A topological theory for unoriented \(\mathrm{SL}(4)\) foams. (English) Zbl 1536.05191
Summary: Unoriented \(\mathrm{SL}(3)\) foams are two-dimensional CW complexes with generic singularities embedded in 3- and 4-manifolds. They naturally come up in the Kronheimer-Mrowka \(\mathrm{SO}(3)\) gauge theory for 3-orbifolds and, in the oriented case, in a categorification of the Kuperberg bracket quantum invariant. The present paper studies the more technically complicated case of \(\mathrm{SL}(4)\) foams. Combinatorial evaluation of unoriented \(\mathrm{SL}(4)\) foams is defined and state spaces for it are studied. In particular, over a suitably localized ground ring, the state space of any web is free of the rank given by the number of its 4-colorings.
MSC:
| 05C15 | Coloring of graphs and hypergraphs |
| 05A30 | \(q\)-calculus and related topics |
| 57M15 | Relations of low-dimensional topology with graph theory |
| 57K16 | Finite-type and quantum invariants, topological quantum field theories (TQFT) |
| 18N25 | Categorification |
References:
| [1] | Aganagic M.: Homological knot invariants from mirror symmetry. arXiv:2207.14104 (2022) |
| [2] | Anno, R.; Vinoth, N., Exotic \(t\)-structures for two-block Springer fibres, Trans. Am. Math. Soc, 376, 3, 1523-1552, 2023 · Zbl 1524.14037 · doi:10.1090/tran/8765 |
| [3] | Abouzaid, M.; Smith, I., Khovanov homology from Floer cohomology, J. Am. Math. Soc., 32, 1, 1-79, 2019 · Zbl 1401.57005 · doi:10.1090/jams/902 |
| [4] | Bernstein, J.; Frenkel, I.; Khovanov, M., A categorification of the Temperley-Lieb algebra and Schur quotients of \(U({\mathfrak{sl} }_2)\) via projective and Zuckerman functors, Sel. Math., 5, 2, 199-241, 1999 · Zbl 0981.17001 · doi:10.1007/s000290050047 |
| [5] | Blanchet, C.; Habegger, N.; Masbaum, G.; Vogel, P., Topological quantum field theories derived from the Kauffman bracket, Topology, 34, 4, 883-927, 1995 · Zbl 0887.57009 · doi:10.1016/0040-9383(94)00051-4 |
| [6] | Boozer, D., Computer bounds for Kronheimer-Mrowka foam evaluation, Exp. Math., 2021 · Zbl 1527.05058 · doi:10.1080/10586458.2021.1982078 |
| [7] | Boozer, D.: The combinatorial and gauge-theoretic foam evaluation functors are not the same. arXiv:2304.07659 (2023) |
| [8] | Chudnovsky, M.; Edwards, K.; Kawarabayashi, K.; Seymour, P., Edge-colouring seven-regular planar graphs, J. Combin. Theory Ser. B, 115, 276-302, 2015 · Zbl 1319.05051 · doi:10.1016/j.jctb.2014.11.005 |
| [9] | Cautis, S.; Kamnitzer, J., Knot homology via derived categories of coherent sheaves. II. \( \mathfrak{sl}_m\) case, Invent. Math., 174, 1, 165-232, 2008 · Zbl 1298.57007 · doi:10.1007/s00222-008-0138-6 |
| [10] | Cautis, S.; Kamnitzer, J.; Morrison, S., Webs and quantum skew Howe duality, Math. Ann., 360, 1-2, 351-390, 2014 · Zbl 1387.17027 · doi:10.1007/s00208-013-0984-4 |
| [11] | Gukov, S.; Schwarz, A.; Vafa, C., Khovanov-Rozansky homology and topological strings, Lett. Math. Phys., 74, 1, 53-74, 2005 · Zbl 1105.57011 · doi:10.1007/s11005-005-0008-8 |
| [12] | Guenin, B.: Packing \({T}\)-joins and edge-colouring in planar graphs. Math. Oper. Res. (2014) (To appear) |
| [13] | Kempe, AB, On the geographical problem of the four colours, Am. J. Math., 2, 3, 193-200, 1879 · doi:10.2307/2369235 |
| [14] | Khovanov, M., sl(3) link homology, Algebr. Geom. Topol., 4, 1045-1081, 2004 · Zbl 1159.57300 · doi:10.2140/agt.2004.4.1045 |
| [15] | Khovanov, M.: Universal construction of topological theories in two dimensions. arXiv:2007.03361 (2020) |
| [16] | Kim, D.: Graphical calculus on representations of quantum lie algebras. Ph.D. thesis, University of California, Davis. arXiv:math/0310143 (2003) |
| [17] | Khovanov, M., Kitchloo, N.: A deformation of Robert-Wagner foam evaluation and link homology. Contemp. Math. arXiv:2004.14197 (2020) (To appear) |
| [18] | Kronheimer, PB; Mrowka, TS, Exact triangles for \(SO(3)\) instanton homology of webs, J. Topol., 9, 3, 774-796, 2016 · Zbl 1371.57027 · doi:10.1112/jtopol/jtw010 |
| [19] | Kronheimer, PB; Mrowka, TS, A deformation of instanton homology for webs, Geom. Topol., 23, 3, 1491-1547, 2019 · Zbl 1444.57021 · doi:10.2140/gt.2019.23.1491 |
| [20] | Kronheimer, PB; Mrowka, TS, Tait colorings, and an instanton homology for webs and foams, J. Eur. Math. Soc. (JEMS), 21, 1, 55-119, 2019 · Zbl 1423.57050 · doi:10.4171/JEMS/831 |
| [21] | Khovanov, M.; Rozansky, L., Matrix factorizations and link homology, Fund. Math., 199, 1, 1-91, 2008 · Zbl 1145.57009 · doi:10.4064/fm199-1-1 |
| [22] | Khovanov, M.; Robert, L-H, Foam evaluation and Kronheimer-Mrowka theories, Adv. Math., 376, 2021 · Zbl 1455.57019 · doi:10.1016/j.aim.2020.107433 |
| [23] | Khovanov, M.; Robert, L-H, Conical \(SL(3)\) foams, J. Comb. Algebra, 6, 1-2, 79-108, 2022 · Zbl 1492.05034 · doi:10.4171/jca/61 |
| [24] | Manolescu, C., Link homology theories from symplectic geometry, Adv. Math., 211, 1, 363-416, 2007 · Zbl 1117.57012 · doi:10.1016/j.aim.2006.09.007 |
| [25] | Mrudul, M.T.: Webs and Foams of Simple Lie Algebras. Ph.D. thesis. Columbia University (2023) |
| [26] | Mackaay, M.; Stošić, M.; Vaz, P., \( \mathfrak{sl} (N)\)-link homology \((N\ge 4)\) using foams and the Kapustin-Li formula, Geom. Topol., 13, 2, 1075-1128, 2009 · Zbl 1202.57017 · doi:10.2140/gt.2009.13.1075 |
| [27] | Mackaay, M.; Vaz, P., The universal \({\rm sl}_3\)-link homology, Algebra Geom. Topol., 7, 1135-1169, 2007 · Zbl 1170.57011 · doi:10.2140/agt.2007.7.1135 |
| [28] | Queffelec, H.; Rose, DEV, The \(\mathfrak{sl}_n\) foam 2-category: a combinatorial formulation of Khovanov-Rozansky homology via categorical skew Howe duality, Adv. Math., 302, 1251-1339, 2016 · Zbl 1360.57025 · doi:10.1016/j.aim.2016.07.027 |
| [29] | Robert, L-H; Wagner, E., A closed formula for the evaluation of foams, Quantum Topol., 11, 3, 411-487, 2020 · Zbl 1476.57054 · doi:10.4171/qt/139 |
| [30] | Seidel, P.; Smith, I., A link invariant from the symplectic geometry of nilpotent slices, Duke Math. J., 134, 3, 453-514, 2006 · Zbl 1108.57011 · doi:10.1215/S0012-7094-06-13432-4 |
| [31] | Stroppel, C.; Sussan, J., A Lie theoretic categorification of the coloured Jones polynomial, J. Pure Appl. Algebra, 226, 10, 2022 · Zbl 1497.57014 · doi:10.1016/j.jpaa.2022.107043 |
| [32] | Sussan, J.: Category O and sl(k) link invariants. ProQuest LLC, Ann Arbor, MI, 2007. Thesis (Ph.D.)-Yale University. https://www.proquest.com/docview/304773953. arXiv:math/0701045 |
| [33] | Tutte, WT, A ring in graph theory, Proc. Camb. Philos. Soc., 43, 26-40, 1947 · Zbl 0031.41803 · doi:10.1017/s0305004100023173 |
| [34] | Webster, B., Knot invariants and higher representation theory, Mem. Am. Math. Soc., 250, 1191, v+141, 2017 · Zbl 1446.57001 · doi:10.1090/memo/1191 |
| [35] | Witten, E.: Two lectures on the Jones polynomial and Khovanov homology. In: Lectures on Geometry, Clay Lect. Notes, pp. 1-27. Oxford University Press, Oxford (2017). arXiv:1401.6996 · Zbl 1393.57004 |
| [36] | Wu, H., A colored \(\mathfrak{sl} (N)\) homology for links in \(S^3\), Diss. Math., 499, 1-217, 2014 · Zbl 1320.57020 · doi:10.4064/dm499-0-1 |
| [37] | Yonezawa, Y., Quantum \((\mathfrak{sl}_n,\wedge V_n)\) link invariant and matrix factorizations, Nagoya Math. J., 204, 69-123, 2011 · Zbl 1271.57033 · doi:10.1215/00277630-1431840 |
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