Path integral on star graph

In this paper we study path integral for a single spinless particle on a star graph with NN edges, whose vertex is known to be described by U(N)U(N) family of boundary conditions. After carefully studying the free particle case, both at the critical and off-critical levels, we propose a new path integral formulation that correctly captures all the scale-invariant subfamily of boundary conditions realized at fixed points of boundary renormalization group flow. Our proposal is based on the folding trick, which maps a scalar-valued wave function on star graph to an NN-component vector-valued wave function on half-line. All the parameters of scale-invariant subfamily of boundary conditions are encoded into the momentum independent weight factors, which appear to be associated with the two distinct path classes on half-line that form the cyclic group Z2Z2. We show that, when bulk interactions are edge-independent, these weight factors are generally given by an NN-dimensional unitary representation of Z2Z2. Generalization to momentum dependent weight factors and applications to worldline formalism are briefly discussed.

► We propose the new path integral formulation on star graph with NN edges.
► U(N)U(N) family of boundary conditions is well-described by weight factors.
► The scale-invariant weight factor is given by NN-dimensional unitary representation of Z2Z2.
► Generalization to momentum dependent weight factors is briefly discussed.