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Register a new user${cal N}=4$ supersymmetric Yang-Mills thermodynamics to order $lambda^2$
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We calculate the resummed perturbative free energy of ${cal N}=4$ supersymmetric Yang-Mills in four spacetime dimensions ($text{SYM}_{4,4}$) through second order in the t Hooft coupling $lambda$ at finite temperature and zero chemical potential. Our final result is ultraviolet finite and all infrared divergences generated at three-loop level are canceled by summing over $text{SYM}_{4,4}$ ring diagrams. Non-analytic terms at ${cal O}({lambda}^{3/2}) $ and $ {cal O}({lambda}^2 loglambda )$ are generated by dressing the $A_0$ and scalar propagators. The gauge-field Debye mass $m_D$ and the scalar thermal mass $M_D$ are determined from their corresponding finite-temperature self-energies. Based on this, we obtain the three-loop thermodynamic functions of $text{SYM}_{4,4}$ to ${cal O}(lambda^2)$. We compare our final result with prior results obtained in the weak- and strong-coupling limits and construct a generalized Pad{e} approximant that interpolates between the weak-coupling result and the large-$N_c$ strong-coupling result. Our results suggest that the ${cal O}(lambda^2)$ weak-coupling result for the scaled entropy density is a quantitatively reliable approximation to the scaled entropy density for $0 leq lambda lesssim 2$.
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For all types of N=4 anti-de Sitter (AdS) supersymmetry in three dimensions, we construct manifestly supersymmetric actions for Abelian vector multiplets and explain how to extend the construction to the non-Abelian case. Manifestly N=4 supersymmetric Yang-Mills (SYM) actions are explicitly given in the cases of (2,2) and critical (4,0) AdS supersymmetries. The N=4 vector multiplets and the corresponding actions are then reduced to (2,0) AdS superspace, in which only N=2 supersymmetry is manifest. Using the off-shell structure of the N=4 vector multiplets, we provide complete N=4 SYM actions in (2,0) AdS superspace for all types of N=4 AdS supersymmetry. In the case of (4,0) AdS supersymmetry, which admits a Euclidean counterpart, the resulting N=2 action contains a Chern-Simons term proportional to q/r, where r is the radius of AdS_3 and q is the R-charge of a chiral scalar superfield. The R-charge is a linear inhomogeneous function of X, an expectation value of the N=4 Cotton superfield. Thus our results explain the mysterious structure of N=4 supersymmetric Yang-Mills theories on S^3 discovered in arXiv:1401.7952. In the case of (3,1) AdS supersymmetry, which has no Euclidean counterpart, the SYM action contains both a Chern-Simons term and a chiral mass-like term. In the case of (2,2) AdS supersymmetry, which admits a Euclidean counterpart, the SYM action has no Chern-Simons and chiral mass-like terms.
Non-perturbative investigations of $mathcal N = 4$ supersymmetric Yang--Mills theory formulated on a space-time lattice have advanced rapidly in recent years. Large-scale numerical calculations are currently being carried out based on a construction that exactly preserves a single supersymmetry at non-zero lattice spacing. A recent development is the creation of an improved lattice action through a new procedure to regulate flat directions in a manner compatible with this supersymmetry, by modifying the moduli equations. In this proceedings I briefly summarize this new procedure and discuss the parameter space of the resulting improved action that is now being employed in numerical calculations.
We present some of the latest results from our numerical investigations of N=4 supersymmetric Yang--Mills theory formulated on a space-time lattice. Based on a construction that exactly preserves a single supersymmetry at non-zero lattice spacing, we recently developed an improved lattice action that is now being employed in large-scale calculations. Here we update our studies of the static potential using this new action, also applying tree-level lattice perturbation theory to improve the analysis of the potential itself. Considering relatively weak couplings, we obtain results for the Coulomb coefficient that are consistent with continuum perturbation theory.
This is a pedagogical review on the integrability-based approach to the three-point function in N=4 supersymmetric Yang-Mills theory. We first discuss the computation of the structure constant at weak coupling and show that the result can be recast as a sum over partitions of the rapidities of the magnons. We then introduce a non-perturbative framework, called the hexagon approach, and explain how one can use the symmetries (i.e. superconformal and gauge symmetries) and integrability to determine the structure constants. This article is based on the lectures given in Les Houches Summer School Integrability: From statistical systems to gauge theory in June 2016.
We compute the six-particle maximally-helicity-violating (MHV) and next-to-MHV (NMHV) amplitudes in planar maximally supersymmetric Yang-Mills theory through seven loops and six loops, respectively, as an application of the extended Steinmann relations and using the cosmic Galois coaction principle. Starting from a minimal space of functions constructed using these principles, we identify the amplitude by matching its symmetries and predicted behavior in various kinematic limits. Through five loops, the MHV and NMHV amplitudes are uniquely determined using only the multi-Regge and leading collinear limits. Beyond five loops, the MHV amplitude requires additional data from the kinematic expansion around the collinear limit, which we obtain from the Pentagon Operator Product Expansion, and in particular from its single-gluon bound state contribution. We study the MHV amplitude in the self-crossing limit, where its singular terms agree with previous predictions. Analyzing and plotting the amplitudes along various kinematical lines, we continue to find remarkable stability between loop orders.
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