HOMFLY polynomials, stable pairs and motivic Donaldson–Thomas invariants

Diaconescu, Duiliu-Emanuel; Hua, Zheng; Soibelman, Yan

doi:10.4310/cntp.2012.v6.n3.a1

Cited by 23 publications

(34 citation statements)

References 61 publications

(219 reference statements)

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“…It is also interesting to relate our results to other developments, e.g., to the connection-albeit in a different context-between (uncolored) HOMFLY-PT polynomials and Donaldson-Thomas invariants in [36]. Then, the generating functions (12) take the form of combinations of q-series that appear in Nahm's conjecture [37], which indicates their relation to integrable systems and conformal field theory.…”

Section: Discussionmentioning

confidence: 62%

BPS states, knots, and quivers

Kucharski

Reineke

Stošić³

et al. 2017

Phys. Rev. D

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We argue how to identify the supersymmetric quiver quantum mechanics description of BPS states, which arise in string theory in brane systems representing knots. This leads to a surprising relation between knots and quivers: to a given knot, we associate a quiver, so that various types of knot invariants are expressed in terms of characteristics of a moduli space of representations of the corresponding quiver. This statement can be regarded as a novel type of categorification of knot invariants, and among its various consequences we find that Labastida-Mariño-Ooguri-Vafa (LMOV) invariants of a knot can be expressed in terms of motivic Donaldson-Thomas invariants of the corresponding quiver; this proves integrality of LMOV invariants (once the corresponding quiver is identified), conjectured originally based on string theory and M-theory arguments.

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Section: Discussionmentioning

confidence: 62%

BPS states, knots, and quivers

Kucharski

Reineke

Stošić³

et al. 2017

Phys. Rev. D

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“…As shown in [44], the variables (s, t) used in refined Chern-Simons theory are the same as those used in the refined vertex formalism [46], hence they are related to (q, y) by s = qy and t = qy −1 . Therefore, as conjectured in [23], one expects a relation of the form V (n) µ 1 ,...,µ ℓ (q, y) = w (n) µ 1 ,...,µ ℓ (s, t)W (n−2) µ 1 ,...,µ ℓ (s, t) s=qy, t=qy −1 (4.10)…”

Section: The Final Formulamentioning

confidence: 81%

“…In string theory these conjectures have been shown to follow from large N duality for conifold transitions in [24,23]. The physical derivation leads to a colored refined generalization of these conjectures formulated in [23] and proven by Maulik in [59] for the unrefined case. In the present context, these conjectures relate the refined stable pair theory of Y ξ to refined colored invariants of (ℓ, (n − 2)ℓ)-torus links.…”

Section: Refined Stable Pair Theory Via Torus Linksmentioning

confidence: 83%

BPS States, Torus Links and Wild Character Varieties

Diaconescu

Donagi

Pantev

2018

Commun. Math. Phys.

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A string theoretic framework is constructed relating the cohomology of wild character varieties to refined stable pair theory and torus link invariants. Explicit conjectural formulas are derived for wild character varieties with a unique irregular point on the projective line. For this case the string theoretic construction leads to a conjectural colored generalization of existing results of Hausel, Mereb and Wong as well as Shende, Treumann and Zaslow.

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“…Refined GV invariants have been discussed in the literature [CKK14, CDDP15, NO14, GHKPK17, references therein] as refined Pandharipande-Thomas (Donaldson-Thomas) invariants for non-compact toric Calabi-Yau threefolds. However, mathematical understanding of their open analogues are still immature although the Poincaré polynomials of uncolored HOMFLY-PT homology of torus knots have been related to motivic Donaldson-Thomas invariants in [DHS12]. Actually, upon the reduction on the cigar of the Taub-NUT in (1.3), BPS states can be understood as D6-D4-D2-D0 bound states.…”

Section: Discussionmentioning

confidence: 99%

Refined large N duality for knots

Kameyama

Nawata

2020

J. Knot Theory Ramifications

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We formulate large N duality of U(N ) refined Chern-Simons theory with a torus knot/link in S 3 . 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 Ω-background. This form enables us to relate refined Chern-Simons invariants of a torus knot/link in S 3 to refined BPS invariants in the resolved conifold. Assuming that the extra U(1) 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 nontorus knots.Dedicated to John H. Schwarz on his 75th birthday 1 Charges of J1, J2 and SR are normalized to be half-integers. 2 The brane setting gives rise to U(N ) gauge group instead of SU(N ) gauge group. However, the U(1) part merely provides the correction due to the framing number as well as the linking number of a knot/link, which play no role in this paper. Therefore, the difference between U(N ) and SU(N ) invariants will be ignored in this paper. 3 We denote the unreduced invariants by rCS λ (Tm,n; a, q, t) and the reduced invariants by rCS λ (Tm,n; a, q, t)where they are related by rCS λ (Tm,n; a, q, t) = rCS λ ( ; a, q, t) rCS λ (Tm,n; a, q, t) .where the function g λ can be determined by using the unknot invariants and it turns out to be the Macdonald norm defined in Appendix A. At the unrefined limit q = t, it reduces to the generating function of SU(N ) quantum invariants J SU(N ),λ (T m,n ; q) first obtained in [OV00] Z def,SU(N ) = λ J SU(N ),λ (T m,n ; q) s λ (x) , where s λ (x) are the Schur functions.

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HOMFLY polynomials, stable pairs and motivic Donaldson–Thomas invariants

Cited by 23 publications

References 61 publications

BPS states, knots, and quivers

BPS states, knots, and quivers

BPS States, Torus Links and Wild Character Varieties

Refined large N duality for knots

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