201 research outputs found
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
Matrix Integrals and Feynman Diagrams in the Kontsevich Model
We review some relations occurring between the combinatorial intersection
theory on the moduli spaces of stable curves and the asymptotic behavior of the
't Hooft-Kontsevich matrix integrals. In particular, we give an alternative
proof of the Witten-Di Francesco-Itzykson-Zuber theorem --which expresses
derivatives of the partition function of intersection numbers as matrix
integrals-- using techniques based on diagrammatic calculus and combinatorial
relations among intersection numbers. These techniques extend to a more general
interaction potential.Comment: 52 pages; final versio
Matrix recursion for positive characteristic diagrammatic Soergel bimodules for affine Weyl groups
Let be an affine Weyl group, and let be a field of characteristic
. The diagrammatic Hecke category for over is a
categorification of the Hecke algebra for with rich connections to modular
representation theory. We explicitly construct a functor from to
a matrix category which categorifies a recursive representation , where is the rank of the
underlying finite root system. This functor gives a method for understanding
diagrammatic Soergel bimodules in terms of other diagrammatic Soergel bimodules
which are "smaller" by a factor of . It also explains the presence of
self-similarity in the -canonical basis, which has been observed in small
examples. By decategorifying we obtain a new lower bound on the -canonical
basis, which corresponds to new lower bounds on the characters of the
indecomposable tilting modules by the recent -canonical tilting character
formula due to Achar-Makisumi-Riche-Williamson.Comment: 62 pages, many figures, best viewed in colo
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