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

+ 6 like - 0 dislike

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
asked Oct 22, 2017 in Theoretical Physics by john mangual (310 points) [ no revision ]
retagged Nov 3, 2017

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