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The $3$-fold K-theoretic DT/PT vertex correspondence holds

Nikolas Kuhn, Henry Liu and Felix Thimm

Geometry & Topology 30 (2026) 71–154
Bibliography
1 D Abramovich, M Olsson, A Vistoli, Tame stacks in positive characteristic, Ann. Inst. Fourier (Grenoble) 58 (2008) 1057 MR2427954
2 M Aganagic, A Klemm, M Mariño, C Vafa, The topological vertex, Comm. Math. Phys. 254 (2005) 425 MR2117633
3 J Alper, J Hall, D Rydh, A Luna étale slice theorem for algebraic stacks, Ann. of Math. 191 (2020) 675 MR4088350
4 D Aranha, A A Khan, A Latyntsev, H Park, C Ravi, Virtual localization revisited, Adv. Math. 479 (2025) 110434 MR4933128
5 S V Beentjes, J Calabrese, J V Rennemo, A proof of the Donaldson–Thomas crepant resolution conjecture, Invent. Math. 229 (2022) 451 MR4448990
6 D Borisov, D Joyce, Virtual fundamental classes for moduli spaces of sheaves on Calabi–Yau four-folds, Geom. Topol. 21 (2017) 3231 MR3692967
7 T Bridgeland, Hall algebras and curve-counting invariants, J. Amer. Math. Soc. 24 (2011) 969 MR2813335
8 J Bryan, A Gholampour, The quantum McKay correspondence for polyhedral singularities, Invent. Math. 178 (2009) 655 MR2551767
9 J Bryan, D Steinberg, Curve counting invariants for crepant resolutions, Trans. Amer. Math. Soc. 368 (2016) 1583 MR3449219
10 J Bryan, C Cadman, B Young, The orbifold topological vertex, Adv. Math. 229 (2012) 531 MR2854183
11 J Calabrese, On the crepant resolution conjecture for Donaldson–Thomas invariants, J. Algebraic Geom. 25 (2016) 1 MR3419955
12 Y Cao, M Kool, S Monavari, K-theoretic DT/PT correspondence for toric Calabi–Yau 4-folds, Comm. Math. Phys. 396 (2022) 225 MR4499016
13 Y Cao, M Kool, S Monavari, A Donaldson–Thomas crepant resolution conjecture on Calabi–Yau 4-folds, Trans. Amer. Math. Soc. 376 (2023) 8225 MR4657232
14 H l Chang, J Li, Semi-perfect obstruction theory and Donaldson–Thomas invariants of derived objects, Comm. Anal. Geom. 19 (2011) 807 MR2880216
15 N Chriss, V Ginzburg, Representation theory and complex geometry, Birkhäuser (1997) MR1433132
16 I Ciocan-Fontanine, M Kapranov, Virtual fundamental classes via dg-manifolds, Geom. Topol. 13 (2009) 1779 MR2496057
17 B Davison, A T Ricolfi, The local motivic DT/PT correspondence, J. Lond. Math. Soc. 104 (2021) 1384 MR4332481
18 N Fasola, S Monavari, A T Ricolfi, Higher rank K-theoretic Donaldson–Thomas theory of points, Forum Math. Sigma 9 (2021) MR4228267
19 B Feigin, M Jimbo, T Miwa, E Mukhin, Finite type modules and Bethe ansatz for quantum toroidal 𝔤𝔩1, Comm. Math. Phys. 356 (2017) 285 MR3694029
20 D Gaiotto, M Rapčák, Miura operators, degenerate fields and the M2-M5 intersection, J. High Energy Phys. 2022 (2022) 086 MR4403458
21 P Gross, Tensor generators on schemes and stacks, Algebr. Geom. 4 (2017) 501 MR3683505
22 D Huybrechts, M Lehn, Framed modules and their moduli, Internat. J. Math. 6 (1995) 297 MR1316305
23 D Huybrechts, M Lehn, The geometry of moduli spaces of sheaves, E31, Vieweg & Sohn (1997) MR1450870
24 D Huybrechts, R P Thomas, Deformation-obstruction theory for complexes via Atiyah and Kodaira–Spencer classes, Math. Ann. 346 (2010) 545 MR2578562
25 L Illusie, Complexe cotangent et déformations, I, 239, Springer (1971) MR491680
26 A Iqbal, C Vafa, N Nekrasov, A Okounkov, Quantum foam and topological strings, J. High Energy Phys. 2008 (2008) 011, 47 MR2425292
27 A Iqbal, C Kozçaz, C Vafa, The refined topological vertex, J. High Energy Phys. 2009 (2009) 069, 58 MR2607441
28 H Jenne, G Webb, B Young, The combinatorial PT–DT correspondence, Sém. Lothar. Combin. 85B (2021) 89 MR4311970
29 D Joyce, Enumerative invariants and wall-crossing formulae in abelian categories, preprint (2021) arXiv:2111.04694
30 D Joyce, Y Song, A theory of generalized Donaldson–Thomas invariants, 1020, Amer. Math. Soc. (2012) MR2951762
31 Y H Kiem, J Li, A wall crossing formula of Donaldson–Thomas invariants without Chern–Simons functional, Asian J. Math. 17 (2013) 63 MR3038725
32 Y H Kiem, M Savvas, Localizing virtual structure sheaves for almost perfect obstruction theories, Forum Math. Sigma 8 (2020) MR4190061
33 Y H Kiem, M Savvas, K-theoretic generalized Donaldson–Thomas invariants, Int. Math. Res. Not. 2022 (2022) 2123 MR4366004
34 B Kim, A Kresch, T Pantev, Functoriality in intersection theory and a conjecture of Cox, Katz, and Lee, J. Pure Appl. Algebra 179 (2003) 127 MR1958379
35 Y Kononov, A Okounkov, A Osinenko, The 2-leg vertex in K-theoretic DT theory, Comm. Math. Phys. 382 (2021) 1579 MR4232774
36 N Kuhn, The Atiyah class on algebraic stacks, Forum Math. Sigma 12 (2024) MR4825109
37 N Kuhn, Y Tanaka, A blowup formula for virtual enumerative invariants on projective surfaces, preprint (2021) arXiv:2107.08155
38 S G Langton, Valuative criteria for families of vector bundles on algebraic varieties, Ann. of Math. 101 (1975) 88 MR364255
39 G Laumon, L Moret-Bailly, Champs algébriques, 39, Springer (2000) MR1771927
40 Y P Lee, Quantum K-theory, I : Foundations, Duke Math. J. 121 (2004) 389 MR2040281
41 J Li, B Wu, Good degeneration of Quot-schemes and coherent systems, Comm. Anal. Geom. 23 (2015) 841 MR3385781
42 J Li, C C M Liu, K Liu, J Zhou, A mathematical theory of the topological vertex, Geom. Topol. 13 (2009) 527 MR2469524
43 H Liu, Asymptotic representations of shifted quantum affine algebras from critical K-theory, PhD thesis, Columbia University (2021) MR4257419
44 H Liu, Quasimaps and stable pairs, Forum Math. Sigma 9 (2021) MR4247638
45 H Liu, Semistable refined Vafa–Witten invariants, preprint (2023) arXiv:2309.03673
46 H Liu, The 4-fold Pandharipande–Thomas vertex, J. Geom. Phys. 198 (2024) 105104 MR4692498
47 J Lo, Polynomial Bridgeland stable objects and reflexive sheaves, Math. Res. Lett. 19 (2012) 873 MR3008421
48 C Manolache, Virtual pull-backs, J. Algebraic Geom. 21 (2012) 201 MR2877433
49 D Maulik, N Nekrasov, A Okounkov, R Pandharipande, Gromov–Witten theory and Donaldson–Thomas theory, I, Compos. Math. 142 (2006) 1263 MR2264664
50 D Maulik, N Nekrasov, A Okounkov, R Pandharipande, Gromov–Witten theory and Donaldson–Thomas theory, II, Compos. Math. 142 (2006) 1286 MR2264665
51 D Maulik, R Pandharipande, R P Thomas, Curves on K3 surfaces and modular forms, J. Topol. 3 (2010) 937 MR2746343
52 D Maulik, A Oblomkov, A Okounkov, R Pandharipande, Gromov–Witten/Donaldson–Thomas correspondence for toric 3-folds, Invent. Math. 186 (2011) 435 MR2845622
53 T Mochizuki, Donaldson type invariants for algebraic surfaces: transition of moduli stacks, 1972, Springer (2009) MR2508583
54 H Nakajima, K Yoshioka, Perverse coherent sheaves on blowup, III : Blow-up formula from wall-crossing, Kyoto J. Math. 51 (2011) 263 MR2793270
55 N Nekrasov, 𝔅𝔓𝔖ℭ𝔉𝔗 correspondence : instantons at crossroads and gauge origami, from: "String-Math 2015" (editors S Li, B H Lian, W Song, S T Yau), Proc. Sympos. Pure Math. 96, Amer. Math. Soc. (2017) 183 MR3751819
56 N Nekrasov, A Okounkov, Membranes and sheaves, Algebr. Geom. 3 (2016) 320 MR3504535
57 A Oblomkov, EGL formula for DT/PT theory of local curves, preprint (2019) arXiv:1901.03014
58 J Oh, R P Thomas, Counting sheaves on Calabi–Yau 4-folds, I, Duke Math. J. 172 (2023) 1333 MR4583653
59 A Okounkov, On the crossroads of enumerative geometry and geometric representation theory, from: "Proceedings of the International Congress of Mathematicians" (editors B Sirakov, P N de Souza, M Viana), World Sci. (2018) 839 MR3966746
60 A Okounkov, N Reshetikhin, C Vafa, Quantum Calabi–Yau and classical crystals, from: "The unity of mathematics" (editors P Etingof, V Retakh, I M Singer), Progr. Math. 244, Birkhäuser (2006) 597 MR2181817
61 M Olsson, Sheaves on Artin stacks, J. Reine Angew. Math. 603 (2007) 55 MR2312554
62 H Park, Virtual pullbacks in Donaldson–Thomas theory of Calabi–Yau 4-folds, preprint (2021) arXiv:2110.03631
63 M Rapčák, Y Soibelman, Y Yang, G Zhao, Cohomological Hall algebras and perverse coherent sheaves on toric Calabi–Yau 3-folds, preprint (2020) arXiv:2007.13365
64 F Sala, Symplectic structures on moduli spaces of framed sheaves on surfaces, Cent. Eur. J. Math. 10 (2012) 1455 MR2925615
65 A Sheshmani, Higher rank stable pairs and virtual localization, Comm. Anal. Geom. 24 (2016) 139 MR3514557
66 P Belmans, A J de Jong, others, The Stacks project, electronic reference (2005–)
67 J Stoppa, R P Thomas, Hilbert schemes and stable pairs : GIT and derived category wall crossings, Bull. Soc. Math. France 139 (2011) 297 MR2869309
68 Y Tanaka, R P Thomas, Vafa–Witten invariants for projective surfaces, I : Stable case, J. Algebraic Geom. 29 (2020) 603 MR4158461
69 R P Thomas, A holomorphic Casson invariant for Calabi–Yau 3-folds, and bundles on K3 fibrations, J. Differential Geom. 54 (2000) 367 MR1818182
70 R P Thomas, Equivariant K-theory and refined Vafa–Witten invariants, Comm. Math. Phys. 378 (2020) 1451 MR4134951
71 R P Thomas, A K-theoretic Fulton class, from: "Facets of algebraic geometry, II" (editors P Aluffi, D Anderson, M Hering, M Mustaţă, S Payne), London Math. Soc. Lecture Note Ser. 473, Cambridge Univ. Press (2022) 367 MR4381920
72 Y Toda, Curve counting theories via stable objects, I : DT/PT correspondence, J. Amer. Math. Soc. 23 (2010) 1119 MR2669709
73 Y Toda, Curve counting theories via stable objects, II : DT/ncDT flop formula, J. Reine Angew. Math. 675 (2013) 1 MR3021446
74 Y Toda, Hall algebras in the derived category and higher-rank DT invariants, Algebr. Geom. 7 (2020) 240 MR4087861
75 B Totaro, The resolution property for schemes and stacks, J. Reine Angew. Math. 577 (2004) 1 MR2108211
76 C A Weibel, Homotopy algebraic K-theory, from: "Algebraic K-theory and algebraic number theory" (editors M R Stein, R K Dennis), Contemp. Math. 83, Amer. Math. Soc. (1989) 461 MR991991