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Twisted Donaldson invariants

Part of: Higher algebraic $K$-theory Quantum field theory; related classical field theories Infinite-dimensional manifolds

Published online by Cambridge University Press:  04 February 2021

TSUYOSHI KATO ,
HIROFUMI SASAHIRA  and
HANG WANG
TSUYOSHI KATO
Affiliation:
Department of Mathematics, Graduate School of Science, Kyoto University, Kyoto, 606-8502, Japan. e-mail: tkato@math.kyoto-u.ac.jp
HIROFUMI SASAHIRA
Affiliation:
Faculty of Mathematics, Kyushu University, 744, Motooka, Nishi-ku, Fukuoka, 819-0395, Japan. e-mail: hsasahira@math.kyushu-u.ac.jp
HANG WANG
Affiliation:
School of Mathematical Sciences, East China Normal University, South Lian Hua Road 5005 Minhang district, Shanghai200241, P. R. China. e-mail: wanghang@math.ecnu.edu.cn
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Abstract

Fundamental group of a manifold gives a deep effect on its underlying smooth structure. In this paper we introduce a new variant of the Donaldson invariant in Yang–Mills gauge theory from twisting by the Picard group of a 4-manifold in the case when the fundamental group is free abelian. We then generalise it to the general case of fundamental groups by use of the framework of non commutative geometry. We also verify that our invariant distinguishes smooth structures between some homeomorphic 4-manifolds.

MSC classification

Secondary: 58B34: Noncommutative geometry (à la Connes) 81T13: Yang-Mills and other gauge theories 19D55: $K$-theory and homology; cyclic homology and cohomology
Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 171 , Issue 3 , November 2021 , pp. 515 - 568
Copyright
© Cambridge Philosophical Society 2021

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Footnotes

†

Supported by JSPS KAKENHI Grant Number JP17H02841

‡

Supported by JSPS KAKENHI Grant Number JP19K03493

§

Supported by grants NSFC-11801178 and Shanghai Rising-Star Program 19QA1403200.

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