The basic object of invastigation of this paper is the calulation of Nekrasov's partition function for a specidic class of superconformal field theories. In specific the authors try and success to calculate the partition function of 5d gauge theories such as the class $T_N$ theories which are constructed by geometrical engineering, in specific by compactification of M5-branes in toric Calabi-Yau threefolds.Such theories can be easily manipulated through the so called webs of 5-branes. The main calculational tool is the refined topological vertex of Ibal et.al.
Idealy, in any quantum field theory, supersymmetric or not,the main object of interest is the partition function
$$Z=\int [D\phi] e^{S[\phi]}$$
but the partition function is extremelly hard to evaluate exactly. Nekrasov showed in the early 2000's that for supersymmetric gauge theories that contain 8 or more supersymmetries such a task can be possible if one puts the theory on the so-called $\Omega$ background. Later in 2007, localization also became on popular technique on how to calculate the partition function exactly in some specific backgrounds (e.g. products of various spheres). Despite that there exist many theories that they have no Lagrangian description or their Lagrangian description is not known for example the$T_N$ theories or Argyres-Douglas theories. Now, as Yuji Tachikawa put it in Strings 2014, there are two ways to proceed: either try to find the Lagrangian (if it exists) or use other tools coming from string/M-theory. Now, the authors of this paper proceed with the second way. They compute the 5d Nekrasov partiiton function of the $T_N$ theory using tools from topological string theory. They obtain the 5d $T_N$ theory by lifting the known 4d theory. The 4d theory is obtained by by wrapping N M5-branes on a sphere with 3 punctures. Now since this theory has no (known) Lagrangian description, the authors use geometric engineering. They engineer it by considering M-theory on non-compact (toric for this paper) Calabi-Yau threefolds. The theory has a BPS spectrum that corresponds to M2-branes on 2-cycles. The theory lives on $\mathbb{R}^4 \times S^1$. The other dimensions are the ones of the CY threefold. Then by taking a slice of the 6d CY they create the web of 5-branes where the power of the refined topological vertex comes into play. The partition function is relatively easy to be found in terms of the relevant Kahler moduli, and the Young diagrams of the web of 5-branes. The paper discusses first the $Sp(1)=SU(2)$ theory with a different number of flavors and makes sure it agrees with the known results. Then it proceeds to the calculation of the partition function of the $T_N$ theory. The authors then find that the partition function of the $T_3$ theory is not quite what they were expecting since the theory does not agree with the corresponding linear quiver.The reason is that a special extra factor appears. The topological string partition function is written as $Z_{top} = Z_{extra} \times Z_{Nekrasov}$ where the Nekrasov part agrees with the corresponding one of the linear quiver. A similar situation is found to happen with the SU(2) theory with four flavors. Now, the authors find that the reason behind this disagreement is the fact that in these theories there are M2-branes whose electric charge, a 2-cycle integral of the harmonic form $\omega$, and their contrucutions must be decoupled in the 5d gauge theory under investigation. The authors find that upon elimination of these contributions the results are in a perfect agreement with each other and they explicitly test it for example for the case of the $T_3$ theory. It turns out that the decoupled factoe is always a $U(1)$ factors. The main proposal of this paper is the relation $Z_{top}=Z_{U(1)} \times Z_{T_N}$ where $Z_{U(1)}$ is the factor that must be decoupled.
In brief the authors have evaluated the 5d Nekrasov partition function of the $T_N$ theory using the refined topological vertex and they perform a non-trivial check for the T_N theory. They recognise some extra contributions in their calculations from decoupled M2-branes when they try to compare the topological string partition function and the Nekrasov functions. This decoupled factor is always a prefactor of the topological string. They propose the formula we mentioned just in the previous paragraph.
This paper is original and actually makes a good advance towards our understanding of the $T_N$ theories. Both computationaly but also intuitionaly. The authors have revealed a new relation between the 5d Nekrasov partition functions of $U(2)$ and $SU(2)$ gauge theories. Although this article is very technical and requires advanced ability with the tools of topological string theory the authors cite all relevant references needed in order a non-expert to be able to follow it in a good pace. I would strongly recommend it.
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