Overview
This study introduces a definition for the lift of a partial cohomological field theory and the corresponding lift of local polyvector fields. The work extends a previously proposed lift procedure by Della Vedova, Lorenzoni, and Savoldi. The framework developed allows for the systematic generation of examples of non-semisimple homogeneous partial cohomological field theories. These generated theories are characterized by integrable systems that possess a second Hamiltonian structure, which supports a conjecture by Buryak et al. regarding an explicit formula for the second Poisson bracket in non-semisimple contexts. Furthermore, the research identifies relationships between these lift constructions and Morimoto theory, specifically concerning the lift of geometric structures to the Weil bundle of infinitely near points associated with a local algebra.
Research Context
The concept of a cohomological field theory (CohFT) is central to this research. CohFTs are mathematical structures that connect quantum field theory with Gromov-Witten theory and integrable systems. Partial cohomological field theories represent a specific class or generalization within this broader framework. The study builds upon existing methods for lifting mathematical structures, referencing a procedure developed by Della Vedova, Lorenzoni, and Savoldi. A key aspect of the context is the investigation of bi-Hamiltonian structures, which are relevant in the study of integrable systems. The work addresses the non-semisimple case, distinguishing it from simpler or known semisimple scenarios. The research also directly addresses a conjecture by Buryak et al. concerning the explicit formulation of a second Poisson bracket, specifically in non-semisimple cases. Additionally, the study connects its findings to Morimoto theory, which deals with the lift of geometric structures to Weil bundles.
Approach
The approach involves defining the lift of a partial cohomological field theory with respect to a Frobenius algebra. Concurrently, the corresponding lift of local polyvector fields is defined. This methodology is presented as an extension of an existing lift procedure. The defined lift procedures are then applied to systematically produce examples of non-semisimple homogeneous partial cohomological field theories. The properties of the integrable systems associated with these theories are then analyzed to determine the presence of a second Hamiltonian structure. The results obtained from this systematic generation are used to confirm a specific conjecture by Buryak et al. about an explicit formula for the second Poisson bracket in non-semisimple cases. Finally, the research explores and presents relationships between these lift constructions and the Morimoto theory of the lift of geometric structures to the Weil bundle of infinitely near points associated with a local algebra.
Findings
- A definition for the lift of a partial cohomological field theory with respect to a Frobenius algebra was established.
- A corresponding definition for the lift of local polyvector fields was developed.
- These definitions extend the lift procedure previously proposed by Della Vedova, Lorenzoni, and Savoldi.
- The defined procedures systematically produce examples of non-semisimple homogeneous partial cohomological field theories.
- The integrable systems associated with these generated theories possess a second Hamiltonian structure.
- The findings confirm the conjecture by Buryak et al. regarding an explicit formula for the second Poisson bracket in new non-semisimple cases.
- Relations between these lift constructions and the Morimoto theory of the lift of geometric structures to the Weil bundle of infinitely near points associated with a local algebra were identified.
Why This Matters
This research provides a systematic method for constructing examples of non-semisimple homogeneous partial cohomological field theories. The confirmation of the Buryak et al. conjecture on the explicit formula for the second Poisson bracket in non-semisimple cases directly contributes to the theoretical understanding of integrable systems. The established connections with Morimoto theory integrate these new lift constructions within a broader framework of geometric structures.