String Theory Journal Club

Supersymmetric theories and topological field theories are examples of exactly solvable quantum field theories. Although for very different reasons, it is possible to compute observables non-perturbatively. Surprisingly, string theory predicts that a certain class of supersymmetric theories characterised by a directed graph, called a quiver, are dual to certain topological field theories, which encode knot invariants. This duality is now known as the knots-quivers correspondence. We will explore this duality in two ways: First, we show that several quivers may be associated to the same knot. We find simple symmetry transformations relating all the quivers associated to the same knot. We further discover that this rich web of dualities is constructed from simple combinatorial objects known as permutohedra. Second, A-polynomials encode the leading classical behaviour of knot invariants. We then extend the notion of A-polynomials to quivers, and provide explicit formulas for various examples.