Higher-point integrands in $\mathcal{N} = 4$ super Yang-Mills theory

We derive a nonperturbative, convergent operator product expansion (OPE) for null-integrated operators on the same null plane in a CFT. The objects appearing in the expansion are light-ray operators, whose matrix elements can be computed by the generalized Lorentzian inversion formula. For example, a product of average null energy (ANEC) operators has an expansion in the light-ray operators that appear in the stress-tensor OPE. An important application is to collider event shapes. The light-ray OPE gives a nonperturbative expansion for event shapes in special functions that we call celestial blocks. As an example, we apply the celestial block expansion to energy-energy correlators in mathcal{N} = 4 Super Yang-Mills theory. Using known OPE data, we find perfect agreement with previous results both at weak and strong coupling, and make new predictions at weak coupling through 4 loops (NNNLO).

By constructing a nilpotent extended Becchi-Rouet-Stora-Tyutin (BRST) operator $\overline{s}$ that involves the $N=2$ global supersymmetry transformations of one chirality, we show that the standard $N=2$ off-shell super Yang-Mills action can be represented as an exact BRST term $\overline{s}\ensuremath{\Psi},$ if the gauge fermion $\ensuremath{\Psi}$ is allowed to depend on the inverse powers of supersymmetry (SUSY) ghosts. By using this nonanalytical structure of the gauge fermion (via inverse powers of supersymmetry ghosts), we give field redefinitions in terms of composite fields of SUSY ghosts and $N=2$ fields and we show that Witten's topological Yang-Mills (TYM) theory can be obtained from the ordinary Euclidean $N=2$ super Yang-Mills (SYM) theory directly by using such field redefinitions. In other words, TYM theory is obtained as a change of variables (without twisting). As a consequence it is found that physical and topological interpretations of $N=2$ SYM theory are intertwined together due to the requirement of analyticity of global SUSY ghosts. Moreover, after an instanton-inspired truncation of the model is used, we show that the given field redefinitions yield the Baulieu-Singer formulation of topological Yang-Mills theory.

We consider the hexagonal Wilson loop dual to the six-point MHV amplitude in planar $ \mathcal{N} = 4 $ super Yang-Mills theory. We apply constraints from the operator product expansion in the near-collinear limit to the symbol of the remainder function at three loops. Using these constraints, and assuming a natural ansatz for the symbol’s entries, we determine the symbol up to just two undetermined constants. In the multi-Regge limit, both constants drop out from the symbol, enabling us to make a non-trivial confirmation of the BFKL prediction for the leading-log approximation. This result provides a strong consistency check of both our ansatz for the symbol and the duality between Wilson loops and MHV amplitudes. Furthermore, we predict the form of the full three-loop remainder function in the multi-Regge limit, beyond the leading-log approximation, up to a few constants representing terms not detected by the symbol. Our results confirm an all-loop prediction for the real part of the remainder function in multi-Regge 3 → 3 scattering. In the multi-Regge limit, our result for the remainder function can be expressed entirely in terms of classical polylogarithms. For generic six-point kinematics other functions are required.

Quantum metamorphosis of conformal symmetry in N=4 super Yang–Mills theory

We study the operator product expansion (OPE) limit of correlation functions in field theories which possess string theory duals, from the point of view of the string worldsheet. We show how the interesting (single-trace) terms in the OPE of the field theory arise in this limit from the OPE of the worldsheet theory of the string dual, using a dominant saddle point which appears in computations of worldsheet correlation functions in the space-time OPE limit. The worldsheet OPE generically contains only non-physical operators, but all the non-physical contributions are resummed by the saddle point to a contribution similar to that of a physical operator, which exactly matches the field theory expectations. We verify that the OPE limit of the worldsheet theory does not have any other contributions to the OPE limit of space-time correlation functions. Our discussion is completely general and applies to any local field theory (conformal at high energies) that has a weakly coupled string theory dual (with arbitrary curvature). As a first application, we compare our results to a proposal of R. Gopakumar for the string theory dual of free gauge theories.

We consider conformal mathcal{N}=2 super Yang-Mills theories with gauge group SU(N) and Nf = 2N fundamental hypermultiplets in presence of a circular 1/2-BPS Wilson loop. It is natural to conjecture that the matrix model which describes the expectation value of this system also encodes the one-point functions of chiral scalar operators in presence of the Wilson loop. We obtain evidence of this conjecture by successfully comparing, at finite N and at the two-loop order, the one-point functions computed in field theory with the vacuum expectation values of the corresponding normal-ordered operators in the matrix model. For the part of these expressions with transcendentality ζ(3), we also obtain results in the large-N limit that are exact in the ’t Hooft coupling λ.

Operator product expansion in two-dimensional quantum gravity

We show, within the framework of the massive Euclidean $${\varphi^4}$$ -quantum field theory in four dimensions, that the Wilson operator product expansion (OPE) is not only an asymptotic expansion at short distances as previously believed, but even converges at arbitrary finite distances. Our proof rests on a detailed estimation of the remainder term in the OPE, of an arbitrary product of composite fields, inserted as usual into a correlation function with further “spectator fields”. The estimates are obtained using a suitably adapted version of the method of renormalization group flow equations. Convergence follows because the remainder is seen to become arbitrarily small as the OPE is carried out to sufficiently high order, i.e. to operators of sufficiently high dimension. Our results hold for arbitrary, but finite, loop orders. As an interesting side-result of our estimates, we can also prove that the “gradient expansion” of the effective action is convergent.

The background field method for N=2 super Yang-Mills theories in harmonic superspace

Finiteness preserving mass terms in N=4 super Yang-Mills theory

We perform Monte Carlo calculation of correlation functions in 4d N = 4 super Yang-Mills theory on R S 3 in the planar limit. In order to circumvent the well-known problem of lattice SUSY, we adopt the idea of a novel large-N reduction, which reduces the calculation to that of corresponding correlation functions in the plane-wave matrix model or the BMN matrix model. This model is a 1d gauge theory with 16 supersymmetries, which can be simulated in a manner similar to the recent studies of the D0-brane system. We study two-point and three-point functions of chiral primary operators at various coupling constant, and find that they agree with the free theory results up to overall constant factors. The ratio of the overall factors for two-point and three-point functions agrees with the prediction of the AdS/CFT correspondence.

Certain aspects of the AdS/CFT correspondence are studied in detail. We investigate the one-loop mass shift to certain two-impurity string states in light-cone string field theory on a plane wave background. We find that there exist logarithmic divergences in the sums over intermediate mode numbers which cancel between the cubic Hamiltonian and quartic contact term. We argue that generically, every order in intermediate state impurities contributes to the mass shift at leading perturbative order. The same mass shift is also computed using an improved 3-string vertex proposed by Dobashi and Yoneya. The result is found to agree with gauge theory at leading order and is close but not quite in agreement at subleading order. We extend the analysis to include discrete light-cone quantization, considering states with up to three units of p+. We study the (apparently) first-order phase transition in the weakly coupled plane-wave matrix model at finite temperature. We analyze the effect of interactions by computing the relevant parts of the effective potential for the Polyakov loop operator to three loop order. We show that the phase transition is indeed of first order. We also compute the 2-loop correction to the Hagedorn temperature. Finally, correlation functions of 1/4 BPS Wilson loops with the infinite family of 1/2 BPS chiral primary operators are computed in N=4 super Yang-Mills theory by summing planar ladder diagrams. The correlation functions are also computed in the strong-coupling limit using string theory; the result is found to agree with the extrapolation of the planar ladders. The result is related to similar correlators of 1/2 BPS loops by a simple re-scaling of the coupling constant, discovered by Drukker for the case of the 1/4 BPS loop VEV.

Aspects of the D=4, N=4 superconformal symmetry relevant to the AdS/CFT duality and integrability are reviewed. These include the Lie superalgebra psu(2,2|4), its representations, conformal transformations and correlation functions in N=4 super Yang-Mills theory as well as an illustration of the AdS5xS5 superspace on which the dual string theory is formulated.

The large linear superconformal algebra (generated by four spin- currents, seven spin-1 currents, four spin- currents and one spin-2 current) found by Sevrin, Troost and Van Proeyen (and other groups) was realized in the superconformal coset theory previously. The lowest 16 higher spin currents of spins , , and are obtained by starting with the operator product expansions (OPEs) between the four spin- currents from the above large linear superconformal algebra and the lowest higher spin-1 current which is the same as the one in the Wolf space coset theory. These OPEs determine the four higher spin- currents and the next six higher spin-2 currents are obtained from the OPEs between the above four spin- currents associated with the supersymmetry and these four higher spin- currents. The four higher spin- currents can be determined by calculating the OPEs between the above four spin- currents and the higher spin-2 currents. Similarly, the higher spin-3 current is obtained from the OPEs between the four spin- currents and the higher spin- currents. The explicit relations between the above 16 higher spin currents and the corresponding 16 higher spin currents which were found in the extension of large nonlinear superconformal algebra previously are given. By examining the OPEs between the 16 currents from the large linear superconformal algebra and the 16 higher spin currents, the match with the findings of Beccaria, Candu and Gaberdiel is also given. The next 16 higher spin currents of spins , , and occur from the OPEs between the above lowest 16 higher spin currents.

This paper studies various properties of amplitudes in 3d super Yang Mills theory. First we explain how to obtain the amplitudes of 3d super Yang Mills theories from 4d super Yang Mills theories and obtain their helicity structure. Next, we use a 3d BFCW recursion relation to show that the tree amplitudes and loop integrands of maximal 3d super Yang Mills have dual conformal covariance (although not invariance, so that the amplitudes themselves are not dual conformal). Finally, we argue that the one-loop amplitudes of maximal 3d super Yang-Mills can be reduced to scalar box diagrams and evaluate these diagrams using dimensional regularization. We find that the one-loop MHV amplitudes vanish and the one-loop non-MHV amplitudes are finite.