Download this article
 Download this article For screen
For printing
Recent Issues

Volume 30
Issue 3, 835–1201
Issue 2, 389–833
Issue 1, 1–388

Volume 29, 9 issues

Volume 28, 9 issues

Volume 27, 9 issues

Volume 26, 8 issues

Volume 25, 7 issues

Volume 24, 7 issues

Volume 23, 7 issues

Volume 22, 7 issues

Volume 21, 6 issues

Volume 20, 6 issues

Volume 19, 6 issues

Volume 18, 5 issues

Volume 17, 5 issues

Volume 16, 4 issues

Volume 15, 4 issues

Volume 14, 5 issues

Volume 13, 5 issues

Volume 12, 5 issues

Volume 11, 4 issues

Volume 10, 4 issues

Volume 9, 4 issues

Volume 8, 3 issues

Volume 7, 2 issues

Volume 6, 2 issues

Volume 5, 2 issues

Volume 4, 1 issue

Volume 3, 1 issue

Volume 2, 1 issue

Volume 1, 1 issue

The Journal
About the journal
Ethics and policies
Peer-review process
 
Submission guidelines
Submission form
Editorial board
 
Subscriptions
 
ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
 
Author index
To appear
 
Other MSP journals
The level structure in quantum $K$-theory and mock theta functions

Yongbin Ruan, Ming Zhang and Yaoxiong Wen

Geometry & Topology 30 (2026) 835–882
Bibliography
1 D Abramovich, T Graber, A Vistoli, Algebraic orbifold quantum products, from: "Orbifolds in mathematics and physics" (editors A Adem, J Morava, Y Ruan), Contemp. Math. 310, Amer. Math. Soc. (2002) 1 MR1950940
2 D Abramovich, A Corti, A Vistoli, Twisted bundles and admissible covers, Comm. Algebra 31 (2003) 3547 MR2007376
3 D Abramovich, T Graber, M Olsson, H H Tseng, On the global quotient structure of the space of twisted stable maps to a quotient stack, J. Algebraic Geom. 16 (2007) 731 MR2357688
4 D Abramovich, T Graber, A Vistoli, Gromov–Witten theory of Deligne–Mumford stacks, Amer. J. Math. 130 (2008) 1337 MR2450211
5 M Aganagic, A Okounkov, Quasimap counts and Bethe eigenfunctions, Mosc. Math. J. 17 (2017) 565 MR3734654
6 M Aganagic, E Frenkel, A Okounkov, Quantum q-Langlands correspondence, Trans. Moscow Math. Soc. 79 (2018) 1 MR3881458
7 S Agnihotri, Quantum cohomology and the Verlinde algebra, PhD thesis, University of Oxford (1995)
8 K Behrend, Y Manin, Stacks of stable maps and Gromov–Witten invariants, Duke Math. J. 85 (1996) 1 MR1412436
9 P Belkale, Quantum generalization of the Horn conjecture, J. Amer. Math. Soc. 21 (2008) 365 MR2373354
10 W Chen, Y Ruan, Orbifold Gromov–Witten theory, from: "Orbifolds in mathematics and physics" (editors A Adem, J Morava, Y Ruan), Contemp. Math. 310, Amer. Math. Soc. (2002) 25 MR1950941
11 D Cheong, I Ciocan-Fontanine, B Kim, Orbifold quasimap theory, Math. Ann. 363 (2015) 777 MR3412343
12 I Ciocan-Fontanine, B Kim, Moduli stacks of stable toric quasimaps, Adv. Math. 225 (2010) 3022 MR2729000
13 I Ciocan-Fontanine, B Kim, Wall-crossing in genus zero quasimap theory and mirror maps, Algebr. Geom. 1 (2014) 400 MR3272909
14 I Ciocan-Fontanine, B Kim, D Maulik, Stable quasimaps to GIT quotients, J. Geom. Phys. 75 (2014) 17 MR3126932
15 T H Coates, Riemann–Roch theorems in Gromov–Witten theory, PhD thesis, University of California, Berkeley (2003) MR2705177
16 T Coates, A Givental, Quantum cobordisms and formal group laws, from: "The unity of mathematics" (editors P Etingof, V Retakh, I M Singer), Progr. Math. 244, Birkhäuser (2006) 155 MR2181805
17 T Coates, A Givental, Quantum Riemann–Roch, Lefschetz and Serre, Ann. of Math. 165 (2007) 15 MR2276766
18 I Dolgachev, Lectures on invariant theory, 296, Cambridge Univ. Press (2003) MR2004511
19 D Gepner, Fusion rings and geometry, Comm. Math. Phys. 141 (1991) 381 MR1133272
20 A Givental, On the WDVV equation in quantum K-theory, Michigan Math. J. 48 (2000) 295 MR1786492
21 A B Givental, Symplectic geometry of Frobenius structures, from: "Frobenius manifolds" (editors C Hertling, M Marcolli), Aspects Math. E36, Friedr. Vieweg (2004) 91 MR2115767
22 A Givental, Permutation-equivariant quantum K-theory, II : Fixed point localization, preprint (2015) arXiv:1508.04374
23 A Givental, Permutation-equivariant quantum K-theory, III : Lefschetz’ formula on 0,n∕Sn and adelic characterization, preprint (2015) arXiv:1508.06697
24 A Givental, Permutation-equivariant quantum K-theory, IV : Dq-modules, preprint (2015) arXiv:1509.00830
25 A Givental, Permutation-equivariant quantum K-theory, V : Toric q-hypergeometric functions, preprint (2015) arXiv:1509.03903
26 A Givental, Permutation-equivariant quantum K-theory, VI : Mirrors, preprint (2015) arXiv:1509.07852
27 A Givental, Permutation-equivariant quantum K-theory, VII : General theory, preprint (2015) arXiv:1510.03076
28 A Givental, Permutation-equivariant quantum K-theory, VIII : Explicit reconstruction, preprint (2015) arXiv:1510.06116
29 A Givental, Permutation-equivariant quantum K-theory, I : Definitions : Elementary K-theory of 0,n∕Sn, Mosc. Math. J. 17 (2017) 691 MR3734658
30 A Givental, Permutation-equivariant quantum K-theory, IX : Quantum Hirzebruch–Riemann–Roch in all genera, preprint (2017) arXiv:1709.03180
31 A Givental, Permutation-equivariant quantum K-theory, XI : Quantum Adams–Riemann–Roch, preprint (2017) arXiv:1711.04201
32 A Givental, Permutation-equivariant quantum K-theory, X : Quantum Hirzebruch–Riemann–Roch in genus 0, Symmetry Integrability Geom. Methods Appl. 16 (2020) MR4089511
33 A Givental, V Tonita, The Hirzebruch–Riemann–Roch theorem in true genus-0 quantum K-theory, from: "Symplectic, Poisson, and noncommutative geometry" (editors T Eguchi, Y Eliashberg, Y Maeda), Math. Sci. Res. Inst. Publ. 62, Cambridge Univ. Press (2014) 43 MR3380674
34 K Intriligator, Fusion residues, Modern Phys. Lett. A 6 (1991) 3543 MR1138873
35 H Jockers, P Mayr, A 3d gauge theory/quantum K-theory correspondence, Adv. Theor. Math. Phys. 24 (2020) 327 MR4125364
36 A Kapustin, B Willett, Wilson loops in 3-dimensional N = 6 supersymmetric Chern–Simons theory and their string theory duals, preprint (2013) arXiv:1302.2164
37 T Kawasaki, The Riemann–Roch theorem for complex V -manifolds, Osaka Math. J. 16 (1979) 151 MR527023
38 F F Knudsen, D Mumford, The projectivity of the moduli space of stable curves : I : Preliminaries on “det” and “Div”, Math. Scand. 39 (1976) 19 MR437541
39 P Koroteev, P P Pushkar, A V Smirnov, A M Zeitlin, Quantum K-theory of quiver varieties and many-body systems, Selecta Math. 27 (2021) MR4308933
40 Y P Lee, Quantum K-theory, I : Foundations, Duke Math. J. 121 (2004) 389 MR2040281
41 A Marian, D Oprea, Counts of maps to Grassmannians and intersections on the moduli space of bundles, J. Differential Geom. 76 (2007) 155 MR2312051
42 A Marian, D Oprea, The level-rank duality for non-abelian theta functions, Invent. Math. 168 (2007) 225 MR2289865
43 A Marian, D Oprea, GL Verlinde numbers and the Grassmann TQFT, Port. Math. 67 (2010) 181 MR2662866
44 A Okounkov, Lectures on K-theoretic computations in enumerative geometry, from: "Geometry of moduli spaces and representation theory" (editors R Bezrukavnikov, A Braverman, Z Yun), IAS/Park City Math. Ser. 24, Amer. Math. Soc. (2017) 251 MR3752463
45 A Okounkov, A Smirnov, Quantum difference equation for Nakajima varieties, Invent. Math. 229 (2022) 1203 MR4462626
46 P P Pushkar, A V Smirnov, A M Zeitlin, Baxter Q-operator from quantum K-theory, Adv. Math. 360 (2020) MR4035952
47 F Qu, Virtual pullbacks in K-theory, Ann. Inst. Fourier (Grenoble) 68 (2018) 1609 MR3887429
48 Y Ruan, M Zhang, Verlinde/Grassmannian correspondence and rank 2 δ-wall-crossing, Peking Math. J. 6 (2023) 217 MR4552643
49 C Su, G Zhao, C Zhong, On the K-theory stable bases of the Springer resolution, Ann. Sci. Éc. Norm. Supér. 53 (2020) 663 MR4118526
50 V Tonita, Twisted orbifold Gromov–Witten invariants, Nagoya Math. J. 213 (2014) 141 MR3161407
51 V Tonita, A virtual Kawasaki–Riemann–Roch formula, Pacific J. Math. 268 (2014) 249 MR3207609
52 V Tonita, Twisted K-theoretic Gromov–Witten invariants, Math. Ann. 372 (2018) 489 MR3856819
53 V Tonita, H H Tseng, Quantum orbifold Hirzebruch–Riemann–Roch theorem in genus zero, preprint (2013) arXiv:1307.0262
54 H H Tseng, Orbifold quantum Riemann–Roch, Lefschetz and Serre, Geom. Topol. 14 (2010) 1 MR2578300
55 C Vafa, Topological mirrors and quantum rings, from: "Essays on mirror manifolds" (editor S T Yau), Int. Press (1992) 96 MR1191421
56 E Witten, The Verlinde algebra and the cohomology of the Grassmannian, from: "Geometry, topology, & physics" (editor S T Yau), Conf. Proc. Lecture Notes Geom. Topology IV, Int. Press (1995) 357 MR1358625