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 B ›› 2022, Vol. 42 ›› Issue (3): 1160-1172.doi: 10.1007/s10473-022-0320-8

• Articles • Previous Articles Next Articles

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 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 B, 2022, 42(3): 1160-1172.

Trendmd

References

[1] Corlette K. Flat G-bundles with canonical metrics. J Differential Geom, 1988, 28(3):361-382
[2] Donaldson S K. Floer homology groups in Yang-Mills theory. Vol 147. Cambridge:Cambridge University Press, 2002
[3] Donaldson S K, Kronheimer P B. The geometry of four-manifolds. New York:Oxford University Press, 1990
[4] Gagliardo M, Uhlenbeck K K. Geometric aspects of the Kapustin-Witten equations. J Fixed Point Theory Appl, 2012, 11:185-198
[5] Haydys H. Fukaya-Seidel category and gauge theory. J Symplectic Geom, 2015, 13:151-207
[6] Hitchin N J. The self-duality equations on a Riemann surface. Proc London Math Soc, 1987, 55(3):59-126
[7] Huang T. A lower bound on the solutions of Kapustin-Witten equations. Lett Math Phys, 2016. DOI:https://doi.org/10.1007/s11005-016-0910-2
[8] Huang T. An energy gap for complex Yang-Mills equations. SIGMA Symmetry Integrability Geom Methods Appl, 2017, 13:Paper No 061, 15 pp
[9] Huang T. On a topology property for moduli space of Kapustin-Witten equations. Forum Math, 2019, 31(5):1119-1138
[10] Kobayashi S. Differential geometry of complex vector bundles. Publications of the Mathematical Society of Japan. Vol 15. Princeton, NJ:Princeton University Press, 1987
[11] Mares B. Some Analytic Aspects of Vafa-Witten Twisted N=4 Supersymmetric Yang-Mills theory[D]. MIT, 2010
[12] Tanaka Y. Some boundedness property of solutions to the Vafa-Witten equations on closed four-manifolds. Quart J Math, 2017, 68(4):1203-1225
[13] Taubes C H. Compactness theorems for SL(2; C) generalizations of the 4-dimensional anti-self dual equations. arXiv:1307.6447v4
[14] Taubes C H. The zero loci of Z/2 harmonic spinors in dimension 2, 3 and 4. arXiv:1407.6206
[15] Taubes C H. PSL(2; C) connections on 3-manifolds with L² bounds on curvature. Camb J Math, 2014, 1:239-397
[16] Taubes C H. The behavior of sequences of solutions to the Vafa-Witten equations. arXiv:1702.04610
[17] Uhlenbeck K K. Connctions with L^p bounds on curvature. Comm Math Phys, 1982, 83:31-42
[18] Uhlenbeck K K. The Chern classes of Sobolev connections. Comm Math Phys, 1985, 101:445-457
[19] Vafa C, Witten E. A Strong Coupling Test of S-Dualy. Nucl Phys B, 1994, 431:3-77
[20] Wehrheim K. Uhlenbeck Compactness. Vol 1. European Mathematical Society, 2004
[21] Witten E. Fivebranes and Knots. Quantum Topol, 2012, 3:1-137

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

Abstract

Cite this article

share this article

References

Related Articles 2

Metrics

Comments

Recommended 0

[1]	Changyu GUO, Changlin XIANG. REGULARITY OF P-HARMONIC MAPPINGS INTO NPC SPACES [J]. Acta mathematica scientia,Series B, 2021, 41(2): 633-645.
[2]	Debin LIU, Pan ZHANG. HERMITIAN-EINSTEIN METRICS FOR HIGGS BUNDLES OVER COMPLETE HERMITIAN MANIFOLDS [J]. Acta mathematica scientia,Series B, 2020, 40(1): 211-225.