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Register a new userTailoring Three-Point Functions and Integrability IV. Theta-morphism
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We compute structure constants in N=4 SYM at one loop using Integrability. This requires having full control over the two loop eigenvectors of the dilatation operator for operators of arbitrary size. To achieve this, we develop an algebraic description called the Theta-morphism. In this approach we introduce impurities at each spin chain site, act with particular differential operators on the standard algebraic Bethe ansatz vectors and generate in this way higher loop eigenvectors. The final results for the structure constants take a surprisingly simple form. For some quantities we conjecture all loop generalizations. These are based on the tree level and one loop patterns together and also on some higher loop experiments involving simple operators.
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We compute three-point functions of single trace operators in planar N=4 SYM. We consider the limit where one of the operators is much smaller than the other two. We find a precise match between weak and strong coupling in the Frolov-Tseytlin classical limit for a very general class of classical solutions. To achieve this match we clarify the issue of back-reaction and identify precisely which three-point functions are captured by a classical computation.
In this paper, we develop a new method of computing three-point functions in the SU(2) sector of the $mathcal{N}=4$ super Yang-Mills theory in the semi-classical regime at weak coupling, which closely parallels the strong coupling analysis. The structure threading two disparate regimes is the so-called monodromy relation, an identity connecting the three-point functions with and without the insertion of the monodromy matrix. We shall show that this relation can be put to use directly for the semi-classical regime, where the dynamics is governed by the classical Landau-Lifshitz sigma model. Specifically, it reduces the problem to a set of functional equations, which can be solved once the analyticity in the spectral parameter space is specified. To determine the analyticity, we develop a new universal logic applicable at both weak and strong couplings. As a result, compact semi-classical formulas are obtained for a general class of three-point functions at weak coupling including the ones whose semi-classical behaviors were not known before. In addition, the new analyticity argument applied to the strong coupling analysis leads to a modification of the integration contour, producing the results consistent with the recent hexagon bootstrap approach. This modification also makes the Frolov-Tseytlin limit perfectly agree with the weak coupling form.
Adapting the powerful integrability-based formalism invented previously for the calculation of gluon scattering amplitudes at strong coupling, we develop a method for computing the holographic three point functions for the large spin limit of Gubser-Klebanov- Polyakov (GKP) strings. Although many of the ideas from the gluon scattering problem can be transplanted with minor modifications, the fact that the information of the external states is now encoded in the singularities at the vertex insertion points necessitates several new techniques. Notably, we develop a new generalized Riemann bilinear identity, which allows one to express the area integral in terms of appropriate contour integrals in the presence of such singularities. We also give some general discussions on how semiclassical vertex operators for heavy string states should be constructed systematically from the solutions of the Hamilton-Jacobi equation.
We compute non-extremal three-point functions of scalar operators in $mathcal{N}=4$ super Yang-Mills at tree-level in $g_{YM}$ and at finite $N_c$, using the operator basis of the restricted Schur characters. We make use of the diagrammatic methods called quiver calculus to simplify the three-point functions. The results involve an invariant product of the generalized Racah-Wigner tensors ($6j$ symbols). Assuming that the invariant product is written by the Littlewood-Richardson coefficients, we show that the non-extremal three-point functions satisfy the large $N_c$ background independence; correspondence between the string excitations on $AdS_5 times S^5$ and those in the LLM geometry.
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.
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