As> iVinf^ b^ where dx denotes the degree of the vertex x. In particular, we derive lower bounds of eigenvalues for convex subgraphs which consist of lattice points in an d-dimensional Riemannian manifolds M with convex boundary. The techniques involve both the (discrete) heat kernels of graphs and improved estimates of the (continuous) heat kernels of Riemannian manifolds. We prove eigenvalue lower bounds for convex subgraphs of the form ce2/(dD (M)) 2 where e denotes the distance between two closest lattice points, D (M) denotes the diameter of the manifold M and c is a constant (independent of the dimension d and the number of vertices in 5, but depending on the how" dense" the lattice points are). This eigenvalue bound is useful for bounding the rates of convergence for various random walk problems. Since many enumeration problems can be approximated by considering random walks in convex subgraphs of some appropriate host graph, the eigenvalue inequalities here have many applications.

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We conjecture that the level k U(N) Chern-Simons theory coupled to free anticommuting scalar matter in the fundamental is dual to non-minmal higher-spin Vasiliev gravity in dS4 with parity-violating phase \theta0 = \pi N/2k and Neumann boundary conditions for the scalar. Related conjectures are made for fundamental commuting spinor matter and critical theories. This generalizes a recent conjecture relating the minimal Type A Vasiliev theory in dS4 to the Sp(N) model with fundamental real anti-commuting scalars.

We give a definition of character sheaves on the group compactification which is equivalent to Lusztig's definition in [G. Lusztig, Parabolic character sheaves, II, Mosc. Math. J. 4 (4) (2004) 869896]. We also prove some properties of the character sheaves on the group compactification.

Abelian Chern-Simons theory, characterized by the so-called K matrix, has been quite successful in characterizing and classifying Abelian fractional quantum Hall effect as well as symmetry protected topological (SPT) phases, especially for bosonic SPT phases. However, there are still some puzzles in dealing with fermionic SPT (fSPT) phases. In this paper, we utilize the Abelian Chern-Simons theory to study the fSPT phases protected by arbitrary Abelian total symmetry G_f. Comparing to the bosonic SPT phases, fSPT phases with Abelian total symmetry Gf have three new features: (1) they may support gapless Majorana fermion edge modes, (2) some nontrivial bosonic SPT phases may be trivialized if G_f is a nontrivial extension of bosonic symmetry G_b by \mathbb{Z}_f^2, (3) certain intrinsic fSPT phases can only be realized in interacting fermionic system. We obtain edge theories for various fSPT phases, which can also be regarded as conformal field theories with proper symmetry anomaly. In particular, we discover the construction of Luttinger liquid edge theories with central charge n−1 for type-III bosonic SPT phases protected by (\mathbb{Z}_n)^3 symmetry and the Luttinger liquid edge theories for intrinsically interacting fSPT protected by unitary Abelian symmetry. The ideas and methods used in these examples could be generalized to derive the edge theories of fSPT phases with arbitrary unitary Abelian total symmetry G_f.