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

Question

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?

This post imported from StackExchange MathOverflow at 2017-04-03 17:31 (UTC), posted by SE-user spietro

Answer 1

You do not have to use the "free energies"(F= $\log Z$) in order to see the fact that the partition function $Z_{\text{full}}$ is composed of three parts although two of them we pack together sometimes and then we have two parts. I would probably write down the full partition function as
$$
Z_{\text{full}} = Z_{\text{classical}} \times Z_{\text{1-loop}} \times Z_{\text{instantons}}
$$
and according to your notation $Z_{\text{perturbative}} = Z_{\text{classical}} \times Z_{\text{1-loop}}$. You ask what are the perturbative and what is the full part. The definition is tautological, the full partition function is a product of two pieces. The first piece, the perturbative, also sometimes called classical, is due to perturbation theory. We begin with some low energy effective action (mathematically a map from the space of "fields" to a field (no pun intended) usually $\mathbb{C}$. This contribution is obtained by expanding out the path integral in terms of divergent series, some sort of Taylor expansion, and each term corresponds to some Feynman diagrams. This is the usual process of perturbative quantum field theory which you can find in almost any quantum field theory textbook. To do this one  and get reliable results has to have a small "coupling constant". The problem arises when this coupling constant is not small. The Seiberg-Witten theory, that is the $\mathcal{N}=2$ SU(2) or SO(3) (and in general U(N) theory with N>1) is an example of a confining theory (like real life QCD). This means that below some energy scale $\Lambda$ the coupling of the theory (the strength of the interactions of the particles if you wish) becomes so big that we cannot rely on perturbative expansion of the path integral. For most quantum field theories one has to rely to lattice simulations or AdS/CFT to learn something about them. What is special in SW theory is that one can determine a modular function called the pre-potential which fully controls the non-perturbative aspects of the theory. This is where the $Z_{\text{instanton}}$ part comes in. Seiberg and Witten gave a formula for the full solution of the path integral but it was only until Nekrasov (elaborating in older ideas he had with Shatashvili mainly but a few others too) came up with the solution in terms of the $\Omega$-background. I do not think that the instanton part is the genus zero free energy of the topological string. The free energy of the gauge theory is expanded in terms of the genus but you kind of get the whole answer. In specific you can write
$$
\log Z = (a, \epsilon_1, \epsilon_2; \Lambda) = \sum_{n,g \geq 0} (\epsilon_1 + \epsilon_2)^n (\epsilon_1 \epsilon_2)^{g-1} F^{(n/2,g)}(a,\Lambda)
$$
where $a$ is a coordinate of the Cartan torus of the gauge group modulo Weyl group and $\Lambda$ is the dynamically generated scale usually related to the instantons as $ \sum_n \Lambda^{4n} = \sum_n q^n = \sum_{c_2} q^{c_2} $ since you want to sum over all c_2(E). Of course $E$ is the corresponding vector bundle over which you study the ASD equations. So, the (0,0) term of the above expansion would correspond to Seiberg-Witten prepotential and the rest of the terms, all given in terms of modular forms, would be higher order corrections corresponding to higher genus topological string amplitudes.

Finally, for a supersymmetric field theory there are various indices you can define one of them being the Hirzebruch $\chi_y$-genus but I know that this usually appears in the $\mathcal{N}=4$ Vafa-Witten theory. I am not sure if what you say is correct, I would have to check.Usually the exponents that appear in those formal variables in the indices are "charges" of various symmetries the theory has. You can consider mass as some kind of charge as well but, again, maybe you should say explicitly where in your reference is that thing for your second question so I can see exactly what is stated there.

Comments

@conformal_gk Thanks for this excellent answer :) This cleared up for me the general structure of these partition functions in this context.  As far as the SUSY indices you wanted sources on, I actually posted another question (https://www.physicsoverflow.org/38751) where I elaborate on what I was confused about.  Thanks again!

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@scpietromonaco sure, I will look at that question too