“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

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

are metaplectically connected to the secondary invariants Chern-Simons type, which is the Atiyah-Patodi formula, that the Witten equation entails are -invariants of such operators for spaces that fibre over . 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 , with arclength on 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 and is the phase where is the determinant of the holonomy of . Let be a four-fold Calabi-Yau manifold, and a Hermitian vector bundle with Hermitian connection over and

given by

on . Now, since and super-extends over , letting and refer to the characteristic forms corresponding to the -class of and Chern character of , we can derive

Let us get to the Witten formula now. Letting

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

and

Then clearly is self-adjoint, elliptic, and and and now write

where is arclength on . Then we have

hence Witten's formula for ) is locally computable on . Put

and let

denote the heat kernel of . Then for the operator and we get the of determinant line bundle global anomaly relation

Now, with , we have and the super-Dubrovin relation

and 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

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

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