Abstract
We find a direct relation between quiver representation theory and open topological string theory on a class of toric Calabi-Yau manifolds without compact four-cycles, also referred to as strip geometries. We show that various quantities that characterize open topological string theory on these manifolds, such as partition functions, Gromov-Witten invariants, or open BPS invariants, can be expressed in terms of characteristics of the moduli space of representations of the corresponding quiver. This has various deep consequences; in particular, expressing open BPS invariants in terms of motivic Donaldson-Thomas invariants, immediately proves integrality of the former ones. Taking advantage of the relation to quivers we also derive explicit expressions for classical open BPS invariants for an arbitrary strip geometry, which lead to a large set of number theoretic integrality statements. Furthermore, for a specific framing, open topological string partition functions for strip geometries take form of generalized q-hypergeometric functions, which leads to a novel representation of these functions in terms of quantum dilogarithms and integral invariants. We also study quantum curves and A-polynomials associated to quivers, various limits thereof, and their specializations relevant for strip geometries. The relation between toric manifolds and quivers can be regarded as a generalization of the knots-quivers correspondence to more general Calabi-Yau geometries.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
M. Aganagic, A. Klemm, M. Mariño and C. Vafa, The Topological vertex, Commun. Math. Phys. 254 (2005) 425 [hep-th/0305132] [INSPIRE].
M. Aganagic, R. Dijkgraaf, A. Klemm, M. Mariño and C. Vafa, Topological strings and integrable hierarchies, Commun. Math. Phys. 261 (2006) 451 [hep-th/0312085] [INSPIRE].
E. Witten, Quantum Field Theory and the Jones Polynomial, Commun. Math. Phys. 121 (1989) 351 [INSPIRE].
E. Witten, Chern-Simons gauge theory as a string theory, Prog. Math. 133 (1995) 637 [hep-th/9207094] [INSPIRE].
H. Ooguri and C. Vafa, Knot invariants and topological strings, Nucl. Phys. B 577 (2000) 419 [hep-th/9912123] [INSPIRE].
J.M.F. Labastida and M. Mariño, Polynomial invariants for torus knots and topological strings, Commun. Math. Phys. 217 (2001) 423 [hep-th/0004196] [INSPIRE].
J.M.F. Labastida, M. Mariño and C. Vafa, Knots, links and branes at large N, JHEP 11 (2000) 007 [hep-th/0010102] [INSPIRE].
J.M.F. Labastida and M. Marino, A new point of view in the theory of knot and link invariants, J. Knot Theor. Ramifications 11 (2002) 173 [math/0104180].
P. Ramadevi and T. Sarkar, On link invariants and topological string amplitudes, Nucl. Phys. B 600 (2001) 487 [hep-th/0009188] [INSPIRE].
A. Mironov, A. Morozov, A. Morozov, P. Ramadevi, V.K. Singh and A. Sleptsov, Checks of integrality properties in topological strings, JHEP 08 (2017) 139 [arXiv:1702.06316] [INSPIRE].
P. Kucharski, M. Reineke, M. Stosic and P. Sulkowski, BPS states, knots and quivers, Phys. Rev. D 96 (2017) 121902 [arXiv:1707.02991] [INSPIRE].
P. Kucharski, M. Reineke, M. Stosic and P. Sulkowski, Knots-quivers correspondence, arXiv:1707.04017 [INSPIRE].
P. Kucharski and P. Sulkowski, BPS counting for knots and combinatorics on words, JHEP 11 (2016) 120 [arXiv:1608.06600] [INSPIRE].
W. Luo and S. Zhu, Integrality structures in topological strings I: framed unknot, arXiv:1611.06506 [INSPIRE].
M. Stosic and P. Wedrich, Rational links and DT invariants of quivers, arXiv:1711.03333 [INSPIRE].
M. Panfil, M. Stosic and P. Sulkowski, Donaldson-Thomas invariants, torus knots and lattice paths, Phys. Rev. D 98 (2018) 026022 [arXiv:1802.04573] [INSPIRE].
T. Ekholm, P. Kucharski and P. Longhi, Physics and geometry of knots-quivers correspondence, arXiv:1811.03110 [INSPIRE].
M. Aganagic and C. Vafa, Mirror symmetry, D-branes and counting holomorphic discs, hep-th/0012041 [INSPIRE].
M. Aganagic, A. Klemm and C. Vafa, Disk instantons, mirror symmetry and the duality web, Z. Naturforsch. A 57 (2002) 1 [hep-th/0105045] [INSPIRE].
J.A. Harvey and G.W. Moore, On the algebras of BPS states, Commun. Math. Phys. 197 (1998) 489 [hep-th/9609017] [INSPIRE].
M. Kontsevich and Y. Soibelman, Cohomological Hall algebra, exponential Hodge structures and motivic Donaldson-Thomas invariants, Commun. Num. Theor. Phys. 5 (2011) 231 [arXiv:1006.2706] [INSPIRE].
A. Iqbal and A.-K. Kashani-Poor, The Vertex on a strip, Adv. Theor. Math. Phys. 10 (2006) 317 [hep-th/0410174] [INSPIRE].
G. Gasper and M. Rahman, Basic Hypergeometric Series, Encyclopedia of Mathematics, Cambridge University Press (2004).
G. Bonelli, A. Tanzini and J. Zhao, Vertices, Vortices and Interacting Surface Operators, JHEP 06 (2012) 178 [arXiv:1102.0184] [INSPIRE].
T. Kimura and Y. Sugimoto, Quantum mirror curve of periodic chain geometry, arXiv:1810.01885 [INSPIRE].
T. Mainiero, Algebraicity and Asymptotics: An explosion of BPS indices from algebraic generating series, arXiv:1606.02693 [INSPIRE].
P. Smolinski, From topological strings to quantum invariants of knots and quivers, MSc Thesis, University of Warsaw (2017).
A.I. Efimov, Cohomological Hall algebra of a symmetric quiver, Compos. Math. 148 (2012) 1133 [arXiv:1103.2736].
S. Garoufalidis, P. Kucharski and P. Sulkowski, Knots, BPS states and algebraic curves, Commun. Math. Phys. 346 (2016) 75 [arXiv:1504.06327] [INSPIRE].
E. Basor, B. Conrey and K.E. Morrison, Knots and ones, arXiv:1703.00990 [INSPIRE].
M. Alim, S. Cecotti, C. Cordova, S. Espahbodi, A. Rastogi and C. Vafa, BPS Quivers and Spectra of Complete N = 2 Quantum Field Theories, Commun. Math. Phys. 323 (2013) 1185 [arXiv:1109.4941] [INSPIRE].
J. Manschot, B. Pioline and A. Sen, On the Coulomb and Higgs branch formulae for multi-centered black holes and quiver invariants, JHEP 05 (2013) 166 [arXiv:1302.5498] [INSPIRE].
R. Eager, S.A. Selmani and J. Walcher, Exponential Networks and Representations of Quivers, JHEP 08 (2017) 063 [arXiv:1611.06177] [INSPIRE].
M. Gabella, P. Longhi, C.Y. Park and M. Yamazaki, BPS Graphs: From Spectral Networks to BPS Quivers, JHEP 07 (2017) 032 [arXiv:1704.04204] [INSPIRE].
S. Zhu, Topological strings, quiver varieties and Rogers-Ramanujan identities, arXiv:1707.00831 [INSPIRE].
R. Gopakumar and C. Vafa, M theory and topological strings. 1., hep-th/9809187 [INSPIRE].
R. Gopakumar and C. Vafa, M theory and topological strings. 2., hep-th/9812127 [INSPIRE].
A. Okounkov, N. Reshetikhin and C. Vafa, Quantum Calabi-Yau and classical crystals, Prog. Math. 244 (2006) 597 [hep-th/0309208] [INSPIRE].
S. Gukov and P. Sulkowski, A-polynomial, B-model and Quantization, JHEP 02 (2012) 070 [arXiv:1108.0002] [INSPIRE].
M. Reineke, Cohomology of quiver moduli, functional equations, and integrality of Donaldson-Thomas type invariants, Compos. Math. 147 (2011) 943.
M. Reineke, Degenerate Cohomological Hall algebra and quantized Donaldson-Thomas invariants for m-loop quivers, Doc. Math. 17 (2012) 1 [arXiv:1102.3978].
S. Meinhardt and M. Reineke, Donaldson-Thomas invariants versus intersection cohomology of quiver moduli, [arXiv:1411.4062].
H. Franzen and M. Reineke, Semi-Stable Chow-Hall Algebras of Quivers and Quantized Donaldson-Thomas Invariants, [arXiv:1512.03748].
W. Nahm, Conformal field theory and torsion elements of the Bloch group, in Proceedings, Les Houches School of Physics: Frontiers in Number Theory, Physics and Geometry II: On Conformal Field Theories, Discrete Groups and Renormalization, Les Houches, France, March 9–21, 2003, pp. 67-132 (2007) [DOI:https://doi.org/10.1007/978-3-540-30308-4_2] [hep-th/0404120] [INSPIRE].
W. Koepf, P.M. Rajković and S.D. Marinković, Properties of q-holonomic functions, J. Differ. Equ. Appl. 13 (2007) 621.
S. Garoufalidis, A.D. Lauda and T.T.Q. Lê, The colored HOMFLYPT function is q-holonomic, Duke Math. J. 167 (2018) 397 [arXiv:1604.08502] [INSPIRE].
M. Aganagic and C. Vafa, Large N Duality, Mirror Symmetry and a Q-deformed A-polynomial for Knots, arXiv:1204.4709 [INSPIRE].
M. Aganagic, T. Ekholm, L. Ng and C. Vafa, Topological Strings, D-Model and Knot Contact Homology, Adv. Theor. Math. Phys. 18 (2014) 827 [arXiv:1304.5778] [INSPIRE].
S. Gukov, S. Nawata, I. Saberi, M. Stosic and P. Sulkowski, Sequencing BPS Spectra, JHEP 03 (2016) 004 [arXiv:1512.07883] [INSPIRE].
A. Schwarz, V. Vologodsky and J. Walcher, Framing the Di-Logarithm (over Z), Proc. Symp. Pure Math. 90 (2015) 113 [arXiv:1306.4298] [INSPIRE].
N. Halmagyi, A. Sinkovics and P. Sulkowski, Knot invariants and Calabi-Yau crystals, JHEP 01 (2006) 040 [hep-th/0506230] [INSPIRE].
Open Access
This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1811.03556
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made.
The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.
To view a copy of this licence, visit https://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Panfil, M., Sułkowski, P. Topological strings, strips and quivers. J. High Energ. Phys. 2019, 124 (2019). https://doi.org/10.1007/JHEP01(2019)124
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP01(2019)124