The mixed spin P-fields (MSP for short) theory sets up a geometric platform to relate Gromov-Witten invariants of the quintic three-fold and Fan-Jarvis-Ruan-Witten invariants of the quintic polynomial in five variables. It starts with Wittens vision and the P-fields treatment of GW invariants and FJRW invariants. Then it brie y discusses the master space technique and its application to the set-up of the MSP moduli. Some key results in MSP theory are explained and some examples are provided.

Landau-Ginzburg/Calabi-Yau correspondence claims the equivalence between the Gromov-Witten theory and the Fan-Jarvis-Ruan-Witten theory. The authors survey recently developed wall-crossing approach to the Landau-Ginzburg/Calabi-Yau correspondence for a quintic threefold.

The authors describe the relationships between categories of B-branes in different phases of the non-Abelian gauged linear sigma model. The relationship is described explicitly for the model proposed by Hori and Tong with non-Abelian gauge group that connects two non-birational Calabi-Yau varieties studied by Rødland. A grade restriction rule for this model is derived using the hemisphere partition function and it is used to map B-type D-branes between the two Calabi-Yau varieties.

This is a survey article for the mathematical theory of Witten’s Gauged Linear Sigma Model, as developed recently by the authors. Instead of developing the theory in the most general setting, in this paper the authors focus on the description of the moduli.

The author reviews some recent developments in Chern-Simons theory on a hyperbolic 3-manifold M with complex gauge group G. The author focuses on the case of G = SL(N, C) and M being a knot complement: M = S ³ \ K. 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 for G = SL(N, C). He also reviews various applications and open questions regarding the cluster partition function and some of its relation with string theory.

The gauged linear sigma model (GLSM for short) is a 2d quantum field theory introduced by Witten twenty years ago. Since then, it has been investigated extensively in physics by Hori and others. Recently, an algebro-geometric theory (for both abelian and nonabelian GLSMs) was developed by the author and his collaborators so that he can start to rigorously compute its invariants and check against physical predications. The abelian GLSM was relatively better understood and is the focus of current mathematical investigation. In this article, the author would like to look over the horizon and consider the nonabelian GLSM. The nonabelian case possesses some new features unavailable to the abelian GLSM. To aid the future mathematical development, the author surveys some of the key problems inspired by physics in the nonabelian GLSM.

Let (M, g) be an n-dimensional Riemannian manifold and T ² M be its second-order tangent bundle equipped with a lift metric $\tilde g$. In this paper, first, the authors construct some Riemannian almost product structures on (T ² M, $\tilde g$) and present some results concerning these structures. Then, they investigate the curvature properties of (T ² M, $\tilde g$). Finally, they study the properties of two metric connections with nonvanishing torsion on (T ² M, $\tilde g$): The H-lift of the Levi-Civita connection of g to T ² M, and the product conjugate connection defined by the Levi-Civita connection of $\tilde g$ and an almost product structure.

In this paper, the classical Galois theory to the H*-Galois case is developed. Let H be a semisimple and cosemisimple Hopf algebra over a field k, A a left H-module algebra, and A/A ^H a right H*-Galois extension. The authors prove that, if A ^H is a separable k-algebra, then for any right coideal subalgebra B of H, the B-invariants A ^B = {a ∈ A | b · a = ε(b)a, ∀b ∈ B} is a separable k-algebra. They also establish a Galois connection between right coideal subalgebras of H and separable subalgebras of A containing A ^H as in the classical case. The results are applied to the case H = (kG)* for a finite group G to get a Galois 1-1 correspondence.

The L-factor of irreducible χ₁ × χ₂ ⋊ σ defined by Piatetski-Shapiro is computed by using non-split Bessel functional.