scholarly journals BPS states in the Ω-background and torus knots

2014 ◽  
Vol 2014 (4) ◽  
Author(s):  
K. Bulycheva ◽  
A. Gorsky
Keyword(s):  
Author(s):  
Masaya Kameyama ◽  
Satoshi Nawata

We formulate large [Formula: see text] duality of [Formula: see text] refined Chern–Simons theory with a torus knot/link in [Formula: see text]. By studying refined BPS states in M-theory, we provide the explicit form of low-energy effective actions of Type IIA string theory with D4-branes on the [Formula: see text]-background. This form enables us to relate refined Chern–Simons invariants of a torus knot/link in [Formula: see text] to refined BPS invariants in the resolved conifold. Assuming that the extra [Formula: see text] global symmetry acts on BPS states trivially, the duality predicts graded dimensions of cohomology groups of moduli spaces of M2–M5 bound states associated to a torus knot/link in the resolved conifold. Thus, this formulation can be also interpreted as a positivity conjecture of refined Chern–Simons invariants of torus knots/links. We also discuss about an extension to non-torus knots.


2020 ◽  
Vol 2020 (9) ◽  
Author(s):  
Piotr Kucharski

Abstract We introduce and explore the relation between quivers and 3-manifolds with the topology of the knot complement. This idea can be viewed as an adaptation of the knots-quivers correspondence to Gukov-Manolescu invariants of knot complements (also known as FK or $$ \hat{Z} $$ Z ̂ ). Apart from assigning quivers to complements of T(2,2p+1) torus knots, we study the physical interpretation in terms of the BPS spectrum and general structure of 3d $$ \mathcal{N} $$ N = 2 theories associated to both sides of the correspondence. We also make a step towards categorification by proposing a t-deformation of all objects mentioned above.


2016 ◽  
Vol 25 (03) ◽  
pp. 1640003
Author(s):  
Ross Elliot ◽  
Sergei Gukov

The goal of this paper is twofold. First, we find a natural home for the double affine Hecke algebras (DAHA) in the physics of BPS states. Second, we introduce new invariants of torus knots and links called hyperpolynomials that address the “problem of negative coefficients” often encountered in DAHA-based approaches to homological invariants of torus knots and links. Furthermore, from the physics of BPS states and the spectra of singularities associated with Landau–Ginzburg potentials, we also describe a rich structure of differentials that act on homological knot invariants for exceptional groups and uniquely determine the latter for torus knots.


2021 ◽  
Vol 2021 (6) ◽  
Author(s):  
Brice Bastian ◽  
Thomas W. Grimm ◽  
Damian van de Heisteeg

Abstract We study the charge-to-mass ratios of BPS states in four-dimensional $$ \mathcal{N} $$ N = 2 supergravities arising from Calabi-Yau threefold compactifications of Type IIB string theory. We present a formula for the asymptotic charge-to-mass ratio valid for all limits in complex structure moduli space. This is achieved by using the sl(2)-structure that emerges in any such limit as described by asymptotic Hodge theory. The asymptotic charge-to-mass formula applies for sl(2)-elementary states that couple to the graviphoton asymptotically. Using this formula, we determine the radii of the ellipsoid that forms the extremality region of electric BPS black holes, which provides us with a general asymptotic bound on the charge-to-mass ratio for these theories. Finally, we comment on how these bounds for the Weak Gravity Conjecture relate to their counterparts in the asymptotic de Sitter Conjecture and Swampland Distance Conjecture.


2021 ◽  
Vol 2021 (6) ◽  
Author(s):  
Vivek Kumar Singh ◽  
Rama Mishra ◽  
P. Ramadevi

Abstract Weaving knots W(p, n) of type (p, n) denote an infinite family of hyperbolic knots which have not been addressed by the knot theorists as yet. Unlike the well known (p, n) torus knots, we do not have a closed-form expression for HOMFLY-PT and the colored HOMFLY-PT for W(p, n). In this paper, we confine to a hybrid generalization of W(3, n) which we denote as $$ {\hat{W}}_3 $$ W ̂ 3 (m, n) and obtain closed form expression for HOMFLY-PT using the Reshitikhin and Turaev method involving $$ \mathrm{\mathcal{R}} $$ ℛ -matrices. Further, we also compute [r]-colored HOMFLY-PT for W(3, n). Surprisingly, we observe that trace of the product of two dimensional $$ \hat{\mathrm{\mathcal{R}}} $$ ℛ ̂ -matrices can be written in terms of infinite family of Laurent polynomials $$ {\mathcal{V}}_{n,t}\left[q\right] $$ V n , t q whose absolute coefficients has interesting relation to the Fibonacci numbers $$ {\mathrm{\mathcal{F}}}_n $$ ℱ n . We also computed reformulated invariants and the BPS integers in the context of topological strings. From our analysis, we propose that certain refined BPS integers for weaving knot W(3, n) can be explicitly derived from the coefficients of Chebyshev polynomials of first kind.


2014 ◽  
Vol 14 (2) ◽  
pp. 769-793 ◽  
Author(s):  
Peter Feller
Keyword(s):  

Magnetic relaxation of a magnetic field embedded in a perfectly conducting incompressible fluid to minimum energy magnetostatic equilibrium states is considered. It is supposed that the magnetic field is confined to a single flux tube which may be knotted. A local non-orthogonal coordinate system, zero-framed with respect to the knot, is introduced, and the field is decomposed into toroidal and poloidal ingredients with respect to this system. The helicity of the field is then determined; this vanishes for a field that is either purely toroidal or purely poloidal. The magnetic energy functional is calculated under the simplifying assumptions that the tube is axially uniform and of circular cross-section. The case of a tube with helical axis is first considered, and new results concerning kink mode instability and associated bifurcations are obtained. The case of flux tubes in the form of torus knots is then considered and the ‘ground-state’ energy function ͞m ( h ) (where h is an internal twist parameter) is obtained; as expected, ͞m ( h ), which is a topological invariant of the knot, increases with increasing knot complexity. The function ͞m ( h ) provides an upper bound on the corresponding function m ( h ) that applies when the above constraints on tube structure are removed. The technique is applicable to any knot admitting a parametric representation, on condition that points of vanishing curvature are excluded.


1997 ◽  
Vol 490 (1-2) ◽  
pp. 121-144 ◽  
Author(s):  
J.G. Russo ◽  
A.A. Tseytlin
Keyword(s):  

1998 ◽  
Vol 07 (05) ◽  
pp. 639-650 ◽  
Author(s):  
K. NAKAMURA ◽  
Y. NAKANISHI ◽  
Y. UCHIDA

The Δ-unknotting number for a knot is defined to be the minimum number of Δ-unknotting operations which deform the knot into the trivial knot. We determine the Δ-unknotting numbers for torus knots, positive pretzel knots, and positive closed 3-braids.


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