We review some recent developments in Chern-Simons theory on a hyperbolic 3-manifold M with complex gauge group G. We focus on the case G=SL(N,C) and with M a knot complement. The main result presented in this note is the cluster partition function, a computational tool that uses cluster algebra techniques to evaluate the Chern-Simons path integral. We also review various applications and open questions regarding the cluster partition function and some of its relation with string theory.

Based on three general guiding principles, i.e., no double occupancy constraint, accurate description of antiferromagnetism at half-filling, and the precise sign structure of the t-J model, a new ground state wave function has been constructed recently [Weng, New J. Phys. 13, 103039 (2011)]. In this paper, we specifically study such kind of variational ground state for the one-hole-doped anisotropic two-leg t-J ladder using variational Monte Carlo (VMC) method. The results are then systematically compared with those recently obtained by density matrix renormalization group (DMRG) simulation. An excellent agreement is found between the VMC and DMRG results, including a "quantum critical point" at the anisotropy parameter α = α_c ≈ 0.7 (with the parameters t/J=3), and the emergence of charge modulation and momentum (Fermi point) reconstruction at α > α_c due to the quantum interference of the sign structure. In particular, the wave function indicates that a Landau's quasiparticle description remains valid at α < α_c but fails at α > α_c due to the breakdown of the one-to-one correspondence of momentum and translational symmetry of the hole. The explicit form of the wave function provides a direct understanding on how the many-body strong correlation effect takes place non-perturbatively in a doped Mott insulator, which sheds interesting light on the two-dimensional case where the same type of wave function was proposed to describe the cuprate superconductor.

We study 2d and 3d gravity theories on spacetimes with causal (timelike or null) codimension one boundaries while allowing for variations in the position of the boundary. We construct the corresponding solution phase space and specify boundary degrees freedom by analysing boundary (surface) charges labelling them. We discuss Y and W freedoms and change of slicing in the solution space. For D dimensional case we find D + 1 surface charges, which are generic functions over the causal boundary. We show that there exist solution space slicings in which the charges are integrable. For the 3d case there exists an integrable slicing where charge algebra takes the form of Heisenberg ⊕ A3 where A3 is two copies of Virasoro at Brown-Henneaux central charge for AdS3 gravity and BMS3 for the 3d flat space gravity.

We study tree level scattering amplitudes of four massless states in the double scaled little string theory, and compare them to perturbative loop amplitudes in six-dimensional super-Yang-Mills theory. The little string amplitudes are computed from correlators in the cigar coset CFT and in N = 2 minimal models. The results are expressed in terms of integrals of conformal blocks and evaluated numerically in the α ′ expansion. We find striking agreements with up to 2-loop scattering amplitudes of massless gluons in 6D SU(k) SYM at a ℤ_k invariant point on the Coulomb branch. We comment on the issue of UV divergence at higher loop orders in the gauge theory and discuss the implication of our results.

Mirror principle is a general method developed in [LLY1]-[LLY4] to compute characteristic classes and characteristic numbers on moduli spaces of stable maps in terms of hypergeometric type series. The counting of the numbers of curves in Calabi-Yau manifolds from mirror symmetry corresponds to the computation of Euler numbers. This principle computes quite general Hirzebruch multiplicative classes such as the total Chern classes.