Quantum mechanics of bipartite ribbon graphs: Integrality, Lattices and Kronecker coefficients

Abstract

We define solvable quantum mechanical systems on a Hilbert space spanned by bipartite ribbon graphs with a fixed number of edges. The Hilbert space is also an associative algebra, where the product is derived from permutation group products. The existence and structure of this Hilbert space algebra has a number of consequences. The algebra product, which can be expressed in terms of integer ribbon graph reconnection coefficients, is used to define solvable Hamiltonians with eigenvalues expressed in terms of normalized characters of symmetric group elements and degeneracies given in terms of Kronecker coefficients, which are tensor product multiplicities of symmetric group representations. The square of the Kronecker coefficient for a triple of Young diagrams is shown to be equal to the dimension of a sub-lattice in the lattice of ribbon graphs. This leads to an answer to the long-standing question of a combinatorial interpretation of the Kronecker coefficients. As avenues for future research, we discuss applications of the ribbon graph quantum mechanics in algorithms for quantum computation. We also describe a quantum membrane interpretation of these quantum mechanical systems.

String theory for builders: New mathematical perspectives on algorithms and complexity in quantum physics

“…In a series of recent papers, Dr Sanjaye Ramgoolam of Queen Mary University of London and a team of international collaborators started a research programme to take a fresh look at simple mathematical models of string theory and duality [1][2][3][4][5][6] with a view to wider gravity. One study in the early 1990s, for example, identified certain mathematical model universes within string theory which exhibited a striking 'duality' with ordinary quantum theories, with no explicit strings or gravity [10].…”

“…It is precisely this kind of large particle limit in quantum field theories that features in the AdS/CFT correspondence. In their papers, Dr Ramgoolam and Dr Ben Geloun developed quantum mechanical models where the quantum states are combinations of ribbon graphs [1,2,6]. In this framework, they found new algorithms for computing group theoretic quantities, called 'Kronecker coefficients', which are of particular interest in complexity theory.…”