From hypertoric geometry to bordered Floer homology via the \(m = 1\) amplituhedron. (English) Zbl 1537.57022
“We relate the Fukaya category of the symmetric power of a genus zero surface to deformed category \(\mathcal{O}\) of a cyclic hypertoric variety by establishing an isomorphism between algebras defined by Ozsváth-Szabó in Heegaard-Floer theory [P. Ozsváth and Z. Szabó, Adv. Math. 328, 1088–1198 (2018; Zbl 1417.57015)] and Braden-Licata-Proudfoot-Webster in hypertoric geometry [T. Braden et al., Adv. Math. 225, No. 4, 2002–2049 (2010; Zbl 1205.14066)]. The proof extends work of Karp-Williams on sign variation and the combinatorics of the \(m = 1\) amplituhedron”, which appear in theoretical physics to compute scattering amplitudes.
“S. N. Karp and L. K. Williams [Int. Math. Res. Not. 2019, No. 5, 1401–1462 (2019; Zbl 1429.52024)] gave a cell decomposition of the \(m = 1\) amplituhedron using images of a collection of distinguished cells of the totally nonnegative Grassmannian, which provides an identification of the amplituhedron with the bounded faces of a cyclic hyperplane arrangement.”
Lauda, Licata, and Manion “use the algebras associated to cyclic arrangements to construct categorical actions of \(\mathfrak{gl}(1|1)\), and generalize our isomorphism to give a conjectural algebraic description of the Fukaya category of a complexified hyperplane complement.”
“S. N. Karp and L. K. Williams [Int. Math. Res. Not. 2019, No. 5, 1401–1462 (2019; Zbl 1429.52024)] gave a cell decomposition of the \(m = 1\) amplituhedron using images of a collection of distinguished cells of the totally nonnegative Grassmannian, which provides an identification of the amplituhedron with the bounded faces of a cyclic hyperplane arrangement.”
Lauda, Licata, and Manion “use the algebras associated to cyclic arrangements to construct categorical actions of \(\mathfrak{gl}(1|1)\), and generalize our isomorphism to give a conjectural algebraic description of the Fukaya category of a complexified hyperplane complement.”
Reviewer: Mee Seong Im (Annapolis)
MSC:
| 57K16 | Finite-type and quantum invariants, topological quantum field theories (TQFT) |
| 57K18 | Homology theories in knot theory (Khovanov, Heegaard-Floer, etc.) |
| 53D37 | Symplectic aspects of mirror symmetry, homological mirror symmetry, and Fukaya category |
| 14M25 | Toric varieties, Newton polyhedra, Okounkov bodies |
| 17B37 | Quantum groups (quantized enveloping algebras) and related deformations |
| 81U99 | Quantum scattering theory |
Keywords:
categorification; amplituhedron; hypertoric variety; \(\mathfrak{gl}(1|1)\); cyclic hyperplane; Ozsváth-Szabó algebra; complexified hyperplane complementSoftware:
HFKcalcReferences:
| [1] | Arkani-Hamed, N., Bai, Y., Lam, T.: Positive geometries and canonical forms. J. High Energy Phys. 11, 039, front matter+121, 2017. arXiv:1703.04541 · Zbl 1383.81273 |
| [2] | Arkani-Hamed, N.; Trnka, J., The Amplituhedron, JHEP, 2014, 30, 1029-8479, 2014 · Zbl 1468.81075 |
| [3] | Arkani-Hamed, N.; Thomas, H.; Trnka, J., Unwinding the amplituhedron in binary, JHEP, 2018, 16, 1029-8479, 2018 · Zbl 1384.81130 |
| [4] | Ando, T., Totally positive matrices, Linear Algebra Appl., 90, 165-219, 1987 · Zbl 0613.15014 · doi:10.1016/0024-3795(87)90313-2 |
| [5] | Auroux, D.: Fukaya categories and bordered Heegaard-Floer homology. In: Proceedings of the International Congress of Mathematicians. Volume II, pp. 917-941. Hindustan Book Agency, New Delhi (2010). arXiv:1003.2962 · Zbl 1275.53082 |
| [6] | Bielawski, R.; Dancer, AS, The geometry and topology of toric hyperkähler manifolds, Commun. Anal. Geom., 8, 4, 727-760, 2000 · Zbl 0992.53034 · doi:10.4310/CAG.2000.v8.n4.a2 |
| [7] | Braden, T.; Licata, A.; Phan, C.; Proudfoot, N.; Webster, B., Localization algebras and deformations of Koszul algebras, Selecta Math. (N.S.), 17, 3, 533-572, 2011 · Zbl 1259.16035 · doi:10.1007/s00029-011-0058-y |
| [8] | Braden, T.; Licata, A.; Proudfoot, N.; Webster, B., Gale duality and Koszul duality, Adv. Math., 225, 4, 2002-2049, 2010 · Zbl 1205.14066 · doi:10.1016/j.aim.2010.04.011 |
| [9] | Braden, T.; Licata, A.; Proudfoot, N.; Webster, B., Hypertoric category \(\cal{O} \), Adv. Math., 231, 3-4, 1487-1545, 2012 · Zbl 1284.16029 · doi:10.1016/j.aim.2012.06.019 |
| [10] | Braden, T., Licata, A., Proudfoot, N., Webster, B.: Quantizations of conical symplectic resolutions II: category \(\cal{O}\) and symplectic duality. Astérisque384, 75-179, 2016. with an appendix by I. Losev, arXiv:1407.0964 · Zbl 1360.53001 |
| [11] | Björner, A.; Las Vergnas, M.; Sturmfels, B.; White, N.; Ziegler, GM, Oriented Matroids of Encyclopedia, volume 46 of Mathematics and its Applications, 1999, Cambridge: Cambridge University Press, Cambridge · Zbl 0944.52006 |
| [12] | Braden, T.; Proudfoot, N.; Webster, B., Quantizations of conical symplectic resolutions I: local and global structure, Astérisque, 384, 1-73, 2016 · Zbl 1360.53001 |
| [13] | Calabi, E., Métriques kählériennes et fibrés holomorphes, Ann. Sci. École Norm. Sup. (4), 12, 2, 269-294, 1979 · Zbl 0431.53056 · doi:10.24033/asens.1367 |
| [14] | Douglas, CL; Manolescu, C., On the algebra of cornered Floer homology, J. Topol., 7, 1, 1-68, 2014 · Zbl 1315.57034 · doi:10.1112/jtopol/jtt013 |
| [15] | Eguchi, T.; Hanson, AJ, Self-dual solutions to Euclidean gravity, Ann. Phys., 120, 1, 82-106, 1979 · Zbl 0409.53020 · doi:10.1016/0003-4916(79)90282-3 |
| [16] | Ellis, AP; Petkova, I.; Vértesi, V., Quantum \(\mathfrak{gl}_{1|1}\) and tangle Floer homology, Adv. Math., 350, 130-189, 2019 · Zbl 1441.57013 · doi:10.1016/j.aim.2019.04.023 |
| [17] | Fan, Z.; Li, Y., A geometric setting for quantum \(\mathfrak{osp} (1|2)\), Trans. Am. Math. Soc., 367, 11, 7895-7916, 2015 · Zbl 1378.17027 · doi:10.1090/S0002-9947-2015-06266-7 |
| [18] | Forge, D.; Ramírez Alfonsín, JL, On counting the \(k\)-face cells of cyclic arrangements, Eur. J. Combin., 22, 3, 307-312, 2001 · Zbl 0984.52009 · doi:10.1006/eujc.2000.0462 |
| [19] | Gibbons, GW; Hawking, SW, Gravitational multi-instantons, Phys. Lett. B, 78, 4, 430-432, 1978 · doi:10.1016/0370-2693(78)90478-1 |
| [20] | Gantmacher, FR; Krein, MG, Oscillation Matrices and Small Oscillations of Mechanical Systems, 1941, Gostekhizdat: Moscow-Leningrad, Gostekhizdat · Zbl 0060.03303 |
| [21] | Goto, R.: On toric hyper-Kähler manifolds given by the hyper-Kähler quotient method. In: Infinite Analysis, Part A, B (Kyoto, 1991), volume 16 of Advances in Mathematical Physics, pp. 317-338. World Scientific Publishing, River Edge (1992) · Zbl 0924.53023 |
| [22] | Hilbert, D., Ueber die Theorie der algebraischen Formen, Math. Ann., 36, 4, 473-534, 1890 · JFM 22.0133.01 · doi:10.1007/BF01208503 |
| [23] | Hochster, M.: Topics in the homological theory of modules over commutative rings. Published for the Conference Board of the Mathematical Sciences by the American Mathematical Society, Providence, R.I., 1975. Expository lectures from the CBMS Regional Conference held at the University of Nebraska, Lincoln, Neb., June 24-28, 1974, Conference Board of the Mathematical Sciences Regional Conference Series in Mathematics, No. 24 |
| [24] | Intriligator, K.; Seiberg, N., Mirror symmetry in three-dimensional gauge theories, Phys. Lett. B, 387, 3, 513-519, 1996 · doi:10.1016/0370-2693(96)01088-X |
| [25] | Kleshchev, A., Affine highest weight categories and affine quasihereditary algebras, Proc. Lond. Math. Soc. (3), 110, 4, 841-882, 2015 · Zbl 1360.16010 · doi:10.1112/plms/pdv004 |
| [26] | Khovanov, M.; Seidel, P., Quivers, Floer cohomology, and braid group actions, J. Am. Math. Soc., 15, 1, 203-271, 2002 · Zbl 1035.53122 · doi:10.1090/S0894-0347-01-00374-5 |
| [27] | Karp, SN; Williams, LK, The \(m=1\) amplituhedron and cyclic hyperplane arrangements, Int. Math. Res. Not. IMRN, 5, 1401-1462, 2019 · Zbl 1429.52024 · doi:10.1093/imrn/rnx140 |
| [28] | Lauda, A.D., Licata, A., Manion, A.: Strands algebras and the affine highest weight property for equivariant hypertoric categories (2021). arXiv:2107.06480 |
| [29] | Lauda, A.D., Manion, A.: Ozsváth-Szabó bordered algebras and subquotients of category \({\cal{O}} \). Adv. Math.376, 107455, 59 (2021). arXiv:1910.03770 · Zbl 1460.57016 |
| [30] | Lipshitz, R., Ozsváth, P.S., Thurston, D.P.: Bordered Heegaard Floer homology. Mem. Am. Math. Soc.254(1216):viii+279 (2018). arXiv:0810.0687 |
| [31] | Lekili, Y.; Polishchuk, A., Homological mirror symmetry for higher-dimensional pairs of pants, Compos. Math., 156, 7, 1310-1347, 2020 · Zbl 1467.14099 · doi:10.1112/S0010437X20007150 |
| [32] | Manion, A., Khovanov-Seidel quiver algebras and Ozsváth-Szabó’s bordered theory, J. Algebra, 488, 110-144, 2017 · Zbl 1409.16010 · doi:10.1016/j.jalgebra.2017.05.029 |
| [33] | Manion, A., On the decategorification of Ozsváth and Szabó’s bordered theory for knot Floer homology, Quantum Topol., 10, 1, 77-206, 2019 · Zbl 1478.57014 · doi:10.4171/qt/123 |
| [34] | Manion, A.: Trivalent vertices and bordered knot Floer homology in the standard basis, (2020). arXiv:2012.07184 |
| [35] | Manion, A., Marengon, M., Willis, M.: Generators, relations, and homology for Ozsváth-Szabó’s Kauffman-states algebras. Nagoya Math. J. (2020). arXiv:1903.05654 |
| [36] | Manion, A.; Marengon, M.; Willis, M., Strands algebras and Ozsváth and Szabó’s Kauffman-states functor, Algebr. Geom. Topol., 20, 7, 3607-3706, 2020 · Zbl 1482.57011 · doi:10.2140/agt.2020.20.3607 |
| [37] | Manion, A., Rouquier, R.: Higher representations and cornered Heegaard Floer homology (2020). arXiv:2009.09627 |
| [38] | Ozsváth, PS; Szabó, Z., Kauffman states, bordered algebras, and a bigraded knot invariant, Adv. Math., 328, 1088-1198, 2018 · Zbl 1417.57015 · doi:10.1016/j.aim.2018.02.017 |
| [39] | Ozsváth, P. S., Szabó, Z.: Algebras with matchings and knot Floer homology (2019). arXiv:1912.01657 |
| [40] | Ozsváth, PS; Szabó, Z., Bordered knot algebras with matchings, Quantum Topol., 10, 3, 481-592, 2019 · Zbl 1439.57032 · doi:10.4171/qt/127 |
| [41] | Ozsváth, P. S., Szabó, Z.: Knot Floer homology calculator, 2019. https://web.math.princeton.edu/ szabo/HFKcalc.html |
| [42] | Ozsváth, P. S., Szabó, Z.: Algebras with matchings and link Floer homology, 2020. arXiv:2004.07309 |
| [43] | Postnikov, A.: Total positivity, Grassmannians, and networks, 2006. arXiv:math/0609764 |
| [44] | Proudfoot, N.J.: Hyperkahler Analogues of Kahler Quotients. ProQuest LLC, Ann Arbor (2004). Thesis (Ph.D.)-University of California, Berkeley |
| [45] | Ramírez Alfonsín, JL, Cyclic arrangements and Roudneff’s conjecture in the space, Inform. Process. Lett., 71, 5-6, 179-182, 1999 · Zbl 1003.52012 · doi:10.1016/S0020-0190(99)00115-5 |
| [46] | Sartori, A., Categorification of tensor powers of the vector representation of \(U_q(\mathfrak{gl} (1|1))\), Selecta Math. (N.S.), 22, 2, 669-734, 2016 · Zbl 1407.17012 · doi:10.1007/s00029-015-0202-1 |
| [47] | Schoenberg, I., Über variationsvermindernde lineare Transformationen, Math. Z., 32, 1, 321-328, 1930 · JFM 56.0106.06 · doi:10.1007/BF01194637 |
| [48] | Shannon, RW, Simplicial cells in arrangements of hyperplanes, Geom. Dedic., 8, 2, 179-187, 1979 · Zbl 0423.51013 · doi:10.1007/BF00181486 |
| [49] | Shan, P.; Varagnolo, M.; Vasserot, E., Koszul duality of affine Kac-Moody algebras and cyclotomic rational double affine Hecke algebras, Adv. Math., 262, 370-435, 2014 · Zbl 1333.17020 · doi:10.1016/j.aim.2014.05.012 |
| [50] | Tian, Y., A categorification of \(U_T(\mathfrak{sl} (1|1))\) and its tensor product representations, Geom. Topol., 18, 3, 1635-1717, 2014 · Zbl 1305.18053 · doi:10.2140/gt.2014.18.1635 |
| [51] | Tian, Y., Categorification of Clifford algebras and \({ {U}}_q(\mathfrak{sl} (1|1))\), J. Symplectic Geom., 14, 2, 541-585, 2016 · Zbl 1355.57006 · doi:10.4310/JSG.2016.v14.n2.a5 |
| [52] | Zarev, R.: Bordered Sutured Floer Homology. ProQuest LLC, Ann Arbor (2011). Thesis (Ph.D.)—Columbia University |
| [53] | Ziegler, GM, Higher Bruhat orders and cyclic hyperplane arrangements, Topology, 32, 2, 259-279, 1993 · Zbl 0782.06003 · doi:10.1016/0040-9383(93)90019-R |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
