“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 $X$

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

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

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

with

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

given by

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

Let us get to the Witten formula now. Letting

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

and

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

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

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

and let

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

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

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

for

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

with

Let us see where the mathematics leads us next, for