69 research outputs found
Hopf Categories
We introduce Hopf categories enriched over braided monoidal categories. The
notion is linked to several recently developed notions in Hopf algebra theory,
such as Hopf group (co)algebras, weak Hopf algebras and duoidal categories. We
generalize the fundamental theorem for Hopf modules and some of its
applications to Hopf categories.Comment: 47 pages; final version to appear in Algebras and Representation
Theor
Knot Theory: from Fox 3-colorings of links to Yang-Baxter homology and Khovanov homology
This paper is an extended account of my "Introductory Plenary talk at Knots
in Hellas 2016" conference We start from the short introduction to Knot Theory
from the historical perspective, starting from Heraclas text (the first century
AD), mentioning R.Llull (1232-1315), A.Kircher (1602-1680), Leibniz idea of
Geometria Situs (1679), and J.B.Listing (student of Gauss) work of 1847. We
spend some space on Ralph H. Fox (1913-1973) elementary introduction to diagram
colorings (1956). In the second section we describe how Fox work was
generalized to distributive colorings (racks and quandles) and eventually in
the work of Jones and Turaev to link invariants via Yang-Baxter operators, here
the importance of statistical mechanics to topology will be mentioned. Finally
we describe recent developments which started with Mikhail Khovanov work on
categorification of the Jones polynomial. By analogy to Khovanov homology we
build homology of distributive structures (including homology of Fox colorings)
and generalize it to homology of Yang-Baxter operators. We speculate, with
supporting evidence, on co-cycle invariants of knots coming from Yang-Baxter
homology. Here the work of Fenn-Rourke-Sanderson (geometric realization of
pre-cubic sets of link diagrams) and Carter-Kamada-Saito (co-cycle invariants
of links) will be discussed and expanded.
Dedicated to Lou Kauffman for his 70th birthday.Comment: 35 pages, 31 figures, for Knots in Hellas II Proceedings, Springer,
part of the series Proceedings in Mathematics & Statistics (PROMS
Canonical quantization of non-commutative holonomies in 2+1 loop quantum gravity
In this work we investigate the canonical quantization of 2+1 gravity with
cosmological constant in the canonical framework of loop quantum
gravity. The unconstrained phase space of gravity in 2+1 dimensions is
coordinatized by an SU(2) connection and the canonically conjugate triad
field . A natural regularization of the constraints of 2+1 gravity can be
defined in terms of the holonomies of . As a first step
towards the quantization of these constraints we study the canonical
quantization of the holonomy of the connection on the
kinematical Hilbert space of loop quantum gravity. The holonomy operator
associated to a given path acts non trivially on spin network links that are
transversal to the path (a crossing). We provide an explicit construction of
the quantum holonomy operator. In particular, we exhibit a close relationship
between the action of the quantum holonomy at a crossing and Kauffman's
q-deformed crossing identity. The crucial difference is that (being an operator
acting on the kinematical Hilbert space of LQG) the result is completely
described in terms of standard SU(2) spin network states (in contrast to
q-deformed spin networks in Kauffman's identity). We discuss the possible
implications of our result.Comment: 19 pages, references added. Published versio
Approximating the Turaev-Viro Invariant of Mapping Tori is Complete for One Clean Qubit
The Turaev-Viro invariants are scalar topological invariants of
three-dimensional manifolds. Here we show that the problem of estimating the
Fibonacci version of the Turaev-Viro invariant of a mapping torus is a complete
problem for the one clean qubit complexity class (DQC1). This complements a
previous result showing that estimating the Turaev-Viro invariant for arbitrary
manifolds presented as Heegaard splittings is a complete problem for the
standard quantum computation model (BQP). We also discuss a beautiful analogy
between these results and previously known results on the computational
complexity of approximating the Jones polynomial.Comment: 16 pages, 8 figures, presented at TQC '11. Added reference
Kitaev's quantum double model from a local quantum physics point of view
A prominent example of a topologically ordered system is Kitaev's quantum
double model for finite groups (which in particular
includes , the toric code). We will look at these models from
the point of view of local quantum physics. In particular, we will review how
in the abelian case, one can do a Doplicher-Haag-Roberts analysis to study the
different superselection sectors of the model. In this way one finds that the
charges are in one-to-one correspondence with the representations of
, and that they are in fact anyons. Interchanging two of such
anyons gives a non-trivial phase, not just a possible sign change. The case of
non-abelian groups is more complicated. We outline how one could use
amplimorphisms, that is, morphisms to study the superselection
structure in that case. Finally, we give a brief overview of applications of
topologically ordered systems to the field of quantum computation.Comment: Chapter contributed to R. Brunetti, C. Dappiaggi, K. Fredenhagen, J.
Yngvason (eds), Advances in Algebraic Quantum Field Theory (Springer 2015).
Mainly revie
SL(2,R) Chern-Simons, Liouville, and Gauge Theory on Duality Walls
We propose an equivalence of the partition functions of two different 3d
gauge theories. On one side of the correspondence we consider the partition
function of 3d SL(2,R) Chern-Simons theory on a 3-manifold, obtained as a
punctured Riemann surface times an interval. On the other side we have a
partition function of a 3d N=2 superconformal field theory on S^3, which is
realized as a duality domain wall in a 4d gauge theory on S^4. We sketch the
proof of this conjecture using connections with quantum Liouville theory and
quantum Teichmuller theory, and study in detail the example of the
once-punctured torus. Motivated by these results we advocate a direct
Chern-Simons interpretation of the ingredients of (a generalization of) the
Alday-Gaiotto-Tachikawa relation. We also comment on M5-brane realizations as
well as on possible generalizations of our proposals.Comment: 53+1 pages, 14 figures; v2: typos corrected, references adde
Quantum Gravity in 2+1 Dimensions: The Case of a Closed Universe
In three spacetime dimensions, general relativity drastically simplifies,
becoming a ``topological'' theory with no propagating local degrees of freedom.
Nevertheless, many of the difficult conceptual problems of quantizing gravity
are still present. In this review, I summarize the rather large body of work
that has gone towards quantizing (2+1)-dimensional vacuum gravity in the
setting of a spatially closed universe.Comment: 61 pages, draft of review for Living Reviews; comments, criticisms,
additions, missing references welcome; v2: minor changes, added reference
Constructing the extended Haagerup planar algebra
We construct a new subfactor planar algebra, and as a corollary a new
subfactor, with the `extended Haagerup' principal graph pair. This completes
the classification of irreducible amenable subfactors with index in the range
, which was initiated by Haagerup in 1993. We prove that the
subfactor planar algebra with these principal graphs is unique. We give a skein
theoretic description, and a description as a subalgebra generated by a certain
element in the graph planar algebra of its principal graph. In the skein
theoretic description there is an explicit algorithm for evaluating closed
diagrams. This evaluation algorithm is unusual because intermediate steps may
increase the number of generators in a diagram.Comment: 45 pages (final version; improved introduction
Holomorphic Blocks in Three Dimensions
We decompose sphere partition functions and indices of three-dimensional N=2
gauge theories into a sum of products involving a universal set of "holomorphic
blocks". The blocks count BPS states and are in one-to-one correspondence with
the theory's massive vacua. We also propose a new, effective technique for
calculating the holomorphic blocks, inspired by a reduction to supersymmetric
quantum mechanics. The blocks turn out to possess a wealth of surprising
properties, such as a Stokes phenomenon that integrates nicely with actions of
three-dimensional mirror symmetry. The blocks also have interesting dual
interpretations. For theories arising from the compactification of the
six-dimensional (2,0) theory on a three-manifold M, the blocks belong to a
basis of wavefunctions in analytically continued Chern-Simons theory on M. For
theories engineered on branes in Calabi-Yau geometries, the blocks offer a
non-perturbative perspective on open topological string partition functions.Comment: 124 pages, 21 figures. v3: Typos correcte
The Spin Foam Approach to Quantum Gravity
This article reviews the present status of the spin foam approach to the
quantization of gravity. Special attention is payed to the pedagogical
presentation of the recently introduced new models for four dimensional quantum
gravity. The models are motivated by a suitable implementation of the path
integral quantization of the Plebanski formulation of gravity on a simplicial
regularization. The article also includes a self-contained treatment of the 2+1
gravity. The simple nature of the latter provides the basis and a perspective
for the analysis of both conceptual and technical issues that remain open in
four dimensions.Comment: To appear in Living Reviews in Relativit
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