The mathematician David Nadler is at KITP, talking about work with David Ben-Zvi on three dimensional topological field theories. The talk starts with the now familiar statement: there is a six dimensional field theory out there that contains all of mathematics. This must be apparent to the classical geometers and representation theorists who work on the Langlands correspondence!
Nadler thinks about compactifying ($N=4$ SYM) gauge theory in four dimensions on a circle, in one of two distinct ways, both leading to a three dimensional topological field theory based on $2$-categories of modules. Let us look at the $\chi_{LG}$ case, that is the character theory for a loop group. Here $G$ is our (semisimple) gauge group and $B$ a Borel subgroup.
As a functor, $\chi_{LG}$ assigns a certain $2$-category of (highest weight) module categories to a point, and then character information to an $S^1$. Geometric Langlands is about surface operators in a four dimensional theory, and these have become one dimensional operators in the three dimensional theory. The right category to consider turns out to be what string theorists would call the monoidal category of $A$ branes in $T^{*}(K)$ where $K = I \backslash LG / I$ for $I$ an Iwahori subgroup. If we think of $LG$ as matrices with Laurent series entries, then $I$ is the set of matrices with power series entries such that the constant term is in $B$, the upper triangular matrices. The end aim is to understand $S$ duality (between $A$ and $B$ models) in this rigorous context, using a Grothendieck-Springer resolution for ordering the eigenvalues of a matrix. It depends on an elliptic curve associated to the maximal torus for the Langlands dual of $G$.
One humorous aspect of the talk was Nadler's insistent apology for not knowing the physics jargon. At one point, he even explained his hesitency to use the word point, at which point the physicists said the word point was quite all right (clearly, these are not categorical physicists).
14 years ago
So where is the six dimensional theory? The mathematicians have no doubt realised that the physics has moved on to more twistorial geometries in six dimensions. They probably also know that $S$ duality must be tripled a la $STU$ dualities and triality. There is a sense in which Nadler and Ben-Zvi's construction is only two dimensional, in which case a true tripling brings us to dimension six.
ReplyDeleteI cannot imagine how these classical geometers envisage the six dimensional theory. Certainly, they know it is about cohomology and arithmetic. Without a doubt, they just want to prove fancy theorems like the Riemann hypothesis (which of course they never will, without a more axiomatic point of view). What on earth do they see triality doing? I guess they also realise that noncommutative and nonassociative geometry comes in here ...
As usual, three dualities for three qubits! If each duality is associated to a loop, we are somehow looking at links with three components, as in the entanglement picture for three qubits. For example, a traced $B_3$ link of the form $\nu \nu^{m}$ is a three link Hopf chain, as is a cyclic traced colored $Z$ boson ribbon in $B_6$.
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