  In this work, we present a mathematical theory for Dirac points and interface
modes in honeycomb topological photonic structures consisting of impenetrable
obstacles. Starting from a honeycomb lattice of obstacles attaining
$120^\circ$-rotation symmetry and horizontal reflection symmetry, we apply the
boundary integral equation method to show the existence of Dirac points for the
first two bands at the vertices of the Brillouin zone. We then study interface
modes in a joint honeycomb photonic structure, which consists of two periodic
lattices obtained by perturbing the honeycomb one with Dirac points
differently. The perturbations break the reflection symmetry of the system, as
a result, they annihilate the Dirac points and generate two structures with
different topological phases, which mimics the quantum valley Hall effect in
topological insulators. We investigate the interface modes that decay
exponentially away from the interface of the joint structure in several
configurations with different interface geometries, including the zigzag
interface, the armchair interface, and the rational interfaces. Using the layer
potential technique and asymptotic analysis, we first characterize the band-gap
opening for the two perturbed periodic structures and derive the asymptotic
expansions of the Bloch modes near the band gap surfaces. By formulating the
eigenvalue problem for each joint honeycomb structure using boundary integral
equations over the interface and analyzing the characteristic values of the
associated boundary integral operators, we prove the existence of interface
modes when the perturbation is small.
