Dimofte, Gaiotto and Gukov introduced a powerful invariant, the 3D-index, associated to a suitable ideal triangulation of a 3-manifold with torus boundary components. The 3D-index is a collection of formal power series in with integer coefficients. Our goal is to explain how the 3D-index is a generating serie…
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20 results for “3d-index”
Identifies 3D-index as invariant for cusped hyperbolic 3-manifolds.
problem Invariance of q-series for cusped hyperbolic 3-manifolds.
method Relates Frohman-Kania-Bartoszynska's q-series to Dimofte-Gaiotto-Gukov's 3D-index and tetrahedron index.
result Topological invariance of Frohman-Kania-Bartoszynska's q-series.
The 3D-index connects to Turaev-Viro invariant and knot periods.
problem Understanding the asymptotic expansions of the 3D-index.
method Analyzing the asymptotic behavior of the 3D-index and its connection to the Turaev-Viro invariant and knot periods.
result The asymptotic expansions of the 3D-index match to all orders with the Turaev-Viro invariant of a knot, explaining the Volume Conjecture.
Proves formula for 3D index change with Dehn filling.
problem Transforming 3D index under Dehn filling.
method Relative 3D index, gluing principle, inductive framework, q-hypergeometric functions.
result Rigorous proof of Gang-Yonekura formula.
Study of 3d-3d correspondence involving -Weyl algebra and 3d-index.
problem Understanding the action of a -Weyl algebra on the 3d-index of knots.
method Investigation of the -Weyl algebra's module action on the 3d-index, conjecturing structural properties.
result Bilinear factorization, pair of linear -difference equations, and rational function matrix for the 3d-index determination.
Defines a map connecting 3d-index and skein module.
problem Connecting mathematical physics predictions with topological quantum field theory.
method Defines a map from skein module to Laurent series ring.
result The map fulfills a supersymmetry prediction and is part of a conjectural topological quantum field theory.
The 3D index of Dimofte-Gaiotto-Gukov a partially defined function on the set of ideal triangulations of 3-manifolds with torii boundary components. For a fixed tuple of integers, the index takes values in the set of -series with integer coefficients. Our goal is to give an axiomatic definition of the tetra…
Refined 3D indexhep-th
Refined 3D index uses surgery and gradings to distinguish 3-manifolds.
problem Distinguishing 3-manifolds and gauge theories phases.
method Dehn surgery presentation, ideal triangulation, and enhanced flavor symmetries.
result Invariance of refined index under various transformations.
Study of asymptotics of meromorphic 3D-index as q approaches 1.
problem Understanding the asymptotic behavior of a meromorphic function related to 3D-index.
method Developed a conjectural asymptotic approximation using stationary phase analysis of a circle-valued angle structure integral.
result Found connections to angle structures and volume optimization.
In this paper we will promote the 3D index of an ideal triangulation T of an oriented cusped 3-manifold M (a collection of q-series with integer coefficients, introduced by Dimofte-Gaiotto-Gukov) to a topological invariant of oriented cusped hyperbolic 3-manifolds. To achieve our goal we show that (a) T admits an index…
Using the locally compact abelian group $\BT \times \BZ$, we assign a meromorphic function to each ideal triangulation of a 3-manifold with torus boundary components. The function is invariant under all 2--3 Pachner moves, and thus is a topological invariant of the underlying manifold. If the ideal triangulation has a …
The abstract discusses resurgent functions in quantum knot invariants.
problem Understanding the asymptotic expansion of quantum knot invariants.
method Using resurgent functions and -series to conjecture and compute knot invariants.
result Explicit computations match conjectured values for specific knots.
We propose a new algorithm for Dehn surgery problem, finding exceptional Dehn filling slopes for a given hyperbolic 3-manifold with a torus boundary, using a quantum invariant called "3D index". The invariant is defined using an ideal triangulation of the cusped 3-manifold. We test the algorithm for many examples.
Holomorphic functions from knot complements link to quantum modular forms.
problem Analyzing holomorphic functions from knot complements.
method Matrix-valued holomorphic functions, cocycles, and quantum modularity.
result Identifies a matrix-valued holomorphic quantum modular form.
Study angle structures on 3-manifolds, linking to representation theory.
problem Understanding spaces of angle structures on 3-manifolds.
method Cohomology groups and geometric bijections.
result Establishes a bijection between angle structures and obstruction classes.
The tetrahedral index connects to a q-Bessel function, revealing new mathematical techniques.
problem Exploring connections between the tetrahedral index and Hahn-Exton q-Bessel function.
method Establishing a correspondence between the tetrahedral index and the q-Bessel function.
result New techniques and conjectures in q-hypergeometric theory.
Twisted Neumann--Zagier matrices for quantum invariants.
problem Constructing quantum invariants from ideal triangulations.
method Define and compute twisted Neumann--Zagier matrices from combinatorics.
result Twisted 1-loop invariant equals adjoint twisted Alexander polynomial.
We use the 3d-3d correspondence together with the DGG construction of theories labelled by 3-manifolds M to define a non-perturbative state-integral model for SL(n,C) Chern-Simons theory at any level k, based on ideal triangulations. The resulting partition functions generalize a widely studied k=1 state-integ…
Quantum modularity proved for a knot manifold.
problem Proving quantum modularity for a specific closed hyperbolic 3-manifold.
method Using factorization of state integrals and proving quantum modularity for functions and -series.
result Quantum modularity for the closed manifold provides a unification of volume conjecture and Witten's asymptotic expansion conjecture.
The trivial flat connection's Chern-Simons theory is resurgent, revealing its structure.
problem Understanding the resurgent structure of Chern-Simons theory at the trivial flat connection.
method Analyzing an extended square matrix of -series to describe the resurgent structure and Stokes constants.
result The resurgent structure and Stokes constants of the Chern-Simons series are completely described.