Abstract
In Special Geometry there are two inequivalent notions of “Kodaira type” for a singular fiber: one associated with its local monodromy and one with its Hwang-Oguiso characteristic cycle. When the two Kodaira types are not equal the geometry is subtler and its deformation space gets smaller (“partially frozen” singularities). The paper analyzes the physical interpretation of the Hwang-Oguiso invariant in the context of 4d \( \mathcal{N} \) = 2 QFT and describes the surprising phenomena which appear when it does not coincide with the monodromy type. The Hwang-Oguiso multiple fibers are in one-to-one correspondence with the partially frozen singularities in M-theory compactified on a local elliptic K3: a chain of string dualities relates the two geometric set-ups. Paying attention to a few subtleties, this correspondence explains in purely geometric terms how the “same” Kodaira elliptic fiber may have different deformations spaces. The geometric computation of the number of deformations agrees with the physical expectations. At the end we briefly outline the implications of the Hwang-Oguiso invariants for the classification program of 4d \( \mathcal{N} \) = 2 SCFTs.
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References
E. Witten, Toroidal compactification without vector structure, JHEP 02 (1998) 006 [hep-th/9712028] [INSPIRE].
Y. Tachikawa, Frozen singularities in M and F theory, JHEP 06 (2016) 128 [arXiv:1508.06679] [INSPIRE].
L. Bhardwaj, D.R. Morrison, Y. Tachikawa and A. Tomasiello, The frozen phase of F-theory, JHEP 08 (2018) 138 [arXiv:1805.09070] [INSPIRE].
N. Seiberg and E. Witten, Electric - magnetic duality, monopole condensation, and confinement in N = 2 supersymmetric Yang-Mills theory, Nucl. Phys. B 426 (1994) 19 [Erratum ibid. 430 (1994) 485] [hep-th/9407087] [INSPIRE].
N. Seiberg and E. Witten, Monopoles, duality and chiral symmetry breaking in N = 2 supersymmetric QCD, Nucl. Phys. B 431 (1994) 484 [hep-th/9408099] [INSPIRE].
P. Argyres, M. Lotito, Y. Lü and M. Martone, Geometric constraints on the space of \( \mathcal{N} \) = 2 SCFTs. Part I: physical constraints on relevant deformations, JHEP 02 (2018) 001 [arXiv:1505.04814] [INSPIRE].
P.C. Argyres and M. Martone, Towards a classification of rank r\( \mathcal{N} \) = 2 SCFTs. Part II. Special Kahler stratification of the Coulomb branch, JHEP 12 (2020) 022 [arXiv:2007.00012] [INSPIRE].
K. Kodaira, On compact analytic surfaces, in Analytic Functions, Princeton University Press (1960) pp. 121–135.
K. Kodaira, On compact analytic surfaces. II, Ann. Math. 77 (1963) 563.
K. Kodaira, On compact analytic surfaces. III, Ann. Math. 78 (1963) 1.
W. Barth, K. Hulek, C. Peters and A. van de Ven, Compact complex surfaces, Second enlarged edition, Springer (2004).
J. de Boer et al., Triples, fluxes, and strings, Adv. Theor. Math. Phys. 4 (2002) 995 [hep-th/0103170] [INSPIRE].
M. Atiyah and E. Witten, M theory dynamics on a manifold of G(2) holonomy, Adv. Theor. Math. Phys. 6 (2003) 1 [hep-th/0107177] [INSPIRE].
Y. Tachikawa, On fractional M5 branes and frozen singularities, https://member.ipmu.jp/yuji.tachikawa/transp/kiastalk.pdf .
R. Donagi and E. Witten, Supersymmetric Yang-Mills theory and integrable systems, Nucl. Phys. B 460 (1996) 299 [hep-th/9510101] [INSPIRE].
R. Donagi, Seiberg-Witten integrable systems, Surveys Diff. Geom. IV (1998) 83.
K. Hori et al., Mirror symmetry, Clay Mathematics Monographs, vol. 1, AMS, Clay Mathematical Institute (2003).
O. Aharony, N. Seiberg and Y. Tachikawa, Reading between the lines of four-dimensional gauge theories, JHEP 08 (2013) 115 [arXiv:1305.0318] [INSPIRE].
J.-M. Hwang and K. Oguiso, Characteristic foliation on the discriminantal hypersurface of a holomorphic Lagrangian fibration, arXiv:0710.2376.
J.-M. Hwang and K. Oguiso, Multiple fibers of holomorphic Lagrangian fibrations, arXiv:0907.4869.
J.-M. Hwang and K. Oguiso, Local structure of principally polarized stable Lagrangian fibrations, arXiv:1007.2043.
A. Borel, R. Friedman and J.W. Morgan, Almost commuting elements in compact Lie groups, math/9907007 [INSPIRE].
E. Witten, Supersymmetric index in four-dimensional gauge theories, arXiv:hep-th/0006010.
D. Matsushita, On singular fibres of complex Lagrangian fibrations, math/9911164.
D. Matsushita, Higher direct images of dualizing sheaves of Lagrangian fibrations, Amer. J. Math. 127 (2005) 243.
J. Sawon, Singular fibres of very general Lagrangian fibrations, arXiv:1905.03386 [https://doi.org/10.1142/S021919972150070X].
M. Caorsi and S. Cecotti, Special Arithmetic of Flavor, JHEP 08 (2018) 057 [arXiv:1803.00531] [INSPIRE].
P.C. Argyres, M. Lotito, Y. Lü and M. Martone, Geometric constraints on the space of \( \mathcal{N} \) = 2 SCFTs. Part II: construction of special Kähler geometries and RG flows, JHEP 02 (2018) 002 [arXiv:1601.00011] [INSPIRE].
P.C. Argyres, M. Lotito, Y. Lü and M. Martone, Expanding the landscape of \( \mathcal{N} \) = 2 rank 1 SCFTs, JHEP 05 (2016) 088 [arXiv:1602.02764] [INSPIRE].
P. Argyres, M. Lotito, Y. Lü and M. Martone, Geometric constraints on the space of \( \mathcal{N} \) = 2 SCFTs. Part III: enhanced Coulomb branches and central charges, JHEP 02 (2018) 003 [arXiv:1609.04404] [INSPIRE].
P.C. Argyres and M. Martone, 4d \( \mathcal{N} \) =2 theories with disconnected gauge groups, JHEP 03 (2017) 145 [arXiv:1611.08602] [INSPIRE].
M. Caorsi and S. Cecotti, Homological classification of 4d \( \mathcal{N} \) = 2 QFT. Rank-1 revisited, JHEP 10 (2019) 013 [arXiv:1906.03912] [INSPIRE].
S. Cecotti, S. Ferrara and L. Girardello, Geometry of Type II Superstrings and the Moduli of Superconformal Field Theories, Int. J. Mod. Phys. A 4 (1989) 2475 [INSPIRE].
D. Gaiotto, G.W. Moore and A. Neitzke, Four-dimensional wall-crossing via three-dimensional field theory, Commun. Math. Phys. 299 (2010) 163 [arXiv:0807.4723] [INSPIRE].
G. ’t Hooft, On the Phase Transition Towards Permanent Quark Confinement, Nucl. Phys. B 138 (1978) 1 [INSPIRE].
G. ’t Hooft, A Property of Electric and Magnetic Flux in Nonabelian Gauge Theories, Nucl. Phys. B 153 (1979) 141 [INSPIRE].
G. ’t Hooft, Topology of the Gauge Condition and New Confinement Phases in Nonabelian Gauge Theories, Nucl. Phys. B 190 (1981) 455 [INSPIRE].
S. Cecotti, Direct and Inverse Problems in Special Geometry, arXiv:2312.02536 [INSPIRE].
P.A. Griffiths, Periods of integrals on algebraic manifolds, III (Some global differential- geometric properties of the period mapping), Publ. IHES 38 (1970) 125.
P. Griffiths, Topics in Transcendental Algebraic Geometry, Princeton University Press (1984).
P. Deligne, Travaux de Griffiths, Séminaire Boubaki Exp. 376, Lect. Notes Math. 180 (1970).
P. Griffiths, Mumford-Tate groups, https://publications.ias.edu/sites/default/files/Trieste.pdf.
M. Green, P. Griffiths and M. Kerr, Mumford-Tate domains, Boll. Unione Mat. Ital. (Serie 9) 3 (2010) 281, https://www.math.wustl.edu/~matkerr/MTD.pdf.
M. Green, P. Griffiths and M. Kerr, Mumford-Tate Groups and Domains: Their Geometry and Arithmetic, Ann. Math. Stud., Princeton University Press (2012).
J. Carlson, S. Müller-Stach and C. Peters, Period Mappings and Period Domains, Second Edition, Cambridge studies in advanced mathematics 168, Cambridge University Press (2017).
W. Schmid, Variation of hodge structure: The singularities of the period mapping, Invent. Math. 22 (1973) 211 [INSPIRE].
J.S. Milne, Algebraic Groups. The Theory of Group Schemes of Finite Type over a Field, Cambridge University Press (2017).
C. Birkenhake and H. Lange, Complex Abelian Varieties, Second Edition, Series of Comprenhensive Studies in Mathematics 302, Springer (2004).
M. Caorsi and S. Cecotti, Geometric classification of 4d \( \mathcal{N} \) = 2 SCFTs, JHEP 07 (2018) 138 [arXiv:1801.04542] [INSPIRE].
S. Cecotti, A. Neitzke and C. Vafa, R-Twisting and 4d/2d Correspondences, arXiv:1006.3435 [INSPIRE].
S. Cecotti and M. Del Zotto, Y systems, Q systems, and 4D \( \mathcal{N} \) = 2 supersymmetric QFT, J. Phys. A 47 (2014) 474001 [arXiv:1403.7613] [INSPIRE].
R. Miranda, The Basic Theory of Elliptic Surfaces, ETS Editrice, Pisa, Italiy (1989).
P.C. Argyres and J.R. Wittig, Infinite coupling duals of N = 2 gauge theories and new rank 1 superconformal field theories, JHEP 01 (2008) 074 [arXiv:0712.2028] [INSPIRE].
J.-P. Serre, Algebraic groups and class fields, Graduate Text in Mathematics 117, Springer (1988).
T. Banks, M.R. Douglas and N. Seiberg, Probing F theory with branes, Phys. Lett. B 387 (1996) 278 [hep-th/9605199] [INSPIRE].
Y. Hamada and C. Vafa, 8d supergravity, reconstruction of internal geometry and the Swampland, JHEP 06 (2021) 178 [arXiv:2104.05724] [INSPIRE].
S. Cecotti, Supersymmetric field theories: geometric structures and dualities, Cambridge University Press (2015).
M. Alim et al., \( \mathcal{N} \) = 2 quantum field theories and their BPS quivers, Adv. Theor. Math. Phys. 18 (2014) 27 [arXiv:1112.3984] [INSPIRE].
S. Cecotti, M. Del Zotto, M. Martone and R. Moscrop, The Characteristic Dimension of Four-Dimensional \( \mathcal{N} \) = 2 SCFTs, Commun. Math. Phys. 400 (2023) 519 [arXiv:2108.10884] [INSPIRE].
S. Cecotti and M. Del Zotto, Galois covers of \( \mathcal{N} \) = 2 BPS spectra and quantum monodromy, Adv. Theor. Math. Phys. 20 (2016) 1227 [arXiv:1503.07485] [INSPIRE].
S. Cecotti, J. Song, C. Vafa and W. Yan, Superconformal Index, BPS Monodromy and Chiral Algebras, JHEP 11 (2017) 013 [arXiv:1511.01516] [INSPIRE].
S. Cecotti, Special Geometry and the Swampland, JHEP 09 (2020) 147 [arXiv:2004.06929] [INSPIRE].
P.C. Argyres and M.R. Douglas, New phenomena in SU(3) supersymmetric gauge theory, Nucl. Phys. B 448 (1995) 93 [hep-th/9505062] [INSPIRE].
P.C. Argyres, M.R. Plesser, N. Seiberg and E. Witten, New N = 2 superconformal field theories in four-dimensions, Nucl. Phys. B 461 (1996) 71 [hep-th/9511154] [INSPIRE].
T. Eguchi and K. Hori, N = 2 superconformal field theories in four-dimensions and A-D-E classification, in the proceedings of the Conference on the Mathematical Beauty of Physics (In Memory of C. Itzykson), Saclay, France, June 05–07 (1996) [hep-th/9607125] [INSPIRE].
J.A. Minahan and D. Nemeschansky, An N = 2 superconformal fixed point with E(6) global symmetry, Nucl. Phys. B 482 (1996) 142 [hep-th/9608047] [INSPIRE].
J.A. Minahan and D. Nemeschansky, Superconformal fixed points with E(n) global symmetry, Nucl. Phys. B 489 (1997) 24 [hep-th/9610076] [INSPIRE].
Acknowledgments
I thank Mario Martone for sharing with me his insights about the physics at the discriminant locus, and Michele Del Zotto for correcting many of my misconceptions (the residual ones are my fault). I thank Yuji Tachikawa for clarifications about frozen singularities in M-theory.
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Cecotti, S. Hwang-Oguiso invariants and frozen singularities in special geometry. J. High Energ. Phys. 2024, 12 (2024). https://doi.org/10.1007/JHEP04(2024)012
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DOI: https://doi.org/10.1007/JHEP04(2024)012