SciPost Phys. 15, 156 (2023) ·
published 12 October 2023

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We consider $\mathcal{N}=4$ supersymmetric gauge theories on the squashed threesphere with six preserved supercharges. We first discuss how Wilson and vortex loops preserve up to four of the supercharges and we find squashing independence for the expectation values of these 2/3BPS loops. We then show how the additional supersymmetries facilitate the analytic matching of partition functions and loop operator expectation values to those in the mirror dual theory, allowing one to lift all the results that were previously established on the round sphere to the squashed sphere. Additionally, on the squashed sphere with four preserved supercharges, we numerically evaluate the partition functions of ABJM and its dual superYangMills at low ranks of the gauge group. We find matching values of their partition functions, prompting us to conjecture the general equality on the squashed sphere. From the numerics we also observe the squashing dependence of the LeeYang zeros and of the nonperturbative corrections to the all order large $N$ expression for the ABJM partition function.
SciPost Phys. 14, 028 (2023) ·
published 6 March 2023

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We consider the partition function for Euclidean $SU(N)$ super YangMills on a squashed sevensphere. We show that the localization locus of the partition function has instanton membrane solutions wrapping the six ``fixed" threespheres on the $\mathbb{S}^7$. The ADHM variables of these instantons are fields living on the membrane world volume. We compute their contribution by localizing the resulting threedimensional supersymmetric field theory. In the roundsphere limit the individual instanton contributions are singular, but the singularities cancel when adding the contributions of all six threespheres. The full partition function on the ${\mathbb S}^7$ is welldefined even when the square of the effective YangMills coupling is negative. We show for an $SU(2)$ gauge theory in this regime that the bare negative tension of the instanton membranes is canceled off by contributions from the instanton partition function, indicating the existence of tensionless membranes. We provide evidence that this phase is distinct from the usual weakly coupled super YangMills and, in fact, is gravitational.
SciPost Phys. 13, 058 (2022) ·
published 21 September 2022

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We study $\mathcal{N}=2$ theories on fourdimensional manifolds that admit a Killing vector $v$ with isolated fixed points. It is possible to deform these theories by coupling positiondependent background fields to the flavor current multiplet. The partition function of the deformed theory only depends on the value of the background scalar fields at the fixed points of $v$. For a single adjoint hypermultiplet, the partition function becomes independent of the supergravity as well as the flavor background if the scalars attain special values at the fixed points. For these special values, supersymmetry at the fixed points enhances from the DonaldsonWitten twist to the Marcus twist or the VafaWitten twist of $\mathcal{N}=4$ SYM. Our results explain the recently observed squashing independence of $\mathcal{N}=2^*$ theory on the squashed sphere and provide a new squashing independent point. Interpreted through the AGTcorrespondence, this implies the $b$independence of torus onepoint functions of certain local operators in Liouville/Toda CFT. The positiondependent deformations imply relations between correlators of partially integrated operators in any $\mathcal{N}=2$ SCFT with flavor symmetries.
SciPost Phys. 12, 025 (2022) ·
published 17 January 2022

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We consider massdeformed theories with ${\cal N}\geq2$ supersymmetry on round and squashed threespheres. By embedding the supersymmetric backgrounds in extended supergravity we show that at special values of mass deformations the supersymmetry is enhanced on the squashed spheres. When the $3d$ partition function can be obtained by a limit of a $4d$ index we also show that for these special mass deformations only the states annihilated by extra supercharges contribute to the index. By using an equivalence between partition functions on squashed spheres and ellipsoids, we explain the recently observed squashing independence of the partition function of massdeformed ABJ(M) theory on the ellipsoid. We provide further examples of such simplification for various $3d$ supersymmetric theories.
SciPost Phys. 10, 063 (2021) ·
published 10 March 2021

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We study the free energy of fourdimensional CFTs on deformed spheres. For generic nonsupersymmetric CFTs only the coefficient of the logarithmic divergence in the free energy is physical, which is an extremum for the round sphere. We then specialize to $\mathcal{N}=2$ SCFTs where one can preserve some supersymmetry on a compact manifold by turning on appropriate background fields. For deformations of the round sphere the $c$ anomaly receives corrections proportional to the supersymmetric completion of the (Weyl)$^2$ term, which we determine up to one constant by analyzing the scale dependence of various correlators in the stresstensor multiplet. We further show that the double derivative of the free energy with respect to the marginal couplings is proportional to the twopoint function of the bottom components of the marginal chiral multiplet placed at the two poles of the deformed sphere. We then use anomaly considerations and counterterms to parametrize the finite part of the free energy which makes manifest its dependence on the K\"ahler potential. We demonstrate these results for a theory with a vector multiplet and a massless adjoint hypermultiplet using results from localization. Finally, by choosing a special value of the hypermultiplet mass where the free energy is independent of the deformation, we derive an infinite number of constraints between various integrated correlators in $\mathcal{N}=4$ super YangMills with any gauge group and at all values of the coupling, extending previous results.