The Witten Super-Dubrovin 'Compactification Chern-Simons' Formula

“But in my opinion, all things in nature occur mathematically” ~ René Descartes

In my last two posts on the Witten Equation, I showed that in the context of Picard-Lefschetz Theory, it entails the non-forking, super-stability and hyper-categoricity of M-theory, making it, up to isomorphism, the only unification theory possible between Einstein's ToGR and Quantum Field Theory, as well as derived the Dubrovin connection and related it to Kähler-Witten integral and showed that it is a necessary condition for the action-principle of any such unified 'Theory-of-Everything'. Let's delve deeper. Keep in mind the Kähler-Witten integral of X

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

with X the Calabi-Yau smooth projective variety corresponding to the Witten-potential. This is key because the \eta -invariants of the differential operators in the Dubrovin meromorphic flat connection \nabla on

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

{\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

are metaplectically connected to the secondary invariants Chern-Simons type, which is the Atiyah-Patodi formula, that the Witten equation entails are \eta -invariants of such operators for spaces that fibre over {S_X}^1. It is key to homeomorphically derive, from the Dubrovin connection, the corresponding Witten-fibrated Atiyah-Patodi-Singer super-Dubrovin connection

i{\nabla _{\not \partial /\not \partial u}}

that acts on sections of an infinite-dimensional Hermitian vector bundle over {S_X}^1, with u arclength on {S_X}^1 because, as we shall see in future posts, it is essential for the principal action of M-theoretic braneworld cosmology.

First, note we have

\begin{array}{c}\eta \left( {i{{\widetilde \nabla }_{\not \partial /\not \partial u}}} \right) \equiv 2{\widehat c_1}\left( \nabla \right)\left[ {{S_X}^1} \right]\\{\rm{mod}}\,\mathbb{Z}\end{array}

with

{\widehat c_1}\left( \nabla \right)\left[ {{S_X}^1} \right]

the Kähler-invariant associated to the first Chern class and the connection \nabla and is the phase \theta where {e^{2\pi i\theta }} is the determinant of the holonomy of \nabla . Let {N^{4k - 1}} be a four-fold Calabi-Yau manifold, and E a Hermitian vector bundle with Hermitian connection over {N^{4k - 1}} and

A:{\Lambda ^{{\rm{ev}}}}\left( N \right) \otimes E \to {\Lambda ^{{\rm{ev}}}}\left( N \right) \otimes E

given by

A = \left( {d * + {{\left( { - 1} \right)}^p} * d} \right)

on {\Lambda ^{2p}}\left( N \right). Now, since {N^{4k - 1}} = \not \partial {M^{4k}} and E super-extends over {M^{4k}}, letting {P_L}\left( R \right) and {P_{{\rm{ch}}}}\left( \Omega \right) refer to the characteristic forms corresponding to the L-class of {M^{4k}} and Chern character of E, we can derive

\begin{array}{c}\eta \left( A \right) \equiv \int_{{M^{4k}}} {{P_L}} \left( R \right)\Lambda \,{P_{{\rm{ch}}}}\left( \Omega \right)\\{\rm{mod}}\,\mathbb{Z}\end{array}

Let us get to the Witten formula now. Letting

{Y^{4k - 2}} \to {N^{4k - 1}}{ \to _\pi }{S_X}^1

be a Calabi-Yau submersion, and with \widetilde d\widetilde *   refer to the exterior differentiation, with the Hodge \widetilde * -operator, let

\beta = \left\{ {\begin{array}{*{20}{c}}{\left( { - 1} \right)p\widetilde {\, * }}\\{\widetilde * }\end{array}} \right.\quad \left\{ {\begin{array}{*{20}{c}}{{\Lambda ^{2p}}\left( {{Y^{4k - 2}}} \right)}\\{{\Lambda ^{2p - 1}}\left( {{Y^{4k - 2}}} \right)}\end{array}} \right.

and

{{\rm A}^\dagger } = \left\{ {\begin{array}{*{20}{c}}{{{\left( { - 1} \right)}^{p + 1}}\left( {\widetilde d\beta + \beta \widetilde d} \right)}\\{\widetilde d\beta - \beta \widetilde d}\end{array}} \right.

\left\{ {\begin{array}{*{20}{c}}{{\Lambda ^{2p}}\left( {{Y^{4k - 2}}} \right)}\\{{\Lambda ^{2p - 1}}\left( {{Y^{4k - 2}}} \right)}\end{array}} \right.

Then clearly {{\rm A}^\dagger }  is self-adjoint, elliptic, and {\beta ^2} = - 1 and {{\rm A}^\dagger }\beta = - \beta {{\rm A}^\dagger } and now write

\left( {\begin{array}{*{20}{c}}\phi \\\omega \end{array}} \right) = \phi + du\Lambda \omega \in {\Lambda ^{2p}}\left( {{X^{4k - 1}}} \right)

where u is arclength on {S_X}^1. Then we have

A = \left( {\begin{array}{*{20}{c}}{\beta {\mkern 1mu} {{\not \partial }_u}}&{{{\rm{A}}^\dagger }}\\{{{\rm{A}}^\dagger }}&{{{\not \partial }_u}\beta }\end{array}} \right)

hence Witten's formula for {\lim _{\delta \to }}\eta \left( {{A_\delta }} \right) ) is locally computable on {S_X}^1. Put

{\dot {\rm A}^\dagger } = \not \partial {{\rm A}^\dagger }/{\not \partial _u}

and let

{e^{ - {{\rm A}^{\dagger 2\varepsilon }}}}

denote the heat kernel of {{\rm A}^{\dagger 2}}. Then for the operator {{\rm A}^\dagger }_\delta and {\wp _\delta } we get the of determinant line bundle  global anomaly relation

\frac{1}{\pi }\int_{{S_X}^1} {\mathop {\lim }\limits_{\varepsilon \to 0} } \,{\rm{tr}}\left( {\frac{\beta }{2}{{\rm A}^{\dagger - 1}}{{\dot {\rm A}}^\dagger }{e^{ - {{\rm A}^{\dagger 2\varepsilon }}}}} \right)du \equiv \mathop {\lim }\limits_{\delta \to 0} \eta \left( {{A_\delta }} \right)

Now, with \alpha \equiv {\widetilde * ^{ - 1}}\not \partial \,\widetilde * , we have \alpha \beta - \beta \alpha and the super-Dubrovin relation

{\dot \nabla ^\dagger }_{\not \partial /{{\not \partial }_u}} = {\not \partial _u} + \frac{1}{2}\alpha

and {\dot \nabla ^\dagger }\beta = 0 holding. Therefore, we can deduce

{A_\delta } = \left( {\begin{array}{*{20}{c}}{{\delta _i}{{\dot \nabla }^\dagger }_{\not \partial /{{\not \partial }_u}}}&{{C_\delta }}\\{{C_\delta }}&{ - {\delta _i}{{\dot \nabla }^\dagger }_{\not \partial /{{\not \partial }_u}}}\end{array}} \right)

for

{C_\delta } = \left( {\begin{array}{*{20}{c}}{ - \delta \beta \frac{\alpha }{2}}&{{{\rm A}^\dagger }}\\{{{\rm A}^\dagger }}&{\delta \beta \frac{\alpha }{2}}\end{array}} \right)

finally arriving at the Witten super-Dubrovin compactification Chern-Simons formula

\begin{array}{l}2\left[ {{{\widehat c}_1}\left( {{{\dot \nabla }^{\dagger * }}} \right) - {c_1}\left( {{{\dot \nabla }^{\dagger - 1}}} \right)} \right] = \\2\left[ {{{\widehat c}_1}\left( {{{\dot \nabla }^{\dagger * }}} \right) - {{\widehat c}_1}\left( {{{\rm A}^\dagger }\left( {{{\dot \nabla }^{\dagger * }}} \right)} \right)} \right]\\ = \frac{{ - 1}}{{\pi i}}\int_{{S_X}^1} {{\rm{tr}}\,\left( {\frac{{1 - i\beta }}{2}{{\rm A}^{\dagger - 1}}{{\dot \nabla }^\dagger }_{\not \partial /{{\not \partial }_u}}} \right)} \,du\end{array}

with

{\dot \nabla ^\dagger } - {{\rm A}^\dagger }\left( {{{\dot \nabla }^{\dagger - 1}}} \right) = \, - \frac{{1 - i\beta }}{2}{{\rm A}^{\dagger - 1}}{\dot \nabla ^\dagger }{{\rm A}^\dagger }

Let us see where the mathematics leads us next, for

"We are servants rather than masters in mathematics" ~ Charles Hermite

 

 

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