Large \(N\) duality, Lagrangian cycles, and algebraic knots. (English) Zbl 1309.57011
This paper marks a major advance in large \(N\) duality between the Chern-Simons theory on \(S^3\) and the Gromov-Witten theory on the resolved conifold. Witten showed that the Chern-Simons theory on a \(3\)-manifold \(M\) can be interpreted as a theory of (degenerate) holomorphic curves on the cotangent bundle \(T^*M\) represented by ribbon graphs in \(M\). For \(M=S^3\) Gopakumar and Vafa conjectured that as \(T^*S^3\) undergoes a geometric transition into the resolved conifold (the \(\mathcal{O}(-1)\oplus\mathcal{O}(-1)\) bundle over \(\mathbb{C}P^1\)) this degenerate theory is transformed into an equivalent theory of genuine holomorphic curves there. Their conjecture, named large \(N\) duality, has since been confirmed directly, for the partition functions and for the Wilson loop expectation values of the unknot, and indirectly, e.g. by checking its integrality predictions for the coefficients of HOMFLY polynomials of arbitrary links.
In this paper the authors extend the direct verification to torus knots, and an indirect one to algebraic knots, by deriving for them from large \(N\) duality a conjecture of Oblomkov-Shende that relates BPS states from Donaldson-Thomas like invariants to HOMFLY polynomials. The underlying advance is a novel general construction of Lagrangian submanifolds in the resolved conifold that correspond to algebraic knots in \(S^3\), which enables direct localization computations in the case of torus knots.
Given a knot \(K\) in \(S^3\) it is expected that the holomorphic curves of the dual theory on the resolved conifold are the open ones with boundaries on a Lagrangian submanifold \(L_K\). This \(L_K\) should be “related” to the conormal bundle \(N_K^*\subset T^*S^3\) via the conifold transition. However, \(N_K^*\) intersects the zero section, which shrinks into a singular point under the conifold transition which is then resolved. And naively taking \(N_K^*\) through the transition “relates” it to a singular set, which is neither Lagrangian nor a submanifold, except when \(K\) is the unknot. This issue plagued the theory for years preventing computations of open Gromov-Witten invariants in other examples. Mariño and Vafa suggested early on that \(N_K^*\) should first be deformed away from the zero section, and only then put through the transition. A construction along these lines was carried out by the reviewer using symplectic methods, however the resulting \(L_K\) were only totally real, and lacked symmetries needed for computations.
In this paper the authors suggest a more general type of deformation, which in the case of algebraic knots results in explicitly describable Lagrangian submanifolds. For torus knots these \(L_K\) allow formal virtual localization computations analogous to the ones for the unknot (rigorous theory of open Gromov-Witten invariants is still unavailable), which confirm the duality. The proof is an open Gromov-Witten reflection of the Chern-Simons \(S\)-matrix formula that relates HOMFLY polynomials of torus knots to the colored invariants of the unknot.
Moreover, rigid holomorphic cylinders that connect the zero section in \(T^*S^3\) to the deformed \(N_K^*\) are constructed, and they are shown to be transformed by the conifold transition into singular holomorphic disks with the boundary on \(L_K\). This completes the geometric picture of the duality, and opens a path towards proving it by deformation arguments.
In this paper the authors extend the direct verification to torus knots, and an indirect one to algebraic knots, by deriving for them from large \(N\) duality a conjecture of Oblomkov-Shende that relates BPS states from Donaldson-Thomas like invariants to HOMFLY polynomials. The underlying advance is a novel general construction of Lagrangian submanifolds in the resolved conifold that correspond to algebraic knots in \(S^3\), which enables direct localization computations in the case of torus knots.
Given a knot \(K\) in \(S^3\) it is expected that the holomorphic curves of the dual theory on the resolved conifold are the open ones with boundaries on a Lagrangian submanifold \(L_K\). This \(L_K\) should be “related” to the conormal bundle \(N_K^*\subset T^*S^3\) via the conifold transition. However, \(N_K^*\) intersects the zero section, which shrinks into a singular point under the conifold transition which is then resolved. And naively taking \(N_K^*\) through the transition “relates” it to a singular set, which is neither Lagrangian nor a submanifold, except when \(K\) is the unknot. This issue plagued the theory for years preventing computations of open Gromov-Witten invariants in other examples. Mariño and Vafa suggested early on that \(N_K^*\) should first be deformed away from the zero section, and only then put through the transition. A construction along these lines was carried out by the reviewer using symplectic methods, however the resulting \(L_K\) were only totally real, and lacked symmetries needed for computations.
In this paper the authors suggest a more general type of deformation, which in the case of algebraic knots results in explicitly describable Lagrangian submanifolds. For torus knots these \(L_K\) allow formal virtual localization computations analogous to the ones for the unknot (rigorous theory of open Gromov-Witten invariants is still unavailable), which confirm the duality. The proof is an open Gromov-Witten reflection of the Chern-Simons \(S\)-matrix formula that relates HOMFLY polynomials of torus knots to the colored invariants of the unknot.
Moreover, rigid holomorphic cylinders that connect the zero section in \(T^*S^3\) to the deformed \(N_K^*\) are constructed, and they are shown to be transformed by the conifold transition into singular holomorphic disks with the boundary on \(L_K\). This completes the geometric picture of the duality, and opens a path towards proving it by deformation arguments.
Reviewer: Sergiy Koshkin (Houston)
MSC:
| 57M27 | Invariants of knots and \(3\)-manifolds (MSC2010) |
| 81T30 | String and superstring theories; other extended objects (e.g., branes) in quantum field theory |
| 57M25 | Knots and links in the \(3\)-sphere (MSC2010) |
| 81T45 | Topological field theories in quantum mechanics |
Keywords:
Chern-Simons; torus knots; open Gromov-Witten invariants; resolved conifold; conifold transition; HOMFLY polynomials; Lagrangian submanifold; S-matrix formula; virtual localizationReferences:
| [1] | Aganagic M., Klemm A., Mariño M., Vafa C.: The topological vertex. Commun. Math. Phys. 254(2), 425-478 (2005) · Zbl 1114.81076 · doi:10.1007/s00220-004-1162-z |
| [2] | Aganagic M., Marino M., Vafa C.: All loop topological string amplitudes from Chern-Simons theory. Commun. Math. Phys. 247, 467-512 (2004) · Zbl 1055.81055 · doi:10.1007/s00220-004-1067-x |
| [3] | Aganagic M., Ooguri H., Vafa C., Yamazaki M.: Wall Crossing and M-theory. Publ. Res. Inst. Math. Sci. Kyoto 47, 569 (2011) · Zbl 1230.14084 · doi:10.2977/PRIMS/44 |
| [4] | Aganagic, M., Vafa, C.: Mirror symmetry, D-branes and counting holomorphic discs. http://arxiv.org/abs/hep-th/0012041v1, 2000 · Zbl 1031.57022 |
| [5] | Brini, A., Eynard, B., Marino, M.: Torus knots and mirror symmetry. http://arxiv.org/abs/1105.2012v1 [hep-th], 2011 · Zbl 1256.81086 |
| [6] | Campillo A., Delgado F., Gussein-Zade S.: The Alexander polynomial of a plane curve singularity via the ring of functions on it. Duke Math. J. 117(1), 125-156 (2003) · Zbl 1028.32013 · doi:10.1215/S0012-7094-03-11712-3 |
| [7] | Cecotti, S., Neitzke, A., Vafa, C.: R-Twisting and 4d/2d Correspondences. 2010. http://arxiv.org/abs/1006.3435v2 [hep-th], 2010 · Zbl 1036.81515 |
| [8] | Diaconescu D.-E., Florea B.: Large N duality for compact Calabi-Yau threefolds. Adv. Theor. Math. Phys. 9(1), 31-128 (2005) · Zbl 1169.14322 |
| [9] | Diaconescu D.-E., Florea B.: Localization and gluing of topological amplitudes. Commun. Math. Phys. 257, 119-149 (2005) · Zbl 1097.14044 · doi:10.1007/s00220-005-1323-8 |
| [10] | Diaconescu D.-E., Florea B., Grassi A.: Geometric transitions and open string instantons. Adv. Theor. Math. Phys. 6, 619-642 (2003) |
| [11] | Diaconescu D.-E., Florea B., Grassi A.: Geometric transitions, del Pezzo surfaces and open string instantons. Adv. Theor. Math. Phys. 6, 643-702 (2003) |
| [12] | Dijkgraaf, R., Vafa, C., Verlinde, E.: M-theory and a topological string duality. http://arxiv.org/abs/hep-th/0602087v1, 2006 · Zbl 1071.53541 |
| [13] | Gopakumar, R., Vafa, C.: M-theory and topological strings. I. 1998. http://arxiv.org/abs/hep-th/9809187v1, 1998 · Zbl 0922.32015 |
| [14] | Gopakumar R., Vafa C.: On the gauge theory/geometry correspondence. Adv. Theor. Math. Phys. 3(5), 1415-1443 (1999) · Zbl 0972.81135 |
| [15] | Govindarajan S., Jayaraman T., Sarkar T.: Disc instantons in linear sigma models. Nucl. Phys. B646, 498-523 (2002) · Zbl 0999.81073 · doi:10.1016/S0550-3213(02)00901-X |
| [16] | Hellerman S., Kachru S., Lawrence A.E., McGreevy J.: Linear sigma models for open strings. JHEP 0207, 002 (2002) · doi:10.1088/1126-6708/2002/07/002 |
| [17] | Hori, K.: Linear models of supersymmetric D-branes. http://arxiv.org/abs/hep-th/0012179v1, 2000 · Zbl 1022.81045 |
| [18] | Iqbal A., Kozcaz C., Vafa C.: The refined topological vertex. JHEP 10, 069 (2009) · doi:10.1088/1126-6708/2009/10/069 |
| [19] | Katz S., Liu C.-C.M.: Enumerative geometry of stable maps with Lagrangian boundary conditions and multiple covers of the disc. Adv. Theor. Math. Phys. 5(1), 1-49 (2001) · Zbl 1026.32028 |
| [20] | Katz S.H., Klemm A., Vafa C.: M theory, topological strings and spinning black holes. Adv. Theor. Math. Phys. 3, 1445-1537 (1999) · Zbl 0985.81081 |
| [21] | Kawai T.: String and vortex. Publ. Res. Inst. Math. Sci. Kyoto 40, 1063-1091 (2004) · Zbl 1255.81208 · doi:10.2977/prims/1145475503 |
| [22] | Koshkin S.: Conormal bundles to knots and the Gopakumar-Vafa conjecture. Adv. Theor. Math. Phys. 11(4), 591-634 (2007) · Zbl 1134.81397 |
| [23] | Labastida J.M.F., Marino M., Vafa C.: Knots, links and branes at large N. JHEP 11, 007 (2000) · Zbl 0990.81545 · doi:10.1088/1126-6708/2000/11/007 |
| [24] | Li, J., Song, Y.: Open string instantons and relative stable morphisms. In: The interaction of finite-type and Gromov-Witten invariants (BIRS 2003), Volume 8 of Geom. Topol. Monogr., Coventry: Geom. Topol. Publ., 2006, pp. 49-72 · Zbl 1107.14303 |
| [25] | Liu C.-C.M., Liu K., Zhou J.: A proof of a conjecture of Mariño-Vafa on Hodge integrals. J. Diff. Geom. 65(2), 289-340 (2003) · Zbl 1077.14084 |
| [26] | Liu, C.-C.M., Liu, K., Zhou, J.: A formula of two-partition Hodge integrals. J. Amer. Math. Soc. 20(1), 149-184 (electronic) (2007) · Zbl 1111.14053 |
| [27] | Liu, K., Peng, P.: New Structure of Knot Invariants. http://arxiv.org/abs/1012.2636v1 [math.GT], 2010 · Zbl 1247.81391 |
| [28] | Liu, K., Peng, P.: Proof of the Labastida-Marino-Ooguri-Vafa conjecture. http://arxiv.org/abs/0704.1526v3 [math.GA], 2009 · Zbl 1217.81129 |
| [29] | Liu K., Peng P.: On a proof of the Labastida-Marino-Ooguri-Vafa conjecture. Math. Res. Lett. 17, 493-506 (2010) · Zbl 1223.81155 |
| [30] | Marino, M., Vafa, C.: Framed knots at large N. http://arxiv.org/abs/hep-th/0108064v1, 2001 · Zbl 1042.81071 |
| [31] | Mayr P.: N=1 mirror symmetry and open / closed string duality. Adv. Theor. Math. Phys. 5, 213-242 (2002) · Zbl 1022.81046 |
| [32] | McDuff, D., Salamon, D.: Introduction to symplectic topology. Oxford Mathematical Monographs. New York: The Clarendon Press / Oxford University Press, second edition, 1998 · Zbl 1066.53137 |
| [33] | Oblomkov A., Shende V.: The Hilbert scheme of a plane curve singularity and the HOMFLY polynomial of its link. Duke Math. J 161(7), 1277-1303 (2012) · Zbl 1256.14025 · doi:10.1215/00127094-1593281 |
| [34] | Okounkov A., Pandharipande R.: Hodge integrals and invariants of the unknot. Geom. Topol. 8, 675-699 (2004) · Zbl 1062.14035 · doi:10.2140/gt.2004.8.675 |
| [35] | Ooguri H., Vafa C.: Knot invariants and topological strings. Nucl. Phys. B577, 419-438 (2000) · Zbl 1036.81515 · doi:10.1016/S0550-3213(00)00118-8 |
| [36] | Pandharipande R., Thomas R.P.: Curve counting via stable pairs in the derived category. Invent. Math. 178(2), 407-447 (2009) · Zbl 1204.14026 · doi:10.1007/s00222-009-0203-9 |
| [37] | Pandharipande R., Thomas R.P.: Stable pairs and BPS invariants. J. Amer. Math. Soc. 23(1), 267-297 (2010) · Zbl 1250.14035 · doi:10.1090/S0894-0347-09-00646-8 |
| [38] | Smith I., Thomas R.: Symplectic surgeries from singularities. Turkish J. Math. 27(1), 231-250 (2003) · Zbl 1031.57022 |
| [39] | Smith I., Thomas R.P., Yau S.-T.: Symplectic conifold transitions. J. Diff. Geom. 62(2), 209-242 (2002) · Zbl 1071.53541 |
| [40] | Stevan S.: Chern-Simons Invariants of Torus Links. Ann. H. Poincare 11, 1201-1224 (2010) · Zbl 1208.81149 · doi:10.1007/s00023-010-0058-z |
| [41] | Strominger A., Yau S.-T., Zaslow E.: Mirror symmetry is T duality. Nucl. Phys. B479, 243-259 (1996) · Zbl 0896.14024 · doi:10.1016/0550-3213(96)00434-8 |
| [42] | Taubes C.H.: Lagrangians for the Gopakumar-Vafa conjecture. Adv. Theor. Math. Phys. 5, 139-163 (2001) · Zbl 1022.53057 |
| [43] | Witten E.: Quantum Field Theory and the Jones Polynomial. Commun. Math. Phys. 121, 351 (1989) · Zbl 0667.57005 · doi:10.1007/BF01217730 |
| [44] | Witten, E.: Chern-Simons gauge theory as a string theory. In: The Floer memorial volume, Volume 133 of Progr. Math., Basel: Birkhäuser, 1995, pp. 637-678 · Zbl 0844.58018 |
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