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Frontiers in Complex DynamicsIn Celebration of John Milnor's 80th Birthday$
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Araceli Bonifant, Misha Lyubich, and Scott Sutherland

Print publication date: 2014

Print ISBN-13: 9780691159294

Published to Princeton Scholarship Online: October 2017

DOI: 10.23943/princeton/9780691159294.001.0001

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Invariants of four-manifolds with flows via cohomological field theory

Invariants of four-manifolds with flows via cohomological field theory

Chapter:
(p.645) Invariants of four-manifolds with flows via cohomological field theory
Source:
Frontiers in Complex Dynamics
Author(s):

Hugo Garcia-Compeân

Roberto Santos-Silva

Alberto Verjovsky

, Araceli Bonifant, Mikhail Lyubich, Scott Sutherland
Publisher:
Princeton University Press
DOI:10.23943/princeton/9780691159294.003.0023

This chapter argues that the Jones–Witten invariants can be generalized for smooth, nonsingular vector fields with invariant probability measure on three-manifolds, thus giving rise to new invariants of dynamical systems. After a short survey of cohomological field theory for Yang–Mills fields, Donaldson–Witten invariants are generalized to four-dimensional manifolds with non-singular smooth flows generated by homologically non-trivial p-vector fields. The chapter studies the case of Kähler manifolds by using the Witten's consideration of the strong coupling dynamics of N = 1 supersymmetric Yang–Mills theories. The whole construction is performed by implementing the notion of higher-dimensional asymptotic cycles. In the process Seiberg–Witten invariants are also described within this context. Finally, the chapter gives an interpretation of the asymptotic observables of four-manifolds in the context of string theory with flows.

Keywords:   Jones–Witten invariants, cohomological field theory, dynamical systems, Yang–Mills fields, Donaldson–Witten invariants, four-dimensional manifolds, Kähler manifolds, higher-dimensional asymptotic cycles, Seiberg–Witten invariants

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