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• In physics, you don't have to go around making trouble for yourself - nature does it for you ~ Frank Wilczek ... Why I always say: 'lucky we have mathematicians'

In my last four posts, I derived two propositions about quantum cohomology, with mirror-symmetry for Calabi-Yau manifolds being an isomorphism of variations of Hodge structures, with the A-model, defined by the genus zero Gromov–Witten theory of $X$ , isomorphic to the B-model of variations of Hodge structures associated to deformation of complex structures of the mirror $Y$

• "It was mathematics, the non-empirical science par excellence, wherein the mind appears to play only with itself, that turned out to be the science of sciences, delivering the key to those laws of nature and the universe that are concealed by appearances" ~ Hannah Arendt

In this post, I will begin the first part of a study of the cohomological-Dubrovin connection in the context of mirror-symmetry for Calabi-Yau Gromov–Witten theory. In particular, I will derive two propositions about quantum cohomology. In my last three posts, I finally derived, after studying the Witten Equation as well as the Landau-Ginzburg/Calabi-Yau correspondence, 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.

• Mathematics is the music of reason ~
James Joseph Sylvester

In my last post in this series on the Witten-equation, I showed that the genus-zero Landau-Ginzburg/Calabi-Yau correspondence amounts to the assertion that there is a degree-preserving isomorphism between the two state spaces and that after certain identifications, the small $J$-functions coincide, thus, the 'Picard-Lefschetz' Witten relation

• 'Edward Witten is often likened to Einstein; one colleague reached even further back for a comparison, suggesting that Witten possessed the greatest mathematical mind since Newton' ~ John Horgan

This is a series of posts that hopefully will culminate in a book I am working on. The aim is to show that the Witten equation in the context of Picard-Lefschetz Theory, leads to the non-forking, super-stability and hyper-categoricity of M-theory, thus making it the 'only-game-in-town' indeed. First, I will analyse the Witten nonlinear elliptic system of PDEs associated with a quasi-homogeneous polynomial-super-potential by showing its depth via the Landau-Ginzburg/Calabi-Yau correspondence.

• Ex nihilo nihil fit ~ Parmenides

In this post, in order to solve the Wheeler-deWitt problem-of-time discussed by me, and since string/M-theoretic braneworld cosmology offers the only solution in a unified C*-Heisenberg-algebraic sense, I must first analyse a field theory on a D-brane sigma worldvolume and study the Dirac-Born-Infeld action

• Go down deep enough into anything and you will find mathematics ~ Dean Schlicter

The unification problem in physics has at its heart the problem of time: quantum physics, via the Heisenberg Uncertainty Principle between time and energy is not compatible with the Einstein notion of time as coupled with space that gives rise to a 'smooth' spacetime continuum. John Wheeler and Bryce DeWitt did successfully resolve this incompatibility via the Wheeler-DeWitt equation: however, at two high prices. One, time ceases to play any dynamical role in the universe since it occurs idly in the equation, and secondly, by metaplectic geometry, there can be no quantum cosmological universal metric definable. In this post, I will try and analyse the metric problem and leave my analysis of the Page-Wootters 'solution' to the WdW problem of time for another post.

• Anyone can count the seeds in an apple, but no one can count the apples in a seed ~ Anonymous

How to make a space-like brane time-like, and why it matters. In this post, I will derive the Dirac-Born-Infeld S-brane action for Euclidean D-world-volumes in the S-brane context of super-condensation of non-BPS branes. Space-like branes are a class of time-dependent solutions of string/M-theory with topological defects localized in (P + 1)-dimensional space-like surfaces and exist at a moment in time, and are time-like super-tachyonic kink solutions of unstable D(P + 1)-branes in string theory and provide the topology of the throat-bulk. They are also deeply explanatory in quantum cosmology.

• The imagination of nature is far, far greater than the imagination of man ~ Richard Feynman

In this post, I will derive universal expansionary acceleration in a braneworld context without any mention of exotic matter, hence further strengthening the explanatory power and success of M-theory

• I know that two and two make four - and should be glad to prove it too if I could - though I must say if by any sort of process I could convert 2 and 2 into five it would give me much greater pleasure ~ George Gordon Byron

Great news for me personally - after nearly 20 years of researching M-Theory: see bottom link on M-Theory and my last formula in this post involving supersymmetry. I ended my last post by showing how Peccei-Quinn invariance leads to the compactification smoothness required for M-theory to isomorphically embed an 'Einsteinian-Minkowski' 4-D space-time in a Calabi-Yau fourfold in a way necessitated by quantum gravity.

• If a man's wit be wandering, let him study mathematics ~ Francis Bacon

In my last two posts, I launched a series where M-Theory can be Calabi-Yau fourfold-compactified, let me make a connection with PQ-symmetries in this post.

• I accept no principles of physics which are not also accepted in mathematics ~ René Descartes

Continuing with my M-theoretic Calabi-Yau fourfold compactification series, let me note that a C-'fourfolding' of M-Theory, as I will show ultimately, is equivalent to a proof of its testability and predictive power.