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Register a new userExact partition functions of Higgsed 5d $T_N$ theories
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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 $E_8$.
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We analyse the computation of the partition function of 5d $T_N$ theories in Higgs branches using the topological vertex. The theories are realised by a web of $(p,q)$ 5-branes whose dual description may be given by an M-theory compactification on a certain local non-toric Calabi-Yau threefold. We explicitly show how it is possible to directly apply the topological vertex to the non-toric geometry. Using this novel technique, which considerably simplifies the computation by the existing method, we are able to compute the partition function of the higher rank $E_6$, $E_7$ and $E_8$ theories. Moreover we show how in some specific cases similar results can be extended to the computation of the partition function of 5d $T_N$ theories in the Higgs branch using the refined topological vertex. These cases require a modification of the refined topological vertex.
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 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.
We discuss a general procedure to obtain 1/2 BPS partition functions for generic N=1 quiver gauge theories. These functions count the gauge invariant operators (bosonic and fermionic), charged under all the global symmetries (mesonic and baryonic), in the chiral ring of a given quiver gauge theory. In particular we discuss the inclusion of the spinor degrees of freedom in the partition functions.
We compute partition functions of Chern-Simons type theories for cylindrical spacetimes $I times Sigma$, with $I$ an interval and $dim Sigma = 4l+2$, in the BV-BFV formalism (a refinement of the Batalin-Vilkovisky formalism adapted to manifolds with boundary and cutting-gluing). The case $dim Sigma = 0$ is considered as a toy example. We show that one can identify - for certain choices of residual fields - the physical part (restriction to degree zero fields) of the BV-BFV effective action with the Hamilton-Jacobi action computed in the companion paper [arXiv:2012.13270], without any quantum corrections. This Hamilton-Jacobi action is the action functional of a conformal field theory on $Sigma$. For $dim Sigma = 2$, this implies a version of the CS-WZW correspondence. For $dim Sigma = 6$, using a particular polarization on one end of the cylinder, the Chern-Simons partition function is related to Kodaira-Spencer gravity (a.k.a. BCOV theory); this provides a BV-BFV quantum perspective on the semiclassical result by Gerasimov and Shatashvili.
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