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Register a new user3d Expansions of 5d Instanton Partition Functions
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We propose a set of novel expansions of Nekrasovs instanton partition functions. Focusing on 5d supersymmetric pure Yang-Mills theory with unitary gauge group on $mathbb{C}^2_{q,t^{-1}} times mathbb{S}^1$, we show that the instanton partition function admits expansions in terms of partition functions of unitary gauge theories living on the 3d subspaces $mathbb{C}_{q} times mathbb{S}^1$, $mathbb{C}_{t^{-1}} times mathbb{S}^1$ and their intersection along $mathbb{S}^1$. These new expansions are natural from the BPS/CFT viewpoint, as they can be matched with $W_{q,t}$ correlators involving an arbitrary number of screening charges of two kinds. Our constructions generalize and interpolate existing results in the literature.
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We derive the partition function of 5d ${cal N}=1$ gauge theories on the manifold $S^3_b times Sigma_{frak g}$ with a partial topological twist along the Riemann surface, $Sigma_{frak g}$. This setup is a higher dimensional uplift of the two-dimensional A-twist, and the result can be expressed as a sum over solutions of Bethe-Ansatz-type equations, with the computation receiving nontrivial non-perturbative contributions. We study this partition function in the large $N$ limit, where it is related to holographic RG flows between asymptotically locally AdS$_6$ and AdS$_4$ spacetimes, reproducing known holographic relations between the corresponding free energies on $S^{5}$ and $S^{3}$ and predicting new ones. We also consider cases where the 5d theory admits a UV completion as a 6d SCFT, such as the maximally supersymmetric ${cal N}=2$ Yang-Mills theory, in which case the partition function computes the 4d index of general class ${cal S}$ theories, which we verify in certain simplifying limits. Finally, we comment on the generalization to ${cal M}_3 times Sigma_{frak g}$ with more general three-manifolds ${cal M}_3$ and focus in particular on ${cal M}_3=Sigma_{frak g}times S^{1}$, in which case the partition function relates to the entropy of black holes in AdS$_6$.
We analyze intersecting surface defects inserted in interacting four-dimensional N = 2 supersymmetric quantum field theories. We employ the realization of a class of such systems as the infrared fixed points of renormalization group flows from larger theories, triggered by perturbed Seiberg-Witten monopole-like configurations, to compute their partition functions. These results are cast into the form of a partition function of 4d/2d/0d coupled systems. Our computations provide concrete expressions for the instanton partition function in the presence of intersecting defects and we study the corresponding ADHM model.
We present a general prescription by which we can systematically compute exact partition functions of five-dimensional supersymmetric theories which arise in Higgs branches of the $T_N$ theory. The theories may be realized by webs of 5-branes whose dual geometries are non-toric. We have checked our method by calculating the partition functions of the theories realized in various Higgs branches of the $T_3$ theory. A particularly interesting example is the $E_8$ theory which can be obtained by Higgsing the $T_6$ theory. We explicitly compute the partition function of the $E_8$ theory and find the agreement with the field theory result as well as the enhancement of the global symmetry to $E_8$.
A central object in any axionic theory is its periodic potential, which is typically generated by instantons. The goal of this paper is to understand what physically happens to the theory when we lose control of the potentials instanton expansion. We argue, using the Yang-Lee theory of phase transitions, that the theory breaks down in the classic sense: states become light. However, these states are not necessarily light for all values of the axion and there can be large regions where the effective description remains valid. We find alternative expressions for the effective potential in terms of the properties of these light states, which remain useful even when the instanton expansion breaks down, and thus initiate a push beyond the lamppost of large instanton actions. Most of these questions are motivated by the axionic Weak Gravity Conjecture, which we reformulate without reference to instanton actions. We also comment on its ability to constrain large-field axion inflation.
An exact formula for partition functions in 3d field theories was recently suggested by Jafferis, and Hama, Hosomichi, and Lee. These functions are expressed in terms of specific $q$-hypergeometric integrals whose key building block is the double sine function (or the hyperbolic gamma function). Elliptic hypergeometric integrals, discovered by the second author, define 4d superconformal indices. Using their reduction to the hyperbolic level, we describe a general scheme of reducing 4d superconformal indices to 3d partition functions which imply an efficient way of getting 3d $mathcal{N}=2$ supersymmetric dualities for both SYM and CS theories from the parent 4d $mathcal{N}=1$ dualities for SYM theories. As an example, we consider explicitly the duality pattern for 3d $mathcal{N}=2$ SYM and CS theories with SP(2N) gauge group with the antisymmetric tensor matter.
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