scholarly journals The colored Jones polynomials as vortex partition functions

2021 ◽  
Vol 2021 (12) ◽  
Author(s):  
Masahide Manabe ◽  
Seiji Terashima ◽  
Yuji Terashima
Keyword(s):  
Building Block ◽  
Wilson Loop ◽  
Gauge Theories ◽  
Partition Functions ◽  
Expectation Values ◽  
Abelian Gauge ◽  
Jones Polynomials ◽  
Knot Diagrams ◽  
Colored Jones Polynomials ◽  
Chern Simons

Abstract We construct 3D $$ \mathcal{N} $$ N = 2 abelian gauge theories on $$ \mathbbm{S} $$ S 2 × $$ \mathbbm{S} $$ S 1 labeled by knot diagrams whose K-theoretic vortex partition functions, each of which is a building block of twisted indices, give the colored Jones polynomials of knots in $$ \mathbbm{S} $$ S 3. The colored Jones polynomials are obtained as the Wilson loop expectation values along knots in SU(2) Chern-Simons gauge theories on $$ \mathbbm{S} $$ S 3, and then our construction provides an explicit correspondence between 3D $$ \mathcal{N} $$ N = 2 abelian gauge theories and 3D SU(2) Chern-Simons gauge theories. We verify, in particular, the applicability of our constructions to a class of tangle diagrams of 2-bridge knots with certain specific twists.

scholarly journals Wilson loop algebras and quantum K-theory for Grassmannians

2020 ◽  
Vol 2020 (10) ◽  
Author(s):  
Hans Jockers ◽  
Peter Mayr ◽  
Urmi Ninad ◽  
Alexander Tabler
Keyword(s):  
Wilson Loop ◽  
Gauge Theories ◽  
Wilson Line ◽  
Quantum Cohomology ◽  
Three Dimensional ◽  
Loop Algebra ◽  
Loop Algebras ◽  
Verlinde Algebra ◽  
K Theory ◽  
Chern Simons

Abstract We study the algebra of Wilson line operators in three-dimensional $$ \mathcal{N} $$ N = 2 supersymmetric U(M ) gauge theories with a Higgs phase related to a complex Grassmannian Gr(M, N ), and its connection to K-theoretic Gromov-Witten invariants for Gr(M, N ). For different Chern-Simons levels, the Wilson loop algebra realizes either the quantum cohomology of Gr(M, N ), isomorphic to the Verlinde algebra for U(M ), or the quantum K-theoretic ring of Schubert structure sheaves studied by mathematicians, or closely related algebras.

scholarly journals Three-dimensional 𝒩 = 2 supersymmetric gauge theories and partition functions on Seifert manifolds: A review

2019 ◽  
Vol 34 (23) ◽  
pp. 1930011 ◽  
Author(s):  
Cyril Closset ◽  
Heeyeon Kim
Keyword(s):  
Correlation Functions ◽  
Gauge Theories ◽  
Three Dimensional ◽  
Partition Functions ◽  
Exact Results ◽  
Strongly Coupled ◽  
Seifert Manifolds ◽  
Recent Developments ◽  
Supersymmetric Gauge ◽  
Chern Simons

We give a pedagogical introduction to the study of supersymmetric partition functions of 3D [Formula: see text] supersymmetric Chern–Simons-matter theories (with an [Formula: see text]-symmetry) on half-BPS closed three-manifolds — including [Formula: see text], [Formula: see text], and any Seifert three-manifold. Three-dimensional gauge theories can flow to nontrivial fixed points in the infrared. In the presence of 3D [Formula: see text] supersymmetry, many exact results are known about the strongly-coupled infrared, due in good part to powerful localization techniques. We review some of these techniques and emphasize some more recent developments, which provide a simple and comprehensive formalism for the exact computation of half-BPS observables on closed three-manifolds (partition functions and correlation functions of line operators). Along the way, we also review simple examples of 3D infrared dualities. The computation of supersymmetric partition functions provides exceedingly precise tests of these dualities.

scholarly journals EXACT RENORMALIZATION GROUP IN ALGEBRAIC NONCOVARIANT GAUGES

2001 ◽  
Vol 16 (11) ◽  
pp. 2125-2130
Author(s):  
M. SIMIONATO
Keyword(s):  
Perturbation Theory ◽  
Renormalization Group ◽  
Light Cone ◽  
Wilson Loop ◽  
Gauge Theories ◽  
Mass Term ◽  
Abelian Gauge ◽  
Light Cone Gauge ◽  
Infrared Limit ◽  
Exact Renormalization Group

I study a class of Wilsonian formulations of non-Abelian gauge theories in algebraic noncovariant gauges where the Wilsonian infrared cutoff Λ is inserted as a mass term for the propagating fields. In this way the Ward-Takahashi identities are preserved to all scales. Nevertheless the BRS-invariance in broken and the theory is gauge-dependent and unphysical at Λ≠ 0. Then I discuss the infrared limit Λ→0. I show that the singularities of the axial gauge choice are avoided in planar gauge and in light-cone gauge. Finally the rectangular Wilson loop of size 2L×2T is evaluated at lowest order in perturbation theory and a noncommutativity between the limits Λ→0 and T→∞ is pointed out.

scholarly journals Non-invertible global symmetries and completeness of the spectrum

2021 ◽  
Vol 2021 (9) ◽  
Author(s):  
Ben Heidenreich ◽  
Jacob McNamara ◽  
Miguel Montero ◽  
Matthew Reece ◽  
Tom Rudelius ◽  
...  
Keyword(s):  
Gauge Theories ◽  
Cosmic Strings ◽  
Global Symmetry ◽  
Complete Spectrum ◽  
Abelian Gauge ◽  
Analogous Statement ◽  
Gauge Groups ◽  
Four Dimensions ◽  
Chern Simons ◽  
General Gauge

Abstract It is widely believed that consistent theories of quantum gravity satisfy two basic kinematic constraints: they are free from any global symmetry, and they contain a complete spectrum of gauge charges. For compact, abelian gauge groups, completeness follows from the absence of a 1-form global symmetry. However, this correspondence breaks down for more general gauge groups, where the breaking of the 1-form symmetry is insufficient to guarantee a complete spectrum. We show that the correspondence may be restored by broadening our notion of symmetry to include non-invertible topological operators, and prove that their absence is sufficient to guarantee a complete spectrum for any compact, possibly disconnected gauge group. In addition, we prove an analogous statement regarding the completeness of twist vortices: codimension-2 objects defined by a discrete holonomy around their worldvolume, such as cosmic strings in four dimensions. We discuss how this correspondence is modified in various, more general contexts, including non-compact gauge groups, Higgsing of gauge theories, and the addition of Chern-Simons terms. Finally, we discuss the implications of our results for the Swampland program, as well as the phenomenological implications of the existence of twist strings.

NON-ABELIAN GAUGE THEORY FOR CHERN–SIMONS PATH INTEGRAL ON R3

2012 ◽  
Vol 21 (04) ◽  
pp. 1250039 ◽  
Author(s):  
ADRIAN P. C. LIM
Keyword(s):  
Gauge Theory ◽  
Gauge Group ◽  
Path Integral ◽  
Wilson Loop ◽  
Wiener Measure ◽  
Abelian Gauge ◽  
Abelian Gauge Theory ◽  
Chern Simons

In a prequel to this article, we used abstract Wiener measure to define the Chern–Simons path integral over ℝ3. In this sequel, we compute the Wilson Loop observable for the non-abelian gauge group and compare with current knot literature.

scholarly journals ON THE CONSISTENCY OF THE EXACT RENORMALIZATION GROUP APPROACH APPLIED TO GAUGE THEORIES IN ALGEBRAIC NONCOVARIANT GAUGES

2000 ◽  
Vol 15 (30) ◽  
pp. 4811-4848 ◽  
Author(s):  
M. SIMIONATO
Keyword(s):  
Wilson Loop ◽  
Gauge Theories ◽  
Mass Term ◽  
Group Approach ◽  
Abelian Gauge ◽  
Light Cone Gauge ◽  
Renormalization Group Approach ◽  
Infrared Limit ◽  
Brst Invariance ◽  
Exact Renormalization Group

We study a class of Wilsonian formulations of non-Abelian gauge theories in algebraic noncovariant gauges where the Wilsonian infrared cutoff Λ is inserted as a mass term for the propagating fields. In this way the Ward–Takahashi identities are preserved to all scales. Nevertheless BRST-invariance in broken and the theory is gauge-dependent and unphysical at Λ≠0. Then we discuss the infrared limit Λ→0. We show that the singularities of the axial gauge choice are avoided in planar gauge and light-cone gauge. In addition the issue of on-shell divergences is addressed in some explicit example. Finally the rectangular Wilson loop of size 2L×2T is evaluated at lowest order in perturbation theory and a noncommutativity between the limits Λ→0 and T→∞ is pointed out.

scholarly journals Connecting mirror symmetry in 3D and 2D via localization

2014 ◽  
Vol 29 (32) ◽  
pp. 1530004 ◽  
Author(s):  
Heng-Yu Chen ◽  
Hsiao-Yi Chen ◽  
Jun-Kai Ho
Keyword(s):  
Mirror Symmetry ◽  
Gauge Theories ◽  
Three Dimensional ◽  
High Energy ◽  
Partition Functions ◽  
Two Dimensional ◽  
Abelian Gauge ◽  
Fusion Procedure ◽  
Mirror Pair ◽  
Simple Limit

We explicitly apply localization results to study the interpolation between three- and two-dimensional mirror symmetries for Abelian gauge theories with four supercharges. We first use the ellipsoid [Formula: see text] partition functions to verify the mirror symmetry between a pair of general three-dimensional 𝒩 = 2 Abelian Chern–Simons quiver gauge theories. These expressions readily factorize into holomorphic blocks and their antiholomorphic copies, so we can also obtain the partition functions on S1×S2 via fusion procedure. We then demonstrate S1×S2 partition functions for the three-dimensional Abelian gauge theories can be dimensionally reduced to the S2 partition functions of 𝒩 = (2, 2) GLSM and Landau–Ginzburg model for the corresponding two-dimensional mirror pair, as anticipated previously in M. Aganagic et al., J. High Energy Phys.0107, 022 (2001). We also comment on the analogous interpolation for the non-Abelian gauge theories and compute the K-theory vortex partition function for a simple limit to verify the prediction from holomorphic block.

scholarly journals Wilson loops in 5d long quiver gauge theories

2020 ◽  
Vol 2020 (9) ◽  
Author(s):  
Christoph F. Uhlemann
Keyword(s):  
Field Theory ◽  
Fixed Points ◽  
Wilson Loop ◽  
Gauge Theories ◽  
Internal Space ◽  
Wilson Loops ◽  
Expectation Values ◽  
Holographic Duals ◽  
Two Parameter ◽  
Supergravity Solution

Abstract Quiver gauge theories with a large number of nodes host a wealth of Wilson loop operators. Expectation values are obtained, using supersymmetric localization, for Wilson loops in the antisymmetric representations associated with each individual gauge node, for a sample of 5d long quiver gauge theories whose UV fixed points have holographic duals in Type IIB. The sample includes the TN theories and the results are uniformly given in terms of Bloch-Wigner functions. The holographic representation of the Wilson loops is identified. It comprises, for each supergravity solution, a two-parameter family of D3-branes which exactly reproduce the field theory results and identify points in the internal space with the faces of the associated 5-brane web. The expectation values of (anti)fundamental Wilson loops exhibit an enhanced scaling for many operators, which matches between field theory and supergravity.

scholarly journals Tests of Seiberg-like dualities in three dimensions

2020 ◽  
Vol 2020 (8) ◽  
Author(s):  
Anton Kapustin ◽  
Brian Willett ◽  
Itamar Yaakov
Keyword(s):  
Gauge Theories ◽  
Three Dimensions ◽  
Simons Theory ◽  
Partition Functions ◽  
Seiberg Duality ◽  
Monopole Operators ◽  
Real Mass ◽  
Supersymmetric Gauge ◽  
The Relationship ◽  
Chern Simons

Abstract We use localization techniques to study several duality proposals for supersymmetric gauge theories in three dimensions reminiscent of Seiberg duality. We compare the partition functions of dual theories deformed by real mass terms and FI parameters. We find that Seiberg-like duality for $$ \mathcal{N} $$ N = 3 Chern-Simons gauge theories proposed by Giveon and Kutasov holds on the level of partition functions and is closely related to level-rank duality in pure Chern-Simons theory. We also clarify the relationship between the Giveon-Kutasov duality and a duality in theories of fractional M2 branes and propose a generalization of the latter. Our analysis also confirms previously known results concerning decoupled free sectors in $$ \mathcal{N} $$ N = 4 gauge theories realized by monopole operators.

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