Edward Witten | Knot Invariants From Gauge Theory in Three, Four, and Five Dimensions 

It's always a pleasure to listen to him.

Given a Morse functional F : U → R with finitely many critical points (solutions of the KW equations) and a gradient-like flow for F (HW equations) , one can construct the Morse-Witten flow category C_{MW} (F) as follows. The objects of C_{MW](F) are the critical points of F (solutions to the KW equations); the grading (fermion number) of an object is the index of the critical point. For critical points x, y of relative index n + 1, let Mf(x, y) be the parametrized moduli space from x to y, which is the (n + 1)-dimensional subspace of U consisting of all the points that flow up to x and flow down to y (instantons from y to x). Let M˚(x, y) be the quotient of Mf(x, y) by the natural R-action. Define M(x, y), the space of morphisms from x to y, to be the unparametrized moduli space from x to y, which is obtained by compactifying M˚(x, y) by adding all the broken flowlines from x to y.

Quoting Mr. Gumby from Monty Python, “my BRANE hurts!“ Piece.

Is it significant that chiral _Type IIB String Theory_ is the only flavor which has S-duality with itself, and is related through _K-theory_ through the geometric _Langlangs Program_ through _Modular Forms_ (which were used by Sir Andrew Wiles in his proof of _Fermat's Last Theorem_ ), to _Knot Theory_ and from there to _Quantum Field Theory,_ with the only combination of temporal and spatial dimensions (t, s) in which it is possible to tie a persistent knot* is (1, 3) which suggests that this is the fundamental reason which acts as a constraint that would filter out all possible unrelated _Theories of Everything,_ which would also need to include all observed phenomena represented as quantised fields in a symmetric model, where these are likely aspects of a pervasive single unified field in a sufficient number of infinite complexified dimensions (in order to support P-symmetric fields that describe Dark Matter), but only as an intermediate step, as this gauge group would then be decomposed to a finite one, that is coupled to a split-signature Spin group that is the unification of a Spin group Spin(1, 3) which is isomorphic to the _Lorentz group_ Sl(2, ℂ) that is a _Lie group_ (where a group is a set with operations, and a _Lie group_ includes a _Differential Manifold_ which defines operations that support Noether's symmetries which yield conservation laws fundamental to physics in (1, 3) space-time, such as conservation of energy), which then leaves as a "remainder" a Spin group which needs to be sufficient to describe the _Pati-Salam_ model (if not more elaborated gauge groups should space-time SUSY be desired, which would be entailed by a _N = 4 super Yang-Mills theory_ as covered in this lecture); and it might be convenient to conjecture a less compacted 5 dimensional arena which does not concern itself with gravitation (and that emerges within the implied set of dimensional measures operating over (1, 3) as the "Metric" actually being the space of connections implied from the Horizontal vector space that carries Spin(1, 3) by going in an unconventional reverse direction "down" the Levi-Civita connection), based off the work on the anti de Sitter / Conformal Field Theory correspondence by Juan Maldecena, to "view" 4D + gravity from the "vantage" of 5D without gravity, where the math is simpler (similar to mapping SU(n) _Yang-Mills theories_ to U(n) to make the calculations easier, using the _Seiberg-Witten invariants_ ); so, that we could have Selectrons and Squarks exist purely mathematically in 5D and everything physically modeled in terms of Rank 7/2, 3, 5/2, 2, 3/2, 1, 1/2, 0 Tensors within 4D where "gravitons" aren't Spin 2 but Spin 3, and the Spin 7/2 "anti-gravitons" are responsible for the accelerating expansion of the Cosmos, and the (1, 3) Section that is recovered from its gauge group, thereby accounting for Dark Energy, and Spin 5/2 are Supersymmetric Fermions which are related to their Spin 1/2 Superpartners, and Spin 2 are Supersymmetric Bosons which are related to their Spin 1 Superpartners, and Spin 0 is just the Higgs field, with the rest of the mass being given by the Spin 3 and Spin 7/2 fields operating in opposition to each other in a local relativistic context in 4D, which can be regarded as a Hyperforce equivalent to Hypercharge U(1) except where like charges repel here like Matter and Dark matter attracts to yield what is observed within the Section as gravitation, and the unlike Supersymmetric superpartners repel each other, influencing large scale idiosyncracies historically kludged by the now redundant Cosmological constant; and all the associated PDEs within these Tensors become more tractable in 5D via AdS/CFT as it just becomes Spin 2, 3/2, 1, 1/2, 0 as no gravitation needs to be modeled within what becomes a 5D _Kaluza-Klein Unified Field Theory_ in which U(1)ₑₘ is swapped to U(1)ₕ such that this hyperforce has reverse polarity of electromagnetism in the context of how it repels "like" Superpartner particles to produce the phenomenon of _Dark Energy_ within the _Principal Fiber Bundle_ before a Section of it is recovered as space-time and leaves this artefact of an accelerating expansion which isn't a property of our physical Universe, but of the mathematical Cosmos and its SUSY as it makes sense through fibers that are at right angles to a psuedoreality defined conveniently to be 5D that maps via AdS/CFT to (1, 3) reality, and where the problem with defining the _Theory of Everything_ arises from imposing a design on the Universe in the form it is dimensionally observed, rather than allow the math to take the least path of resistance which also ends up elegantly unparameterised as 4D is the sum of (1, 3) rather than some "magic number" needed to get the model to work, and a "Swampland" accounts for fine tunings on an Anthropic basis as physics is reified into existence from pure atemporal mathematics? *Informally, it is self evident that there exists no "over and under" with which to cross the braids to form a knot in 2 or fewer spatial dimensions (this is analogous to the spatial restrictions explored in Edwin Abbott's _Flatland: A Romance of Many Dimensions_ ), furthermore 4 or more will mean that you always have some adjacent hyperspace through which braids could slip their bonds (this is harder to visualise however a "cheat code" could be imagined such that a braid will pass through another braid of the same colour, and that is somewhat analogous of the colour changing protagonist of Yasushi Suzuki's _Ikaruga_ videogame having missiles "phase through" their matter when they are of a matching colour), and where it is obvious that an extra temporal dimension would allow for process reversal (or travel back to a point in time before the knot got knotted), and where a formal proof of this (1, 3) persistent knot being the fundamental constraint filtering out almost all varieties of _Theories of Everything_ is a conjecture that will be left as an exercise for the sufficiently motivated reader.

I like this direction

A framed flow category whose objects are the solutions of the Kapustin-Witten equations and whose morphisms are the solutions of the Haydys-Witten equations, leads to the Khovanov Homotopy.

nice talk

Khovanov homotopy demands six, seven or eight dimensions?