arXiv research

New surface observables yield 2-knot invariants in nonabelian theories.

We introduce a $\mathbb{C}/\mathbb{Z}$-valued invariant of a foliated manifold with a stable framing and with a partially flat vector bundle. This invariant can be expressed in terms of integration in differential $K$-theory, or alternatively, in terms of $η$-invariants of Dirac operators and local correction terms. In…

Polynomial invariant derived from birack labelling of knots.

Maximal Laplacian algebras applied to invariant theory solved inverse problems.

Deser and Nepomechie established a relationship between masslessness and rigid conformal invariance by coupling to a background metric and demanding local Weyl invariance, a method which applies neither to massive theories nor theories which rely upon gauge invariances for masslessness. We extend this method to describ…

Machine learning knot invariants with physics applications.

Study Vassiliev invariants for virtual knots, expanding quantum theory.

We study scale invariant but not necessarily conformal invariant deformations of non-relativistic conformal field theories from the dual gravity viewpoint. We present the corresponding metric that solves the Einstein equation coupled with a massive vector field. We find that, within the class of metric we study, when w…

The study extends knot theory to knotoids using two approaches.

Program connects birational invariants with G-equivariant ones using Gromov-Witten theory.

We define a bordism invariant for the fiberwise intersection theory. Under some certain conditions, this invariant is an obstruction for the theory.

New connections found between knot invariants and Rozansky-Witten theory.

New invariant computed for Brieskorn homology spheres.

Quantum cocycle invariants derived from Yang-Baxter cohomology.

The theory of link-homotopy, introduced by Milnor, is an important part of the knot theory, with Milnor's mu-bar-invariants being the basic set of link-homotopy invariants. Skein relations for knot and link invariants played a crucial role in the recent developments of knot theory. However, while skein relations for Al…

This is a survey of our recent work with Tom Mrowka on Seiberg-Witten gauge theory and index theory for manifolds with periodic ends. We explain how this work leads to a new invariant, which is related to the classical Rohlin invariant of homology 3-spheres and to the Furuta-Ohta invariant originating in Yang-Mills gau…

Unified framework for Morita invariant cohomology of Lie groupoids.

Researchers compute invariants for knots and links in lens spaces using large N and k limits.

Diagrammatic method calculates knot invariant related to Chern-Simons theory.

We study the Goussarov-Habiro finite type invariants theory for framed string links in homology balls. Their degree 1 invariants are computed: they are given by Milnor's triple linking numbers, the mod 2 reduction of the Sato-Levine invariant, Arf and Rochlin's $μ$ invariant. These invariants are seen to be naturally r…

A theory of finite type invariants for arbitrary compact oriented 3-manifolds is proposed, and illustrated through many examples arising from both classical and quantum topology. The theory is seen to be highly non-trivial even for manifolds with large first betti number, encompassing much of the complexity of Ohtsuki'…

Algorithm computes knot invariants for surgeries on prime knots.

This study defines finite-type invariants for curves on surfaces and reveals the construction of these finite-type invariants for stable homeomorphism classes of curves on compact oriented surfaces without boundaries. These invariants are a higher-order generalisation of a part of Arnold's invariants that are first-ord…

Develops invariants for webs and foams using Seiberg-Witten theory.

Observable structures of a topological field theory of AKSZ type are analyzed. From a double (or multiple) complex structure of observable algebras, new topological invariants are constructed. Especially, Donaldson polynomial invariants and their generalizations are constructed from a topological field theory of AKSZ t…

New skein theory for Links-Gould polynomial simplifies link evaluations.

Geometric invariant theory introduces stability conditions mirroring abelian category theory.

Power series invariant of hyperbolic 3-manifolds matches knot invariants.

Study of Seiberg-Witten invariants for 4-manifolds with group actions.

This is the second of a series of two technical papers devoted to the analysis of holonomy invariants in strict higher gauge theory with end applications in higher Chern--Simons theory. We provide a definition of trace over a crossed module such to yield surface knot invariants upon application to 2-holonomies. We show…

This paper analyzes $Δ_a$ invariants in non-perturbative complex Chern-Simons theory.

Global theory of relative invariants and equivariant line bundles established.

Formula calculates invariant for 3-manifolds with torus boundaries.

Unified geometric perspectives on PDEs, torsion invariants, and moduli theory.

New concept of partial law invariance connects decision theory and financial risk management.

New invariants from Seiberg-Witten theory for 3-spheres with involution.

New invariants for 3-manifolds derived from equivariant Cerf theory.

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…

The paper extends invariant theory to non-compact and non-reductive actions, classifying four regimes.

Using the theory of involutive Heegaard Floer knot theory developed by Hendricks-Manolescu, we define two involutive analogs of the Upsilon knot concordance invariant of Ozsvath-Stipsicz-Szabo. These involutive invariants are piecewise linear functions defined on the interval [0,2]. Each is a concordance invariant and …

New invariant unifies two theories of 3-manifolds, recovering quantum invariants.

Data science enhances knot theory by analyzing invariant relations.

A new invariant for pure braids is defined and shown not to be trivial.

The Alexander polynomial is linked to Bott-Cattaneo-Rossi invariants via Chern-Simons theory.

Category theory generalizes finite type invariants using diagrams systems.

This is lecture notes for a course given at the PCMI Summer School "Quantum Field Theory and Manifold Invariants" (July 1 -- July 5, 2019). I describe basics of gauge-theoretic approach to construction of invariants of manifolds. The main example considered here is the Seiberg--Witten gauge theory. However, I tried to …

A three dimensional supergravity theory which generalizes the super IG theory of Witten and resembles the model discussed recently by Mann and Papadopoulos is displayed. The partition function is computed, and is shown to be a three-manifold invariant generalizing the Casson invariant.

Revisits Vafa-Witten theory, deriving new invariants and homologies.