Researchers compute Chern-Simons invariants for a specific type of knot orbifolds.
problem Calculating Chern-Simons invariants for hyperbolic knot orbifolds.
method Used Schläfli formula for generalized Chern-Simons function on cone-manifold structures.
result Explicit formulae for the invariants of hyperbolic J(2n,−2m) knot orbifolds were derived. Formulae for volume and Chern-Simons invariant of hyperbolic knot orbifolds.
problem Computing volume and Chern-Simons invariant for hyperbolic knot orbifolds.
method Extending Neumann's methods to derive explicit formulae.
result Explicit formulae for volume and Chern-Simons invariant.
The abstract discusses conjectures about Chern-Simons invariants of 3-manifolds.
problem Conjectures about the reciprocity of Chern-Simons invariants of 3-manifolds.
method Supporting evidence through Galois descent of a K3-group. result The conjectures hold under the condition of Galois descent of a K3-group. New approach connects 3D Chern-Simons theory to spectral networks.
problem Understanding Chern-Simons invariants in 3D manifolds.
method Constructing equivalences between bundles and spectral networks.
result New formulas for Chern-Simons invariants of 3D manifolds.
We calculate the asymptotic behavior of hyperbolic volume and Chern-Simons invariant.
problem Calculating the asymptotic behavior of hyperbolic volume and Chern-Simons invariant.
method Renormalization of the Chern-Simons invariant using asymptotics along an equidistance foliation.
result The leading coefficient introduces a complex-valued quantity consisting of mean curvature and torsion 2-form.
New invariant from non-acyclic flat connections.
problem Constructing a higher-loop perturbative invariant.
method Integral of a Chern-Simons volume form over moduli space of flat connections.
result Generalization of Chern-Simons invariant to non-acyclic connections.
We give an efficient simplicial formula for the volume and Chern-Simons invariant of a boundary-parabolic PSL(2,C)-representation of a tame 3-manifold. If the representation is the geometric representation of a hyperbolic 3-manifold, our formula computes the volume and Chern-Simons invariant directly from an ideal tria…
Hennings and Chern-Simons invariants match for certain quantum groups.
problem Matching invariants for quantum groups and Chern-Simons theory.
method Comparing invariants for arbitrary simple Lie algebras at roots of unity.
result Agreement of Hennings and Chern-Simons invariants for integer homology three-spheres.
New formula calculates volumes and Chern-Simons invariants for closed 3-manifolds.
problem Computing volumes and Chern-Simons invariants for non-parabolic representations.
method Introducing deformed Ptolemy varieties to extend Zickert's formula.
result Volume and Chern-Simons invariants computed for closed 3-manifolds.
Computes Chern-Simons invariants for 3-manifolds with specific properties.
problem Computing Chern-Simons invariants for complex 3-manifolds.
method Analyzes 3-manifolds as doubles of compression bodies, computes critical values for structure groups.
result Moduli space of flat connections is connected without reducibles.
Diagrammatic method calculates knot invariant related to Chern-Simons theory.
problem Calculating knot invariants in Chern-Simons theory.
method Diagrammatic approach for compact oriented 3-manifolds.
result Diagrammatic method for computing knot invariant d. Study calculates volumes and Chern-Simons invariants for 2-bridge knot orbifolds.
problem Calculating volumes and Chern-Simons invariants for 2-bridge knot orbifolds.
method Using the unpublished approach of Mednykh and Rasskazov, derive Riley-Mednykh polynomial and explicit formulae.
result Explicit formulae for volume and Chern-Simons invariant of 2-bridge knot orbifolds.
We introduce certain relative differential characters which we call Cheeger-Chern-Simons characters. These combine the well-known Cheeger-Simons characters with Chern-Simons forms. In the same way as the Cheeger-Simons characters generalize Chern-Simons invariants of oriented closed manifolds, the Cheeger-Chern-Simons …
Chern-Simons invariants of closed oriented Riemannian 3-manifolds are introduced and studied from the basics. Their first-order variation is the Cotton tensor. The properties of the Cotton tensor: symmetry, conformal covariance, trace- and divergence-freedom, are recovered as corollaries of the Chern-Simons invariant…
The distribution of Chern-Simons invariants on 3-manifolds resembles quadratic residues.
problem Distribution of Chern-Simons invariants on 3-manifolds.
method Analyzing the values of the Chern-Simons function on conjugacy classes of representations.
result Chern-Simons invariants tend to become equidistributed on the circle with white noise fluctuations.
This work connects knot invariants to Chern-Simons theories via factorization homology.
problem Understanding knot invariants in Chern-Simons theories.
method Constructing a filtered E3-algebra and proving an equality between factorization homology trace and Reshetikhin-Turaev link invariant. result Established a connection between knot invariants and Chern-Simons theories.
Formula calculates invariant for 3-manifolds with torus boundaries.
problem Calculating an invariant for 3-manifolds with torus boundaries.
method Generalized Chern-Simons invariant and provided a gluing formula.
result A gluing formula for the invariant d(M,ρ). Quantizes Chern-Simons invariant for tangle exteriors.
problem Geometric quantization of Chern-Simons invariant for tangles.
method Defining a sequence of invariants ZNψ using modules over quantum sl2 and holonomy R-matrices. result Directly recovers Chern-Simons invariant when N=1. We calculate the Chern-Simons invariants of the hyperbolic orbifolds of the knot with Conway's notation C(2n,3) using the Schläfli formula for the generalized Chern-Simons function on the family of C(2n,3) cone-manifold structures. We present the concrete and explicit formula of them. We apply the general instruct…
The paper extends arithmetic Chern-Simons invariants to real quadratic fields and calculates mod 2 Dijkgraaf-Witten invariants.
problem Calculating arithmetic Dijkgraaf-Witten invariants for real quadratic number fields.
method Using modified étale cohomology groups and fundamental groups, explicit formulas are derived for real quadratic fields.
result Explicit formulas for mod 2 arithmetic Dijkgraaf-Witten invariants for real quadratic fields are provided.
Study calculates Kashaev invariants for twice-iterated torus knots.
problem Calculating Kashaev invariants for complex knots.
method Asymptotic analysis and topological interpretations.
result Obtained formulas linking invariants to Chern-Simons and Reidemeister torsion.
Geometrically constructs dilogarithm from Chern-Simons theory.
problem Dilogarithm function and its properties.
method Spin Chern-Simons invariant of C*-connections.
result Geometric proofs of dilogarithm identities and branching structure.
The contribution of reducible connections to the U(N) Chern-Simons invariant of a Seifert manifold M can be expressed in some cases in terms of matrix integrals. We show that the U(N) evaluation of the LMO invariant of any rational homology sphere admits a matrix model representation which agrees with the Chern-Simon…
A function Γ_Y relates Chern-Simons functional to homology cobordism invariants.
problem Understanding the relationship between Chern-Simons functional and homology cobordism.
method Constructing a function Γ_Y on homology spheres and relating it to Chern-Simons functional and invariants.
result Γ_Y recovers the Frøyshov invariant and is related to Fintushel-Stern's R-invariant.
The Alexander polynomial is linked to Bott-Cattaneo-Rossi invariants via Chern-Simons theory.
problem Expressing Alexander polynomial of long knots in terms of invariants.
method Using a previously established formula relating Bott-Cattaneo-Rossi invariants to the Alexander polynomial and Chern-Simons theory.
result Relating Bott-Cattaneo-Rossi invariants to the Alexander polynomial and Chern-Simons theory.
Computes entanglement entropy using Chern-Simons theory and symmetric webs.
problem Determining if a product state implies unlinked components.
method Using symmetric webs to compute colored link invariants and write multi-partite entangled states.
result Written down multi-partite entangled states of any given link.
The paper connects Chern-Simons invariants to mixed Tate motives in hyperbolic 3-manifolds.
problem Understanding the relationship between Chern-Simons invariants and mixed Tate motives in hyperbolic 3-manifolds.
method Constructing a mixed Tate motive over the invariant trace field whose image equals the Chern-Simons invariant and complex volume.
result The mixed Hodge realization of the motive is a quotient of the path torsor of the augmented character variety.
We define an extended Bloch group and show it is naturally isomorphic to H_3(PSL(2,C)^δ;Z). Using the Rogers dilogarithm function this leads to an exact simplicial formula for the universal Cheeger-Chern-Simons class on this homology group. It also leads to an independent proof of the analytic relationship between volu…
The paper defines a Chern-Simons invariant for stably trivial vector bundles and uses it to obstruct conformal immersions.
problem Obstructing conformal immersions of Riemannian manifolds.
method Defining a Chern-Simons invariant for stably trivial vector bundles and using it to derive an obstruction.
result An obstruction for conformally immersing a n-dimensional Riemannian manifold in a translation manifold of dimension n+1.
We present an alternate proof of the Bismut-Zhang localization formula for η-invariants without using the analytic techniques developed by Bismut-Lebeau. A Riemann-Roch property for Chern-Simons currents, which is of independent interest, is established in due course.
Machine learning knot invariants with physics applications.
problem Understanding relations between knot invariants in physics.
method Machine learning and theoretical physics (Chern-Simons theory, gauge theories).
result New analytic results from Big Data experiments.
This paper makes certain observations regarding some conjectures of Milnor and Ramakrishnan in hyperbolic geometry and algebraic K-theory. As a consequence of our observations, we obtain new results and conjectures regarding the rationality and irrationality of Chern-Simons invariants of hyperbolic 3-manifolds.
The paper explores flat extensions of connections and their relation to Chern-Simons invariants.
problem Understanding flat extensions of principal connections and their implications.
method Introducing flat extensions and relating them to Chern-Simons invariants.
result Flat extensions of connections are linked to the vanishing of Chern-Simons invariants.
Kauffman knot polynomial invariants are discovered in classical abelian Chern-Simons field theory. A topological invariant tI(L) is constructed for a link L, where I is the abelian Chern-Simons action and t a formal constant. For oriented knotted vortex lines, tI satisf…
This paper analyzes Δa invariants in non-perturbative complex Chern-Simons theory.
problem Analyzing the structure and properties of Δa invariants. method Reviewing foundations and analyzing structure for a subclass of integer homology spheres.
result The Δa invariants are not homology cobordism invariants. The volume conjecture is extended to all orders for hyperbolic 3-manifolds using complex Chern-Simons theory.
problem Extending the volume conjecture to all orders for hyperbolic 3-manifolds.
method Deriving formulas for the perturbative expansion of the partition function of complex Chern-Simons theory and comparing it to Witten-Reshetikhin-Turaev invariants.
result The conjecture that the perturbative expansion of the partition function of complex Chern-Simons theory matches the Witten-Reshetikhin-Turaev invariants at roots of unity in the limit of infinitely many invariants.
The universal perturbative invariants of rational homology spheres can be extracted from the Chern-Simons partition function by combining perturbative and nonperturbative results. We spell out the general procedure to compute these invariants, and we work out in detail the case of Seifert spaces. By extending some prev…
We define a Chern-Simons invariant for a certain class of infinite volume hyperbolic 3-manifolds. We then prove an expression relating the Bergman tau function on a cover of the Hurwitz space, to the lifting of the function F defined by Zograf on Teichmüller space, and another holomorphic function on the cover of the…
We compute the Chern-Simons transgressed forms of some modularly invariant characteristic forms, which are related to the elliptic genera. We study the modularity properties of these secondary characteristic forms and the relations among them. We also compute the Chern-Simons forms of some vector bundles over free loop…
In the late 1980s Witten used the Chern-Simons form of a connection to construct new invariants of 3-manifolds and knots, recovering in particular the Jones invariants. Since then the associated topological quantum field theory (TQFT) has served as a key example in understanding the structure of TQFTs in general. We su…
Study shows exponential growth of knot polynomial tied to Chern-Simons invariant.
problem Asymptotic behavior of colored Jones polynomials of figure-eight knot.
method Analyzes growth rate of polynomial evaluated at specific points.
result Growth rate determined by Chern-Simons invariant of an affine representation.
We develop several methods that allow us to compute all-loop partition functions in perturbative Chern-Simons theory with complex gauge group G_C, sometimes in multiple ways. In the background of a non-abelian irreducible flat connection, perturbative G_C invariants turn out to be interesting topological invariants, wh…
We compute the vacuum expectation values of torus knot operators in Chern-Simons theory, and we obtain explicit formulae for all classical gauge groups and for arbitrary representations. We reproduce a known formula for the HOMFLY invariants of torus links and we obtain an analogous formula for Kauffman invariants. We …
We define an extended Bloch group and show it is isomorphic to H3(PSL(2,C)δ;Z). Using the Rogers dilogarithm function this leads to an exact simplicial formula for the universal Cheeger-Simons class on this homology group. It also leads to an independent proof of the analytic relationship between volume and Chern-S…
We extend the Chern-Simons perturbative invariant of Axelrod and Singer to non-acyclic connections. We construct a solution of the quantum master equation on the space of functions on the cohomology of the connection. We prove that this solution is well defined up to master homotopy. We discuss also invariants of links…
The paper provides formulas linking knot invariants to deformation quantization.
problem Deformation quantization of the space of connections on a 2-manifold.
method Using Chern-Simons gauge theory in 3D, the paper derives explicit formulas for star products.
result Explicit formulas connect knot invariants to deformation quantization and gauge theory.
Quantum invariants of Seifert fibered homology spheres are resummated and related to classical Chern-Simons values.
problem Resummation and classification of quantum invariants for Seifert fibered homology spheres.
method Analysis of q-series and Ohtsuki series, asymptotic expansions, Borel transform, and classification of moduli spaces.
result The limit of the resummation equals the WRT quantum invariant and classifies components of the moduli space.
Lecture notes on Lie groups and Chern-Simons theory for grad students.
problem Understanding Chern-Simons theory and its applications.
method Explains Lie groups and their relation to Chern-Simons theory.
result Motivated new fields in knot theory and topology.