Fano Manifolds, U-Duality, Quantum Cohomology and M-Theory

By George Shiber Friday, February 5, 2016Friday, October 26, 2018 In Grand Unification Physics, M Theory, Mathematical Physics, Mathematics, Philosophy Of Physics, Physics, Super String Theory Fano Manifolds, Gromov–Witten

If you want to appreciate why Edward Witten is considered the greatest mathematical mind in physics since Newton, scroll to the end. In my last post, I defined the quantum algebraic connection form

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

via

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

and showed 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 $\left( {{{\widetilde {\not R}}_1},...,{{\widetilde {\not R}}_u}} \right)$ .

and deduced that

$\nabla = d + {\Omega ^h}$

is gauge equivalent to a connection

$\widehat \nabla = d\,{\widehat \Omega ^h}$

with

${\widehat \Omega ^h} = \frac{1}{h}\widehat \omega$

$\widehat \omega = {Q_0}\omega Q_0^{ - 1}$

for the holomorphic Dubrovin equivalence relation map

${U_0}:{U_0} \to G{L_{s + 1}}\mathbb{C}$

with $L$ the Laurent expansion. The proof is given here.

In this post, I will work in the context of Fano manifolds to thread all my last few posts on quantum cohomology together in the context of U-duality in M-theory and hopefully Edward Witten‘s historic genius would be reflected in the context of M-theoretic compactification as derived from quantum-cohomological arguments. A Fano manifold is a Kähler manifold $M$ whose Kähler $2$ –form represents the first Chern class ${c_1}M$ of the manifold and has a nicely-behaved quantum cohomology. Start with a deformation

${\widetilde A_{QC}} = K\left[ {{b_1},...,{b_r}} \right]/\left( {{{\widetilde {\not R}}_i},...,{{\widetilde {\not R}}_u}} \right)$

of the cohomology algebra

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

Now with flag manifolds and Fano toric manifolds, the connection form ${\Omega ^h}$ satisfies $\left| {{q_i}} \right| \ge 2$ , with ${q_i}$ defined via

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

Proposition: Assume that $\Omega _1^h,...,\Omega _r^h$ are polynomial in ${q_1},...,{q_r}$ with $\left| {{q_1}} \right|,...,\left| {{q_r}} \right| \ge 4$ . Then

${L_ + } = {Q_0}\left( {I + h{Q_1} + {h^2}{Q_2} + ...} \right)$

satisfies

$\left\{ {\begin{array}{*{20}{c}}{{Q_0} = \exp X}\\{{X_{\alpha ,\beta }} = 0}\\{\alpha \ge \beta - 1}\end{array}} \right.$

$\left\{ {\begin{array}{*{20}{c}}{i \ge 1}\\{{{\left( {{Q_i}} \right)}_{\alpha ,\beta }} = 0}\\{\alpha \ge \beta - i - 1}\end{array}} \right.$

Hence, ${Q_i} = 0$ , that is, ${L_ + }$ is a polynomial in $h$

Proof: given that

$\begin{array}{c}h\,{\Omega ^h} = \omega + h{\theta ^{(0)}} + {h^2}{\theta ^{(1)}} + \\{h^{p + 1}}{\theta ^{(p)}}\end{array}$

holds, it follows from the homogeneity and polynomiality properties that $\theta _i^{(j)}$ satisfies

$\left\{ {\begin{array}{*{20}{c}}{{{\left( {\theta _i^{(j)}} \right)}_{\alpha ,\beta }}}\\{\alpha \ge \beta - j - 1}\end{array}} \right.$

Therefore, ${\Omega ^h}$ takes values in the Lie algebra consisting of loops of the form

$\sum\nolimits_{i \in \mathbb{Z}} {{h^i}} {\rm{ }}{\widetilde A_{QC}}$

whose coefficients satisfy

$\left\{ {\begin{array}{*{20}{c}}{{{\left( {{{\widetilde A}_{QC}}} \right)}_{\alpha ,\beta }} = 0}\\{\alpha \ge \beta - i - 1}\end{array}} \right.$

when $i \ge 0$ and

$\left\{ {\begin{array}{*{20}{c}}{{{\left( {{A_{QC}}} \right)}_{\alpha ,\beta }} = 0}\\{\alpha \ge \beta - i + 1}\end{array}} \right.$

when $i < 0$ . Hence $L$ and ${L_ - }$ , ${L_ + }$ take values in the corresponding loop group. So

${\left( {{L_ + }} \right)^{ - 1}}d{L_ + } = \sum\nolimits_{i \ge 0} {{h^i}} {\widetilde A_{QC}}^{\exp ({q_i})}$

holds, where ${\left( {{{\widetilde A}_{QC}}} \right)_{\alpha ,\beta }} = 0$ for $\alpha \ge \beta - i - 1$ , from which the stated properties of ${L_ + }$ follow, yielding Fano-Laurent identity

$\frac{1}{h}{Q_0}\omega Q_0^{ - 1}{L_ + } = {L_ + }{\Omega ^h} - d{L_ + }$

which I will now link to U-Duality via the Kähler-Witten integral of $M \equiv X$

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

with ${a_{1,...,}}{a_n} \in {H_X}$ and ${\rm{e}}{{\rm{v}}_k}:{X_{g,n,d}} \to X$ being the evaluation map at the $k$ -th marked point and ${\psi _1},...,{\psi _n} \in {H^2}\left( {{X_{g,n,d}};\mathbb{Q}} \right)$ being the universal cotangent line classes. Now we can derive

$\left\langle {{a_1}\psi _1^{{i_1}},...,{a_n}\psi _n^{{i_n}}} \right\rangle \,_{g,n,d}^X = \int_{{{\left[ {{X_{g,n.d}}} \right]}^{{\rm{vir}}}}} {\prod\limits_{k = 1}^{k = n} {{\rm{ev}}{\mkern 1mu} _k^*} } \left( {{a_k}} \right) \cup \psi {\mkern 1mu} _k^{ik}$

Realize first that M-theory includes not only strings but also D-branes in order to support compactification in braneworld-cosmology as well as algebraic-geometric D-orbifoidal, and topological reasons. What U-duality is in a deep sense, is a symmetry between strings and branes. Hence in the limit of small string coupling $\lambda$ the second quantized string Hilbert space takes the form

${{\rm H}^{Hil}}{S^ * }\left( {{{\rm H}^{Hil}}_{string}} \right) \otimes {S^ * }\left( {{{\rm H}^{Hil}}_{brane}} \right)$

so U-duality allows, via

${e^{\int_{{{\left[ {{X_{g,n.d}}} \right]}^{{\rm{vir}}}}} {{e^{{{\left( {\psi \left( {h\,{\Omega ^h}} \right)} \right)}^4}}}} }}\smallint \left[ {{X_{g,n.d}}} \right]{\rm{ e}}{{\rm{v}}_k}{e^{h\left( {{\Omega ^h}} \right)}}{\left( {\psi \,\left( \omega \right)} \right)^4}$

a compactification of M-theory on a four-torus ${\Gamma ^4} = {\mathbb{R}^4}/L$ , hence causally connecting with Einsteinian space-time due to charge lattice

${\Upsilon ^{4,4}} \oplus {K^0}\left( {{\Gamma ^4}} \right)$

having rank16, and so forms an irreducible spinor representation under the U-duality group $SO\left( {4,4,\mathbb{Z}} \right)$ and the corresponding Fock space is given by

${\widetilde {\not F}_q}\mathop \otimes \limits_{n = 0}^\infty {S_{{q^n}}}\left( {{\mathbb{R}^8}} \right) \otimes \wedge \mathop \oplus \limits_{N \ge 0} {q^N}\widetilde {\not F}\left( N \right)$

So the Hilbert space of strings with momenta $p \in {\Gamma ^{4,4}}$ has the form

${{\rm H}^{Hil}}_{string}\left( p \right) = \widetilde {\not F}\left( {\frac{1}{2}{p^2}} \right)$

with the cohomology of moduli spaces being

$\mathop \oplus \limits_{N \ge 0} {q^N}{{\rm{H}}^{Hil}}^{,\dagger }\left( {{{\widetilde {\rm{H}}}^{Hil}}_N\left( {{\Gamma ^4}} \right)} \right) = {\widetilde {\not F}_q}$

noting the number 4, and by

${e^{\int_{\left[ {{X_{g,n.d}}} \right]} {{\rm{vir}}} }}h{\widetilde A_{QC}}^{{e^{\left( {h{{\widetilde \Omega }^h}} \right)}}}\mathop \otimes \limits_{n = 0}^\infty {e^{\int_{\left[ {{X_{g,n.d}}} \right]} {{\rm{vir}}} }}{h^{{e^{{{\left( {\psi \left( \omega \right)} \right)}^4}}}}}$

we get the 4-D-Witten compactificational projection of M-theory onto Einsteinian space-time as revealed by

$\begin{array}{c}{\widetilde {\rm H}^{Hil}}_{Brane}{\left( \mu \right)^4} = \widetilde {\not F}{\left( {{\mu ^2}/2} \right)^4} \cong \\{\widetilde {\rm H}^{Hil}}_{String}{\left( p \right)^4}\end{array}$