Study of 3d-3d correspondence involving q-Weyl algebra and 3d-index.
problem Understanding the action of a q-Weyl algebra on the 3d-index of knots. method Investigation of the q-Weyl algebra's module action on the 3d-index, conjecturing structural properties. result Bilinear factorization, pair of linear q-difference equations, and rational function matrix for the 3d-index determination. 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 q1/2 with integer coefficients. Our goal is to explain how the 3D-index is a generating serie…
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.
The study connects knot complements to 3d theories via half-index calculations.
problem Understanding the relationship between knot complements and 3d theories.
method Using half-index calculations and inverted Habiro series, the study realizes knot complements as homological blocks.
result The colored Jones polynomial is derived from choosing specific poles in the half-index integral expression.
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.
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 r torii boundary components. For a fixed 2r tuple of integers, the index takes values in the set of q-series with integer coefficients. Our goal is to give an axiomatic definition of the tetra…
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.
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.
Proposes a new effective central charge for 3d N=2 theories.
problem Understanding the effective central charge in 3d N=2 theories.
method Analyzes the superconformal index to propose a new quantity and discusses its properties and computation.
result Proposes a new effective central charge for 3d N=2 theories.
We test the 3d-3d correspondence for theories that are labelled by Lens spaces. We find a full agreement between the index of the 3d N=2 "Lens space theory" T[L(p,1)] and the partition function of complex Chern-Simons theory on L(p,1). In particular, for p=1, we show how the familiar S3 partition func…
Study of symmetries in 3D Lie groups, determining index and moduli space properties.
problem Understanding symmetries in 3D Lie groups and their moduli space.
method Computed full isometry groups of left-invariant metrics on 3D Lie groups.
result Determined index of symmetry and properties of moduli space.
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…
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.
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 …
Study finds minimal hypersurfaces grow linearly in index, contrary to 3D.
problem Understanding index growth of minimal hypersurfaces.
method Partitioning methods for compact Lie groups, applied to hypersurfaces.
result Linear index growth for hypersurfaces of fixed topological type.
Study finds index of symmetry for solvable 3D Lie groups with left-invariant metrics.
problem Determining the index of symmetry for solvable 3D Lie groups with left-invariant metrics.
method Examined all solvable three-dimensional Lie groups, combined with previous work on unimodular groups.
result Index of symmetry is positive for every solvable 3D Lie algebra with a left-invariant metric, and is never 2.
New graph invariant measures embeddability in 3D.
problem Measuring embeddability of graphs in 3D.
method Defining freeness index to measure embeddability of graph complements.
result Cubic graphs satisfying orientable cycle double cover conjecture have freeness index at least two.
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.
The abstract discusses resurgent functions in quantum knot invariants.
problem Understanding the asymptotic expansion of quantum knot invariants.
method Using resurgent functions and q-series to conjecture and compute knot invariants. result Explicit computations match conjectured values for specific knots.
We use the 3d-3d correspondence together with the DGG construction of theories Tn[M] 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…
We describe rules for building 2d theories labeled by 4-manifolds. Using the proposed dictionary between building blocks of 4-manifolds and 2d N=(0,2) theories, we obtain a number of results, which include new 3d N=2 theories T[M_3] associated with rational homology spheres and new results for Vafa-Witten partition fun…
The study classifies translating and self-expanding solitons in 3D space.
problem Characterizing the topology and index of solitons in mean curvature flow.
method Analyzing the spectrum and index of expanding and translating solitons in R3. result Translating and self-expanding solitons have finite topology under certain conditions.
Study proves rigidity of capillary surfaces in curved 3D spaces.
problem Proving rigidity of capillary surfaces in curved 3D spaces.
method Local rigidity result for infinitesimally rigid capillary surfaces in Riemannian 3-manifolds with mean convex boundary.
result Bounds on genus, boundary components, and area of compact capillary minimal surfaces with low index.
Non-autoregressive method speeds up protein folding prediction 23 times.
problem Generating protein sequences with higher order interactions.
method Discrete diffusion conditioned on 3D structure using ProteinMPNN.
result 23 times speed up in inference without performance loss.
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.
The study finds an infinite number of minimal surfaces in 3D spheres.
problem Finding minimal surfaces in 3D spheres.
method Two-parameter min-max scheme in lens spaces, Heegaard foliations flipping.
result Constructs an infinite number of minimal surfaces in S3. 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.
Proof that stable minimal surfaces in 3D are flat.
problem Classification of stable minimal surfaces in R3. method Index theory for Dirac operators on twisted spinor bundles.
result Every complete two-sided stable minimal surface in R3 is flat. Novel boundary integral equations for Dirac operators in 3D Lipschitz domains.
problem Developing equations for Dirac operators in complex 3D domains.
method First-kind boundary integral equations, generalized Garding inequalities, Fredholm operators, finite dimensional kernels, Betti numbers.
result Finite dimensional kernels equal to the sum of Betti numbers, explaining the bilinear forms.
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.
A new method calculates the minimum volume swept by a sphere's homotopy in 3D space.
problem Finding the minimum volume swept by a sphere's homotopy in 3D space.
method Cable system approach to define and compute cable indices.
result A linear-time algorithm computes all cable indices and achieves the lower bound for the swept volume.
We study 4d superconformal indices for a large class of N=1 superconformal quiver gauge theories realized combinatorially as a bipartite graph or a set of "zig-zag paths" on a two-dimensional torus T^2. An exchange of loops, which we call a "double Yang-Baxter move", gives the Seiberg duality of the gauge theory, and t…
We provide a physical definition of new homological invariants Ha(M3) of 3-manifolds (possibly, with knots) labeled by abelian flat connections. The physical system in question involves a 6d fivebrane theory on M3 times a 2-disk, D2, whose Hilbert space of BPS states plays the role of a basic build…
New findings confirm parallels to De Giorgi's conjecture for phase transitions in higher dimensions.
problem Understanding phase transitions with bounded index in higher-dimensional spaces.
method Establishing parallels to De Giorgi's conjecture for general solutions of bounded Morse index.
result Finite index solutions to the Allen--Cahn equation in R4 are one-dimensional, and this holds for all 4≤n≤7. The exploitation of large-scale population data has the potential to improve healthcare by discovering and understanding patterns and trends within this data. To enable high throughput analysis of cardiac imaging data automatically, a pipeline should comprise quality monitoring of the input images, segmentation of the …
Novel link classification connects quadratic forms and knot theory.
problem Classifying isotopy classes of links in 3D space.
method Established a correspondence between quadratic forms and isotopy classes of links.
result Class numbers of quadratic number fields measure link distinguishability.
We identify a large class R of three-dimensional N=2 superconformal field theories. This class includes the effective theories T_M of M5-branes wrapped on 3-manifolds M, discussed in previous work by the authors, and more generally comprises theories that admit a UV description as abelian Chern-Simons-matter theories w…
We study knots in 3d Chern-Simons theory with complex gauge group SL(N,C), in the context of its relation with 3d N=2 theory (the so-called 3d-3d correspondence). The defect has either co-dimension 2 or co-dimension 4 inside the 6d (2,0) theory, which is compactified on a 3-manifold M^. …
The paper proves rigidity and stability properties of self-shrinking surfaces in 3D space.
problem Rigidity and stability of self-shrinking surfaces in R3. method Analyzing the mean curvature flow and L-index of self-shrinkers. result No stable two-dimensional self-shrinker in R3 exists without properness. 3D dual field theories for Virasoro minimal models constructed using Seifert fiber spaces.
problem Constructing 3D dual field theories for Virasoro minimal models.
method 3D-3D correspondence and Seifert fiber spaces.
result 3D dual field theories constructed for Virasoro minimal models.
3D flying wings created for any angle asymptotic cones.
problem Creating 3D steady gradient Ricci solitons with any angle asymptotic cones.
method Constructing 3D flying wings for any angle asymptotic cones.
result 3D flying wings constructed for any angle asymptotic cones.
Autonomous driving requires 3D perception of vehicles and other objects in the in environment. Much of the current methods support 2D vehicle detection. This paper proposes a flexible pipeline to adopt any 2D detection network and fuse it with a 3D point cloud to generate 3D information with minimum changes of the 2D d…
Smooth 3D flows from non-smooth starting points.
problem Creating smooth Ricci flows from non-smooth initial conditions.
method Generalized singular Ricci flow applied to 3D complete manifolds.
result Existence of smooth Ricci flows starting from non-smooth initial conditions.
3D Axial-Attention improves lung nodule classification accuracy.
problem Limited 3D attention in existing methods.
method Proposes 3D Axial-Attention network with 3D positional encoding.
result 3D Axial-Attention achieves state-of-the-art performance.
We show that the smooth geometry of a hyperbolic 3-manifold emerges from a classical spin system defined on a 2d discrete lattice, and moreover show that the process of this "dimensional oxidation" is equivalent with the dimensional reduction of a supersymmetric gauge theory from 4d to 3d. More concretely, we propose a…
3D steady gradient Ricci solitons are all O(2)-symmetric.
problem Characterizing 3D steady gradient Ricci solitons.
method Analyzing asymptotic behavior and using O(2) symmetry.
result All 3D steady gradient Ricci solitons are O(2)-symmetric.
DreamFusion uses text-to-image diffusion models to create 3D images efficiently.
problem Lack of large-scale 3D datasets and efficient architectures for 3D synthesis.
method Adapting a 2D diffusion model to 3D synthesis using a loss based on probability density distillation.
result A 3D model can be optimized from a 2D diffusion model, allowing for text-to-3D synthesis.