# Checking modularity-like transformation property

Assume $$M$$ is a 4 manifold. Let $$Z_v$$ be partition function of fixed magnetic flux $$v$$ with all instanton configuration summed over where $$v\in H^2(M,Z/nZ)$$. $$\tau$$ denotes complex parameter on upper half plane. In the following the dot product is interpreted as cup product by $$H^2(M,Z)\otimes H^2(M,Z)\to H^4(M,Z)=Z.$$

Assume the following relations hold $$Z_v(\tau+1)=exp(2\pi i(-\frac{c}{24}+h_v))Z_v(\tau)$$ where $$h_v=v\cdot v(\frac{1}{2N}-2)$$ counts instanton number for $$SU(N)/Z_N$$ theory as one has to take dual group of $$SU(N)$$. Denote $$T:\tau\to\tau+1$$

Assume $$S:\tau\to\frac{-1}{\tau}$$. $$b_2=dim H^2(M,Z)$$ is the second betti number. $$Z_u(\frac{-1}{\tau})=\pm N^{-b_2/2}\sum_v exp(2\pi i (v\cdot u)/N)Z_v(\tau).$$

Assume $$\sum_v exp(\frac{2\pi i v\cdot v}{N})=N^{b_2}\delta_{u,0}. \tag{\star}$$

The paper claims $$(ST)^3=1$$ results $$Z_u$$ transforming up to a constant. If one wants to get rid of that constant, one demands the equstion $$exp(2\pi i c/24)^3=\pm N^{-b_2/2}\sum_v(-1)^{v^2}(exp(2\pi i/N))^{v^2/2}$$ where $$v^2=v\cdot v$$.

I checked the following. Dropping $$N^{-b_2/2}$$ proportionality after $$S$$ transformation, I obtain $$exp(2\pi i\frac{c}{24})^3Z_u((ST)^3\tau)=N^{-3b_2/2}\sum_{v_1,v_2,v_3}Z_{v_3}(\tau)\prod_{i\leq 3}exp(2\pi i (\frac{v_i\cdot v_{i-1}}{N}+h_{v_i}))$$ where $$v_0=u$$. I do not see how to use $$\sum_v exp(\frac{2\pi i v\cdot v}{N})=N^{b_2}\delta_{u,0}$$ to reduce summation here. My guess is somehow I contract $$v_2$$ part to reduce summation to $$v$$. I could try to sum over $$u$$ again to replace $$v_1$$ summation by $$\delta_{v_1,0}$$ but this still left me with two more summations.

$$\textbf{Q:}$$ How do I obtain relation $$exp(2\pi i c/24)^3=\pm N^{-b_2/2}\sum_v(-1)^{v^2}(exp(2\pi i/N))^{v^2/2}$$ via $$(\star)$$ here? Outline will be sufficient. The paper seems to claim that one can reduce 3 summations by $$(\star)$$.

Ref. https://arxiv.org/abs/hep-th/9408074 (pdf page 51-52)

I am only using $$\hat{Z}_v$$ expression of eq 3.14, 3.15, 3.16 (pdf page 47-48) to derive eq 3.20(pdf page 52)