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A PERTURBATION AND GENERIC SMOOTHNESS OF THE VAFA–WITTEN MODULI SPACES ON CLOSED SYMPLECTIC FOUR-MANIFOLDS

Published online by Cambridge University Press:  13 July 2018

YUUJI TANAKA
YUUJI TANAKA*
Affiliation:
Mathematical Institute, University of Oxford, Radcliffe Observatory Quarter, Woodstock Road, Oxford, OX2 6GG, UK e-mail: tanaka@maths.ox.ac.uk
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Abstract

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We prove a Freed–Uhlenbeck style generic smoothness theorem for the moduli space of solutions to the Vafa–Witten equations on a closed symplectic four-manifold by using a method developed by Feehan for the study of the PU(2)-monopole equations on smooth closed four-manifolds. We introduce a set of perturbation terms to the Vafa–Witten equations, and prove that the moduli space of solutions to the perturbed Vafa–Witten equations on a closed symplectic four-manifold for the structure group SU(2) or SO(3) is a smooth manifold of dimension zero for a generic choice of the perturbation parameters.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 61 , Issue 2 , May 2019 , pp. 471 - 486
Copyright
Copyright © Glasgow Mathematical Journal Trust 2018 

References

REFERENCES

1. Donaldson, S. K. and Kronheimer, P. B., The Geometry of Four-Manifolds (Oxford University Press, New York, 1990).Google Scholar
2. Feehan, P. M. N., Generic metrics, irreducible rank-one PU(2) monopoles, and transversality, Comm. Anal. Geom. 8 (2000), 905–967.Google Scholar
3. Feehan, P. M. N. and Leness, T. G., PU(2) monopoles. I: Regularity, Uhlenbeck compactness, and transversality, J. Differ. Geom. 49 (1998), 265–410.Google Scholar
4. Haydys, A., Fukaya–Seidel category and gauge theory, J. Symplectic Geom. 13 (2015), 151–207.Google Scholar
5. Lawson, H. B. and Michaelshon, M-L., Spin geometry (Princeton University Press, Princeton, NJ, 1989).Google Scholar
6. Mares, B., Some analytic aspects of Vafa–Witten twisted $\mathcal{N}$ = 4 supersymmetric Yang–Mills theory, PhD Thesis (M.I.T., 2010).Google Scholar
7. Morgan, J. W., The Seiberg–Witten equations and application to the topology of smooth four-manifolds (Princeton University Press, Princeton, NJ, 1996).Google Scholar
8. Tanaka, Y., Some boundedness property of the Vafa–Witten equations on closed four-manifolds, Q. J. Math. 68 (2017), 1203–1225.Google Scholar
9. Teleman, A., Moduli spaces of PU(2)-monopoles, Asian J. Math. 4 (2000), 391–435.Google Scholar
10. Vafa, C. and Witten, E., A strong coupling test of S-duality, Nucl. Phys. B 432 (1994), 484–550.Google Scholar
11. Witten, E., Fivebranes and Knots, Quantum Topol. 3 (2012), 1–137.Google Scholar