Universal Structures in ℂ-Linear Enumerative Invariant Theories
An enumerative invariant theory in algebraic geometry, differential geometry, or representation theory, is the study of invariants which "count" -(semi)stable objects with fixed topological invariants ⟦⟧ = α in some geometric problem, by means of a virtual class [ℳˢˢα()]ᵥᵢᵣₜ in some homol...
Gespeichert in:
| Veröffentlicht in: | Symmetry, Integrability and Geometry: Methods and Applications |
|---|---|
| Datum: | 2022 |
| Hauptverfasser: | , , |
| Format: | Artikel |
| Sprache: | Englisch |
| Veröffentlicht: |
Інститут математики НАН України
2022
|
| Online Zugang: | https://nasplib.isofts.kiev.ua/handle/123456789/211720 |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Zitieren: | Universal Structures in ℂ-Linear Enumerative Invariant Theories. Jacob Gross, Dominic Joyce and Yuuji Tanaka. SIGMA 18 (2022), 068, 61 pages |
Institution
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862736004062904320 |
|---|---|
| author | Gross, Jacob Joyce, Dominic Tanaka, Yuuji |
| author_facet | Gross, Jacob Joyce, Dominic Tanaka, Yuuji |
| citation_txt | Universal Structures in ℂ-Linear Enumerative Invariant Theories. Jacob Gross, Dominic Joyce and Yuuji Tanaka. SIGMA 18 (2022), 068, 61 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | An enumerative invariant theory in algebraic geometry, differential geometry, or representation theory, is the study of invariants which "count" -(semi)stable objects with fixed topological invariants ⟦⟧ = α in some geometric problem, by means of a virtual class [ℳˢˢα()]ᵥᵢᵣₜ in some homology theory for the moduli spaces ℳˢᵗα() ⊆ ℳˢˢα() of -(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 ℳ, ℳᵖˡ, where the second author (see https://people.maths.ox.ac.uk/~joyce/hall.pdf) gives ∗(ℳ) the structure of a graded vertex algebra, and ∗(ℳᵖˡ) a graded Lie algebra, closely related to ∗(ℳ). The virtual classes [ℳˢˢα()]ᵥᵢᵣₜ take values in ∗(ℳᵖˡ). In most such theories, defining [ℳˢˢα()]ᵥᵢᵣₜ when ℳˢᵗα() ≠ ℳssα() (in gauge theory, when the moduli space contains reducibles) is a difficult problem. We conjecture that there is a natural way to define invariants [ℳˢˢα()]ᵢₙᵥ in homology over Q, with [ℳˢˢα()]ᵢₙᵥ = [ℳˢˢα()]ᵥᵢᵣₜ when ℳˢᵗα() = ℳˢˢα(), and that these invariants satisfy a universal wall-crossing formula under change of stability condition , written using the Lie bracket on ∗(ℳᵖˡ). 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].
|
| first_indexed | 2026-04-17T16:31:25Z |
| format | Article |
| fulltext | |
| id | nasplib_isofts_kiev_ua-123456789-211720 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-04-17T16:31:25Z |
| publishDate | 2022 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Gross, Jacob Joyce, Dominic Tanaka, Yuuji 2026-01-09T12:45:06Z 2022 Universal Structures in ℂ-Linear Enumerative Invariant Theories. Jacob Gross, Dominic Joyce and Yuuji Tanaka. SIGMA 18 (2022), 068, 61 pages 1815-0659 2020 Mathematics Subject Classification: 14D20; 17B69; 16G20 arXiv:2005.05637 https://nasplib.isofts.kiev.ua/handle/123456789/211720 https://doi.org/10.3842/SIGMA.2022.068 An enumerative invariant theory in algebraic geometry, differential geometry, or representation theory, is the study of invariants which "count" -(semi)stable objects with fixed topological invariants ⟦⟧ = α in some geometric problem, by means of a virtual class [ℳˢˢα()]ᵥᵢᵣₜ in some homology theory for the moduli spaces ℳˢᵗα() ⊆ ℳˢˢα() of -(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 ℳ, ℳᵖˡ, where the second author (see https://people.maths.ox.ac.uk/~joyce/hall.pdf) gives ∗(ℳ) the structure of a graded vertex algebra, and ∗(ℳᵖˡ) a graded Lie algebra, closely related to ∗(ℳ). The virtual classes [ℳˢˢα()]ᵥᵢᵣₜ take values in ∗(ℳᵖˡ). In most such theories, defining [ℳˢˢα()]ᵥᵢᵣₜ when ℳˢᵗα() ≠ ℳssα() (in gauge theory, when the moduli space contains reducibles) is a difficult problem. We conjecture that there is a natural way to define invariants [ℳˢˢα()]ᵢₙᵥ in homology over Q, with [ℳˢˢα()]ᵢₙᵥ = [ℳˢˢα()]ᵥᵢᵣₜ when ℳˢᵗα() = ℳˢˢα(), and that these invariants satisfy a universal wall-crossing formula under change of stability condition , written using the Lie bracket on ∗(ℳᵖˡ). 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]. This research was supported by the Simons Collaboration on Special Holonomy in Geometry, Analysis, and Physics. The third author was partially supported by JSPS Grant-in-Aid for Scientific Research numbers JP16K05125 and JP21K03246. The authors would like to thank Arkadij Bojko, Yalong Cao, Frances Kirwan, and Markus Upmeier for helpful conversations, and the anonymous referees for careful proofreading. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Universal Structures in ℂ-Linear Enumerative Invariant Theories Article published earlier |
| spellingShingle | Universal Structures in ℂ-Linear Enumerative Invariant Theories Gross, Jacob Joyce, Dominic Tanaka, Yuuji |
| title | Universal Structures in ℂ-Linear Enumerative Invariant Theories |
| title_full | Universal Structures in ℂ-Linear Enumerative Invariant Theories |
| title_fullStr | Universal Structures in ℂ-Linear Enumerative Invariant Theories |
| title_full_unstemmed | Universal Structures in ℂ-Linear Enumerative Invariant Theories |
| title_short | Universal Structures in ℂ-Linear Enumerative Invariant Theories |
| title_sort | universal structures in ℂ-linear enumerative invariant theories |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/211720 |
| work_keys_str_mv | AT grossjacob universalstructuresinclinearenumerativeinvarianttheories AT joycedominic universalstructuresinclinearenumerativeinvarianttheories AT tanakayuuji universalstructuresinclinearenumerativeinvarianttheories |