In upcoming work of Ben-Zvi-Sakellaridis-Venkatesh, (see for instance these [notes](https://www.msri.org/workshops/918/schedules/28233/documents/50487/assets/88599) or this [lecture](https://youtu.be/pixkOp_KK-M?t=5499)) some important aspects of the Langlands correspondence are stated in the language of topological quantum field theory. We have a TQFT i.e. a functor $A\_G$, and it takes for example, a curve $C$ over $\mathbb{F}\_{q}$ and returns the vector space $A\_G(C)$ of unramified automorphic forms which live on $\mathrm{Bun}\_G(C)$. It can further take a local field $F\_x$ (localization of the function field of $C$ at some point $x$ of $C$) and return the category $A\_G(F)$ of smooth representations of $G(F\_x)$, and we can put them together and apply $A\_G$ to $C\setminus S$, $S$ a finite set of places, to get $A\_G(C\setminus S)$, the vector space of automorphic forms with prescribed ramification at $S$. In this picture, we are taking $C$ and the formal puncture disc around $x$ to be the analogues of $2$-manifolds and circles in 2D TQFT.


It has been mentioned that this is in some way related to (or perhaps inspired by) the work of [Kapustin and Witten](https://arxiv.org/abs/hep-th/0604151) on gauge theory and geometric Langlands (over $\mathbb{C}$). In this work Kapustin and Witten construct a topological field theory on a manifold of the form $M=\Sigma\times C$ where $C$ is the Riemann surface of geometric Langlands, and, upon taking $C$ to be small compared to $\Sigma$, they show that the effective field theory on $\Sigma$ can be described by a sigma model of maps from $\Sigma$ to $\mathcal{M}\_{H}(G)$, the moduli of semistable Higgs bundles on $C$. Then by some construction involving the "canonical coisotropic brane" in section 11 of that paper they make contact with the Langlands program by describing how to build a sheaf of D-modules on $\mathrm{Bun}\_G$. Then S-duality on the gauge theory makes ${}^{L}G$ appear and the hyperkahler nature of $\mathcal{M}\_{H}({}^{L}G)$/nonabelian Hodge correspondence connects this with the moduli of vector bundles with flat connection and the geometric Langlands correspondence appears as homological mirror symmetry between $\mathcal{M}\_{H}(G)$ and $\mathcal{M}\_{H}({}^{L}G)$.


The work of Kapustin-Witten, in their paper is a "topological field theory" in that it is independent of the metric. The language of TQFT as a functor, as far as I know, does not show up in the paper (although it does get a little mention in this later [survey](https://people.math.harvard.edu/%7Egaitsgde/Jerusalem_2010/MathPhysicsSeminar/barcelona.pdf) by Kapustin).


In the talks of Ben-Zvi linked to above, the TQFT that they are considering is supposed to actually go up to 4-dimensions, and though the examples I mentioned in the first paragraph are just the 2- and 1-dimensional part. From what I infer this being 4-dimensional is supposed to be somewhat connected to Kapustin-Witten.


How does one reconcile the two? What is the 4-dimensional part of the TQFT of Ben-Zvi-Sakellaridis-Venkatesh? How is this related to the manifold $M=\Sigma\times C$ of Kapustin-Witten? In the MSRI notes linked above, there is a table on page 9 which says the 4-dimensional part should be periods. How is this so? In the same work periods are also supposed to be related to boundary conditions, which pick out a specific object of the output of the TQFT. However in the survey of Kapustin (in section 0.5) boundary conditions are the output of the 0-dimensional part of a 2-dimensional TQFT. What should the 3-dimensional part be?



Relationship between the TQFTs in Kapustin-Witten and Ben-Zvi-Sakellaridis-Venkatesh
A curve $C$ over $\mathbb F\_q$ has dimension $3$ in this perspective (which is why you get a vector space) and a local field has dimension $2$ (which is why you get a category. So one only has to go up one dimension higher to get the four-dimensional theory.


Associated to the "product $C \times S^1$", we would take the dimension of the space of automorphic forms. This is not well-defined as it is infinite-dimensional, but if we consider the space of automorphic forms with prescribed ramification at some finitely many points, we can get a finite number, and this corresponds to a four-manifolds with boundary conditions imposed on some surfaces (the "products" of closed points of $C$, which have dimension $1$, with $S^1$).


The trace of a Hecke operator on the space of automorphic forms is also part of the four-dimensional theory, where we have now inserted a line operator supported on a "1-dimensional submanifold" in our "4-manifold" (a closed point times a point of $S^1$).


Possibly any number that appears in the theory of automorphic forms can be expressed as arising from a four-manifold with suitable boundary conditions imposed.


The relation to Kapustin-Witten here is that a curve over a finite field behaves something like a product of a Riemann surface with a circle, so when we take the product with another circle, we get the product of a Riemann surface with a torus, which fits into the $\Sigma \times C$ picture.


