33 research outputs found
Maslov index, Lagrangians, Mapping Class Groups and TQFT
Given a mapping class f of an oriented surface Sigma and a lagrangian lambda
in the first homology of Sigma, we define an integer n_{lambda}(f). We use
n_{lambda}(f) (mod 4) to describe a universal central extension of the mapping
class group of Sigma as an index-four subgroup of the extension constructed
from the Maslov index of triples of lagrangian subspaces in the homology of the
surface. We give two descriptions of this subgroup. One is topological using
surgery, the other is homological and builds on work of Turaev and work of
Walker. Some applications to TQFT are discussed. They are based on the fact
that our construction allows one to precisely describe how the phase factors
that arise in the skein theory approach to TQFT-representations of the mapping
class group depend on the choice of a lagrangian on the surface.Comment: 31 pages, 11 Figures. to appear in Forum Mathematicu
Ground State Degeneracy in the Levin-Wen Model for Topological Phases
We study properties of topological phases by calculating the ground state
degeneracy (GSD) of the 2d Levin-Wen (LW) model. Here it is explicitly shown
that the GSD depends only on the spatial topology of the system. Then we show
that the ground state on a sphere is always non-degenerate. Moreover, we study
an example associated with a quantum group, and show that the GSD on a torus
agrees with that of the doubled Chern-Simons theory, consistent with the
conjectured equivalence between the LW model associated with a quantum group
and the doubled Chern-Simons theory.Comment: 8 pages, 2 figures. v2: reference added; v3: two appendices adde
Spiders for rank 2 Lie algebras
A spider is an axiomatization of the representation theory of a group,
quantum group, Lie algebra, or other group or group-like object. We define
certain combinatorial spiders by generators and relations that are isomorphic
to the representation theories of the three rank two simple Lie algebras,
namely A2, B2, and G2. They generalize the widely-used Temperley-Lieb spider
for A1. Among other things, they yield bases for invariant spaces which are
probably related to Lusztig's canonical bases, and they are useful for
computing quantities such as generalized 6j-symbols and quantum link
invariants.Comment: 33 pages. Has color figure
A generalized Kac-Ward formula
The Kac-Ward formula allows to compute the Ising partition function on a
planar graph G with straight edges from the determinant of a matrix of size 2N,
where N denotes the number of edges of G. In this paper, we extend this formula
to any finite graph: the partition function can be written as an alternating
sum of the determinants of 2^{2g} matrices of size 2N, where g is the genus of
an orientable surface in which G embeds. We give two proofs of this generalized
formula. The first one is purely combinatorial, while the second relies on the
Fisher-Kasteleyn reduction of the Ising model to the dimer model, and on
geometric techniques. As a consequence of this second proof, we also obtain the
following fact: the Kac-Ward and the Fisher-Kasteleyn methods to solve the
Ising model are one and the same.Comment: 23 pages, 8 figures; minor corrections in v2; to appear in J. Stat.
Mech. Theory Ex
A unified Witten-Reshetikhin-Turaev invariant for integral homology spheres
We construct an invariant J_M of integral homology spheres M with values in a
completion \hat{Z[q]} of the polynomial ring Z[q] such that the evaluation at
each root of unity \zeta gives the the SU(2) Witten-Reshetikhin-Turaev
invariant \tau_\zeta(M) of M at \zeta. Thus J_M unifies all the SU(2)
Witten-Reshetikhin-Turaev invariants of M. As a consequence, \tau_\zeta(M) is
an algebraic integer. Moreover, it follows that \tau_\zeta(M) as a function on
\zeta behaves like an ``analytic function'' defined on the set of roots of
unity. That is, the \tau_\zeta(M) for all roots of unity are determined by a
"Taylor expansion" at any root of unity, and also by the values at infinitely
many roots of unity of prime power orders. In particular, \tau_\zeta(M) for all
roots of unity are determined by the Ohtsuki series, which can be regarded as
the Taylor expansion at q=1.Comment: 66 pages, 8 figure
Ferromagnetic Ordering of Energy Levels for Symmetric Spin Chains
We consider the class of quantum spin chains with arbitrary
-invariant nearest neighbor interactions, sometimes
called for the quantum deformation of , for
. We derive sufficient conditions for the Hamiltonian to satisfy the
property we call {\em Ferromagnetic Ordering of Energy Levels}. This is the
property that the ground state energy restricted to a fixed total spin subspace
is a decreasing function of the total spin. Using the Perron-Frobenius theorem,
we show sufficient conditions are positivity of all interactions in the dual
canonical basis of Lusztig. We characterize the cone of positive interactions,
showing that it is a simplicial cone consisting of all non-positive linear
combinations of "cascade operators," a special new basis of
intertwiners we define. We also state applications to
interacting particle processes.Comment: 23 page
Super-A-polynomials for Twist Knots
We conjecture formulae of the colored superpolynomials for a class of twist
knots where p denotes the number of full twists. The validity of the
formulae is checked by applying differentials and taking special limits. Using
the formulae, we compute both the classical and quantum super-A-polynomial for
the twist knots with small values of p. The results support the categorified
versions of the generalized volume conjecture and the quantum volume
conjecture. Furthermore, we obtain the evidence that the Q-deformed
A-polynomials can be identified with the augmentation polynomials of knot
contact homology in the case of the twist knots.Comment: 22+16 pages, 16 tables and 5 figures; with a Maple program by Xinyu
Sun and a Mathematica notebook in the ancillary files linked on the right; v2
change in appendix B, typos corrected and references added; v3 change in
section 3.3; v4 corrections in Ooguri-Vafa polynomials and quantum
super-A-polynomials for 7_2 and 8_1 are adde