Graph duality as an instrument of Gauge-String correspondence (Q2814216)

In this paper, the authors, within the framework of the CFT/AdS or gauge/gravity correspondence, give a physical interpretation of the Borodin-Olshanski identity among certain probability-like combinatorial quantities associated to the branching graphs of the symmetric groups and of the unitary groups, respectively. This identity is akin to the Schur-Weyl duality and is expected to play a similar role in understanding the CFT/AdS correspondence better.NEWLINENEWLINEMore precisely (see Section IIA and IIB in the paper), out of all the symmetric groups \(\mathrm S_1\subset\mathrm S_2\subset\dots\subset\mathrm S_n\subset\dots\) and their irreducible representations, one can construct the so-called Young graph with certain combinatorial quantities, called a Young Bouquet, attached to all edges. In a similar way, out of all the unitary groups \(\mathrm U(1)\subset\mathrm U(2)\subset\dots\subset\mathrm U(n)\subset\dots\) and their irreducible representations, one can construct the so-called Gelfand-Tsetlin graph with certain combinatorial expressions attached to all edges. The Borodin-Olshanski identity is an equality among these probability-like expressions in a certain limit (see Section IIC and in particular Equation (24) in the paper). The authors physically interpret this combinatorial identity as follows. They find that a Young Bouquet on the symmetric group side of the Borodin-Olshanski identity can be identified on the CFT side of the CFT/AdS correspondence with squares of \(3\)-point functions; while the Gelfand-Tsetlin expressions stemming from the unitary group side of the Borodin-Olshanski identity can be identified on the AdS side of the CFT/AdS correspondence with gravitational scattering probabilites between two AdS regions separated by the CFT domain wall (see Sections III, IV and V in the paper).