Topological string correlators from matrix models

Correlators in symmetric orbifold CFTs are given by a finite sum of admissible branched covers of the 2d spacetime. We consider a Gross-Mende like limit where all operators have large twist, and show that the corresponding branched covers can be described via a Penner-like matrix model. The limiting branched covers are given in terms of the spectral curve for this matrix model, which remarkably turns out to be directly related to the Strebel quadratic differential on the covering space. Interpreting the covering space as the world-sheet of the dual string theory, the spacetime CFT correlator thus has the form of an integral over the entire world-sheet moduli space weighted with a Nambu-Goto-like action. Quite strikingly, at leading order this action can also be written as the absolute value of the Schwarzian of the covering map.Given the equivalence of the symmetric product CFT to tensionless string theory on AdS3, this provides an explicit realisation of the underlying mechanism of gauge-string duality originally proposed in [1] and further refined in [2].

Section: Jhep05(2021)073mentioning

confidence: 98%

Section: Jhep05(2021)073mentioning

confidence: 99%

Section: The Spectral Curve and Feynman Diagramsmentioning

confidence: 99%

Section: Finite N Generalisationmentioning

confidence: 99%

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From symmetric product CFTs to AdS3

Gaberdiel

Gopakumar

Knighton

et al. 2021

“…Recall that any Feynman diagrams in the double-line notation can be drawn on a Riemann surface. 39 We count the dimensions of M gauge g,n by identifying its generators as the edges of the ribbon graph.…”

Section: D5 Feynman Graphs On a Riemann Surfacementioning

confidence: 99%

Multi-trace correlators from permutations as moduli space

Suzuki

2019

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 skeleton 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 .rsuzuki.mp at gmail.com constraints [12,13]. The finite group methods are used to compute various quantities, such as partition functions, two-point functions, and recently extremal n-point functions [14].We ask two questions in this paper. How do we find general non-extremal n-point functions in the finite group methods, at any n and to any orders of 1/N c ? And how do these correlators describe Riemann surfaces?A similar problem was studied in the Hermitian matrix model, which is a simpler version of large N c gauge theory [15]. This matrix model describes two-dimensional gravity in the continuum limit, and its exact free energy is given by the τ -function of the KdV hierarchy [16,17].We will study the tree-level n-point functions of scalar operators in N = 4 SYM with U (N c ) gauge group. We express Feynman graphs in terms of permutations, and describe the space of Wick-contractions in an algebraic manner. We obtain a formula which naturally factorizes into the product of pairs of pants, i.e. three-point functions. This formula is invariant different pants decomposition, and resembles the interaction of n closed strings.Then we perform a skeleton reduction to the Feynman graphs in the n-point functions. Under the skeleton reduction, Feynman graphs become connected metric ribbon graphs. It is known that there is an isomorphism between the space of connected metric ribbon graphs and the decorated moduli space of Riemann surfaces [18][19][20][21]. Thus we define the moduli space of Riemann surfaces in gauge theory M gauge g,n as the space of Wick-contractions in the skeleton-reduced Feynman graphs. These graphs resemble the interaction of open strings which triangulates Σ g,n .The gauge theory moduli space M gauge g,n exhibits two properties. First, it is a proper subset of the moduli space of the decorated arithmetic Riemann surfaces, equivalent to the connected integral ribbon graphs [22,23]. Second, our definition of the skeleton reduction stratifies M gauge g,n by genus, meaning that the diagrams with smaller genera contribute to higher powers of 1/N c . 1In [27] the correlators of gauge theory are used to define an effective two-dimensional worldsheet ...

Hidden symmetries and large N factorisation for permutation invariant matrix observables

Barnes

Padellaro

Ramgoolam

2022

Permutation invariant polynomial functions of matrices have previously been studied as the observables in matrix models invariant under SN, the symmetric group of all permutations of N objects. In this paper, the permutation invariant matrix observables (PIMOs) of degree k are shown to be in one-to-one correspondence with equivalence classes of elements in the diagrammatic partition algebra Pk (N). On a 4-dimensional subspace of the 13-parameter space of SN invariant Gaussian models, there is an enhanced O(N) symmetry. At a special point in this subspace, is the simplest O(N) invariant action. This is used to define an inner product on the PIMOs which is expressible as a trace of a product of elements in the partition algebra. The diagram algebra Pk (N) is used to prove the large N factorisation property for this inner product, which generalizes a familiar large N factorisation for inner products of matrix traces invariant under continuous symmetries.