# SUSY Indices in $\mathcal{N}=2$ Gauge Theories

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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) ;

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