Does this $SU(2)$ Chern-Simons Superconformal Index Example have Modular Properties?

Question

Without any regards to the physics or the geometry used to generate this result, let's examine the formula of Gukov (see p. 32):

$$\mathcal{I}_{SU(2)}(q,t) = \frac{1}{2}\sum_{m \in \mathbb{Z}}\int \frac{dz}{2\pi i \, z}z^{2pm}t^{-2m} q^{-2|m|} (1 - z^{\pm 2} q^{|m|} )\times \frac{(1/t;q)_\infty (z^{\pm 2}q^{|m|}/t;q)_\infty}{ (qt;q)_\infty (z^{\pm 2}q^{|m|+1}t;q)_\infty}$$ The standard notation (for those who use it) is that $z^{\pm}$ is a shorthand: $$f(z^{\pm 2}) = f(z^2)f(z^{-2})$$

While there is element of sarcasm, here this is a real honest-to-goodness superconformal index computation, taken from a serious physics paper. My impression is we spend the good part of 19th century ascribing meaning to $q$-series like these and some of them turned out to have good modular properties.

For now there is basic property:

$$\mathcal{I}(q,t) \in \mathbb{Z}[t]\, [[q]]$$ This is the Lens space index. Let $M_3 = L(p,1)\simeq S^3 / \mathbb{Z}_p$, this has to do with:

• $\mathcal{N}=2$ Chern-Simons theory (so it is possibly supersymmetric)
• The theory is also called $T\big[L(p,1), SU(2)\big]$

This post imported from StackExchange MathOverflow at 2017-11-03 22:20 (UTC), posted by SE-user john mangual