Abstract
We shall interpret the null hexagonal Wilson loop (or, equivalently, six gluon scattering amplitude) in 4D mathcal{N} = 4 Super Yang-Mills, or, precisely, an integral representation of its matrix part, via an ADHM-like instanton construction. In this way, we can apply localisation techniques to obtain combinatorial expressions in terms of Young diagrams. Then, we use our general formula to obtain explicit expressions in several explicit cases. In particular, we discuss those already available in the literature and find exact agreement. Moreover, we are capable to determine explicitly the denominator (poles) of the matrix part, and find some interesting recursion properties for the residues, as well.
Highlights
Most known example of AdS/CFT correspondence [1,2,3], as it lives on the 4d Minkowski boundary of its gravitational dual, the type IIB superstring theory on AdS5 × S5
The proposal was to write the expectation value of the Wilson loops (Wls) as an infinite sum over intermediate excitations on the GKP string vacuum: gluons and their bound states, fermions, antifermions and, scalars. It is reminiscent of the Form Factor (FF) spectral expansion of the correlation functions in integrable 2d quantum field theories, and the pentagonal operator has been identified [25, 28,29,30] with a specific conical twist field [31, 32]
Perhaps the identification of the real ADHM equation counterpart of the matrix equations (2.9)–(2.11) would have some significance as well. Another achievement of this paper may be consider that of a general approach for passing from an integral representation with some group-theoretical structure to a combinatorial sum over Young diagrams
Summary
We rewrite the integral representation of the matrix part of the hexagonal bosonic Wl in such a way to make the connection with the integral representation of instanton partition function in Ω background more obvious. We apply localization technique to present the matrix part as a sum over fixed points, i.e. 2n-tuples of above young diagrams constructed in subsection 2.3. A similar integral arises in a completely different context, the N = 2 SYM theory with gauge group U(n) in Ω-background parameterized by 1 and 2 In the latter case, the k-instanton contribution to the partition function can be written in the form [55, 57]. Upon applying localization techniques for the moduli space of instantons, according to the ADHM construction, the full instanton partition function. This formula can be obtained applying the so called localization technique for moduli space of instantons. In the two subsections we will find an ADHM-like construction such that the corresponding localization formula (see (2.34), (2.35) and (2.37)) gives results compatible with (2.4)
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