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We consider the partition function for Euclidean SU(N) super Yang-Mills on a squashed seven-sphere. We show that the localization locus of the partition function has instanton membrane solutions wrapping the six "fixed" three-spheres on the S7. The ADHM vari-ables of these instantons are fields living on the membrane world volume. We compute their contribution by localizing the resulting three-dimensional supersymmetric field theory. In the round-sphere limit the individual instanton contributions are singular, but the singularities cancel when adding the contributions of all six three-spheres. The full partition function on the S7 is well-defined even when the square of the effective Yang -Mills 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 Yang -Mills and, in fact, is gravitational.

We use the quantum spectral curve to compute the Hagedorn temperature for ABJM theory in terms of the interpolating function h(lambda). At weak coupling we compute this temperature up to eight-loop order, showing that it matches the known tree-level and two-loop results. At strong coupling we compute the dependence numerically, showing that it is consistent with expectations from supergravity and the plane-wave limit for the four leading terms in the strong coupling expansion, up to an overall shift of the zero-point energy for type IIA string theory on AdS4 x DOUBLE-STRUCK CAPITAL CP3. We conjecture an analytic form for this shift to leading order that is consistent with our numerical results.

Using the supergravity dual and the plane-wave limit as a guide, we conjecture the asymptotic large coupling form of the Hagedorn temperature for planar N=4 super Yang-Mills to order 1/root lambda . This is two orders beyond the presently known behavior. Using the quantum spectral curve procedure of Harmark and Wilhelm, we show that our conjectured form is in excellent agreement with the numerical results.

We consider mass-deformed theories with N >= 2 supersymmetry on round and squashed three-spheres. 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 mass-deformed ABJ(M) theory on the ellipsoid. We provide further examples of such simplification for various 3d supersymmetric theories.

We study the free energy of four-dimensional 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 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 stress-tensor multiplet. We further show that the double derivative of the free energy with respect to the marginal couplings is proportional to the two-point 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 counter-terms to parametrize the finite part of the free energy which makes manifest its dependence on the Kahler 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 N = 4 super Yang-Mills with any gauge group and at all values of the coupling, extending previous results.

We consider supersymmetric gauge theories on S⁵ with a negative Yang-Mills coupling in their large N limits. Using localization we compute the partition functions and show that the pure SU(N) gauge theory descends to an SU(N/2)_+N/2 × SU(N/2)_−N/2 × SU(2) Chern-Simons gauge theory as the inverse ’t Hooft coupling is taken to negative infinity for N even. The Yang-Mills coupling of the SU(N/2)_±N/2 is positive and infinite, while that on the SU(2) goes to zero. We also show that the odd N case has somewhat different behavior. We then study the SU(N/2)_N/2 pure Chern-Simons theory. While the eigenvalue density is only found numerically, we show that its width equals 1 in units of the inverse sphere radius, which allows us to find the leading correction to the free energy when turning on the Yang-Mills term. We then consider USp(2N) theories with an antisymmetric hypermultiplet and N_f < 8 fundamental hypermultiplets and carry out a similar analysis. Along the way we show that the one-instanton contribution to the partition function remains exponentially suppressed at negative coupling for the SU(N) theories in the large N limit.

We consider anomaly cancellation for SU(N) x SU(2) x U(1) gauge theories where the left-handed chiral multiplets are in higher SU(2) representations. In particular, if the left-handed quarks and leptons transform under the triplet representation of SU(2) and if the U(1) gauge group is compact then up to an overall scaling there is only one possible nontrivial assignment for the hypercharges if N = 3, and two if N = 9. Otherwise there are infinitely many. We use the Mordell-Weil theorem, Mazur's theorem and the Cremona elliptic curve database which uses Kolyvagin's theorem on the Birch Swinnerton-Dyer conjecture to prove these statements.

Using supersymmetric localization we compute the free energy and BPS Wilson loop vacuum expectation values for planar maximally supersymmetric Yang-Mills theory on S-d in the strong coupling limit for 2 <= d < 6. The same calculation can also be performed in supergravity using the recently found spherical brane solutions. We find excellent agreement between the two sets of results. This constitutes a non-trivial precision test of holography in a non-conformal setting. The free energy of maximal SYM on S-6 diverges in the strong coupling limit which might signify the onset of little string theory. We show how this divergence can be regularized both in QFT and in supergravity. We also consider d = 7 with a small negative 't Hooft coupling and show that the free energy and Wilson loop vacuum expectation value agree with the results from supergravity after addressing some subtleties.

We construct a class of theories with 16 supersymmetries on spheres of dimension nine and less. The gauge coupling and mass terms for the scalar fields depend on the polar angle away from the north pole. Assuming finite coupling on the north pole, this leads to zero coupling at the south pole for d>4 and infinite coupling at the south pole for d<4. The underlying supersymmetry algebra of these theories is shown to be isomorphic to the Poincaré superalgebra in d-dimensions. We also give a localization procedure which leads to nontrivial results for d=2.

We compute the perturbative partition functions for gauge theories with eight supersymmetries on spheres of dimension d <= 5, proving a conjecture by the second author. We apply similar methods to gauge theories with four supersymmetries on spheres with d <= 3. The results are valid for non-integer d as well. We further propose an analytic continuation from d = 3 to d = 4 that gives the perturbative partition function for an N = 1 gauge theory. The results are consistent with the free multiplets and the one-loop beta-functions for general N = 1 gauge theories. We also consider the analytic continuation of an N = 1 preserving mass deformation of the maximally supersymmetric gauge theory and compare to recent holographic results for N = 1* super Yang-Mills. We find that the general structure for the real part of the free energy coming from the analytic continuation is consistent with the holographic results.