19,530 research outputs found
The tensor product in the theory of Frobenius manifolds
We introduce the operation of forming the tensor product in the theory of
analytic Frobenius manifolds. Building on the results for formal Frobenius
manifolds which we extend to the additional structures of Euler fields and flat
identities, we prove that the tensor product of pointed germs of Frobenius
manifolds exists. Furthermore, we define the notion of a tensor product diagram
of Frobenius manifolds with factorizable flat identity and prove the existence
such a diagram and hence a tensor product Frobenius manifold. These diagrams
and manifolds are unique up to equivalence. Finally, we derive the special
initial conditions for a tensor product of semi--simple Frobenius manifolds in
terms of the special initial conditions of the factors.Comment: 41 pages, amslatex, uses xy-te
Notes on topological insulators
This paper is a survey of the -valued invariant of topological
insulators used in condensed matter physics. The -valued
topological invariant, which was originally called the TKNN invariant in
physics, has now been fully understood as the first Chern number. The
invariant is more mysterious, we will explain its equivalent
descriptions from different points of view and provide the relations between
them. These invariants provide the classification of topological insulators
with different symmetries in which K-theory plays an important role. Moreover,
we establish that both invariants are realizations of index theorems which can
also be understood in terms of condensed matter physics.Comment: 62 pages, 3 figure
Geometry of the momentum space: From wire networks to quivers and monopoles
A new nano--material in the form of a double gyroid has motivated us to study
(non-commutative geometry of periodic wire networks and the associated
graph Hamiltonians. Here we present the general abstract framework, which is
given by certain quiver representations, with special attention to the original
case of the gyroid as well as related cases, such as graphene. In these
geometric situations, the non- commutativity is introduced by a constant
magnetic field and the theory splits into two pieces: commutative and
non-commutative, both of which are governed by a geometry.
In the non-commutative case, we can use tools such as K-theory to make
statements about the band structure. In the commutative case, we give geometric
and algebraic methods to study band intersections; these methods come from
singularity theory and representation theory. We also provide new tools in the
study, using -theory and Chern classes. The latter can be computed using
Berry connection in the momentum space. This brings monopole charges and issues
of topological stability into the picture.Comment: 31 pages, 4 figure
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