Quantum Field Theory and the Volume Conjecture
26 Mar 2010-Vol. 541, pp 41-67
TL;DR: The volume conjecture as discussed by the authors states that the asymptotic growth of the colored Jones polynomial of a hyperbolic knot is governed by the volume of the knot complement.
Abstract: The volume conjecture states that for a hyperbolic knot K in the three-sphere S^3 the asymptotic growth of the colored Jones polynomial of K is governed by the hyperbolic volume of the knot complement S^3\K. The conjecture relates two topological invariants, one combinatorial and one geometric, in a very nonobvious, nontrivial manner. The goal of the present lectures is to review the original statement of the volume conjecture and its recent extensions and generalizations, and to show how, in the most general context, the conjecture can be understood in terms of topological quantum field theory. In particular, we consider: a) generalization of the volume conjecture to families of incomplete hyperbolic metrics; b) generalization that involves not only the leading (volume) term, but the entire asymptotic expansion in 1/N; c) generalization to quantum group invariants for groups of higher rank; and d) generalization to arbitrary links in arbitrary three-manifolds.
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TL;DR: In this paper, the authors decompose sphere partition functions and indices of three-dimensional = 2 gauge theories into a sum of products involving a universal set of holomorphic blocks, which are in one-to-one correspondence with the theory's massive vacua.
Abstract: We decompose sphere partition functions and indices of three-dimensional $$ \mathcal{N} $$
= 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 wave-functions 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.
328 citations
TL;DR: In this paper, a 2-parameter family deformation of the A-polynomial that encodes the color dependence of the super polynomial was introduced, which, in suitable limits, reduces to various deformations.
150 citations
TL;DR: In this article, the relation between perturbative knot invariants and the free energies defined by topological string theory on the character variety of the knot was studied beyond the sub-leading order in the perturbation expansion on both sides.
127 citations
TL;DR: The generalized volume conjecture as mentioned in this paper relates asymptotic behavior of the colored Jones polynomials to objects naturally defined on an algebraic curve, the zero locus of the A-polynomial $A(x,y)$.
Abstract: The generalized volume conjecture relates asymptotic behavior of the colored Jones polynomials to objects naturally defined on an algebraic curve, the zero locus of the A-polynomial $A(x,y)$. Another "family version" of the volume conjecture depends on a quantization parameter, usually denoted $q$ or $\hbar$; this quantum volume conjecture (also known as the AJ-conjecture) can be stated in a form of a q-difference equation that annihilates the colored Jones polynomials and $SL(2,\C)$ Chern-Simons partition functions. We propose refinements / categorifications of both conjectures that include an extra deformation parameter $t$ and describe similar properties of homological knot invariants and refined BPS invariants. Much like their unrefined / decategorified predecessors, that correspond to $t=-1$, the new volume conjectures involve objects naturally defined on an algebraic curve $A^{ref} (x,y; t)$ obtained by a particular deformation of the A-polynomial, and its quantization $\hat A^{ref} (\hat x, \hat y; q, t)$. We compute both classical and quantum t-deformed curves in a number of examples coming from colored knot homologies and refined BPS invariants.
101 citations
Book•
18 Dec 2012TL;DR: Decomposition into 3-balls, Ideal Polyhedra, and I-bundles and essential product disks are presented.
Abstract: 1 Introduction.- 2 Decomposition into 3-balls.- 3 Ideal Polyhedra.- 4 I-bundles and essential product disks.- 5 Guts and fibers.- 6 Recognizing essential product disks.- 7 Diagrams without non-prime arcs.- 8 Montesinos links.- 9 Applications.- 10 Discussion and questions.
95 citations
References
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TL;DR: In this paper, it was shown that 2+1 dimensional quantum Yang-Mills theory with an action consisting purely of the Chern-Simons term is exactly soluble and gave a natural framework for understanding the Jones polynomial of knot theory in three dimensional terms.
Abstract: It is shown that 2+1 dimensional quantum Yang-Mills theory, with an action consisting purely of the Chern-Simons term, is exactly soluble and gives a natural framework for understanding the Jones polynomial of knot theory in three dimensional terms. In this version, the Jones polynomial can be generalized fromS
3 to arbitrary three manifolds, giving invariants of three manifolds that are computable from a surgery presentation. These results shed a surprising new light on conformal field theory in 1+1 dimensions.
5,093 citations
TL;DR: In this paper, it was shown that every finite-dimensional Poisson manifold X admits a canonical deformation quantization, and that the set of equivalence classes of associative algebras close to the algebra of functions on X is in one-to-one correspondence with the class of Poisson structures on X modulo diffeomorphisms.
Abstract: I prove that every finite-dimensional Poisson manifold X admits a canonical deformation quantization. Informally, it means that the set of equivalence classes of associative algebras close to the algebra of functions on X is in one-to-one correspondence with the set of equivalence classes of Poisson structures on X modulo diffeomorphisms. In fact, a more general statement is proven (the ‘Formality conjecture’), relating the Lie superalgebra of polyvector fields on X and the Hochschild complex of the algebra of functions on X. Coefficients in explicit formulas for the deformed product can be interpreted as correlators in a topological open string theory, although I do not explicitly use the language of functional integrals.
2,672 citations
TL;DR: By disentangling the hamiltonian constraint equations, 2 + 1 dimensional gravity (with or without a cosmological constant) is shown to be exactly soluble at the classical and quantum levels.
2,636 citations
TL;DR: In this paper, the authors construct topological invariants of compact oriented 3-manifolds and of framed links in such manifolds, where the terms of the sequence are equale to the values of the Jones polynomial of the link in the corresponding roots of 1.
Abstract: The aim of this paper is to construct new topological invariants of compact oriented 3-manifolds and of framed links in such manifolds. Our invariant of (a link in) a closed oriented 3-manifold is a sequence of complex numbers parametrized by complex roots of 1. For a framed link in S 3 the terms of the sequence are equale to the values of the (suitably parametrized) Jones polynomial of the link in the corresponding roots of 1. In the case of manifolds with boundary our invariant is a (sequence of) finite dimensional complex linear operators. This produces from each root of unity q a 3-dimensional topological quantum field theory
1,709 citations
TL;DR: In the case of negative Euler characteristic (genus greater than 1) such a metric gives a hyperbolic structure: any small neighborhood in a surface is isometric to a neighborhood in the hyper-bolic plane, and the surface itself is the quotient of the hyperbola by a discrete group of motions as discussed by the authors.
Abstract: 1. A conjectural picture of 3-manifolds. A major thrust of mathematics in the late 19th century, in which Poincare had a large role, was the uniformization theory for Riemann surfaces: that every conformai structure on a closed oriented surface is represented by a Riemannian metric of constant curvature. For the typical case of negative Euler characteristic (genus greater than 1) such a metric gives a hyperbolic structure: any small neighborhood in the surface is isometric to a neighborhood in the hyperbolic plane, and the surface itself is the quotient of the hyperbolic plane by a discrete group of motions. The exceptional cases, the sphere and the torus, have spherical and Euclidean structures. Three-manifolds are greatly more complicated than surfaces, and I think it is fair to say that until recently there was little reason to expect any analogous theory for manifolds of dimension 3 (or more)—except perhaps for the fact that so many 3-manifolds are beautiful. The situation has changed, so that I feel fairly confident in proposing the
1,641 citations