Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 18 (2022), 068, 61 pages      arXiv:2005.05637
Contribution to the Special Issue on Enumerative and Gauge-Theoretic Invariants in honor of Lothar Göttsche on the occasion of his 60th birthday

Universal Structures in $\mathbb C$-Linear Enumerative Invariant Theories

Jacob Gross a, Dominic Joyce a and Yuuji Tanaka b
a) The Mathematical Institute, Radcliffe Observatory Quarter, Woodstock Road, Oxford, OX2 6GG, UK
b) Department of Mathematics, Faculty of Science, Kyoto University, Kitashirakawa Oiwake-cho, Sakyo-ku, Kyoto 606-8502, Japan

Received November 22, 2021, in final form September 06, 2022; Published online September 23, 2022

An enumerative invariant theory in algebraic geometry, differential geometry, or representation theory, is the study of invariants which `count' $\tau$-(semi)stable objects $E$ with fixed topological invariants $[[E]]=\alpha$ in some geometric problem, by means of a virtual class $[{\mathcal M}_\alpha^{{\rm ss}}(\tau)]_{{\rm virt}}$ in some homology theory for the moduli spaces ${\mathcal M}_\alpha^{{\rm st}}(\tau)\subseteq{\mathcal M}_\alpha^{{\rm ss}}(\tau)$ of $\tau$-(semi)stable objects. Examples include Mochizuki's invariants counting coherent sheaves on surfaces, Donaldson-Thomas type invariants counting coherent sheaves on Calabi-Yau 3- and 4-folds and Fano 3-folds, and Donaldson invariants of 4-manifolds. We make conjectures on new universal structures common to many enumerative invariant theories. Any such theory has two moduli spaces ${\mathcal M}$, ${\mathcal M}^{{\rm pl}}$, where the second author (see gives $H_*({\mathcal M})$ the structure of a graded vertex algebra, and $H_*\big({\mathcal M}^{{\rm pl}}\big)$ a graded Lie algebra, closely related to $H_*({\mathcal M})$. The virtual classes $[{\mathcal M}_\alpha^{{\rm ss}}(\tau)]_{{\rm virt}}$ take values in $H_*\big({\mathcal M}^{{\rm pl}}\big)$. In most such theories, defining $[{\mathcal M}_\alpha^{{\rm ss}}(\tau)]_{{\rm virt}}$ when ${\mathcal M}_\alpha^{{\rm st}}(\tau)\ne{\mathcal M}_\alpha^{{\rm ss}}(\tau)$ (in gauge theory, when the moduli space contains reducibles) is a difficult problem. We conjecture that there is a natural way to define invariants $[{\mathcal M}_\alpha^{{\rm ss}}(\tau)]_{{\rm inv}}$ in homology over $\mathbb Q$, with $[{\mathcal M}_\alpha^{{\rm ss}}(\tau)]_{{\rm inv}}=[{\mathcal M}_\alpha^{{\rm ss}}(\tau)]_{{\rm virt}}$ when ${\mathcal M}_\alpha^{{\rm st}}(\tau)={\mathcal M}_\alpha^{{\rm ss}}(\tau)$, and that these invariants satisfy a universal wall-crossing formula under change of stability condition $\tau$, written using the Lie bracket on $H_*\big({\mathcal M}^{{\rm pl}}\big)$. We prove our conjectures for moduli spaces of representations of quivers without oriented cycles. Versions of our conjectures in algebraic geometry using Behrend-Fantechi virtual classes are proved in the sequel [arXiv:2111.04694].

Key words: invariant; stability condition; vertex algebra; wall crossing formula; quiver.

pdf (1142 kb)   tex (87 kb)  


  1. Abramovich D., Olsson M., Vistoli A., Tame stacks in positive characteristic, Ann. Inst. Fourier (Grenoble) 58 (2008), 1057-1091, arXiv:math.AG/0703310.
  2. Akbulut S., McCarthy J.D., Casson's invariant for oriented homology $3$-spheres. An exposition, Mathematical Notes, Vol. 36, Princeton University Press, Princeton, NJ, 1990.
  3. Álvarez-Cónsul L., García-Prada O., Hitchin-Kobayashi correspondence, quivers, and vortices, Comm. Math. Phys. 238 (2003), 1-33, arXiv:math.DG/0112161.
  4. Arbesfeld N., $K$-theoretic Donaldson-Thomas theory and the Hilbert scheme of points on a surface, Algebr. Geom. 8 (2021), 587-625, arXiv:1905.04567.
  5. Atiyah M.F., Bott R., The Yang-Mills equations over Riemann surfaces, Philos. Trans. Roy. Soc. London Ser. A 308 (1983), 523-615.
  6. Behrend K., Fantechi B., The intrinsic normal cone, Invent. Math. 128 (1997), 45-88, arXiv:alg-geom/9601010.
  7. Benson D.J., Representations and cohomology. II. Cohomology of groups and modules, Cambridge Studies in Advanced Mathematics, Vol. 31, Cambridge University Press, Cambridge, 1991.
  8. Blanc A., Topological K-theory of complex noncommutative spaces, Compos. Math. 152 (2016), 489-555, arXiv:1211.7360.
  9. Boden H.U., Herald C.M., The ${\rm SU}(3)$ Casson invariant for integral homology $3$-spheres, J. Differential Geom. 50 (1998), 147-206, arXiv:math.DG/9809124.
  10. Borcherds R.E., Vertex algebras, Kac-Moody algebras, and the Monster, Proc. Nat. Acad. Sci. USA 83 (1986), 3068-3071.
  11. Borisov D., Joyce D., Virtual fundamental classes for moduli spaces of sheaves on Calabi-Yau four-folds, Geom. Topol. 21 (2017), 3231-3311, arXiv:1504.00690.
  12. Bradlow S.B., Vortices in holomorphic line bundles over closed Kähler manifolds, Comm. Math. Phys. 135 (1990), 1-17.
  13. Bradlow S.B., Special metrics and stability for holomorphic bundles with global sections, J. Differential Geom. 33 (1991), 169-213.
  14. Bradlow S.B., Daskalopoulos G.D., Moduli of stable pairs for holomorphic bundles over Riemann surfaces, Internat. J. Math. 2 (1991), 477-513.
  15. Bridgeland T., Stability conditions on triangulated categories, Ann. of Math. 166 (2007), 317-345, arXiv:math.AG/0703310.
  16. Bridgeland T., Geometry from Donaldson-Thomas invariants, in Integrability, Quantization, and Geometry II. Quantum Theories and Algebraic Geometry, Proc. Sympos. Pure Math., Vol. 103, Amer. Math. Soc., Providence, RI, 2021, 1-66, arXiv:1912.06504.
  17. Cao Y., Gross J., Joyce D., Otability of moduli spaces of ${\rm Spin}(7)$-instantons and coherent sheaves on Calabi-Yau 4-folds, Adv. Math. 368 (2020), 107134, 60 pages, arXiv:1811.09658.
  18. Cao Y., Kool M., Monavari S., $K$-theoretic DT/PT correspondence for toric Calabi-Yau $4$-folds, Commun. Math. Phys., to appear, arXiv:1906.07856.
  19. Cao Y., Leung N.C., Donaldson-Thomas theory for Calabi-Yau $4$-folds, arXiv:1407.7659.
  20. Donaldson S., Segal E., Gauge theory in higher dimensions, II, in Geometry of Special Holonomy and Related Topics, Surv. Differ. Geom., Vol. 16, Int. Press, Somerville, MA, 2011, 1-41, arXiv:0902.3239.
  21. Donaldson S.K., An application of gauge theory to four-dimensional topology, J. Differential Geom. 18 (1983), 279-315.
  22. Donaldson S.K., Polynomial invariants for smooth four-manifolds, Topology 29 (1990), 257-315.
  23. Donaldson S.K., Kronheimer P.B., The geometry of four-manifolds, Oxford Mathematical Monographs, Vol. 1990, The Clarendon Press, Oxford University Press, New York, 1990.
  24. Donaldson S.K., Thomas R.P., Gauge theory in higher dimensions, in The Geometric Universe (Oxford, 1996), Oxford University Press, Oxford, 1998, 31-47.
  25. Ellingsrud G., Göttsche L., Variation of moduli spaces and Donaldson invariants under change of polarization, J. Reine Angew. Math. 467 (1995), 1-49, arXiv:alg-geom/9410005.
  26. Feigin B., Gukov S., ${\rm VOA}[M_4]$, J. Math. Phys. 61 (2020), 012302, 27 pages, arXiv:1806.02470.
  27. Fintushel R., Stern R.J., Donaldson invariants of $4$-manifolds with simple type, J. Differential Geom. 42 (1995), 577-633.
  28. Frenkel E., Ben-Zvi D., Vertex algebras and algebraic curves, Mathematical Surveys and Monographs, Vol. 88, 2nd ed., Amer. Math. Soc., Providence, RI, 2004.
  29. Friedlander E.M., Walker M.E., Semi-topological $K$-theory, in Handbook of $K$-Theory. Vols. 1, 2, Springer, Berlin, 2005, 877-924.
  30. Friedman R., Qin Z., Flips of moduli spaces and transition formulas for Donaldson polynomial invariants of rational surfaces, Comm. Anal. Geom. 3 (1995), 11-83, arXiv:alg-geom/9410007.
  31. García-Prada O., A direct existence proof for the vortex equations over a compact Riemann surface, Bull. London Math. Soc. 26 (1994), 88-96.
  32. García-Prada O., Dimensional reduction of stable bundles, vortices and stable pairs, Internat. J. Math. 5 (1994), 1-52.
  33. Gómez T.L., Algebraic stacks, Proc. Indian Acad. Sci. Math. Sci. 111 (2001), 1-31, arXiv:math.AG/9911199.
  34. Göttsche L., Modular forms and Donaldson invariants for $4$-manifolds with $b_{+}=1$, J. Amer. Math. Soc. 9 (1996), 827-843, arXiv:alg-geom/9506018.
  35. Göttsche L., Kool M., A rank 2 Dijkgraaf-Moore-Verlinde-Verlinde formula, Commun. Number Theory Phys. 13 (2019), 165-201, arXiv:1801.01878.
  36. Göttsche L., Kool M., Virtual refinements of the Vafa-Witten formula, Comm. Math. Phys. 376 (2020), 1-49, arXiv:1703.07196.
  37. Göttsche L., Nakajima H., Yoshioka K., Instanton counting and Donaldson invariants, J. Differential Geom. 80 (2008), 343-390, arXiv:math.AG/0606180.
  38. Göttsche L., Nakajima H., Yoshioka K., $K$-theoretic Donaldson invariants via instanton counting, Pure Appl. Math. Q. 5 (2009), 1029-1111, arXiv:math.AG/0611945.
  39. Göttsche L., Nakajima H., Yoshioka K., Donaldson-Seiberg-Witten from Mochizuki's formula and instanton counting, Publ. Res. Inst. Math. Sci. 47 (2011), 307-359, arXiv:1001.5024.
  40. Göttsche L., Zagier D., Jacobi forms and the structure of Donaldson invariants for $4$-manifolds with $b_+=1$, Selecta Math. (N.S.) 4 (1998), 69-115, arXiv:alg-geom/9612020.
  41. Grojnowski I., Instantons and affine algebras. I. The Hilbert scheme and vertex operators, Math. Res. Lett. 3 (1996), 275-291, arXiv:alg-geom/9506020.
  42. Gross J., The homology of moduli stacks of complexes, arXiv:1907.03269.
  43. Halpern-Leistner D., $\Theta$-stratifications, $\Theta$-reductive stacks, and applications, in Algebraic Geometry: Salt Lake City 2015, Proc. Sympos. Pure Math., Vol. 97, Amer. Math. Soc., Providence, RI, 2018, 349-379, arXiv:1608.04797.
  44. Harder G., Narasimhan M.S., On the cohomology groups of moduli spaces of vector bundles on curves, Math. Ann. 212 (1975), 215-248.
  45. Hitchin N.J., The self-duality equations on a Riemann surface, Proc. London Math. Soc. 55 (1987), 59-126.
  46. Huybrechts D., Lehn M., The geometry of moduli spaces of sheaves, 2nd ed., Cambridge Mathematical Library, Cambridge University Press, Cambridge, 2010.
  47. Joyce D., Compact manifolds with special holonomy, Oxford Mathematical Monographs, Oxford University Press, Oxford, 2000.
  48. Joyce D., Configurations in abelian categories. I. Basic properties and moduli stacks, Adv. Math. 203 (2006), 194-255, arXiv:math.AG/0312190.
  49. Joyce D., Constructible functions on Artin stacks, J. London Math. Soc. 74 (2006), 583-606, arXiv:math.AG/0403305.
  50. Joyce D., Configurations in abelian categories. II. Ringel-Hall algebras, Adv. Math. 210 (2007), 635-706, arXiv:math.AG/0503029.
  51. Joyce D., Configurations in abelian categories. III. Stability conditions and identities, Adv. Math. 215 (2007), 153-219, arXiv:math.AG/0410267.
  52. Joyce D., Motivic invariants of Artin stacks and `stack functions', Q. J. Math. 58 (2007), 345-392, arXiv:math.AG/0509722.
  53. Joyce D., Configurations in abelian categories. IV. Invariants and changing stability conditions, Adv. Math. 217 (2008), 125-204, arXiv:math.AG/0410268.
  54. Joyce D., $D$-manifolds and d-orbifolds: a theory of derived differential geometry, Preliminary version, 2012, available at
  55. Joyce D., An introduction to d-manifolds and derived differential geometry, in Moduli Spaces, London Math. Soc. Lecture Note Ser., Vol. 411, Cambridge University Press, Cambridge, 2014, 230-281, arXiv:1206.4207.
  56. Joyce D., Kuranishi spaces as a 2-category, in Virtual Fundamental Cycles in Symplectic Topology, Math. Surveys Monogr., Vol. 237, Amer. Math. Soc., Providence, RI, 2019, 253-298, arXiv:1510.07444.
  57. Joyce D., Kuranishi spaces and symplectic geometry, Preliminary version of Vols. I, II, available at
  58. Joyce D., Ringel-Hall style vertex algebra and Lie algebra structures on the homology of moduli spaces, Preliminary version, 2020, available at
  59. Joyce D., Enumerative invariants and wall-crossing formulae in abelian categories, arXiv:2111.04694.
  60. Joyce D., Song Y., A theory of generalized Donaldson-Thomas invariants, Mem. Amer. Math. Soc. 217 (2012), iv+199 pages, arXiv:0810.5645.
  61. Joyce D., Tanaka Y., Upmeier M., On orientations for gauge-theoretic moduli spaces, Adv. Math. 362 (2020), 106957, 64 pages, arXiv:1811.01096.
  62. Kac V., Vertex algebras for beginners, 2nd ed., University Lecture Series, Vol. 10, Amer. Math. Soc., Providence, RI, 1998.
  63. Kac V.G., Infinite-dimensional Lie algebras, 3rd ed., Cambridge University Press, Cambridge, 1990.
  64. King A.D., Moduli of representations of finite-dimensional algebras, Quart. J. Math. Oxford Ser. (2) 45 (1994), 515-530.
  65. Kirwan F.C., Cohomology of quotients in symplectic and algebraic geometry, Mathematical Notes, Vol. 31, Princeton University Press, Princeton, NJ, 1984.
  66. Kontsevich M., Soibelman Y., Stability structures, motivic Donaldson-Thomas invariants and cluster transformations, arXiv:0811.2435.
  67. Kotschick D., ${\rm SO}(3)$-invariants for $4$-manifolds with $b^+_2=1$, Proc. London Math. Soc. 63 (1991), 426-448.
  68. Kotschick D., Morgan J.W., ${\rm SO}(3)$-invariants for $4$-manifolds with $b^+_2=1$. II, J. Differential Geom. 39 (1994), 433-456.
  69. Kronheimer P.B., Four-manifold invariants from higher-rank bundles, J. Differential Geom. 70 (2005), 59-112, arXiv:math.GT/0407518.
  70. Kronheimer P.B., Mrowka T.S., Embedded surfaces and the structure of Donaldson's polynomial invariants, J. Differential Geom. 41 (1995), 573-734.
  71. Laarakker T., Monopole contributions to refined Vafa-Witten invariants, Geom. Topol. 24 (2020), 2781-2828, arXiv:1810.00385.
  72. Laumon G., Moret-Bailly L., Champs algébriques, textitErgebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics, Vol. 39, Springer-Verlag, Berlin, 2000.
  73. Lepowsky J., Li H., Introduction to vertex operator algebras and their representations, Progress in Mathematics, Vol. 227, Birkhäuser Boston, Inc., Boston, MA, 2004.
  74. Mariño M., Moore G., The Donaldson-Witten function for gauge groups of rank larger than one, Comm. Math. Phys. 199 (1998), 25-69, arXiv:hep-th/9802185.
  75. Martin S., The Donaldson-Witten function for gauge groups of rank larger than one, arXiv:math.SG/0001002.
  76. May J.P., A concise course in algebraic topology, Chicago Lectures in Mathematics, Vol. 1999, University of Chicago Press, Chicago, IL, 1999.
  77. Metzler D., The Donaldson-Witten function for gauge groups of rank larger than one, arXiv:math.DG/0306176.
  78. Milnor J.W., Stasheff J.D., Characteristic classes, Annals of Mathematics Studies, Vol. 76, Princeton University Press, Princeton, N. J.; University of Tokyo Press, Tokyo, 1974.
  79. Mochizuki T., Donaldson type invariants for algebraic surfaces. Transition of moduli stacks, Lecture Notes in Mathematics, Vol. 1972, Springer-Verlag, Berlin, 2009.
  80. Moore G., Witten E., Integration over the $u$-plane in Donaldson theory, Adv. Theor. Math. Phys. 1 (1997), 298-387, arXiv:alg-geom/9510003.
  81. Mumford D., Fogarty J., Kirwan F., Geometric invariant theory, Ergebnisse der Mathematik und ihrer Grenzgebiete (2), Vol. 34, 3rd ed., Springer-Verlag, Berlin, 1994.
  82. Nakajima H., Instantons on ALE spaces, quiver varieties, and Kac-Moody algebras, Duke Math. J. 76 (1994), 365-416.
  83. Nakajima H., Instantons and affine Lie algebras, Nuclear Phys. B Proc. Suppl., 1996, 154-161, arXiv:alg-geom/9510003.
  84. Nakajima H., Heisenberg algebra and Hilbert schemes of points on projective surfaces, Ann. of Math. 145 (1997), 379-388, arXiv:alg-geom/9507012.
  85. Noohi B., Foundations of topological stacks, I, arXiv:math.AG/0503247.
  86. Noohi B., Homotopy types of topological stacks, Adv. Math. 230 (2012), 2014-2047, arXiv:0808.3799.
  87. Oh J., Thomas R.P., Counting sheaves on Calabi-Yau $4$-folds. I, arXiv:2009.05542.
  88. Okounkov A., Lectures on K-theoretic computations in enumerative geometry, in Geometry of Moduli Spaces and Representation Theory, IAS/Park City Math. Ser., Vol. 24, Amer. Math. Soc., Providence, RI, 2017, 251-380, arXiv:1512.07363.
  89. Olsson M., Algebraic spaces and stacks, American Mathematical Society Colloquium Publications, Vol. 62, Amer. Math. Soc., Providence, RI, 2016.
  90. Pantev T., Toën B., Vaquié M., Vezzosi G., Shifted symplectic structures, Publ. Math. Inst. Hautes Études Sci. 117 (2013), 271-328, arXiv:1111.3209.
  91. Ringel C.M., Hall algebras, in Topics in Algebra, Part 1 (Warsaw, 1988), Banach Center Publ., Vol. 26, PWN, Warsaw, 1990, 433-447.
  92. Romagny M., Group actions on stacks and applications, Michigan Math. J. 53 (2005), 209-236.
  93. Rudakov A., Stability for an abelian category, J. Algebra 197 (1997), 231-245.
  94. Shen J., Cobordism invariants of the moduli space of stable pairs, J. Lond. Math. Soc. 94 (2016), 427-446, arXiv:1409.4576.
  95. Simpson C., The topological realization of a simplicial presheaf, arXiv:q-alg/9609004.
  96. Tanaka Y., Thomas R.P., Vafa-Witten invariants for projective surfaces I: stable case, J. Algebraic Geom. 29 (2020), 603-668, arXiv:1702.08487.
  97. Tanaka Y., Thomas R.P., Vafa-Witten invariants for projective surfaces II: semistable case, Pure Appl. Math. Q. 13 (2017), 517-562, arXiv:1702.08488.
  98. Taubes C.H., Casson's invariant and gauge theory, J. Differential Geom. 31 (1990), 547-599.
  99. Thaddeus M., Stable pairs, linear systems and the Verlinde formula, Invent. Math. 117 (1994), 317-353, arXiv:alg-geom/9210007.
  100. Thomas R.P., A holomorphic Casson invariant for Calabi-Yau 3-folds, and bundles on $K3$ fibrations, J. Differential Geom. 54 (2000), 367-438, arXiv:math.AG/9806111.
  101. Thomas R.P., Equivariant $K$-theory and refined Vafa-Witten invariants, Comm. Math. Phys. 378 (2020), 1451-1500, arXiv:1810.00078.
  102. Toën B., Higher and derived stacks: a global overview, in Algebraic Geometry - Seattle 2005. Part 1, Proc. Sympos. Pure Math., Vol. 80, Amer. Math. Soc., Providence, RI, 2009, 435-487, arXiv:math.AG/0604504.
  103. Toën B., Derived algebraic geometry, EMS Surv. Math. Sci. 1 (2014), 153-240, arXiv:1401.1044.
  104. Toën B., Vaquié M., Moduli of objects in dg-categories, Ann. Sci. École Norm. Sup. (4) 40 (2007), 387-444, arXiv:math.AG/0503269.
  105. Toën B., Vezzosi G., From HAG to DAG: derived moduli stacks, in Axiomatic, Enriched and Motivic Homotopy Theory, NATO Sci. Ser. II Math. Phys. Chem., Vol. 131, Kluwer Acad. Publ., Dordrecht, 2004, 173-216, arXiv:math.AG/0210407.
  106. Toën B., Vezzosi G., Homotopical algebraic geometry. II. Geometric stacks and applications, Mem. Amer. Math. Soc. 193 (2008), x+224 pages, arXiv:math.AG/0404373.
  107. Upmeier M., Homological Lie brackets on moduli spaces and pushforward operations in twisted $K$-theory, Mem. Amer. Math. Soc. (2021), 25 pages, arXiv:2101.10990.
  108. Witten E., Monopoles and four-manifolds, Math. Res. Lett. 1 (1994), 769-796, arXiv:hep-th/9411102.
  109. Zhu Y., Modular invariance of characters of vertex operator algebras, J. Amer. Math. Soc. 9 (1996), 237-302.

Previous article  Next article  Contents of Volume 18 (2022)