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' τ-(semi)stable objects E with fixed topological invariants [[E]]=α in some geometric problem, by means of a virtual class [Mssα(τ)]virt in some homolo...
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| Format: | Journal article |
| Language: | English |
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National Academy of Science of Ukraine
2022
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| _version_ | 1826308999129071616 |
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| author | Gross, J Joyce, D Tanaka, Y |
| author_facet | Gross, J Joyce, D Tanaka, Y |
| author_sort | Gross, J |
| collection | OXFORD |
| description | An enumerative invariant theory in algebraic geometry, differential geometry, or representation theory, is the study of invariants which `count' τ-(semi)stable objects E with fixed topological invariants [[E]]=α in some geometric problem, by means of a virtual class [Mssα(τ)]virt in some homology theory for the moduli spaces Mstα(τ)⊆Mssα(τ) of τ-(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 H∗(M) the structure of a graded vertex algebra, and H∗(Mpl) a graded Lie algebra, closely related to H∗(M). The virtual classes [Mssα(τ)]virt take values in H∗(Mpl). In most such theories, defining [Mssα(τ)]virt when Mstα(τ)≠Mssα(τ) (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(τ)]inv in homology over Q, with [Mssα(τ)]inv=[Mssα(τ)]virt when Mstα(τ)=Mssα(τ), and that these invariants satisfy a universal wall-crossing formula under change of stability condition τ, written using the Lie bracket on H∗(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 | 2024-03-07T07:27:40Z |
| format | Journal article |
| id | oxford-uuid:a06eea03-908a-47a8-a3ad-97dba8a22ac9 |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-03-07T07:27:40Z |
| publishDate | 2022 |
| publisher | National Academy of Science of Ukraine |
| record_format | dspace |
| spelling | oxford-uuid:a06eea03-908a-47a8-a3ad-97dba8a22ac92022-12-01T14:10:01ZUniversal structures in C-linear enumerative invariant theoriesJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:a06eea03-908a-47a8-a3ad-97dba8a22ac9EnglishSymplectic ElementsNational Academy of Science of Ukraine2022Gross, JJoyce, DTanaka, YAn enumerative invariant theory in algebraic geometry, differential geometry, or representation theory, is the study of invariants which `count' τ-(semi)stable objects E with fixed topological invariants [[E]]=α in some geometric problem, by means of a virtual class [Mssα(τ)]virt in some homology theory for the moduli spaces Mstα(τ)⊆Mssα(τ) of τ-(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 H∗(M) the structure of a graded vertex algebra, and H∗(Mpl) a graded Lie algebra, closely related to H∗(M). The virtual classes [Mssα(τ)]virt take values in H∗(Mpl). In most such theories, defining [Mssα(τ)]virt when Mstα(τ)≠Mssα(τ) (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(τ)]inv in homology over Q, with [Mssα(τ)]inv=[Mssα(τ)]virt when Mstα(τ)=Mssα(τ), and that these invariants satisfy a universal wall-crossing formula under change of stability condition τ, written using the Lie bracket on H∗(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]. |
| spellingShingle | Gross, J Joyce, D Tanaka, Y Universal structures in C-linear enumerative invariant theories |
| 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 |
| work_keys_str_mv | AT grossj universalstructuresinclinearenumerativeinvarianttheories AT joyced universalstructuresinclinearenumerativeinvarianttheories AT tanakay universalstructuresinclinearenumerativeinvarianttheories |