We study the space of all kinematically allowed four photon and four graviton S-matrices, polynomial in scattering momenta. We demonstrate that this space is the permutation invariant sector of a module over the ring of polynomials of the Mandelstam invariants s, t and u. We construct these modules for every value of the spacetime dimension D, and so explicitly count and parameterize the most general four photon and four graviton S-matrix at any given derivative order. We also explicitly list the local Lagrangians that give rise to these S-matrices. We then conjecture that the Regge growth of S-matrices in all physically acceptable classical theories is bounded by s 2 at fixed t. A four parameter subset of the polynomial photon S-matrices constructed above satisfies this Regge criterion. For gravitons, on the other hand, no polynomial addition to the Einstein S-matrix obeys this bound for D ≤ 6. For D ≥ 7 there is a single six derivative polynomial Lagrangian consistent with our conjectured Regge growth bound. Our conjecture thus implies that the Einstein four graviton S-matrix does not admit any physically acceptable polynomial modifications for D ≤ 6. A preliminary analysis also suggests that every finite sum of pole exchange contributions to four graviton scattering also violates our conjectured Regge growth bound, at least when D ≤ 6, even when the exchanged particles have low spin.

It has been conjectured that fermions minimally coupled to a Chern-Simons gauge field define a conformal field theory (CFT) that is level-rank dual to Chern-Simons gauged Wilson-Fisher Bosons. The CFTs in question admit relevant deformations parametrized by a real mass. When the mass deformation is positive, the duality of the two deformed theories has previously been checked in detail in the large N limit by comparing explicit all orders results on both sides of the duality. In this paper we perform a similar check for the case of negative mass deformations. In this case the bosonic field condenses triggering the Higgs mechanism. The effective excitations in this phase are massive W bosons. By summing all leading large N graphs involving these W bosons we find an all orders (in the 't Hooft coupling) result for the thermal free energy of the bosonic theory in the condensed phase. Our final answer perfectly matches the previously obtained fermionic free energy under the conjectured duality map. 1 sayantan@aei.mpg.de, sayantan.choudhury@aei.mpg.de 2 anshuman@theory.tifr.res.in 3 indranil.halder@tifr.res.in 4 sachin.jain@iiserpune.ac.in 5 lavneet@theory.tifr.res.in 6 minwalla@theory.tifr.res.in 7 naveensp@theory.tifr.res.in 1 See [10, 11] for further results on correlation functions. 2 See [12-15] for more precise versions of the duality map. 3 See [5,9,10,16,17,20,[23][24][25][33][34][35][36][37][38][39][40][41][42][43][44][45][46] for other large-N computations that provide additional evidence for this duality.

It has recently been demonstrated that the large N limit of a model of fermions charged under the global/gauge symmetry group O(N ) q−1 agrees with the large N limit of the SYK model. In these notes we investigate aspects of the dynamics of the O(N ) q−1 theories that differ from their SYK counterparts. We argue that the spectrum of fluctuations about the finite temperature saddle point in these theories has (q −1) N 2 2 new light modes in addition to the light Schwarzian mode that exists even in the SYK model, suggesting that the bulk dual description of theories differ significantly if they both exist. We also study the thermal partition function of a mass deformed version of the SYK model. At large mass we show that the effective entropy of this theory grows with energy like E ln E (i.e. faster than Hagedorn) up to energies of order N 2 . The canonical partition function of the model displays a deconfinement or Hawking Page type phase transition at temperatures of order 1/ ln N . We derive these results in the large mass limit but argue that they are qualitatively robust to small corrections in J/m.

It has been conjectured that Chern-Simons (CS) gauged 'regular' bosons in the fundamental representation are 'level-rank' dual to CS gauged critical fermions also in the fundamental representation. Generic relevant deformations of these conformal field theories lead to one of two distinct massive phases. In previous work, the large N thermal free energy for the bosonic theory in the unHiggsed phase has been demonstrated to match the corresponding fermionic results under duality. In this note we evaluate the large N thermal free energy of the bosonic theory in the Higgsed phase and demonstrate that our results, again, perfectly match the predictions of duality. Our computation is performed in a unitary gauge by integrating out the physical excitations of the theory -i.e. W bosonsat all orders in the 't Hooft coupling. Our results allow us to construct an exact quantum effective potential forφφ, the lightest gauge invariant scalar operator in the theory. In the zero temperature limit this exact Landau-Ginzburg potential is non-analytic atφφ = 0. The extrema of this effective potential at positiveφφ solve the gap equations in the Higgsed phase while the extrema at negativeφφ solve the gap equations in the unHiggsed phase. Our effective potential is bounded from below only for a certain range of x 6 (the parameter that governs sextic interactions of φ). This observation suggests that the regular boson theory has a stable vacuum only when x 6 lies in this range. 1 anshuman@theory.tifr.res.in 2 indranil.halder@tifr.res.in 3 sachin.jain@iiserpune.ac.in 4 lavneet@theory.tifr.res.in 5 minwalla@theory.tifr.res.in 6 naveensp@theory.tifr.res.in A Previously known results for the large N free energy 32 A.1 Results for the critical fermion theory 33 A.2 Results for regular bosons in the unHiggsed Phase 35 A.3 Duality in the unHiggsed phase and a prediction for the Higgsed phase 36 A.3.1 unHiggsed phase: sgn(X F ) = −sgn(λ B ) 37 A.3.2 Higgsed phase: sgn(X F ) = sgn(λ B ) 37 B The tadpole from W boson loops 38 C The critical boson scaling limit 41 -1 -