full-spectrum tqft

I’m interested in understanding natural[1] combinatorially-explicit constructions of functors:

$$\mathcal{B}_2^{++} \rightarrow \mathcal{HS}$$

Here $ \mathcal{B}_2^{++} $ ought to be a variant [2] on the theme of a 2-bordism category. For example a “first approximation” to an object of this category might be $(m,n) \in \mathcal{Obj}(\mathcal{B_2})$ defined with respect to pairs of integers i.e. $(m,n) \in \mathbb{Z}^{\oplus 2}$ and $\Sigma \in \mathcal{Hom}((a,b), (c,d))$ might similarly to “first order” be something like an equivalence class of Riemann Surfaces with boundary a disjoint sum of $(m,n)$ copies of $S^1$. Popular interpretations involve:

  • $(\textrm{positively oriented}, \textrm{negatively oriented})$
  • arithmetic reciprocity as seen in the degeneration of of abelian varieties to the boundary strata of a toroidal compactification; i.e. degenerate class groups, for e.g. $( \mathbb{Z} \text{ mod } p\mathbb{Z}) \oplus \mathbb{Z}$.

In sufficient generality, one could imagine coloring boundary circles with weight data encoding something like an action of a reductive algebraic group—a la deligne’s theory of weights, simpson’s non-abelian hodge theory program etc.

On the other hand $G \in \mathcal{Obj}(\mathcal{HS})$ is an algebraic group, while $s \in \mathcal{Hom}(G_1, G_2)$ is a geometric “bimodule”; i.e. a holomorphic symplectic variety carrying a $\mathbb{C}^*$-action hamiltonian-commuting with “chiral” $G_1$ and “anti-chiral” $G_2$ actions. This “skew” homogenous ($G_1$, $G_2$)-space can naturally be composed/convolved with another ($G_2$, $G_3$)-space. The basic mechanism is given by the quotient by a “diagonal” $G_2$-action yielding a skew homogenous $(G_1, G_3)$-space. One can and should imagine wanting to explicitly express combinatorial data encoding the data of the group actions, uniquely specifying the notion of “diagonal” and hence the corresponding (quantized) derived quotient.

From my naive point of view it is worth emphasizing the value of taking a quotient in an appropriately derived / homotopic sense, as then one can define the quotient as a certain “localization” functor that enjoys many good properties (essentially, one gets as nice a theory as one could hope to expect).

Constructing these functors appeared totally inaccessible to me until I began to study Conformal Field Theory, where the highest weight representation theory of “chiral vertex algebras” and their appropriate quantized hamiltonian reductions essentially gives the most accessible, combinatorially-explicit examples I have of these derived hamiltonian reductions. A guiding construction is that of derived conformal blocks from conformal field theory. In a formal sense, this is defined by deriving the functor mapping a given module to its (co)invariants of a canonical $\mathbb{C}^*$ action (this is the “conformally invariant” part of conformal field theory).

Let me be totally open: I don’t really have any understanding of any of the “topology” involved. It is disingenuous for me to use words such as “homotopy”, “cycle” and even my use of “symplectic” should be distrusted. The only symplectic varieties I can work with are (co)-adjoint orbits of adjoint actions of Groups on Lie algebras; the only “homotopy” groups I have any constructive understanding of arise via derived categories of sheaves on locally ringed spaces.

[1] whatever that means

[2] $^{++}$ meant here to suggest keeping track of as much combinatorial data as possible