The Orbifold Riemann–Roch Theorem and Todd-Chern Classes

By George Shiber Friday, January 22, 2016Friday, October 26, 2018 In Grand Unification Physics, M Theory, Mathematical Physics, Mathematics, Philosophy Of Physics, Physics, Super String Theory, Supersymmetry Gromov–Witten, orbifold

All the measurements in the world do not balance one theorem by which the science of eternal truths is actually advanced ~ Carl Friedrich Gauss

In this post, I will derive the Orbifold Riemann–Roch Theorem, which I will eventually show is crucial to the meta-stability and categoricity of M-theory. In my last post I showed that the cohomology pairing and the Galois action on $\widehat S\left( \chi \right)$ are uniquely determined by the cohomology framing

$\begin{array}{c}\left( {{{\widehat Z}_{COH}}\left( \alpha \right),{{\widehat Z}_{COH}}\left( \beta \right)} \right) = \\{\left( {{e^{\pi \widehat i\rho }}\alpha ,{e^{\pi \widehat i\mu }}\beta } \right)_{orb}}\end{array}$

$\begin{array}{c}{G^{\widehat S}}\left( \xi \right)\left( {{{\widehat Z}_{COH}}\left( \alpha \right)} \right) = \\{\widehat Z_{COH}}\left( {\left( {\underbrace \oplus _{v \in T}{e^{ - 2\pi \widehat i{\xi _0}}}{e^{2\pi \widehat i{f_v}\left( \xi \right)}}} \right)\alpha } \right)\end{array}$

and hence the Galois actions on $\widehat S\left( \chi \right)$ can be viewed as the monodromy transformations of the flat bundle $F/{H^2}\left( {\chi ,\mathbb{Z}} \right) \to \left( {U/{H^2}\left( {\chi ,\mathbb{Z}} \right)} \right) \times {C^ * }$ in the $\tau$ -direction. The monodromy with respect to $z$ is hence given by

$\begin{array}{c}{\left[ {{{\widehat Z}_{COH}}\left( \alpha \right)} \right]_{{ \bullet _\iota }{ \to ^{\dagger {e^{2\pi \widehat i}}z}}}} = {\widehat Z_{COH}} \cdot \\\left( {{e^{ - 2\pi \left| \mu \right.}}{e^{2\pi \widehat i\rho }}\alpha } \right)\end{array}$

which coincides with the Galois action

${\left( { - 1} \right)^n}{G^{\widehat S}}\left( {\left[ {{K_\chi }} \right]} \right)$

with

${ \bullet _\iota }$

the quantum product I introduced here,

as well as the Serre functor of the Lefschetz category $D\chi$ with $\left[ {{K_\chi }} \right]$ the class of the canonical line bundle and $\chi$ is Calabi–Yau, with with $G\left( \xi \right)$ defined by

$\begin{array}{c}G\left( \xi \right)\left( {{\tau _0} \oplus \underbrace {\widehat \otimes }_{\nu \in T'}{\tau _0}} \right) = \\\left( {{\tau _0} - 2\pi \widehat i{\xi _0}} \right) \oplus \underbrace {\widehat \otimes }_{\nu \in T'}{e^{2\pi \widehat i{f_\nu }\left( \xi \right)}}{\tau _0}\end{array}$

$\begin{array}{c}dG\left( \xi \right)\left( {{\tau _0} \oplus \underbrace {\widehat \otimes }_{\nu \in T'}{\tau _0}} \right) = \\{\tau _0} \oplus \underbrace {\widehat \otimes }_{\nu \in T'}{e^{2\pi \widehat i{f_\nu }\left( \xi \right)}}{\tau _0}\end{array}$

where ${\tau _\nu } \in {H^ * }\left( {{\chi _\nu }} \right)$ and ${\xi _0}$ is the image of $\xi$ in the $\chi$ –quantum D-module: and this is the Galois action of ${H^2}\left( {\chi ,\mathbb{Z}} \right)$ on $\chi$ –quantum D-module. Let me proceed.

By an integral structure in quantum cohomology I mean a $\mathbb{Z}$ -local system ${F_\mathbb{Z}} \to U \times {\mathbb{C}^ * }$ underlying the flat bundle ${\left( {F,\nabla } \right)_{U \times {\mathbb{C}^ * }}}$ induced by an integral lattice $\widehat S{\left( \chi \right)_\mathbb{Z}}$ in the space $\widehat S\left( \chi \right)$ of multi-valued flat sections of QDM- $\left( \chi \right)$ , where QDM stands for quantum D-module. Let me work with the most interesting integral lattices in $\widehat S\left( \chi \right)$ and the $\widetilde \Gamma$ -integral structure which have deep and essential properties. Now let $K\left( \chi \right)$ denote the Grothendieck group of topological orbifold vector bundles on $\chi$ : for an orbifold vector bundle $\widetilde V$ on the inertia stack ${I_\chi }$ , we have an eigenbundle decomposition of

$\widetilde V\left| {_{{\chi _v}}} \right.$

$\widetilde V\left| {_{{\chi _v}}} \right.\underbrace \oplus _{0\, \le f < 1}{\widetilde V_{v,f}}$

with respect to the action of the stabilizer of ${\chi _v}$ acting on ${\widetilde V_{v,f}}$ via

$\exp \left( {2\pi \hat if} \right) \in \mathbb{C}$

Now, let $pr:{I_\chi } \to \chi$ be the Picard-projection. Note, the Chern character $\widetilde {ch}:K\left( \chi \right) \to {H^ * }\left( {{I_\chi }} \right)$ is defined for an orbifold vector bundle $V$ on $\chi$ by

$\widetilde {ch}: = \underbrace \oplus _{v \in T}\sum\limits_{0\, < f < 1} {{e^{2\pi \widehat if}}} \widetilde {ch}\left( {{{\left( {p{r^ * }V} \right)}_{v,f}}} \right)$

with $ch$ being the ordinary Chern character. For an orbifold vector bundle $V$ on $\chi$ , let ${\delta _{v,f,i}}i = 1,...,{I_{v,f}}$ , be the Chern roots of ${\left( {p{r^ * }V} \right)_{v,f}}$ . The Todd class

$\widetilde {Td}:K\left( \chi \right) \to {H^ * }\left( {{I_\chi }} \right)$

is defined by

$\begin{array}{c}\widetilde {Td}\left( V \right) = \underbrace \oplus _{v \in T}\prod\limits_{0 < f,1\, \le i \le {I_{v,f}}} {\frac{1}{{1 - {e^{ - 2\pi \widehat if}}{e^{ - {\delta _{v,f,i}}}}}}} \\ \cdot \prod\limits_{f = 0,1 \le i \le {I_{v,f}}} {\frac{{{\delta _{v,o,i}}}}{{1 - {e^{ - {\delta _{v,0,i}}}}}}} \end{array}$

Now the Chern and Todd characteristic classes appear in the Orbifold Riemann–Roch theorem:

– For a holomorphic orbifold vector bundle $V$ on $\chi$ , the Euler characteristic $\chi \left( V \right)$ is given by

$\begin{array}{c}\chi \left( V \right): = \sum\limits_{i = 0}^{\dim \chi } {{{\left( { - 1} \right)}^i}} \dim {H^i}\left( {\chi ,V} \right)\\ = \int\limits_{{I_\chi }} {\widetilde {ch}\left( V \right)} \cup \widetilde {Td}\left( {T\chi } \right)\end{array}$

Now define a multiplicative characteristic class $\widetilde \Gamma :K\left( \chi \right) \to {H^ * }\left( {{I_\chi }} \right)$ by

$\widetilde \Gamma \left( V \right): = \underbrace \oplus _{v \in T}\prod\limits_{0\, \le f < 1} {\prod\limits_{i = 1}^{{I_v},f} \Gamma } \left( {1 - f + {\delta _{v,f,i}}} \right) \in {H^ * }\left( {{I_\chi }} \right)$

with ${\delta _{v,f,i}}$ is as above and the Gamma function on the right-hand side must be expanded in series at $1 - f > 0$ .

Note that the map $\widetilde {ch}:K\left( \chi \right) \to {H^ * }\left( {{I_\chi }} \right)$ transforms into an isomorphism after being tensored with $\mathbb{C}$ . Define the $K$ -group framing ${{\rm{Z}}_K}:K\left( \chi \right) \to \widetilde S\left( \chi \right)$ of the space $\widetilde S\left( \chi \right)$ of multivalued flat sections of the quantum $D$ -module by the formula

$\begin{array}{c}{{\rm Z}_K}\left( K \right): = {{\rm Z}_{COH}}\left( {\Psi \left( V \right)} \right) = \\L\left( {\tau ,z} \right){z^{ - \mu }}{z^\rho }\Psi \left( V \right)\end{array}$

with

$\begin{array}{c}\Psi \left( V \right): = {\left( {2\pi } \right)^{ - n/2}}\widetilde \Gamma \left( {T\chi } \right) \cup \\{\left( {2\pi \widehat i} \right)^{\deg /2}}in{v^ * }\left( {\widetilde {ch}\left( V \right)} \right)\end{array}$

and $\deg :{H^ * }\left( {{I_\chi }} \right) \to {H^ * }\left( {{I_\chi }} \right)$ is a Lefschetz-grading operator on ${H^ * }\left( {{I_\chi }} \right)$ defined by $\deg = 2k$ on ${H^{2k}}\left( {{I_\chi }} \right)$ and $\cup$ is the cup product in ${H^ * }\left( {{I_\chi }} \right)$ , which is the image $\widetilde S{\left( \chi \right)_\mathbb{Z}}: = {\rm Z}\left( {K\left( \chi \right)} \right)$ of the K-group framing the $\Gamma$ -integral structure.

– Proof of Orbifold Riemann–Roch theorem: The Γ -integral structure $\widetilde S{\left( \chi \right)_\mathbb{Z}}$ satisfies the following three properties

1) $\widetilde S{\left( \chi \right)_\mathbb{Z}}$ is a $\mathbb{Z}$ -lattice in $\widetilde S\left( \chi \right)$

$\widetilde S\left( \chi \right) = \widetilde S{\left( \chi \right)_\mathbb{Z}}{ \otimes _\mathbb{Z}}\mathbb{C}$

2) The Galois action ${G^{\widetilde S}}\left( \xi \right)$ on $\widetilde S\left( \chi \right)$ preserves the lattice $\widetilde S{\left( \chi \right)_\mathbb{Z}}$ and corresponds to the tensor by the line bundle $L_\xi ^ \vee$ in $K{\left( \chi \right)_Z}$

${{\rm Z}_K}\left( {L_\xi ^ \vee \otimes V} \right) = {G^{\widetilde S}}\left( \xi \right)\left( {{{\rm Z}_K}\left( V \right)} \right)$

where ${L_\xi }$ is the line bundle corresponding to $\xi \in {H^2}\left( {\chi ,\mathbb{Z}} \right)$

3) The pairing ${\left( {.,.} \right)_{\widetilde S}}$ on $\widetilde S\left( \chi \right)$ corresponds to the Mukai pairing on $K\left( \chi \right)$ defined by

${\left( {{V_1},{V_2}} \right)_{K\left( \chi \right)}}: = \chi \left( {V_2^ \vee \otimes {V_1}} \right)$

${\left( {{{\rm Z}_K}\left( {{V_1}} \right),{{\rm Z}_K}\left( {{V_2}} \right)} \right)_{\widetilde S}} = {\left( {{V_1},{V_2}} \right)_{K\left( \chi \right)}}$

Now because ${\widetilde \Gamma _\chi } \cup$ and ${\left( {2\pi \widehat i} \right)^{\deg /2}}$ are invertible operators over $\mathbb{C}$ , by the following two relations

we can deduce

$\begin{array}{c} \cup \left( {{e^{\pi \widehat i\left( {{ \bullet _\iota } - \frac{n}{2} + \frac{{\deg }}{2}} \right)}}} \right)\widetilde \Gamma {\left( {T\chi } \right)_v} \cdot \\\widetilde {ch}{\left( {{V_2}} \right)_{inv\left( v \right)}}\end{array}$

which, by

${\left( { - 1} \right)^n}{G^{\widehat S}}\left( {\left[ {{K_\chi }} \right]} \right)$

is equal to

$\frac{1}{{{{\left( {2\pi } \right)}^n}}}{\sum\limits_{v \in \,T} {\int\limits_{{\chi _v}} {\left( {{e^{\pi \widehat i\rho }}\widetilde \Gamma {{\left( {T\chi } \right)}_{inv\left( v \right)}}{{\left( {2\pi \widehat i} \right)}^{\frac{{\deg }}{2}}}\widetilde {ch}{{\left( {{V_1}} \right)}_v}} \right)} } _v}$

$=$

$\begin{array}{c}\left( {{{\rm Z}_K}\left( {{V_1}} \right),{{\rm Z}_K}\left( {{V_2}} \right)} \right) = \\{\left( {{e^{\pi \widehat i\rho }}\Psi \left( {{V_1}} \right),{e^{\pi \widehat i\mu }}\Psi \left( {{V_2}} \right)} \right)_{ORB}}\end{array}$

and to

$\begin{array}{c}\frac{1}{{{{\left( {2\pi } \right)}^n}}}{\sum\limits_{v \in \,T} {\left( {2\pi \widehat i} \right)} ^{\dim {\chi _v}}}\int\limits_{{\chi _v}} {\prod\limits_{f,i} \Gamma } \cdot \\\left( {1 - \overline f + \frac{{{\delta _{v,f,i}}}}{{2\pi \widehat i}}} \right)\Gamma \cdot \\\left( {1 - f - \frac{{{\delta _{v,f,i}}}}{{2\pi \widehat i}}} \right){e^{\frac{\rho }{2}}}\widetilde {ch}{\left( {{V_1}} \right)_v} \cdot \\{e^{\pi \widehat i\left( {{ \bullet _\iota } - \frac{n}{2} + \frac{{\deg }}{2}} \right)}}\widetilde {ch}{\left( {{V_2}} \right)_{inv\left( v \right)}}\end{array}$

and because the lattice $\widetilde S{\left( \chi \right)_\mathbb{Z}} \subset \widetilde S\left( \chi \right)$ defines a $\mathbb{Z}$ -local system ${F_\mathbb{Z}} \to \cup \times {\mathbb{C}^ * }$ underlying the flat vector bundle $\left( {F\left| {_{ \cup \, \times {\mathbb{C}^ * }},\nabla } \right.} \right)$ and given that $\widetilde S{\left( \chi \right)_\mathbb{Z}}$ is invariant under the Galois action, the local system ${F_\mathbb{Z}} \to \cup \times {\mathbb{C}^ * }$ descends to a local system over $\left( { \cup /{H^2}\left( {\chi ,\mathbb{Z}} \right)} \right) \times {\mathbb{C}^ * }$ , hence it follows that the proof of the orbifold Riemann–Roch Theorem is complete.