Davide, Accadia (2023) On the Tautological Ring of Moduli Spaces of Riemann Surfaces. Masters thesis, Concordia University.
Preview |
Text (application/pdf)
319kBAccadia_MASc_F2023.pdf - Accepted Version Available under License Spectrum Terms of Access. |
Abstract
This thesis gives an overview of known results about the relations on the tautological ring of Moduli spaces. First, we introduce the Moduli Space of Stable Curves as an Orbifold of a etale groupoid, a generalization of complex manifold and orbit space of a group. Then we define the Tautological Ring on it as a subring of the Cohomology Ring. Finally, we present the work of Pandharipande, Pixton and Dvonkine in \cite{3-spin}, they discovered a set of relations on the Tautological Ring that is, up to date, the largest known. This set is obtained by Cohomological Field Theories, a tool to compute and glue cohomology classes in a Tautological way, and a group action on CohFTs. Using Teleman’s characterization of Cohomological Field Theories in a specific case, they manage to deduce an explicit formula for a suitable modification of Witten’s 3-spin CohFT. This turns out to vanish non trivially, providing a set of relations.
Davide Accadia
| Divisions: | Concordia University > Faculty of Arts and Science > Mathematics and Statistics |
|---|---|
| Item Type: | Thesis (Masters) |
| Authors: | Davide, Accadia |
| Institution: | Concordia University |
| Degree Name: | M.A. Sc. |
| Program: | Mathematics |
| Date: | September 2023 |
| Thesis Supervisor(s): | Paolo, Rossi |
| Keywords: | Moduli spaces, orbifolds, Tautological Ring. |
| ID Code: | 992935 |
| Deposited By: | Davide Accadia |
| Deposited On: | 16 Nov 2023 20:45 |
| Last Modified: | 16 Nov 2023 20:45 |
References:
Robbin J.W. And Salamon D.(2005), A construction of the Deligne–Mumford orbifoldTavakol Mehdi PhD Thesis (2011), Tautological Rings Of Moduli Spaces of Curves
Pixton Aaron PhD Thesis (2013)l, The Tautological Ring Of The Moduli Spaces of Curves
Zvonkine Dimitri (2014), An introduction to moduli spaces of curves and their intersection theory
Givental Alexander (2001), Semisimple Frobenius Structures At higher Genus
Teleman Constantin (2011), The Structure of 2D Semi-simple Field Theories
Pandharipande R. And Pixton A. (2010),Relations in the tautological ring of the moduli space of curves
Pixton A. (2012), Conjectural Relations in The Tautological Ring of $\overline\mathcalM_g,n$
Getzler E. (1997), Intersection Theory on $\overline\mathcalM_1,4$ and Elliptic Gromov-Witten Invariants
Pandharipande R. And Pixton A. And Zvorkine D. (2015), Relations on $\BarM_g,n$ via 3-spin structures.
Ionel EN (2005). Relations in the tautological ring of $\mathcalM_g$
Graber T. Pandharipande R. (2005), Constructions of nontautological classes on moduli spaces of curves
Petersen D. Tommasi O. (2012), The Gorenstein conjecture fails for the tautological ring of $\overline\mathcalM_2,n$
Morita S. (2003), Generators for the tautological algebra of the moduli space of curves
Behrend K. (2004), On the de Rham Cohomology of Differential and Algebraic Stacks
Fulton W. Introduction to Intersection Theory in Algebraic Geometry
Griffiths P. And Harris J. Principles of Algebraic Geometry
Repository Staff Only: item control page
Download Statistics
Download Statistics