Abstract
We consider regular polystable Higgs pairs (E, ϕ) on compact complex manifolds. We show that a non-trivial Higgs field ϕ ∈ H0(End(E) ⊗ KS) restricts the Ricci curvature of the manifold, generalising previous results in the literature. In particular ϕ must vanish for positive Ricci curvature, while for trivial canonical bundle it must be proportional to the identity. For Kähler surfaces, our results provide a new vanishing theorem for solutions to the Vafa-Witten equations. Moreover they constrain supersymmetric 7-brane configurations in F-theory, giving obstructions to the existence of T-branes, i.e. solutions with [ϕ, ϕ†] ≠ 0. When non-trivial Higgs fields are allowed, we give a general characterisation of their structure in terms of vector bundle data, which we then illustrate in explicit examples.
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References
C. Vafa and E. Witten, A Strong coupling test of S duality, Nucl. Phys. B 431 (1994) 3 [hep-th/9408074] [INSPIRE].
Y. Tanaka and R.P. Thomas, Vafa-Witten invariants for projective surfaces I: stable case, J. Alg. Geom. 29 (2020) 603 [arXiv:1702.08487] [INSPIRE].
Y. Tanaka and R.P. Thomas, Vafa-Witten invariants for projective surfaces II: semistable case, arXiv:1702.08488 [INSPIRE].
N.J. Hitchin, The Selfduality equations on a Riemann surface, Proc. Lond. Math. Soc. 55 (1987) 59.
C.T. Simpson, Constructing Variation of Hodge Structure Using Yang-Mil ls Theory and Applications to Uniformization, J. Am. Math. Soc. 1 (1988) 867.
R. Donagi and M. Wijnholt, Model Building with F-theory, Adv. Theor. Math. Phys. 15 (2011) 1237 [arXiv:0802.2969] [INSPIRE].
C. Beasley, J.J. Heckman and C. Vafa, GUTs and Exceptional Branes in F-theory — I, JHEP 01 (2009) 058 [arXiv:0802.3391] [INSPIRE].
J.J. Heckman, Particle Physics Implications of F-theory, Ann. Rev. Nucl. Part. Sci. 60 (2010) 237 [arXiv:1001.0577] [INSPIRE].
T. Weigand, Lectures on F-theory compactifications and model building, Class. Quant. Grav. 27 (2010) 214004 [arXiv:1009.3497] [INSPIRE].
M. Wijnholt, Higgs Bundles and String Phenomenology, Proc. Symp. Pure Math. 85 (2012) 275 [arXiv:1201.2520] [INSPIRE].
S. Cecotti, C. Cordova, J.J. Heckman and C. Vafa, T-Branes and Monodromy, JHEP 07 (2011) 030 [arXiv:1010.5780] [INSPIRE].
H. Hayashi, T. Kawano, Y. Tsuchiya and T. Watari, Flavor Structure in F-theory Compactifications, JHEP 08 (2010) 036 [arXiv:0910.2762] [INSPIRE].
F. Marchesano, R. Savelli and S. Schwieger, Compact T-branes, JHEP 09 (2017) 132 [arXiv:1707.03797] [INSPIRE].
L. Álvarez-Consul and O. Garcia-Prada, Hitchin-Kobayashi correspondence, quivers, and vortices, Commun. Math. Phys. 238 (2003) 1 [math/0112161] [INSPIRE].
Y. Tanaka, Stable sheaves with twisted sections and the Vafa-Witten equations on smooth projective surfaces, arXiv:1312.2673 [INSPIRE].
R. Donagi, S. Katz and E. Sharpe, Spectra of D-branes with Higgs vevs, Adv. Theor. Math. Phys. 8 (2004) 813 [hep-th/0309270] [INSPIRE].
R. Donagi and M. Wijnholt, Gluing Branes, I, JHEP 05 (2013) 068 [arXiv:1104.2610] [INSPIRE].
F. Takemoto, Stable vector bundles on algebraic surfaces, Nagoya Math. J. 47 (1972) 29.
G.R. Kempf, Pul ling back bundles, Pacific J. Math. 152 (1992) 319.
M. Lübke and A. Teleman, The universal Kobayashi-Hitchin correspondence on Hermitian manifolds, Mem. Am. Math. Soc. 183 (2006) [math/0402341].
W. Barth, K. Hulek, C. Peters and A. Van de Ven, Compact complex surfaces, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. 4 (2nd enlarged ed.), Springer-Verlag, Berlin (2004) [DOI].
G. Harder, Lectures on Algebraic Geometry I: Sheaves, Cohomology of Sheaves, and Applications to Riemann Surfaces, Aspects of Mathematics, Springer Spektrum (2011) [DOI].
H. Jockers and J. Louis, D-terms and F-terms from D7-brane fluxes, Nucl. Phys. B 718 (2005) 203 [hep-th/0502059] [INSPIRE].
L. Martucci, D-branes on general N = 1 backgrounds: Superpotentials and D-terms, JHEP 06 (2006) 033 [hep-th/0602129] [INSPIRE].
F. Marchesano, R. Savelli and S. Schwieger, T-branes and defects, JHEP 04 (2019) 110 [arXiv:1902.04108] [INSPIRE].
A. Collinucci and R. Savelli, T-branes as branes within branes, JHEP 09 (2015) 161 [arXiv:1410.4178] [INSPIRE].
R. Friedman, Algebraic surfaces and holomorphic vector bundles, Universitext, Springer (1998) [DOI].
R. Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, Springer (1977) [DOI].
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Marchesano, F., Moraru, R. & Savelli, R. A vanishing theorem for T-branes. J. High Energ. Phys. 2020, 2 (2020). https://doi.org/10.1007/JHEP11(2020)002
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DOI: https://doi.org/10.1007/JHEP11(2020)002