{c}}{\left[ {{{\not \partial }_i}{P_i}} \right] = \sum\nolimits_{k = 0}^s {{{\left( {\Omega _i^h} \right)}_{kj}}\left[ {{P_k}} \right]} }\\{\left[ {{b_i}{c_j}} \right]\sum\nolimits_{k = 0}^s {{{\left( {{\omega _i}} \right)}_{kj}}\left( {{c_k}} \right)} }\end{array}} \right.$

It follows then that $h\,{\Omega ^h}$ is polynomial in $h$ , so ${\Omega ^h}$ is hence of the form

${\Omega ^h} = \frac{1}{h}\omega + {\theta ^{\left( 0 \right)}} + h\,{\theta ^{\left( 0 \right)}} + ... + h{\theta ^{\left( p \right)}}$

where

$\omega = \sum\nolimits_{i = 1}^r {{\omega _i}} d{t_i}$

and ${\theta ^{\left( 0 \right)}},...,{\theta ^{\left( p \right)}}$ are matrix-valued $1$ -forms, and $p$ is a non-negative integer depending on the relations $\widetilde {{{\not R}_1}},...,\widetilde {{{\not R}_u}}$ .

Which brings me to the proposition of this post: since ${\Omega ^h}$ depends holomorphically on $q = \left( {{q_1},...,{q_r}} \right)$ , for $q$ in some open subset $V$ , then, for any point ${q_0}$ in $V$ , there is a neighbourhood ${U_0}$ of ${q_0}$ on which the connection