A COMPACTNESS THEOREM FOR STABLE FLAT $SL(2,\mathbb{C})$ CONNECTIONS ON 3-FOLDS

doi:10.1007/s10473-022-0320-8

Acta mathematica scientia,Series A ›› 2022, Vol. 42 ›› Issue (3): 1160-1172.doi: 10.1007/s10473-022-0320-8

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A COMPACTNESS THEOREM FOR STABLE FLAT $SL(2,\mathbb{C})$ CONNECTIONS ON 3-FOLDS

Teng HUANG^1,2

1. School of Mathematical Sciences, University of Science and Technology of China, Hefei, China;
2. CAS Key Laboratory of Wu Wen-Tsun Mathematics, University of Science and Technology of China, Hefei, 230026, China

Received:2020-12-23 Revised:2021-06-06 Online:2022-06-26 Published:2022-06-24
Contact: Teng HUANG,E-mail:htmath@ustc.edu.cn E-mail:htmath@ustc.edu.cn
Supported by:
This work was supported in part by NSF of China (11801539) and the Fundamental Research Funds of the Central Universities (WK3470000019), and the USTC Research Funds of the Double First-Class Initiative (YD3470002002).

Abstract

Abstract: Let $Y$ be a closed $3$-manifold such that all flat $SU(2)$-connections on $Y$ are non-degenerate. In this article, we prove a Uhlenbeck-type compactness theorem on $Y$ for stable flat $SL(2,\mathbb{C})$ connections satisfying an $L^{2}$-bound for the real curvature. Combining the compactness theorem and a result from [7], we prove that the moduli space of the stable flat $SL(2,\mathbb{C})$ connections is disconnected under certain technical assumptions.

Key words: Stable flat $SL(2,\mathbb{C})$ connections, Vafa-Witten equations, compactness theorem

CLC Number:

58E15

Teng HUANG. A COMPACTNESS THEOREM FOR STABLE FLAT $SL(2,\mathbb{C})$ CONNECTIONS ON 3-FOLDS[J].Acta mathematica scientia,Series A, 2022, 42(3): 1160-1172.

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A COMPACTNESS THEOREM FOR STABLE FLAT $SL(2,\mathbb{C})$ CONNECTIONS ON 3-FOLDS

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