[Paper Review] Relations in the Tautological Ring and Frobenius Manifolds near the Discriminant

Felix Janda|arXiv (Cornell University)|May 13, 2015

Algebraic Geometry and Number Theory4 citations

TL;DR

This paper establishes that relations derived from generically semisimple cohomological field theories—specifically via pole cancellation in the Givental-Teleman classification—are all expressible in terms of Pixton's generalized Faber-Zagier relations in the tautological ring of the moduli space of curves. It further shows that the local structure of generically semisimple Frobenius manifolds near the non-semisimple locus is modeled on the $A_2 \times A_1^{N-2}$-singularity, supporting the conjecture that Pixton's relations exhaust all tautological relations.

ABSTRACT

For generically semisimple cohomological field theories pole cancellation in the Givental-Teleman classification implies relations between classes in the tautological ring of the moduli space of curves. For the theory of the $A_2$-singularity these are known to be equivalent to Pixton's generalized Faber-Zagier relations. We show that the relations from any other semisimple cohomological field theory can be written in terms of Pixton's relations. This gives large evidence for the conjecture that Pixton's relations are all relations between tautological classes. As part of the proof, we study the structure of an $N$-dimensional generically semisimple Frobenius manifold near smooth points of the non-semisimple locus, giving a local description modeled on the Frobenius manifold corresponding to the $A_2 imes A_1^{N - 2}$-singularity, and give criteria for extending generically semi-simple Frobenius manifolds to cohomological field theories.

Motivation & Objective

• To investigate the structure of relations in the tautological ring of the moduli space of curves arising from generically semisimple cohomological field theories.
• To determine whether relations from arbitrary semisimple cohomological field theories can be expressed in terms of Pixton's generalized Faber-Zagier relations.
• To analyze the local geometry of generically semisimple Frobenius manifolds near the non-semisimple locus and relate it to the $A_2 \times A_1^{N-2}$-singularity.
• To provide criteria for extending generically semisimple Frobenius manifolds to full cohomological field theories.

Proposed method

• Using the Givental-Teleman classification of cohomological field theories, the paper analyzes pole cancellation conditions in the J-function to derive tautological relations.
• It applies the theory of Frobenius manifolds to study the local structure near smooth points of the non-semisimple locus in $N$-dimensional generically semisimple Frobenius manifolds.
• The paper constructs a local model for such Frobenius manifolds based on the $A_2 \times A_1^{N-2}$-singularity, showing equivalence in the Frobenius manifold structure.
• It establishes a correspondence between the tautological relations from any semisimple cohomological field theory and Pixton's relations via the Givental-Teleman formalism.
• The analysis relies on the classification of semisimple cohomological field theories and the structure of the tautological ring, particularly the role of the $A_2$-singularity theory.
• It derives conditions under which a generically semisimple Frobenius manifold can be extended to a full cohomological field theory, based on the behavior near the discriminant locus.

Experimental results

Research questions

• RQ1 Can all tautological relations arising from semisimple cohomological field theories be expressed in terms of Pixton's generalized Faber-Zagier relations?
• RQ2 What is the local geometric structure of a generically semisimple Frobenius manifold near a smooth point of the non-semisimple locus?
• RQ3 How does the Frobenius manifold structure near the non-semisimple locus relate to the $A_2 \times A_1^{N-2}$-singularity?
• RQ4 Under what conditions can a generically semisimple Frobenius manifold be extended to a cohomological field theory?
• RQ5 Is the set of relations from the $A_2$-singularity theory sufficient to generate all tautological relations in the tautological ring?

Key findings

• All tautological relations derived from any generically semisimple cohomological field theory are contained within the relations generated by Pixton's generalized Faber-Zagier relations.
• The local structure of an $N$-dimensional generically semisimple Frobenius manifold near a smooth point of the non-semisimple locus is modeled on the Frobenius manifold associated to the $A_2 \times A_1^{N-2}$-singularity.
• The paper provides explicit criteria for extending a generically semisimple Frobenius manifold to a cohomological field theory, based on the behavior of the Frobenius manifold near the discriminant.
• The $A_2$-singularity theory generates relations that are equivalent to Pixton's relations, providing strong evidence for the conjecture that Pixton's relations are complete in the tautological ring.
• The analysis confirms that pole cancellation in the Givental-Teleman formalism leads precisely to relations expressible in terms of Pixton's relations across all semisimple cohomological field theories.
• The results support the broader conjecture that Pixton's relations are the full set of tautological relations in the tautological ring of the moduli space of curves.