Just to be very general, let $\mathcal{M}_{k}(r,n)$ be the moduli space of $U(r)$ instantons with instanton number $n$ on the ALE resolved space $X_{k}$. Feel free to only comment on specific cases!

Let $d=\text{dim}\mathcal{M}_{k}(r,n)$ and let $\mathcal{T}$ be the tangent bundle of the moduli space. We define the formal bundle

$E_{m}(\mathcal{T}) = m^{d} + m^{d-1} c_{1}(\mathcal{T}) = \ldots + c_{d}(\mathcal{T}).$

Now I'm a mathematician, but I've heard the following slogan from physics:

"Given $\mathcal{N}=2^{*}$ SYM in 4d with $m$ the mass of the adjoint hypermultiplet, if $m \to 0$ we recover $\mathcal{N}=4$ SYM in 4d while if $m \to \infty$ we recover pure $\mathcal{N}=2$ SYM in 4d"

In **(https://arxiv.org/pdf/0808.0884.pdf, Section 4.4)** they define the $\mathcal{N}=2^{*}$ instanton partition function on the ALE space $X_{k}$:

$Z_{X_{k}} = \sum_{n \geq 0} \Lambda^{2rn} \int_{\mathcal{M}_{k}(r,n)}(E_{m})(\mathcal{T})$

where the integral is to be done equivariantly with respect to a natural torus action. Now clearly from the above definition of the formal bundle $E_{m}$, if we let $m \to 0$ we get:

$\lim_{m \to 0} Z_{X_{k}} = \sum_{n \geq 0} \Lambda^{2rn} \int_{\mathcal{M}_{k}(r,n)} c_{d}(\mathcal{T}) = \sum_{n \geq 0} \Lambda^{2rn} e(\mathcal{M}_{k}(r,n))$

where $e( \cdot)$ denotes the topological Euler characteristic. Now, Vafa and Witten famously showed that the instanton partition function of $\mathcal{N}=4$ SYM on an ALE space corresponded to the generating function of the Euler characteristics of the moduli space. Therefore, this seems to agree with the physics I stated above. Moreover, we actually know that the dimension $d$ of the moduli space is $2rn$. Therefore, we can factor $m^{d}$ out of each term in the formal bundle $E_{m}$. We seem to be able to define a new finite parameter $q = (m \Lambda)^{2r}$ and then we can freely let $m \to \infty$:

$\lim_{m \to \infty} Z_{X_{k}} = \sum_{n \geq 0} q^{n} \int_{\mathcal{M}_{k}(r,n)} \, 1$

and this is simply Nekrasov's instanton partition function for pure $\mathcal{N}=2$ SYM, so this also seems consistent.

**First question: Is all of this correct so far? I feel very suspicious about my formula $q = (m \Lambda)^{2r}$ but I can't think of any other way to make this work out the way the above physics slogan claims it should. **

Since this has already been long, I'll make the second part succinct. Essentially, I think I understand what I've done above, modulo some details. **What's been bugging me for some time, is that there are these other SUSY indices like the arithmetic genus, the $\chi_{y}$ genus, and the elliptic genus. How do these fit into this picture!?** I think I can show that starting with $\chi_{y}$ as the index, we get a picture very similar to that above. I'll spare everyone the full formulas, but the $\chi_{y}$ genus is defined to be

$\chi_{y}(\mathcal{M}_{k}(r,n)) = \int_{\mathcal{M}_{k}(r,n)} \prod_{i=1}^{d} x_{i} \frac{1-ye^{-x_{i}}}{1-e^{-x_{i}}}$

where $x_{i}$ are the Chern roots. Imagine I make the analogous partition function to the one above with this index. Then we define the parameter $y$ by $y=e^{-m}$. Notice that when $m \to 0$, then $y \to 1$ and the integrand turns into just a product over the Chern roots which will give the Euler characteristic! This seems consistent with the Vafa-Witten story. However, when $m \to \infty$ we have $y \to 0$ which gives as an index the arithmetic genus, i.e. $\chi_{0}$. Now, clearly this is not merely an integrand of 1 as in Nekrasov's partition function, but there are sources **(https://arxiv.org/pdf/math/0412089.pdf, Page 24)** where the $\chi_{0}$ is apparently the correct index for a pure 4D $\mathcal{N}=2$ SYM theory.

**So what's going on here? This seems painfully similar, yet different, from what I did above with the formal bundle $E_{m}$. How are all these indices related in the physics literature? Specifically, in the "geometric engineering" business, you actually get that Gromov-Witten theory on a related Calabi-Yau threefold engineers a gauge theory in four dimensions whose instanton partition function uses as indices the arithmetic genus, the $\chi_{y}$ genus, and the elliptic genus. See for example, the link immediately above, or the beautiful Vafa, Hollowood, Iqbal paper (https://arxiv.org/pdf/hep-th/0310272.pdf) ;**