# Do all $\mathcal{N}=2$ Gauge Theories "Descend" from String Theory?

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I'm thinking about the beautiful story of "geometrical engineering" by Vafa, Hollowood, Iqbal (https://arxiv.org/pdf/hep-th/0310272.pdf) where various types of $\mathcal{N}=2$ SYM gauge theories on $\mathbb{R}^{4} = \mathbb{C}^{2}$ arise from considering string theory on certain local, toric Calabi-Yau threefolds.

More specifically, the topological string partition function (from Gromov-Witten or Donaldson-Thomas theory via the topological vertex) equals the Yang-Mills instanton partition function which is essentially the generating function of the elliptic genera of the instanton moduli space.  (In various settings you replace elliptic genus with $\chi_{y}$ genus, $\chi_{0}$ genus, Euler characteristic, or something more fancy.)

From my rough understanding of the Yang-Mills side, we can generalize this in a few ways.  Firstly, we can consider more general $\mathcal{N}=2$ quiver gauge theories where I think the field content of the physics is encoded into the vertices and morphisms of a quiver.  And there are various chambers where one can define instanton partition functions in slightly different ways, though they are expected to agree in a non-obvious way.  For a very mathy account see (https://arxiv.org/pdf/1410.2742.pdf)  Of course, the second way to generalize is to consider not $\mathbb{R}^{4}$, but more general four dimensional manifolds like a K3 surface, a four-torus, or the ALE spaces arising from blowing up the singularities of $\mathbb{R}^{4}/\Gamma$.

My questions are the following:

Is it expected that this general class of $\mathcal{N}=2$ quiver gauge theories comes from string theory?  In the sense that their partition functions may equal Gromov-Witten or Donaldson-Thomas theory on some Calabi-Yau threefold.  If not, are there known examples or counter-examples?  Specifically, I'm interested in instantons on the ALE spaces of the resolved $\mathbb{R}^{4}/\Gamma$.

In a related, but slightly different setting, Vafa and Witten (https://arxiv.org/pdf/hep-th/9408074.pdf) showed that the partition function of topologically twisted $\mathcal{N}=4$ SYM theory on these ALE manifolds give rise to a modular form.  Now I know Gromov-Witten and Donaldson-Thomas partition functions often have modularity properties related to S-duality.  So I'm wondering, does this Vafa-Witten partition function equal one of these string theory partition functions?  Or are they related?

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This will not be a complete answer and it will rather be a very short answer since I would have to dig some things up to be more precise. What do you mean by string theory? String theory is not a quantum field theory. Rather, when one considers something more general, M-theory, then you can show that there exist objects called M5-branes and M2-branes which are 6 and 3 dimensional submanifolds of the ambient 11 dimensional spacetime where M-theory lives on. As we know from simple string theory considerations like understanding Chan-Patton factors in D-branes, on such a submanifold lives the so-called 6d (2,0) N=2 SCFT. The number $N$ of M5 branes makes it a $U(N)$ gauge theory which has no Lagrangian description. From there one can start the never ending game of compactification or reductions using various types of geometries. One such example is to use a compact $CY3=X$. Split the ambient space time as $\mathbb{R}^4 \times X$. Then, indeed, the topological string partition function on $X$ would give you some partition function corresponding to some theory on $\mathbb{R}^4$. Make $X$ to be local CY3, e.g. total space of cotangent bundle or some elliptic fibration over a rational surface, and then one obtains the $\mathcal{N}=2$ Nekrasov partition function which is the partition function of the $\mathcal{N}=2$ $U(N)$ gauge theory. Then the corresponding local Gromov-Witten invariants of $X$ are related to the gauge theory partition function indeed. The "funny" thing is that these guys can be shown to be related to quantum $\mathfrak{sl}_N$ invariants of 3-manifolds with knots. There is a nice video of Gukov explaining all these (or much bette his papers) called Homological algebra of knots and BPS states [2]. The quiver gauge theory should arise when you consider a stacky $X$ instead. I would need to do some digging to expand towards that direction but a lot of work has been done towards that direction too. As for the $\mathcal{N}=4$ case you are right. Actually these partition functions which have been explicitly evaluated over some specific manifolds are not just traditional modular forms. For example, over rational surfaces like Hirzebruch surfaces they are written in terms of indefinite theta functions encoding the wall-crossing behaviour as well. Check these guys: https://arxiv.org/pdf/1304.0766.pdf and references within. Indeed, what you expect to me seems correct because of these dualities (I mean the geometrical ones not S-duality which in turn is interesting when we want to consider the Langlands side of the story). At the moment I cannot think of a single reference that would have all the details. I don't think I know one actually so if you find one please let me know as well. Finally, if you are interested in instantons on ALE spaces maybe you should check papers of Bruzzo, Tanzini et. al. where they study sheaves on stacky surfaces. In general on rational surfaces, and especially Hirzebruch and $\mathbb{CP}^2$ one can do a lot. I am quite interested in such spaces myself, so maybe something can be done there ;)

answered Apr 26, 2017 by (3,625 points)
edited Apr 26, 2017
@ArnoldNeumaier thanks for including the links which I am not able to include but please do not alter the format of my answer: I put spaces between paragraphs for a reason. To make the answer more readable ;)

@conformal_gk Thanks a lot :)  As far as what I meant by "string theory" a great reference is that Vafa, Hollowood paper I referenced above.  To my understanding, Nekrasov's instanton partition function is constructed from (equivariant) integrals over the instanton moduli space on $\mathbb{C}^{2}$ with integrand *one*.  These partition functions I think should straight up equal the partition functions of Gromov-Witten theory of the CY three-fold $A_{N} \to \mathbb{P}^{1}$.  More generally, you can replace the integrand of one, with something more exotic like the $\chi_{0}$ genus of the instanton moduli space, the $\chi_{y}$ genus, the elliptic genus, etc.  In that Vafa, Hollowood paper they show that these partition functions *equal* string theory partition functions on a local, toric Calabi-Yau threefolds computed via the topological vertex.

For all of the above on the gauge theory side, the integrals were over the moduli space of instantons on $\mathbb{C}^{2}$.  I'm wondering if we replace $\mathbb{C}^{2}$ by these ALE spaces and compute the instanton partition function, if there would also be some local CY3 whose GW partition function via the topological vertex gives rise to it.

And you mention Bruzzo...I actually referenced one of his papers in my question :)  It's a super beautiful and readable account of the gauge theory side.  He describes how $\mathcal{N}=2$ gauge theories on ALE spaces "factorize" in a sense into roughly copies of the $\mathbb{C}^{2}$ case.  This is known as the "Nekrasov master formula" and it's precisely this which I'm wondering if there's a string theory explanation.

I didn't deliberately change the spacing - this happened without having done anything, and I don't know why. Sorry.

I mentioned it for future reference :) Yeah, something is wrong since my answering and also commenting environment are not working properly.

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