Multi-trace correlators from permutations as moduli space

Ryo Suzuki
2019 Journal of High Energy Physics  
We study the n-point functions of scalar multi-trace operators in the U(N c ) gauge theory with adjacent scalars, such as N = 4 super Yang-Mills, at tree-level by using finite group methods. We derive a set of formulae of the general n-point functions, valid for general n and to all orders of 1/N c . In one formula, the sum over Feynman graphs becomes a topological partition function on Σ 0,n with a discrete gauge group, which resembles closed string interactions. In another formula, a new
more » ... ton reduction of Feynman graphs generates connected ribbon graphs, which resembles open string interaction. We define the moduli space M gauge g,n from the space of skeleton-reduced graphs in the connected n-point function of gauge theory. This moduli space is a proper subset of M g,n stratified by the genus, and its top component gives a simple triangulation of Σ g,n . Our results may be regarded as a variation of the open/closed duality, exploring the n-point functions of a gauge theory at tree-level from various aspects. Correspondingly, this paper consists of two parts. In sections 2 and 3, closed-string-like formulae of the npoint functions are discussed, where one of the formulae has natural pants decomposition. In sections 4 and 5, open-string-like formulae of the n-point functions are discussed, where the skeleton-reduced graphs define the moduli space M gauge g,n through triangulation. 1 This is different from the stratification arising in different situations, such as the Deligne-Mumford compactification of the moduli space [24] (see also [16, 25] ), hermitian matrix model [26] and hexagonization program [8, 9] . 2 Matrix models cover the full moduli space, and describe CFT's in the continuum limit [35] . 13 In the literature, Stab(α•) is also called the automorphism group Aut(α•). 14 Since all operators are assumed to be single-traces, λ k is the number of single-trace operators of length k. 15 At this point n is a free parameter. We will equate n with the number of external single-trace operators in (2.65).
doi:10.1007/jhep05(2019)168 fatcat:achcabpawfellmuwxke3cax4ra