Journal of Holography Applications in Physics
Advancements in Functorial Homological Mirror Symmetry
Document Type : Regular article
Author
Shanghai Institute for Mathematics and Interdisciplinary Sciences (SIMIS); Block A, International Innovation Plaza, No. 657 Songhu Road, 200433 Yangpu District, Shanghai, China
Abstract
Mostly inspired by recent work by Katzarkov, Kontsevich, and Sheshmani, combined with previous work by Aganagic, Ooguri, Saulina and Vafa with regard to BPS black hole microstate counting in terms of topological field theory calculations, we will show how these tools can be applied to concrete setups arising from String Theory, and why the formalism of functorial Homological Mirror Symmetry needs to be further developed. A crucial ingredient will turn out being cobordism techniques for evaluating invariants.
Keywords
Main Subjects
Article PDF
[1] E. Witten, “Two dimensional gravity and intersection theory on moduli space”, Survey in Diff. Geom. 1, 243 (1991).
[2] M. Kontsevich, “Homological Algebra of Mirror Symmetry”, In: Chatterji, S.D. (eds) Proceedings of the International Congress of Mathematicians, Birkhäuser, Basel (1995) DOI:10.1007/978-3-0348-9078-6_11.
[3] A. Strominger, S.-T. Yau, E. Zaslow, “Mirror symmetry is T-duality”, Nuclear Physics B 479(1-2), 243 (1996) DOI:10.1016/0550-3213(96)00434-8.
[4] M. Kontsevich and Y. Soibelman, “Cohomological Hall algebra, exponential Hodge structures and motivic Donaldson-Thomas invariants”, Commun. Num. Theor. Phys. 5(2), 231 (2011) DOI:10.4310/CNTP.2011.v5.n2.a1 [arXiv:1006.2706 [math.AG]].
[5] M. Kontsevich and Y. Soibelman, “Motivic Donaldson-Thomas invariants: Summary of results”, [arXiv:0910.4315 [math.AG]] (2009).
[6] M. Kontsevich and Y. Soibelman, “Stability structures, motivic Donaldson-Thomas invariants and cluster transformations”, [arXiv:0811.2435 [math.AG]] (2008).
[7] L. Katzarkov, M. Kontsevich and T. Pantev, “Hodge theoretic aspects of mirror symmetry”, Proc. Symp. Pure Math. 78, 87 (2008) DOI:10.1090/pspum/078/2483750 [arXiv:0806.0107 [math.AG]].
[8] M. Kontsevich and Y. Soibelman, “Notes on A1-Algebras, A1-Categories and NonCommutative Geometry”, Lect. Notes Phys. 757, 153 (2009) DOI:10.1007/978-3-540- 68030-7_6 [arXiv:math/0606241 [math.RA]].
[9] M. Kontsevich, “Deformation quantization of algebraic varieties”, Lett. Math. Phys. 56, 271 (2006) DOI:10.1023/A:1017957408559 [arXiv:math/0106006 [math.AG]].
[10] Y. j. Chen, M. Kontsevich and A. S. Schwarz, “Symmetries of WDVV equations”, Nucl. Phys. B 730, 352 (2005) DOI:10.1016/j.nuclphysb.2005.09.025 [arXiv:hep-th/0508221 [hep-th]].
[11] C. Fronsdal and M. Kontsevich, “Quantization on curves”, Lett. Math. Phys. 79, 109 (2007) DOI:10.1007/s11005-006-0137-8 [arXiv:math-ph/0507021 [math-ph]].
[12] M. Kontsevich and Y. Soibelman, “Homological mirror symmetry and torus fibrations”, [arXiv:math/0011041 [math.SG]] (2000).
[13] M. Kontsevich and Y. Soibelman, “Deformations of algebras over operads and Deligne’s conjecture”, [arXiv:math/0001151 [math.QA]] (2000).
[14] M. Kontsevich, “Operads and motives in deformation quantization”, Lett. Math. Phys. 48, 35 (1999) DOI:10.1023/A:1007555725247 [arXiv:math/9904055 [math]].
[15] S. Barannikov and M. Kontsevich, “Frobenius Manifolds and Formality of Lie Algebras of Polyvector Fields”, [arXiv:alg-geom/9710032 [math.AG]] (1997).
[16] M. Kontsevich, “Deformation quantization of Poisson manifolds. 1.”, Lett. Math. Phys. 66, 157 (2003) DOI:10.1023/B:MATH.0000027508.00421.bf [arXiv:q-alg/9709040 [math.QA]].
[17] M. Kontsevich and Y. I. Manin, “Quantum cohomology of a product”, Invent. Math. 124, 313 (1996) DOI:10.1007/s002220050055 [arXiv:q-alg/9502009 [math.QA]].
[18] M. Kontsevich and Y. Manin, “Gromov-Witten classes, quantum cohomology, and enumerative geometry”, Commun. Math. Phys. 164, 525 (1994) DOI:10.1007/BF02101490 [arXiv:hep-th/9402147 [hep-th]].
[19] S. Hosono, T. J. Lee, B. H. Lian and S. T. Yau, “Mirror symmetry for double cover Calabi–Yau varieties”, J. Diff. Geom. 127(1), 409 (2024) DOI:10.4310/jdg/1717356161 [arXiv:2003.07148 [math.AG]].
[20] B. H. Lian and S. T. Yau, “Period Integrals of CY and General Type Complete Intersections”, Invent. Math. 191, 1 (2013) DOI:10.1007/s00222-012-0391-6 [arXiv:1105.4872 [math.AG]].
[21] S. Hosono, B. H. Lian, K. Oguiso and S. T. Yau, “Fourier-Mukai partners of a K3 surface of Picard number one”, [arXiv:math/0211249 [math.AG]] (2002).
[22] S. Hosono, B. H. Lian, K. Oguiso and S. T. Yau, “Fourier-Mukai number of a K3 surface”, [arXiv:math/0202014 [math.AG]] (2002).
[23] B. H. Lian, C. H. Liu, K. f. Liu and S. T. Yau, “The S**1 fixed points in Quot schemes and mirror principle computations”, [arXiv:math/0111256 [math.AG]] (2001).
[24] B. H. Lian, L. Kefeng and S. T. Yau, “A Survey of Mirror Principle”, AMS/IP Stud. Adv. Math. 33, 3 (2002).
[25] B. H. Lian, A. Todorov and S. T. Yau, “Maximal unipotent monodromy for complete intersection CY manifolds”, [arXiv:math/0008061 [math.AG]] (2000).
[26] B. H. Lian, C. H. Liu and S. T. Yau, “A Reconstruction of Euler data”, [arXiv:math/0003071 [math.AG]] (2000).
[27] B. H. Lian, K. F. Liu and S. T. Yau, “Mirror principle. 2.”, Asian J. Math. 3, 109 (1999).
[28] B. H. Lian and S. T. Yau, “Differential equations from mirror symmetry”, (1999).
[29] B. H. Lian, K. Liu and S. T. Yau, “The Candelas-de la Ossa-Green-Parkes formula”, Nucl. Phys. B Proc. Suppl. 67, 106 (1998) DOI:10.1016/S0920-5632(98)00126-1.
[30] B. H. Lian, K. F. Liu and S. T. Yau, “Mirror principle. 1.”, Asian J. Math. 1, 729 (1997).
[31] P. Seidel, “Fukaya A∞-structures associated to Lefschetz fibrations. III”, J. Diff. Geom. 117(3), 485 (2021) DOI:10.4310/jdg/1615487005.
[32] P. Seidel, “Formal groups and quantum cohomology”, Geom. Topol. 27(8), 2937 (2023) DOI:10.2140/gt.2023.27.2937 [arXiv:1910.08990 [math.SG]].
[33] P. Seidel, “Fukaya A∞-categories associated to Lefschetz fibrations. II ”, Adv. Theor. Math. Phys. 20(4), 883 (2016) DOI:10.4310/ATMP.2016.v20.n4.a5 [arXiv:1404.1352 [math.SG]].
[34] P. Seidel, “Lagrangian homology spheres in (Am) Milnor fibres”, [arXiv:1202.1955 [math.SG]] (2012).
[35] P. Seidel, “Fukaya A∞-structures associated to Lefschetz fibrations, I”, J. Symplectic Geom. 10(3), 325 (2012) DOI:10.4310/jsg.2012.v10.n3.a1 [arXiv:0912.3932 [math.SG]].
[36] P. Seidel, “Homological mirror symmetry for the quartic surface”, [arXiv:math/0310414 [math.SG]] (2003).
[37] M. Abouzaid, S. Ganatra, H. Iritani and N. Sheridan, “The Gamma and Strominger– Yau–Zaslow conjectures: a tropical approach to periods”, Geom. Topol. 24(5), 2547 (2020) DOI:10.2140/gt.2020.24.2547 [arXiv:1809.02177 [math.AG]].
[38] M. Abouzaid and C. Manolescu, “A sheaf-theoretic model for SL(2,C) Floer homology”, J. Eur. Math. Soc. 22(11), 3641 (2020) DOI:10.4171/jems/994 [arXiv:1708.00289 [math.GT]].
[39] M. Abouzaid, D. Auroux and L. Katzarkov, “Lagrangian fibrations on blowups of toric varieties and mirror symmetry for hypersurfaces”, Publ. Math. Inst. Hautes Étud. Sci. 123, 199 (2016) DOI:10.1007/s10240-016-0081-9.
[40] M. Abouzaid and I. Smith, “Khovanov homology from Floer cohomology”, J. Am. Math. Soc. 32(1), 1 (2019) DOI:10.1090/jams/902 [arXiv:1504.01230 [math.SG]].
[41] M. Abouzaid, D. Auroux, A. I. Efimov, L. Katzarkov and D. Orlov, “Homological mirror symmetry for punctured spheres”, J. Am. Math. Soc. 26(4), 1051 (2013) DOI:10.1090/S0894-0347-2013-00770-5.
[42] R. Bocklandt and M. Abouzaid, “Noncommutative mirror symmetry for punctured surfaces”, Trans. Am. Math. Soc. 368(1), 429 (2016) DOI:10.1090/tran/6375 [arXiv:1111.3392 [math.AG]].
[43] M. Abouzaid and P. Seidel, “Altering symplectic manifolds by homologous recombination”, [arXiv:1007.3281 [math.SG]] (2010).
[44] M. Abouzaid and M. Boyarchenko, “Local structure of generalized complex manifolds”, [arXiv:math/0412084 [math.DG]] (2004).
[45] D. Auroux, A. I. Efimov and L. Katzarkov, “Lagrangian Floer theory for trivalent graphs and homological mirror symmetry for curves”, Selecta Math. 30(5), 95 (2024) DOI:10.1007/s00029-024-00988-6.
[46] L. Cavenaghi, L. Grama and L. Katzarkov, “New look at Milnor Spheres”, [arXiv:2404.19088 [math.DG]] (2024).
[47] R. P. Horja and L. Katzarkov, “Discriminants and toric K-theory”, Adv. Math. 453, 109831 (2024) DOI:10.1016/j.aim.2024.109831 [arXiv:2205.00903 [math.AG]].
[48] F. Haiden, L. Katzarkov and C. Simpson, “Spectral Networks and Stability Conditions for Fukaya Categories with Coeffcients”, Commun. Math. Phys. 405(11), 264 (2024) DOI:10.1007/s00220-024-05138-9 [arXiv:2112.13623 [math.AG]].
[49] L. Katzarkov, E. Lupercio, L. Meersseman and A. Verjovsky, “Quantum (noncommutative) toric geometry: Foundations”, Adv. Math. 391, 107945 (2021) DOI:10.1016/j.aim.2021.107945.
[50] D. Borisov, L. Katzarkov, A. Sheshmani and S. T. Yau, “Strictification and gluing of Lagrangian distributions on derived schemes with shifted symplectic forms”, Adv. Math. 438, 109477 (2024) DOI:10.1016/j.aim.2023.109477 [arXiv:1908.00651 [math.AG]].
[51] A. Kasprzyk, L. Katzarkov, V. Przyjalkowski and D. Sakovics, “Projecting Fanos in the mirror”, [arXiv:1904.02194 [math.AG]] (2019).
[52] M. Ballard, D. Deliu, D. Favero, M. U. Isik and L. Katzarkov, “On the derived categories of degree d hypersurface fibrations”, Math. Ann. 371(1-2), 337 (2018) DOI:10.1007/s00208-017-1613-4.
[53] L. Katzarkov and L. Soriani, “Homological mirror symmetry, coisotropic branes and P=W”, Eur. J. Math. 4(3), 1141 (2018) DOI:10.1007/s40879-018-0273-6.
[54] M. Ballard, D. Favero and L. Katzarkov, “Variation of geometric invariant theory quotients and derived categories”, J. Reine Angew. Math. 2019(746), 235 (2019) DOI:10.1515/crelle-2015-0096.
[55] L. Katzarkov, E. Lupercio, L. Meersseman and A. Verjovsky, “Non-commutative Toric Varieties”, [arXiv:1308.2774 [math.SG]] (2013).
[56] G. Dimitrov, F. Haiden, L. Katzarkov and M. Kontsevich, “Dynamical systems and categories”, (2013) DOI:10.1090/conm/621 [arXiv:1307.8418 [math.CT]].
[57] A. Rebhan, L. Katzarkov, J. Knapp, R. Rashkov and E. Scheidegger, “Strings, gauge fields, and the geometry behind: The legacy of Maximilian Kreuzer”, World Scientific, 2013 ISBN 978-981-4412-54-4 DOI:10.1142/8561.
[58] R. S. Garavuso, L. Katzarkov, M. Kreuzer and A. Noll, “Super Landau-Ginzburg mirrors and algebraic cycles”, JHEP 03, 017 (2011) [erratum: JHEP 08, 063 (2011)] DOI:10.1007/JHEP08(2011)063 [arXiv:1101.1368 [hep-th]].
[59] A. Kapustin, L. Katzarkov, D. Orlov and M. Yotov, “Homological Mirror Symmetry for manifolds of general type”, Open Math. 7(4), 571 (2009) DOI:10.2478/s11533-009- 0056-x [arXiv:1004.0129 [math.AG]].
[60] H. L. Chang, J. Li, W. P. Li and C. C. M. Liu, “An effective theory of GW and FJRW invariants of quintics Calabi–Yau manifolds”, J. Diff. Geom. 120(2), 251 (2022) DOI:10.4310/jdg/1645207466.
[61] J. Li and C. C. M. Liu, “Counting curves in quintic Calabi–Yau threefolds and Landau–Ginzburg models”, Surveys Diff. Geom. 24(1), 173 (2019) DOI:10.4310/sdg.2019.v24.n1.a5.
[62] H. L. Chang, J. Li, W. P. Li and C. C. M. Liu, “An effective theory of GW and FJRW invariants of quintics Calabi-Yau manifolds”, [arXiv:1603.06184 [math.AG]] (2016).
[63] S. Banerjee, P. Longhi and M. Romo, “Modelling A-branes with foliations”, [arXiv:2309.07748 [hep-th]] (2023).
[64] T. J. Lee, B. H. Lian and M. Romo, “Non-commutative resolutions as mirrors of singular Calabi–Yau varieties”, [arXiv:2307.02038 [hep-th]] (2023).
[65] J. Guo and M. Romo, “Hybrid models for homological projective duals and noncommutative resolutions”, Lett. Math. Phys. 112(6), 117 (2022) DOI:10.1007/s11005-022- 01605-3 [arXiv:2111.00025 [hep-th]].
[66] H. Jockers, V. Kumar, J. M. Lapan, D. R. Morrison and M. Romo, “Two-Sphere Partition Functions and Gromov-Witten Invariants”, Commun. Math. Phys. 325, 1139 (2014) DOI:10.1007/s00220-013-1874-z [arXiv:1208.6244 [hep-th]].
[67] L. Katzarkov, M. Kontsevich and T. Pantev, “Hodge theoretic aspects of mirror symmetry”, (2008) [arXiv:0806.0107 [math.AG]].
[68] D. Oprea, “Framed sheaves over treefolds and symmetric obstruction theories”, Doc. Math. 18, 323 (2013) DOI:10.4171/DM/399.
[69] T. Pantev, B. Toën, M. Vaquié and G. Vezzosi, “Shifted symplectic structures”, Publ. Math. Inst. Hautes Étud. Sci. 117, 271 (2013) DOI:10.1007/s10240-013-0054-1.
[70] D. Calaque, “Shifted cotangent stacks are shifted symplectic”, [arXiv:1612.08101 [math.AG]], (2017).
[71] C. Brav, V. Bussi, D. Dupont, D. Joyce and B. Szendroi, “Symmetries and stabilization for sheaves of vanishing cycles”, [arXiv:1211.3259 [math.AG]] (2012).
[72] C. Brav, V. Bussi and D. Joyce, “A Darboux theorem for derived schemes with shifted symplectic structure”, J. Am. Math. Soc. 32(2), 399 (2018) DOI:10.1090/jams/910 [arXiv:1305.6302 [math.AG]].
[73] O. Ben-Bassat, C. Brav, V. Bussi and D. Joyce, “A ‘Darboux theorem’ for shifted symplectic structures on derived Artin stacks, with applications”, Geom. Topol. 19(3), 1287 (2015) DOI:10.2140/gt.2015.19.1287 [arXiv:1312.0090 [math.AG]].
[74] C. Brav and T. Dyckerhoff, “Relative Calabi–Yau structures II: shifted Lagrangians in the moduli of objects”, Selecta Math. 27(4), 63 (2021) DOI:10.1007/s00029-021-00642-5 [arXiv:1812.11913 [math.AG]].
[75] Y. Qin, “Coisotropic branes on tori and Homological mirror symmetry”, [arXiv:2207.10647 [math.SG]] (2022).
[76] Y. Qin, “Shifted symplectic structure on Poisson Lie algebroid and generalized complex geometry”, [arXiv:2407.15598 [math.SG]] (2024).
[77] M. Aganagic, I. Danilenko, Y. Li, V. Shende and P. Zhou, “Quiver Hecke algebras from Floer homology in Coulomb branches”, [arXiv:2406.04258 [math.SG]] (2024).
[78] S. Gukov, P. Koroteev, S. Nawata, D. Pei and I. Saberi, “Branes and DAHA Representations”, Springer, 2023 ISBN 978-3-031-28153-2, 978-3-031-28154-9 DOI:10.1007/978- 3-031-28154-9 [arXiv:2206.03565 [hep-th]].
[79] A. Losev, N. Nekrasov and S. L. Shatashvili, “On four dimensional mirror symmetry”, Fortsch. Phys. 48, 163 (2000).
[80] A. Losev and V. Lysov, “Tropical Mirror”, SIGMA 20, 072 (2024) DOI:10.3842/SIGMA.2024.072 [arXiv:2204.06896 [hep-th]].
[81] A. Losev and V. Lysov, “Tropical Mirror Symmetry: Correlation functions”, [arXiv:2301.01687 [hep-th]] (2023).
[82] A. Losev and V. Lysov, “Tropical mirror for toric surfaces”, [arXiv:2305.00423 [hep-th]] (2023).
[83] E. Frenkel and A. Losev, “Mirror symmetry in two steps: A-I-B”, Commun. Math. Phys. 269, 39 (2006) DOI:10.1007/s00220-006-0114-1 [arXiv:hep-th/0505131 [hep-th]].
[84] R. Donagi, B. A. Ovrut, T. Pantev and D. Waldram, “Standard model bundles on nonsimply connected Calabi-Yau threefolds”, JHEP 08, 053 (2001) DOI:10.1088/1126- 6708/2001/08/053 [arXiv:hep-th/0008008 [hep-th]].
[85] R. Donagi, B. A. Ovrut, T. Pantev and D. Waldram, “Standard model bundles”, Adv. Theor. Math. Phys. 5(3), 563 (2002) DOI:10.4310/ATMP.2001.v5.n3.a5 [arXiv:math/0008010 [math.AG]].
[86] R. Donagi, B. A. Ovrut, T. Pantev and D. Waldram, “Nonperturbative vacua in heterotic M theory”, Class. Quant. Grav. 17, 1049 (2000) DOI:10.1088/0264- 9381/17/5/314.
[87] R. Donagi, B. A. Ovrut, T. Pantev and D. Waldram, “Standard models from heterotic M theory”, Adv. Theor. Math. Phys. 5(1), 93 (2002) DOI:10.4310/ATMP.2001.v5.n1.a4 [arXiv:hep-th/9912208 [hep-th]].
[88] M. Cicoli, I. G. Etxebarria, F. Quevedo, A. Schachner, P. Shukla and R. Valandro, “The Standard Model quiver in de Sitter string compactifications”, JHEP 08, 109 (2021) DOI:10.1007/JHEP08(2021)109 [arXiv:2106.11964 [hep-th]].
[89] L. Katzarkov, M. Kontsevich and A. Sheshmani, “to appear”, (2024).
[90] R. Pandharipande, “Cohomological Field Theory Calculations”, Proc. Int. Cong. of Math. 2018, Rio de Janeiro, Vol. 1, 869 (2018).
[91] M. Aganagic, H. Ooguri, N. Saulina and C. Vafa, “Black Holes, q-Deformed 2d Yang-Mills, and Non-perturbative Topological Strings”, [arXiv:hep-th/0411280 [hepth]] (2004).
[92] G. W. Moore and Y. Tachikawa, “On 2d TQFTs whose values are holomorphic symplectic varieties”, Proc. Symp. Pure Math. 85, 191 (2012) DOI:10.1090/pspum/085/1379.
[93] P. Crooks and M. Mayrand, “The Moore-Tachikawa conjecture via shifted symplectic geometry”, [arXiv:2409.03532 [math.SG]] (2024).
[94] G. W. Moore and G. B. Segal, “D-branes and K-theory in 2D topological field theory”, [arXiv:hep-th/0609042 [hep-th]] (2006).
[95] V. Pasquarella, “Moore-Tachikawa Varieties: Beyond Duality”, JHAP 3(4), 39 (2023) DOI:10.22128/jhap.2023.741.1061 [arXiv:2310.01489 [hep-th]].
[96] C. Teleman, “The structure of 2D semi-simple field theories”, Invent. Math. 188(3), 525 (2012) DOI:10.1007/s00222-011-0352-5.
[97] D. Xie and S. T. Yau, “Three dimensional canonical singularity and five dimensional N = 1 SCFT”, JHEP 2017, 1 (2017).
[98] C. Teleman, “The rôle of Coulomb branches in 2D gauge theory”, J. Eur. Math. Soc. 20(3), 489 (2018).
[99] A. Braverman, M. Finkelberg and H. Nakajima, “Towards a mathematical definition of Coulomb branches of 3-dimensional N = 4 gauge theories, II”, Adv. Theor. Math. Phys. 22(5), 1071 (2018) DOI:10.4310/ATMP.2018.v22.n5.a1 [arXiv:1601.03586 [math.RT]].
[100] P. Safronov, “Shifted Geometric Quantisation”, (2020).
[101] C. F. Doran, A. Harder and A. Thompson, “Mirror symmetry, Tyurin degenerations and fibrations on Calabi-Yau manifolds”, [arXiv:1602.05554 [math.AG]] (2016).
[102] L. Katzarkov, P. S. Pandit and T. C. Spaide, “Calabi-Yau structures, spherical functors, and shifted symplectic structures”, Adv. Math. 335, 279 (2018) DOI:10.1016/j.aim.2018.07.005.
[103] A. Kapustin, L. Rozansky and N. Saulina, “Three-dimensional topological field theory and symplectic algebraic geometry I”, Nucl. Phys. B 816, 295 (2009) DOI:10.1016/j.nuclphysb.2009.01.027 [arXiv:0810.5415 [hep-th]].
[104] C. Teleman, “Gauge theory and mirror symmetry”, [arXiv:1409.6320 [math-ph]] (2014).
[105] N. Sheridan, “On the Homological Mirror Symmetry Conjecture for the Pair of Pants”, J. Diff. Geom. 89(2), 271 (2011).
[106] P. Seidel, “Fukaya Categories and Picard-Lefschetz Theory”, (2008).
[107] B. Webster, “Koszul duality between Higgs and Coulomb categories O”, [arXiv:1611.06541 [math.RT]] (2016).
[2] M. Kontsevich, “Homological Algebra of Mirror Symmetry”, In: Chatterji, S.D. (eds) Proceedings of the International Congress of Mathematicians, Birkhäuser, Basel (1995) DOI:10.1007/978-3-0348-9078-6_11.
[3] A. Strominger, S.-T. Yau, E. Zaslow, “Mirror symmetry is T-duality”, Nuclear Physics B 479(1-2), 243 (1996) DOI:10.1016/0550-3213(96)00434-8.
[4] M. Kontsevich and Y. Soibelman, “Cohomological Hall algebra, exponential Hodge structures and motivic Donaldson-Thomas invariants”, Commun. Num. Theor. Phys. 5(2), 231 (2011) DOI:10.4310/CNTP.2011.v5.n2.a1 [arXiv:1006.2706 [math.AG]].
[5] M. Kontsevich and Y. Soibelman, “Motivic Donaldson-Thomas invariants: Summary of results”, [arXiv:0910.4315 [math.AG]] (2009).
[6] M. Kontsevich and Y. Soibelman, “Stability structures, motivic Donaldson-Thomas invariants and cluster transformations”, [arXiv:0811.2435 [math.AG]] (2008).
[7] L. Katzarkov, M. Kontsevich and T. Pantev, “Hodge theoretic aspects of mirror symmetry”, Proc. Symp. Pure Math. 78, 87 (2008) DOI:10.1090/pspum/078/2483750 [arXiv:0806.0107 [math.AG]].
[8] M. Kontsevich and Y. Soibelman, “Notes on A1-Algebras, A1-Categories and NonCommutative Geometry”, Lect. Notes Phys. 757, 153 (2009) DOI:10.1007/978-3-540- 68030-7_6 [arXiv:math/0606241 [math.RA]].
[9] M. Kontsevich, “Deformation quantization of algebraic varieties”, Lett. Math. Phys. 56, 271 (2006) DOI:10.1023/A:1017957408559 [arXiv:math/0106006 [math.AG]].
[10] Y. j. Chen, M. Kontsevich and A. S. Schwarz, “Symmetries of WDVV equations”, Nucl. Phys. B 730, 352 (2005) DOI:10.1016/j.nuclphysb.2005.09.025 [arXiv:hep-th/0508221 [hep-th]].
[11] C. Fronsdal and M. Kontsevich, “Quantization on curves”, Lett. Math. Phys. 79, 109 (2007) DOI:10.1007/s11005-006-0137-8 [arXiv:math-ph/0507021 [math-ph]].
[12] M. Kontsevich and Y. Soibelman, “Homological mirror symmetry and torus fibrations”, [arXiv:math/0011041 [math.SG]] (2000).
[13] M. Kontsevich and Y. Soibelman, “Deformations of algebras over operads and Deligne’s conjecture”, [arXiv:math/0001151 [math.QA]] (2000).
[14] M. Kontsevich, “Operads and motives in deformation quantization”, Lett. Math. Phys. 48, 35 (1999) DOI:10.1023/A:1007555725247 [arXiv:math/9904055 [math]].
[15] S. Barannikov and M. Kontsevich, “Frobenius Manifolds and Formality of Lie Algebras of Polyvector Fields”, [arXiv:alg-geom/9710032 [math.AG]] (1997).
[16] M. Kontsevich, “Deformation quantization of Poisson manifolds. 1.”, Lett. Math. Phys. 66, 157 (2003) DOI:10.1023/B:MATH.0000027508.00421.bf [arXiv:q-alg/9709040 [math.QA]].
[17] M. Kontsevich and Y. I. Manin, “Quantum cohomology of a product”, Invent. Math. 124, 313 (1996) DOI:10.1007/s002220050055 [arXiv:q-alg/9502009 [math.QA]].
[18] M. Kontsevich and Y. Manin, “Gromov-Witten classes, quantum cohomology, and enumerative geometry”, Commun. Math. Phys. 164, 525 (1994) DOI:10.1007/BF02101490 [arXiv:hep-th/9402147 [hep-th]].
[19] S. Hosono, T. J. Lee, B. H. Lian and S. T. Yau, “Mirror symmetry for double cover Calabi–Yau varieties”, J. Diff. Geom. 127(1), 409 (2024) DOI:10.4310/jdg/1717356161 [arXiv:2003.07148 [math.AG]].
[20] B. H. Lian and S. T. Yau, “Period Integrals of CY and General Type Complete Intersections”, Invent. Math. 191, 1 (2013) DOI:10.1007/s00222-012-0391-6 [arXiv:1105.4872 [math.AG]].
[21] S. Hosono, B. H. Lian, K. Oguiso and S. T. Yau, “Fourier-Mukai partners of a K3 surface of Picard number one”, [arXiv:math/0211249 [math.AG]] (2002).
[22] S. Hosono, B. H. Lian, K. Oguiso and S. T. Yau, “Fourier-Mukai number of a K3 surface”, [arXiv:math/0202014 [math.AG]] (2002).
[23] B. H. Lian, C. H. Liu, K. f. Liu and S. T. Yau, “The S**1 fixed points in Quot schemes and mirror principle computations”, [arXiv:math/0111256 [math.AG]] (2001).
[24] B. H. Lian, L. Kefeng and S. T. Yau, “A Survey of Mirror Principle”, AMS/IP Stud. Adv. Math. 33, 3 (2002).
[25] B. H. Lian, A. Todorov and S. T. Yau, “Maximal unipotent monodromy for complete intersection CY manifolds”, [arXiv:math/0008061 [math.AG]] (2000).
[26] B. H. Lian, C. H. Liu and S. T. Yau, “A Reconstruction of Euler data”, [arXiv:math/0003071 [math.AG]] (2000).
[27] B. H. Lian, K. F. Liu and S. T. Yau, “Mirror principle. 2.”, Asian J. Math. 3, 109 (1999).
[28] B. H. Lian and S. T. Yau, “Differential equations from mirror symmetry”, (1999).
[29] B. H. Lian, K. Liu and S. T. Yau, “The Candelas-de la Ossa-Green-Parkes formula”, Nucl. Phys. B Proc. Suppl. 67, 106 (1998) DOI:10.1016/S0920-5632(98)00126-1.
[30] B. H. Lian, K. F. Liu and S. T. Yau, “Mirror principle. 1.”, Asian J. Math. 1, 729 (1997).
[31] P. Seidel, “Fukaya A∞-structures associated to Lefschetz fibrations. III”, J. Diff. Geom. 117(3), 485 (2021) DOI:10.4310/jdg/1615487005.
[32] P. Seidel, “Formal groups and quantum cohomology”, Geom. Topol. 27(8), 2937 (2023) DOI:10.2140/gt.2023.27.2937 [arXiv:1910.08990 [math.SG]].
[33] P. Seidel, “Fukaya A∞-categories associated to Lefschetz fibrations. II ”, Adv. Theor. Math. Phys. 20(4), 883 (2016) DOI:10.4310/ATMP.2016.v20.n4.a5 [arXiv:1404.1352 [math.SG]].
[34] P. Seidel, “Lagrangian homology spheres in (Am) Milnor fibres”, [arXiv:1202.1955 [math.SG]] (2012).
[35] P. Seidel, “Fukaya A∞-structures associated to Lefschetz fibrations, I”, J. Symplectic Geom. 10(3), 325 (2012) DOI:10.4310/jsg.2012.v10.n3.a1 [arXiv:0912.3932 [math.SG]].
[36] P. Seidel, “Homological mirror symmetry for the quartic surface”, [arXiv:math/0310414 [math.SG]] (2003).
[37] M. Abouzaid, S. Ganatra, H. Iritani and N. Sheridan, “The Gamma and Strominger– Yau–Zaslow conjectures: a tropical approach to periods”, Geom. Topol. 24(5), 2547 (2020) DOI:10.2140/gt.2020.24.2547 [arXiv:1809.02177 [math.AG]].
[38] M. Abouzaid and C. Manolescu, “A sheaf-theoretic model for SL(2,C) Floer homology”, J. Eur. Math. Soc. 22(11), 3641 (2020) DOI:10.4171/jems/994 [arXiv:1708.00289 [math.GT]].
[39] M. Abouzaid, D. Auroux and L. Katzarkov, “Lagrangian fibrations on blowups of toric varieties and mirror symmetry for hypersurfaces”, Publ. Math. Inst. Hautes Étud. Sci. 123, 199 (2016) DOI:10.1007/s10240-016-0081-9.
[40] M. Abouzaid and I. Smith, “Khovanov homology from Floer cohomology”, J. Am. Math. Soc. 32(1), 1 (2019) DOI:10.1090/jams/902 [arXiv:1504.01230 [math.SG]].
[41] M. Abouzaid, D. Auroux, A. I. Efimov, L. Katzarkov and D. Orlov, “Homological mirror symmetry for punctured spheres”, J. Am. Math. Soc. 26(4), 1051 (2013) DOI:10.1090/S0894-0347-2013-00770-5.
[42] R. Bocklandt and M. Abouzaid, “Noncommutative mirror symmetry for punctured surfaces”, Trans. Am. Math. Soc. 368(1), 429 (2016) DOI:10.1090/tran/6375 [arXiv:1111.3392 [math.AG]].
[43] M. Abouzaid and P. Seidel, “Altering symplectic manifolds by homologous recombination”, [arXiv:1007.3281 [math.SG]] (2010).
[44] M. Abouzaid and M. Boyarchenko, “Local structure of generalized complex manifolds”, [arXiv:math/0412084 [math.DG]] (2004).
[45] D. Auroux, A. I. Efimov and L. Katzarkov, “Lagrangian Floer theory for trivalent graphs and homological mirror symmetry for curves”, Selecta Math. 30(5), 95 (2024) DOI:10.1007/s00029-024-00988-6.
[46] L. Cavenaghi, L. Grama and L. Katzarkov, “New look at Milnor Spheres”, [arXiv:2404.19088 [math.DG]] (2024).
[47] R. P. Horja and L. Katzarkov, “Discriminants and toric K-theory”, Adv. Math. 453, 109831 (2024) DOI:10.1016/j.aim.2024.109831 [arXiv:2205.00903 [math.AG]].
[48] F. Haiden, L. Katzarkov and C. Simpson, “Spectral Networks and Stability Conditions for Fukaya Categories with Coeffcients”, Commun. Math. Phys. 405(11), 264 (2024) DOI:10.1007/s00220-024-05138-9 [arXiv:2112.13623 [math.AG]].
[49] L. Katzarkov, E. Lupercio, L. Meersseman and A. Verjovsky, “Quantum (noncommutative) toric geometry: Foundations”, Adv. Math. 391, 107945 (2021) DOI:10.1016/j.aim.2021.107945.
[50] D. Borisov, L. Katzarkov, A. Sheshmani and S. T. Yau, “Strictification and gluing of Lagrangian distributions on derived schemes with shifted symplectic forms”, Adv. Math. 438, 109477 (2024) DOI:10.1016/j.aim.2023.109477 [arXiv:1908.00651 [math.AG]].
[51] A. Kasprzyk, L. Katzarkov, V. Przyjalkowski and D. Sakovics, “Projecting Fanos in the mirror”, [arXiv:1904.02194 [math.AG]] (2019).
[52] M. Ballard, D. Deliu, D. Favero, M. U. Isik and L. Katzarkov, “On the derived categories of degree d hypersurface fibrations”, Math. Ann. 371(1-2), 337 (2018) DOI:10.1007/s00208-017-1613-4.
[53] L. Katzarkov and L. Soriani, “Homological mirror symmetry, coisotropic branes and P=W”, Eur. J. Math. 4(3), 1141 (2018) DOI:10.1007/s40879-018-0273-6.
[54] M. Ballard, D. Favero and L. Katzarkov, “Variation of geometric invariant theory quotients and derived categories”, J. Reine Angew. Math. 2019(746), 235 (2019) DOI:10.1515/crelle-2015-0096.
[55] L. Katzarkov, E. Lupercio, L. Meersseman and A. Verjovsky, “Non-commutative Toric Varieties”, [arXiv:1308.2774 [math.SG]] (2013).
[56] G. Dimitrov, F. Haiden, L. Katzarkov and M. Kontsevich, “Dynamical systems and categories”, (2013) DOI:10.1090/conm/621 [arXiv:1307.8418 [math.CT]].
[57] A. Rebhan, L. Katzarkov, J. Knapp, R. Rashkov and E. Scheidegger, “Strings, gauge fields, and the geometry behind: The legacy of Maximilian Kreuzer”, World Scientific, 2013 ISBN 978-981-4412-54-4 DOI:10.1142/8561.
[58] R. S. Garavuso, L. Katzarkov, M. Kreuzer and A. Noll, “Super Landau-Ginzburg mirrors and algebraic cycles”, JHEP 03, 017 (2011) [erratum: JHEP 08, 063 (2011)] DOI:10.1007/JHEP08(2011)063 [arXiv:1101.1368 [hep-th]].
[59] A. Kapustin, L. Katzarkov, D. Orlov and M. Yotov, “Homological Mirror Symmetry for manifolds of general type”, Open Math. 7(4), 571 (2009) DOI:10.2478/s11533-009- 0056-x [arXiv:1004.0129 [math.AG]].
[60] H. L. Chang, J. Li, W. P. Li and C. C. M. Liu, “An effective theory of GW and FJRW invariants of quintics Calabi–Yau manifolds”, J. Diff. Geom. 120(2), 251 (2022) DOI:10.4310/jdg/1645207466.
[61] J. Li and C. C. M. Liu, “Counting curves in quintic Calabi–Yau threefolds and Landau–Ginzburg models”, Surveys Diff. Geom. 24(1), 173 (2019) DOI:10.4310/sdg.2019.v24.n1.a5.
[62] H. L. Chang, J. Li, W. P. Li and C. C. M. Liu, “An effective theory of GW and FJRW invariants of quintics Calabi-Yau manifolds”, [arXiv:1603.06184 [math.AG]] (2016).
[63] S. Banerjee, P. Longhi and M. Romo, “Modelling A-branes with foliations”, [arXiv:2309.07748 [hep-th]] (2023).
[64] T. J. Lee, B. H. Lian and M. Romo, “Non-commutative resolutions as mirrors of singular Calabi–Yau varieties”, [arXiv:2307.02038 [hep-th]] (2023).
[65] J. Guo and M. Romo, “Hybrid models for homological projective duals and noncommutative resolutions”, Lett. Math. Phys. 112(6), 117 (2022) DOI:10.1007/s11005-022- 01605-3 [arXiv:2111.00025 [hep-th]].
[66] H. Jockers, V. Kumar, J. M. Lapan, D. R. Morrison and M. Romo, “Two-Sphere Partition Functions and Gromov-Witten Invariants”, Commun. Math. Phys. 325, 1139 (2014) DOI:10.1007/s00220-013-1874-z [arXiv:1208.6244 [hep-th]].
[67] L. Katzarkov, M. Kontsevich and T. Pantev, “Hodge theoretic aspects of mirror symmetry”, (2008) [arXiv:0806.0107 [math.AG]].
[68] D. Oprea, “Framed sheaves over treefolds and symmetric obstruction theories”, Doc. Math. 18, 323 (2013) DOI:10.4171/DM/399.
[69] T. Pantev, B. Toën, M. Vaquié and G. Vezzosi, “Shifted symplectic structures”, Publ. Math. Inst. Hautes Étud. Sci. 117, 271 (2013) DOI:10.1007/s10240-013-0054-1.
[70] D. Calaque, “Shifted cotangent stacks are shifted symplectic”, [arXiv:1612.08101 [math.AG]], (2017).
[71] C. Brav, V. Bussi, D. Dupont, D. Joyce and B. Szendroi, “Symmetries and stabilization for sheaves of vanishing cycles”, [arXiv:1211.3259 [math.AG]] (2012).
[72] C. Brav, V. Bussi and D. Joyce, “A Darboux theorem for derived schemes with shifted symplectic structure”, J. Am. Math. Soc. 32(2), 399 (2018) DOI:10.1090/jams/910 [arXiv:1305.6302 [math.AG]].
[73] O. Ben-Bassat, C. Brav, V. Bussi and D. Joyce, “A ‘Darboux theorem’ for shifted symplectic structures on derived Artin stacks, with applications”, Geom. Topol. 19(3), 1287 (2015) DOI:10.2140/gt.2015.19.1287 [arXiv:1312.0090 [math.AG]].
[74] C. Brav and T. Dyckerhoff, “Relative Calabi–Yau structures II: shifted Lagrangians in the moduli of objects”, Selecta Math. 27(4), 63 (2021) DOI:10.1007/s00029-021-00642-5 [arXiv:1812.11913 [math.AG]].
[75] Y. Qin, “Coisotropic branes on tori and Homological mirror symmetry”, [arXiv:2207.10647 [math.SG]] (2022).
[76] Y. Qin, “Shifted symplectic structure on Poisson Lie algebroid and generalized complex geometry”, [arXiv:2407.15598 [math.SG]] (2024).
[77] M. Aganagic, I. Danilenko, Y. Li, V. Shende and P. Zhou, “Quiver Hecke algebras from Floer homology in Coulomb branches”, [arXiv:2406.04258 [math.SG]] (2024).
[78] S. Gukov, P. Koroteev, S. Nawata, D. Pei and I. Saberi, “Branes and DAHA Representations”, Springer, 2023 ISBN 978-3-031-28153-2, 978-3-031-28154-9 DOI:10.1007/978- 3-031-28154-9 [arXiv:2206.03565 [hep-th]].
[79] A. Losev, N. Nekrasov and S. L. Shatashvili, “On four dimensional mirror symmetry”, Fortsch. Phys. 48, 163 (2000).
[80] A. Losev and V. Lysov, “Tropical Mirror”, SIGMA 20, 072 (2024) DOI:10.3842/SIGMA.2024.072 [arXiv:2204.06896 [hep-th]].
[81] A. Losev and V. Lysov, “Tropical Mirror Symmetry: Correlation functions”, [arXiv:2301.01687 [hep-th]] (2023).
[82] A. Losev and V. Lysov, “Tropical mirror for toric surfaces”, [arXiv:2305.00423 [hep-th]] (2023).
[83] E. Frenkel and A. Losev, “Mirror symmetry in two steps: A-I-B”, Commun. Math. Phys. 269, 39 (2006) DOI:10.1007/s00220-006-0114-1 [arXiv:hep-th/0505131 [hep-th]].
[84] R. Donagi, B. A. Ovrut, T. Pantev and D. Waldram, “Standard model bundles on nonsimply connected Calabi-Yau threefolds”, JHEP 08, 053 (2001) DOI:10.1088/1126- 6708/2001/08/053 [arXiv:hep-th/0008008 [hep-th]].
[85] R. Donagi, B. A. Ovrut, T. Pantev and D. Waldram, “Standard model bundles”, Adv. Theor. Math. Phys. 5(3), 563 (2002) DOI:10.4310/ATMP.2001.v5.n3.a5 [arXiv:math/0008010 [math.AG]].
[86] R. Donagi, B. A. Ovrut, T. Pantev and D. Waldram, “Nonperturbative vacua in heterotic M theory”, Class. Quant. Grav. 17, 1049 (2000) DOI:10.1088/0264- 9381/17/5/314.
[87] R. Donagi, B. A. Ovrut, T. Pantev and D. Waldram, “Standard models from heterotic M theory”, Adv. Theor. Math. Phys. 5(1), 93 (2002) DOI:10.4310/ATMP.2001.v5.n1.a4 [arXiv:hep-th/9912208 [hep-th]].
[88] M. Cicoli, I. G. Etxebarria, F. Quevedo, A. Schachner, P. Shukla and R. Valandro, “The Standard Model quiver in de Sitter string compactifications”, JHEP 08, 109 (2021) DOI:10.1007/JHEP08(2021)109 [arXiv:2106.11964 [hep-th]].
[89] L. Katzarkov, M. Kontsevich and A. Sheshmani, “to appear”, (2024).
[90] R. Pandharipande, “Cohomological Field Theory Calculations”, Proc. Int. Cong. of Math. 2018, Rio de Janeiro, Vol. 1, 869 (2018).
[91] M. Aganagic, H. Ooguri, N. Saulina and C. Vafa, “Black Holes, q-Deformed 2d Yang-Mills, and Non-perturbative Topological Strings”, [arXiv:hep-th/0411280 [hepth]] (2004).
[92] G. W. Moore and Y. Tachikawa, “On 2d TQFTs whose values are holomorphic symplectic varieties”, Proc. Symp. Pure Math. 85, 191 (2012) DOI:10.1090/pspum/085/1379.
[93] P. Crooks and M. Mayrand, “The Moore-Tachikawa conjecture via shifted symplectic geometry”, [arXiv:2409.03532 [math.SG]] (2024).
[94] G. W. Moore and G. B. Segal, “D-branes and K-theory in 2D topological field theory”, [arXiv:hep-th/0609042 [hep-th]] (2006).
[95] V. Pasquarella, “Moore-Tachikawa Varieties: Beyond Duality”, JHAP 3(4), 39 (2023) DOI:10.22128/jhap.2023.741.1061 [arXiv:2310.01489 [hep-th]].
[96] C. Teleman, “The structure of 2D semi-simple field theories”, Invent. Math. 188(3), 525 (2012) DOI:10.1007/s00222-011-0352-5.
[97] D. Xie and S. T. Yau, “Three dimensional canonical singularity and five dimensional N = 1 SCFT”, JHEP 2017, 1 (2017).
[98] C. Teleman, “The rôle of Coulomb branches in 2D gauge theory”, J. Eur. Math. Soc. 20(3), 489 (2018).
[99] A. Braverman, M. Finkelberg and H. Nakajima, “Towards a mathematical definition of Coulomb branches of 3-dimensional N = 4 gauge theories, II”, Adv. Theor. Math. Phys. 22(5), 1071 (2018) DOI:10.4310/ATMP.2018.v22.n5.a1 [arXiv:1601.03586 [math.RT]].
[100] P. Safronov, “Shifted Geometric Quantisation”, (2020).
[101] C. F. Doran, A. Harder and A. Thompson, “Mirror symmetry, Tyurin degenerations and fibrations on Calabi-Yau manifolds”, [arXiv:1602.05554 [math.AG]] (2016).
[102] L. Katzarkov, P. S. Pandit and T. C. Spaide, “Calabi-Yau structures, spherical functors, and shifted symplectic structures”, Adv. Math. 335, 279 (2018) DOI:10.1016/j.aim.2018.07.005.
[103] A. Kapustin, L. Rozansky and N. Saulina, “Three-dimensional topological field theory and symplectic algebraic geometry I”, Nucl. Phys. B 816, 295 (2009) DOI:10.1016/j.nuclphysb.2009.01.027 [arXiv:0810.5415 [hep-th]].
[104] C. Teleman, “Gauge theory and mirror symmetry”, [arXiv:1409.6320 [math-ph]] (2014).
[105] N. Sheridan, “On the Homological Mirror Symmetry Conjecture for the Pair of Pants”, J. Diff. Geom. 89(2), 271 (2011).
[106] P. Seidel, “Fukaya Categories and Picard-Lefschetz Theory”, (2008).
[107] B. Webster, “Koszul duality between Higgs and Coulomb categories O”, [arXiv:1611.06541 [math.RT]] (2016).
)
Volume 5, Issue 3
September 2025Pages 64-106
- Receive Date: 16 August 2025
- Revise Date: 30 August 2025
- Accept Date: 30 August 2025