We propose a polynomial time computable graph invariant, which comes out directly from a modified version of discrete Greens function on a graph. It is our hope that it can help to construct fast algorithms for the graph isomorphism testing. We explain the physics behind this discrete Greens function, which is a very basic idea applied to graphs.

The phase space of a particle or a mechanical system contains an intrinsic symplectic structure, and hence, it is a symplectic manifold. Recently, new invariants for symplectic manifolds in terms of cohomologies of differential forms have been introduced by Tseng and Yau. Here, we discuss the physical motivation behind the new symplectic invariants and analyze these invariants for phase space, i.e., the non-compact cotangent bundle.

We quantize a string in the de Sitter background, and we find that the mass spectrum is modified by a term which is quadratic in oscillating numbers, and also proportional to the square of the Hubble constant.

We consider a Chern–Simons theory of planar matter fields interacting with the Chern–Simons gauge field in a SU(N)_global ⊗ U(1)_local invariant fashion. We classify the radially symmetric soliton solutions of the system in terms of the prescribed value of magnetic flux associated with this model. We also prove the uniqueness of the topological solution in a certain condition

Classical two-dimensional Liouville gravity is often considered in conformal gauge which has a residual left and right Virasoro symmetry algebra. We consider an alternate, chiral, gauge which has a residual right Virasoro Kac-Moody algebra, and no left Virasoro algebra. The Kac-Moody zero mode is the left-moving energy. Dirac brackets of the constrained Hamiltonian theory are derived, and the residual symmetries are shown to be generated by integrals of the conserved chiral currents. The central charge and Kac-Moody level are computed. The possible existence of a corresponding quantum theory is discussed.