The mathematical picture language project that we began in 2016 has already yielded interesting results. We also point out areas of mathematics and physics where we hope that it will prove useful in the future.

In recent years, fermionic topological phases of quantum matter has attracted a lot of attention. In a pioneer work by Gu, Wang and Wen, the concept of equivalence classes of fermionic local unitary(FLU) transformations was proposed to systematically understand non-chiral topological phases in 2D fermion systems and an incomplete classification was obtained. On the other hand, the physical picture of fermion condensation and its corresponding super pivotal categories give rise to a generic mathematical framework to describe fermionic topological phases of quantum matter. In particular, it has been pointed out that in certain fermionic topological phases, there exists the so-called q-type anyon excitations, which have no analogues in bosonic theories. In this paper, we generalize the Gu, Wang and Wen construction to include those fermionic topological phases with q-type anyon excitations. We argue that all non-chiral fermionic topological phases in 2+1D are characterized by a set of tensors (N^{ij}_k,F^{ij}_k,F^{ijm,αβ}_{kln,χδ},n_i,d_i), which satisfy a set of nonlinear algebraic equations parameterized by phase factors Ξ^{ijm,αβ}_{kl}, Ξ^{ij}_{kln,χδ}, Ω^{kim,αβ}_{jl} and Ω^{ki}_{jln,χδ}. Moreover, consistency conditions among algebraic equations give rise to additional constraints on these phase factors which allow us to construct a topological invariant partition for an arbitrary triangulation of 3D spin manifold. Finally, several examples with q-type anyon excitations are discussed, including the Fermionic topological phase from Tambara-Yamagami category for \mathbb{Z}_{2N}, which can be regarded as the \mathbb{Z}_{2N} parafermion generalization of Ising fermionic topological phase.

In this paper, by combining modular forms and characteristic forms, we obtain general anomaly cancellation formulas of any dimension. For 4 k+ 2 dimensional manifolds, our results include the gravitational anomaly cancellation formulas of Alvarez-Gaum and Witten in dimensions 2, 6 and 10 (\cite {AW}) as special cases. In dimension 4 k+ 2, we derive anomaly cancellation formulas for index gerbes. In dimension 4 k+ 2, we obtain certain results about eta invariants, which are interesting in spectral geometry.

Motivated by a recent application of quantum graphs to model the anomalous Hall effect we discuss quantum graphs the vertices of which exhibit a preferred orientation. We describe an example of such a vertex coupling and analyze the corresponding band spectra of lattices with square and hexagonal elementary cells showing that they depend heavily on the network topology, in particular, on the degrees of the vertices involved.

No abstract uploaded!