# Mathematics of $\mathcal{N}=2$ Gauge Theory and Instantons

Someone may suggest I post this on PhysicsSE, but I would prefer to not have a physicist answer in jargon I cannot understand. In fact, the reason I'm asking this is that I'm sort of drowning in the physics in this subject! I'm attempting to make mathematical sense of the great papers of Nekrasov (https://arxiv.org/pdf/hep-th/0206161.pdf) and the "geometric engineering" paper of Vafa et. al. (https://arxiv.org/pdf/hep-th/0310272.pdf). I'm hoping someone can mathematically enlighten me here.

Nekrasov's instanton partition function for pure (massless) $\mathcal{N}=2$ $SU(N)$ SUSY gauge theory on $\mathbb{R}^{4} = \mathbb{C}^{2}$ is given by

$$\mathcal{Z}^{inst}_{\mathbb{C}^{2}}(\epsilon_{1}, \epsilon_{2}, \vec{a},Q) = \sum_{k=0}^{\infty} Q^{k} \int_{\mathcal{M}_{k,N}}1 = \sum_{k=0}^{\infty}Q^{k} \chi_{0}( \mathcal{M}_{k,N}),$$

where $\mathcal{M}_{k,N}$ is the moduli space of rank $N$ instantons on $\mathbb{C}^{2}$ with second Chern class $k$, $\chi_{0}$ is the arithmetic genus, and $(\epsilon_{1}, \epsilon_{2}, a_{1}, \ldots, a_{N})$ are coordinates on the Lie algebra of the Cartan torus $(\mathbb{C}^{*})^{2} \times (\mathbb{C}^{*})^{N}$. Remarkably, this equals the partition function of topological string theory on the Calabi-Yau threefold $A_{N-1} \to \mathbb{P}^{1}$, i.e. a non-trivial fibration of the ALE space over the complex line.

Now, using $\mathcal{F} = \log \mathcal{Z}$, I believe the physicists somehow decompose

$$\mathcal{F}^{\text{full}} = \mathcal{F}^{\text{pert}}+\mathcal{F}^{\text{inst}},$$

into "perturbative" and "instanton" parts. I believe Nekrasov's conjecture was that the instanton part was the Seiberg-Witten prepotential. My first question is what are the "full" and "perturbative" parts of the partition function? In particular, I guess on the string theory side, the topological partition function you compute would be the "full" thing, and so the "instanton" part may be just the genus-0 free energy. But then the "perturbative" part would be the higher genus corrections which makes no sense to me. Clearly I could use some clarity here.

Secondly, notice above I wrote the Nekrasov partition function in terms of the arithmetic genus $\chi_{0}$. This is the correct supersymmetric index for a four-dimensional gauge theory. However, one can promote this to the $\chi_{y}$ genus or the elliptic genus $\text{Ell}_{y,q}$. These are the correct indices for five and six dimensional gauge theories, respectively. Here we pick up the extra parameters $y$ and $y,q$. Do the physicists claim that these are "masses" or something, which when set to zero, should recover the lower dimensional theory? And what do these parameters correspond to on the string theory side?

• There is an answer to this on PhysicsOverflow. – Dilaton Apr 3 '17 at 20:06