Hodge Theory and Gromov-Witten Invariants of Calabi-Yau 3-Folds

By George Shiber Sunday, August 14, 2016Thursday, August 15, 2019 In Grand Unification Physics, M Theory, Mathematical Physics, Mathematics, Physics, Super String Theory, Witten Theory Calabi-Yau 3-Folds, Gromov–Witten invariants

I discussed Gromov-Witten Invariants and Hodge integrals on numerous occasions. Here, I shall derive three propositions that play a critical role in flux compactification in M-theory. Hodge integrals analytically arise in Gromov-Witten theory: consider a compact algebraic homogeneous space $X = G/P$ , ${\bar M_{g,n}}$ a moduli stack of genus g, n-pointed, Deligne-Mumford stable curves, cotangent line bundle ${L_i} \to {\bar M_{g,n}}$ with

${\psi _i} = {c_1}\left( {{L_i}} \right) \in {H^ * }\left( {{{\bar M}_{g,n}}\mathbb{Q}} \right)$

and isomorphism:

${\bar M_{g,n}}\left( {X,0} \right) \cong {\bar M_{g,n}} \times X$

then Hodge integrals as a consequence of the Super-Virasoro constraints applied to $P$ arise naturally over stacks of stable maps ${\bar M_{g,n}}\left( {X,\beta } \right)$ for non-singular projective varieties $X$ :

$\int_{{{\left[ {{{\bar M}_{g,n}}\left( {X,\beta } \right)} \right]}^{vir}}} {\prod\limits_{i = 1}^n {\psi _i^{{a_i}}} } \cup e_i^ * \left( {{\gamma _i}} \right) \cup \prod\limits_{j = 1}^g {\lambda _j^{{b_j}}}$

$\begin{array}{l}{c_g}\int_{{{\bar M}_{g,1}}} {\psi _1^{2g - 1}} {\lambda _{g - 1}} = \left( {\sum\limits_{k = 1}^{2g - 1} {\frac{1}{k}} } \right){b_g} - \\\frac{1}{2}\sum\limits_{\begin{array}{*{20}{c}}{{g_1} + {g_2} = g}\\{{g_1},{g_2} > 2}\end{array}} {\frac{{\left( {2{g_1} - 1} \right)!\left( {2{g_2} - 1} \right)!}}{{\left( {2g - 1} \right)!}}} \,{b_{{g_1}}}{b_{{g_2}}}\end{array}$

and:

$\int_{{{\bar M}_{g,n}}} {\psi _1^{{k_1}}} ...\,\psi _n^{{k_n}}{\lambda _g} = \left( {\begin{array}{*{20}{c}}{2g + n - 3}\\{{k_1},...,{k_n}}\end{array}} \right){b_g}$

with:

${b_g} = \left\{ {\begin{array}{*{20}{c}}{1,\quad \quad g = 0}\\{\int_{{{\bar M}_{g,1}}} {\psi _1^{2g - 2}} {\lambda _g},\,\,\quad \;g > 0}\end{array}} \right.$

where the virtual class equality:

${\left[ {{{\bar M}_{g,n}}\left( {X,0} \right)} \right]^{vir}} = {c_{rg}}\left( {{{\bar {\rm E}}^ * }\tilde \otimes {T_X}} \right) \cap \left[ {{{\bar M}_{g,n}}\left( {X,0} \right)} \right]$

yields the Calabi-Yau 3-fold-Gromov-Witten invariant integral:

$\int_{{{\left[ {{{\bar M}_{g,0}}\left( {P,d} \right)} \right]}^{vir}}} {c_{top}^{CY}} \left( {{R^1}{\pi _ * }{\mu ^ * }{N^{\dagger ,3}}} \right)$

Thus, what is crucial is that the degree $0$ –Gromov-Witten invariants of $X$ involve only the classical cohomology ring ${H^ * }\left( {X,\mathbb{Q}} \right)$ and Hodge integrals over ${\bar M_{g,n}}$

where:

${\psi _i}/{/^{co - \tan }}{B_{und}}\left( {{{\bar M}_{g,n}}\left( {X,\beta } \right)} \right)$

and ${e_i}$ are the evaluation maps to $X$ corresponding to the cohomology-ring-markings and:

${\gamma _i} \in {H^ * }\left( {X,\mathbb{Q}} \right)$

An interpretation of the Grothendieck-Riemann-Roch theorem in Gromov-Witten theory via the orbifold Poincaré pairing:

${\left( {a,b} \right)_{orb}}: = \int_{I{X^\dagger }} {a \wedge {I^ * }} b;\quad a,b \in {H^ * }$

yields:

Proposition 1: The class of Hodge integrals over the moduli stacks of maps to $X$ can be uniquely reconstructed from the class of descendent integrals.

Now, consider the differential forms:

${\tilde c_r} \in {\Omega ^{2r}}\left( {{X_{reg}}} \right)$

defined via:

$\sum\limits_i {{\lambda ^i}} {\tilde c_i} = \det \left( {1 + \frac{i}{{2\pi }}\lambda \bar K} \right)$

then given that

${c_I}\left[ X \right] = \int_{{X_{reg}}} {\prod\nolimits_{j = 1}^k {{{\tilde c}_{ij}}} }$

holds, we can define

$1 + \sum\limits_{g \ge 1} {\sum\limits_{i = 0}^g {{t^{2g}}} } {k^i}\int_{{{\bar M}_{g,n}}} {\psi _1^{2g - 2 + i}} {\lambda _{g - i}} = F\left( {t,k} \right)$

and using the orbifold Poincaré pairing, we get

Proposition 2:

$F\left( {t,k} \right) = {\left( {\frac{{t/2}}{{\sin \left( {t/2} \right)}}} \right)^{k + 1}}$

hence, the integrals:

${b_g} = \left\{ {\begin{array}{*{20}{c}}{1,\quad \quad g = 0}\\{\int_{{{\bar M}_{g,1}}} {\psi _1^{2g - 2}} {\lambda _g},\,\,\quad \;g > 0}\end{array}} \right.$

and

allow us to apply proposition 2 to Gromov-Witten theory and derive the integral-formula in Calabi-Yau 3-folds:

$C\left( {g,d} \right) = \int_{\left[ {{{\bar M}_{g,0}}\left( {{P^1},d} \right)} \right]} {{{\tilde c}_{top}}} \left( {\pi {R^1}{\pi _ * }{\mu ^ * }{N^\dagger }} \right)$

which is the contribution to the genus g-Gromov-Witten invariant of a Calabi-Yau 3-fold of multiple covers of a fixed rational curve with normal bundle

Proposition 3:

for $g \ge 2$

$\int_{{{\bar M}_g}} {\lambda _{g - 1}^3} = \frac{{\left| {{B_{2g}}} \right|}}{{2g}}\frac{{\left| {{B_{2g - 2}}} \right|}}{{2g - 2}}\frac{1}{{\left( {2g - 2} \right)!}}$

holds and hence

the genus $g \ge 2$ , degree 0-Gromov-Witten invariant of a Calabi-Yau 3-fold $X$ is:

$\left\langle 1 \right\rangle _{g,0}^X = {\left( { - 1} \right)^g}\frac{\chi }{2}\int_{{{\bar M}_g}} {\lambda _{g - 1}^3}$

with $\chi$ the topological Euler characteristic of $X$ .

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Flux Compactifications of String Theory and Kähler/Calabi–Yau manifolds
Monday, August 22, 2016

[…] in relation to Kähler and Calabi–Yau manifolds. I already discussed the connection with Hodge theory and Gromov-Witten invariants of Calabi-Yau 3-Folds. The key thing to realize is that fluxes present in 10/11-D string theory naturally […]