Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 14 (2018), 037, 27 pages      arXiv:1710.08453

Singular Geometry and Higgs Bundles in String Theory

Lara B. Anderson a, Mboyo Esole b, Laura Fredrickson c and Laura P. Schaposnik de
a) Department of Physics and Department of Mathematics, Virginia Tech, Blacksburg, VA 24061, USA
b) Department of Mathematics, Northeastern University, Boston, MA 02115, USA
c) Department of Mathematics, Stanford University, Stanford, CA 94305, USA
d) Department of Mathematics, University of Illinois at Chicago, 60607 Chicago, USA
e) Department of Mathematics, FU Berlin, 14195 Berlin, Germany

Received November 22, 2017, in final form April 13, 2018; Published online April 18, 2018

This brief survey aims to set the stage and summarize some of the ideas under discussion at the Workshop on Singular Geometry and Higgs Bundles in String Theory, to be held at the American Institute of Mathematics from October 30th to November 3rd, 2017. One of the most interesting aspects of the duality revolution in string theory is the understanding that gauge fields and matter representations can be described by intersection of branes. Since gauge theory is at the heart of our description of physical interactions, it has opened the door to the geometric engineering of many physical systems, and in particular those involving Higgs bundles. This note presents a curated overview of some current advances and open problems in the area, with no intention of being a complete review of the whole subject.

Key words: Higgs bundles; Hitchin fibration; mirror symmetry; F-theory; Calabi-Yau; singular curves; singularities.

pdf (615 kb)   tex (55 kb)


  1. Adams J., Strong real forms and the Kac classification, Atlas of Lie Groups and Representations, 2005, available at
  2. Aganagic M., Frenkel E., Okounkov A., Quantum $q$-Langlands correspondence, arXiv:1701.03146.
  3. Aganagic M., Okounkov A., Elliptic stable envelope, arXiv:1604.00423.
  4. Aharony O., Evtikhiev M., On four dimensional $N=3$ superconformal theories, J. High Energy Phys. 2016 (2016), no. 4, 040, 13 pages, arXiv:1512.03524.
  5. Aharony O., Tachikawa Y., S-folds and 4d ${\mathcal N}=3$ superconformal field theories, J. High Energy Phys. 2016 (2016), no. 6, 044, 26 pages, arXiv:1602.08638.
  6. Aker K., Szabó S., Algebraic Nahm transform for parabolic Higgs bundles on $\mathbb{P}^1$, Geom. Topol. 18 (2014), 2487-2545, math.AG/0610301.
  7. Anderson L.B., Gray J., Raghuram N., Taylor W., Matter in transition, J. High Energy Phys. 2016 (2016), no. 4, 080, 103 pages, arXiv:1512.05791.
  8. Anderson L.B., Heckman J.J., Katz S., T-branes and geometry, J. High Energy Phys. 2014 (2014), no. 5, 080, 68 pages, arXiv:1310.1931.
  9. Anderson L.B., Heckman J.J., Katz S., Schaposnik L.P., T-branes at the limits of geometry, J. High Energy Phys. 2017 (2017), no. 10, 058, 56 pages, arXiv:1702.06137.
  10. Anderson L.B., Taylor W., Geometric constraints in dual F-theory and heterotic string compactifications, J. High Energy Phys. 2014 (2014), no. 8, 025, 81 pages, arXiv:1405.2074.
  11. Arinkin D., Autoduality of compactified Jacobians for curves with plane singularities, J. Algebraic Geom. 22 (2013), 363-388, arXiv:1001.3868.
  12. Arras P., Grassi A., Weigand T., Terminal singularities, Milnor numbers, and matter in F-theory, J. Geom. Phys. 123 (2018), 71-97, arXiv:1612.05646.
  13. Arroyo M.A., The geometry of ${\rm SO}(p,q)$-Higgs bundles, Ph.D. Thesis, Universidad de Salamanca, Spain, 2009.
  14. Aspinwall P.S., Donagi R.Y., The heterotic string, the tangent bundle and derived categories, Adv. Theor. Math. Phys. 2 (1998), 1041-1074, hep-th/9806094.
  15. Aspinwall P.S., Gross M., The ${\rm SO}(32)$ heterotic string on a $K3$ surface, Phys. Lett. B 387 (1996), 735-742, hep-th/9605131.
  16. Baird T.J., Symmetric products of a real curve and the moduli space of Higgs bundles, arXiv:1611.09636.
  17. Balaji V., Barik P., Nagaraj D.S., A degeneration of moduli of Hitchin pairs, Int. Math. Res. Not. 2016 (2016), 6581-6625, arXiv:1308.4490.
  18. Baraglia D., Monodromy of the ${\rm SL}(n)$ and ${\rm GL}(n)$ Hitchin fibrations, arXiv:1612.01583.
  19. Baraglia D., Biswas I., Schaposnik L.P., Automorphisms of $\mathbb{C}^*$ moduli spaces associated to a Riemann surface, SIGMA 12 (2016), 007, 14 pages, arXiv:1508.06587.
  20. Baraglia D., Biswas I., Schaposnik L.P., On the Brauer group of Higgs bundles and connections, in Proceedings of Hitchin 70, Oxford University Press, to appear, arXiv:1609.00454.
  21. Baraglia D., Kamgarpour M., On the image of the parabolic Hitchin map, arXiv:1703.09886.
  22. Baraglia D., Schaposnik L.P., Higgs bundles and $(A,B,A)$-branes, Comm. Math. Phys. 331 (2014), 1271-1300, arXiv:1305.4638.
  23. Baraglia D., Schaposnik L.P., Real structures on moduli spaces of Higgs bundles, Adv. Theor. Math. Phys. 20 (2016), 525-551, arXiv:1309.1195.
  24. Baraglia D., Schaposnik L.P., Monodromy of rank $2$ twisted Hitchin systems and real character varieties, Trans. Amer. Math. Soc., to appear, arXiv:1506.00372.
  25. Baraglia D., Schaposnik L.P., Cayley and Langlands type correspondences for orthogonal Higgs bundles, Trans. Amer. Math. Soc., to appear, arXiv:1708.08828.
  26. Beauville A., Narasimhan M.S., Ramanan S., Spectral curves and the generalised theta divisor, J. Reine Angew. Math. 398 (1989), 169-179.
  27. Beck F., Hitchin and Calabi-Yau integrable systems via variations of Hodge structures, arXiv:1707.05973.
  28. Bena I., Blåbäck J., Savelli R., T-branes and matrix models, J. High Energy Phys. 2017 (2017), no. 6, 009, 15 pages, arXiv:1703.06106.
  29. Bershadsky M., Intriligator K., Kachru S., Morrison D.R., Sadov V., Vafa C., Geometric singularities and enhanced gauge symmetries, Nuclear Phys. B 481 (1996), 215-252, hep-th/9605200.
  30. Bhardwaj L., Classification of 6d ${\mathcal N}=(1,0)$ gauge theories, J. High Energy Phys. 2015 (2015), no. 11, 002, 26 pages, arXiv:1502.06594.
  31. Bhosle U.N., Generalised parabolic bundles and applications to torsionfree sheaves on nodal curves, Ark. Mat. 30 (1992), 187-215.
  32. Bhosle U.N., Generalized parabolic sheaves on an integral projective curve, Proc. Indian Acad. Sci. Math. Sci. 102 (1992), 13-22.
  33. Bhosle U.N., Generalized parabolic bundles and applications. II, Proc. Indian Acad. Sci. Math. Sci. 106 (1996), 403-420.
  34. Bhosle U.N., Generalized parabolic Hitchin pairs, J. Lond. Math. Soc. 89 (2014), 1-23.
  35. Bies M., Mayrhofer C., Pehle C., Weigand T., Chow groups, Deligne cohomology and massless matter in F-theory, arXiv:1402.5144.
  36. Biquard O., Boalch P., Wild non-abelian Hodge theory on curves, Compos. Math. 140 (2004), 179-204, math.DG/0111098.
  37. Biswas I., Calvo L.A., Franco E., García-Prada O., Involutions of the moduli spaces of $G$-Higgs bundles over elliptic curves, arXiv:1612.08364.
  38. Biswas I., Wilkin G., Anti-holomorphic involutive isometry of hyper-Kähler manifolds and branes, J. Geom. Phys. 88 (2015), 52-55, arXiv:1410.6616.
  39. Boalch P., Hyperkähler manifolds and nonabelian Hodge theory of (irregular) curves, arXiv:1203.6607.
  40. Boalch P., Poisson varieties from Riemann surfaces, Indag. Math. (N.S.) 25 (2014), 872-900, arXiv:1309.7202.
  41. Boden H.U., Yokogawa K., Moduli spaces of parabolic Higgs bundles and parabolic $K(D)$ pairs over smooth curves. I, Internat. J. Math. 7 (1996), 573-598, alg-geom/9610014.
  42. Bradlow S.B., Schaposnik L.P., Higgs bundles and exceptional isogenies, Res. Math. Sci. 3 (2016), 14, 28 pages, arXiv:1508.02650.
  43. Branco L.C., Higgs bundles, Lagrangians and mirror symmetry, Ph.D. Thesis, University of Oxford, 2017, arXiv:1803.01611.
  44. Braun A.P., Schäfer-Nameki S., Box graphs and resolutions I, Nuclear Phys. B 905 (2016), 447-479, arXiv:1407.3520.
  45. Casimiro A.C., Ferreira S., Florentino C., Principal Schottky bundles over Riemann surfaces, arXiv:1612.08662.
  46. Cattaneo A., Crepant resolutions of Weierstrass threefolds and non-Kodaira fibres, arXiv:1307.7997.
  47. Cecotti S., Córdova C., Heckman J.J., Vafa C., T-branes and monodromy, J. High Energy Phys. 2011 (2011), no. 7, 030, 93 pages, arXiv:1010.5780.
  48. Collier B., Finite order automorphisms of Higgs bundles: theory and application, Ph.D. Thesis, University of Illinois at Urbana-Champaign, 2016.
  49. Collier B., ${\rm SO}(n,n+1)$-surface group representations and their Higgs bundles, arXiv:1710.01287.
  50. Collinucci A., Giacomelli S., Savelli R., Valandro R., T-branes through 3d mirror symmetry, J. High Energy Phys. 2016 (2016), no. 7, 093, 41 pages, arXiv:1603.00062.
  51. Collinucci A., Giacomelli S., Valandro R., T-branes, monopoles and S-duality, J. High Energy Phys. 2017 (2017), no. 10, 113, 62 pages, arXiv:1703.09238.
  52. Collinucci A., Savelli R., F-theory on singular spaces, J. High Energy Phys. 2015 (2015), no. 9, 100, 39 pages, arXiv:1410.4867.
  53. Del Zotto M., Heckman J.J., Morrison D.R., 6D SCFTs and phases of 5D theories, J. High Energy Phys. 2017 (2017), no. 9, 147, 38 pages, arXiv:1703.02981.
  54. Del Zotto M., Heckman J.J., Tomasiello A., Vafa C., 6d conformal matter, J. High Energy Phys. 2015 (2015), no. 2, 054, 56 pages, arXiv:1407.6359.
  55. Deligne P., Courbes elliptiques: formulaire d'après J. Tate, in Modular Functions of One Variable, IV (Proc. Internat. Summer School, University Antwerp, Antwerp, 1972), Lecture Notes in Math., Vol. 476, Springer, Berlin, 1975, 53-73.
  56. Diaconescu D.-E., Donagi R., Pantev T., Intermediate Jacobians and ADE Hitchin systems, hep-th/0607159.
  57. Diaconescu D.-E., Entin R., Calabi-Yau spaces and five-dimensional field theories and exceptional gauge symmetry, Nuclear Phys. B 538 (1999), 451-484, hep-th/9807170.
  58. Dolgačev I.V., On the purity of the degeneration loci of families of curves, Invent. Math. 8 (1969), 34-54.
  59. Donagi R., Decomposition of spectral covers, Astérisque 218 (1993), 145-175.
  60. Donagi R., Spectral covers, in Current Topics in Complex Algebraic Geometry (Berkeley, CA, 1992/93), Math. Sci. Res. Inst. Publ., Vol. 28, Cambridge University Press, Cambridge, 1995, 65-86, alg-geom/9505009.
  61. Donagi R., Gaitsgory D., The gerbe of Higgs bundles, Transform. Groups 7 (2002), 109-153, math.AG/0005132.
  62. Donagi R., Katz S., Sharpe E., Spectra of D-branes with Higgs vevs, Adv. Theor. Math. Phys. 8 (2004), 815-860, hep-th/0309270.
  63. Donagi R., Markman E., Cubics, integrable systems, and Calabi-Yau threefolds, in Proceedings of the Hirzebruch 65 Conference on Algebraic Geometry (Ramat Gan, 1993), Israel Math. Conf. Proc., Vol. 9, Bar-Ilan University, Ramat Gan, 1996, 199-221, alg-geom/9408004.
  64. Donagi R., Markman E., Spectral covers, algebraically completely integrable, Hamiltonian systems, and moduli of bundles, in Integrable Systems and Quantum Groups (Montecatini Terme, 1993), Lecture Notes in Math., Vol. 1620, Springer, Berlin, 1996, 1-119, alg-geom/9507017.
  65. Donagi R., Pantev T., Langlands duality for Hitchin systems, Invent. Math. 189 (2012), 653-735, arXiv:1311.3624.
  66. Donagi R., Wijnholt M., Gluing branes - I, J. High Energy Phys. 2013 (2013), no. 5, 068, 46 pages, arXiv:1104.2610.
  67. Donagi R., Wijnholt M., Gluing branes - II: flavour physics and string duality, J. High Energy Phys. 2013 (2013), no. 5, 092, 50 pages, arXiv:1112.4854.
  68. Dumas D., Neitzke A., Asymptotics of Hitchin's metric on the Hitchin section, arXiv:1802.07200.
  69. Dumitrescu O., Mulase M., Quantum curves for Hitchin fibrations and the Eynard-Orantin theory, Lett. Math. Phys. 104 (2014), 635-671, arXiv:1310.6022.
  70. Esole M., Fullwood J., Yau S.-T., $D_5$ elliptic fibrations: non-Kodaira fibers and new orientifold limits of F-theory, Commun. Number Theory Phys. 9 (2015), 583-642, arXiv:1110.6177.
  71. Esole M., Jackson S., Jagadeesan R., Noël A.G., Incidence geometry in a Weyl chamber I: ${\rm GL}_n$, arXiv:1508.03038.
  72. Esole M., Jackson S., Jagadeesan R., Noël A.G., Incidence geometry in a Weyl chamber II: ${\rm SL}_n$, arXiv:1601.05070.
  73. Esole M., Jagadeesan R., Kang M.J., The geometry of $G_2$, ${\rm Spin}(7)$, and ${\rm Spin}(8)$-models, arXiv:1709.04913.
  74. Esole M., Jefferson P., Kang M.J., Euler characteristics of crepant resolutions of Weierstrass models, arXiv:1703.00905.
  75. Esole M., Jefferson P., Kang M.J., The geometry of F$_4$-models, arXiv:1704.08251.
  76. Esole M., Kang M.J., Yau S.-T., A new model for elliptic fibrations with a rank one Mordell-Weil group I. Singular fibers and semi-stable degenerations, arXiv:1410.0003.
  77. Esole M., Shao S.-H., Yau S.-T., Singularities and gauge theory phases, Adv. Theor. Math. Phys. 19 (2015), 1183-1247, arXiv:1402.6331.
  78. Esole M., Shao S.-H., Yau S.-T., Singularities and gauge theory phases II, Adv. Theor. Math. Phys. 20 (2016), 683-749, arXiv:1407.1867.
  79. Esole M., Yau S.-T., Small resolutions of $\rm SU(5)$-models in F-theory, Adv. Theor. Math. Phys. 17 (2013), 1195-1253, arXiv:1107.0733.
  80. Faltings G., Stable $G$-bundles and projective connections, J. Algebraic Geom. 2 (1993), 507-568.
  81. Franco E., Jardim M., Mirror symmetry for Nahm branes, arXiv:1709.01314.
  82. Franco E., Jardim M., Marchesi S., Branes in the moduli space of framed instantons, arXiv:1504.05883.
  83. Franco E., Jardim M., Menet G., Brane involutions on irreducible holomorphic symplectic manifolds, arXiv:1606.09040.
  84. Franco E., Peón-Nieto A., The Borel subgroup and branes on the Higgs moduli space, arXiv:1709.03549.
  85. Fredrickson L., Generic ends of the moduli space of ${\rm SL}(n,\mathbb{C})$-Higgs bundles, in preparation.
  86. Fredrickson L., Neitzke A., From $S^1$-fixed points to ${\mathcal W}$-algebra representations, arXiv:1709.06142.
  87. Freed D.S., Special Kähler manifolds, Comm. Math. Phys. 203 (1999), 31-52, hep-th/9712042.
  88. Frenkel E., Ramifications of the geometric Langlands program, in Representation Theory and Complex Analysis, Lecture Notes in Math., Vol. 1931, Springer, Berlin, 2008, 51-135, math.QA/0611294.
  89. Frenkel E., Witten E., Geometric endoscopy and mirror symmetry, Commun. Number Theory Phys. 2 (2008), 113-283, arXiv:0710.5939.
  90. Gaiotto D., S-duality of boundary conditions and the geometric Langlands, arXiv:1609.09030.
  91. Gaiotto D., Moore G.W., Neitzke A., Four-dimensional wall-crossing via three-dimensional field theory, Comm. Math. Phys. 299 (2010), 163-224, arXiv:0807.4723.
  92. García-Etxebarria I., Regalado D., $\mathcal{N}=3$ four dimensional field theories, J. High Energy Phys. 2016 (2016), no. 3, 083, 21 pages, arXiv:1512.06434.
  93. García-Etxebarria I., Regalado D., Exceptional ${\mathcal N}=3$ theories, J. High Energy Phys. 2017 (2017), no. 12, 042, 21 pages, arXiv:1611.05769.
  94. García-Prada O., Wilkin G., Action of the mapping class group on character varieties and Higgs bundles, arXiv:1612.02508.
  95. Giudice A.L., Pustetto A., A compactification of the moduli space of principal Higgs bundles over singular curves, arXiv:1110.0632.
  96. Gothen P.B., Oliveira A.G., The singular fiber of the Hitchin map, Int. Math. Res. Not. 2013 (2013), 1079-1121, arXiv:1012.5541.
  97. Grimm T.W., Hayashi H., F-theory fluxes, chirality and Chern-Simons theories, J. High Energy Phys. 2012 (2012), no. 3, 027, 54 pages, arXiv:1111.1232.
  98. Gu J., Huang M.-X., Kashani-Poor A.-K., Klemm A., Refined BPS invariants of 6d SCFTs from anomalies and modularity, J. High Energy Phys. 2017 (2017), no. 5, 130, 62 pages, arXiv:1701.00764.
  99. Guichard O., Wienhard A., Positivity and higher Teichmüller theory, arXiv:1802.02833.
  100. Gukov S., Quantization via mirror symmetry, Jpn. J. Math. 6 (2011), 65-119, arXiv:1011.2218.
  101. Gukov S., Witten E., Gauge theory, ramification, and the geometric Langlands program, in Current Developments in Mathematics, 2006, Int. Press, Somerville, MA, 2008, 35-180, hep-th/0612073.
  102. Haghighat B., Klemm A., Lockhart G., Vafa C., Strings of minimal 6d SCFTs, Fortschr. Phys. 63 (2015), 294-322, arXiv:1412.3152.
  103. Halverson J., Taylor W., ${\mathbb P}^1$-bundle bases and the prevalence of non-Higgsable structure in 4D F-theory models, J. High Energy Phys. 2015 (2015), no. 9, 086, 59 pages, arXiv:1506.03204.
  104. Hausel T., Compactification of moduli of Higgs bundles, J. Reine Angew. Math. 503 (1998), 169-192, math.AG/9804083.
  105. Hausel T., Geometry of the moduli space of Higgs bundles, math.AG/0107040.
  106. Hausel T., Thaddeus M., Mirror symmetry, Langlands duality, and the Hitchin system, Invent. Math. 153 (2003), 197-229, math.AG/0205236.
  107. Hayashi H., Lawrie C., Morrison D.R., Schäfer-Nameki S., Box graphs and singular fibers, J. High Energy Phys. 2014 (2014), no. 5, 048, 92 pages, arXiv:1402.2653.
  108. Hayashi H., Lawrie C., Schäfer-Nameki S., Phases, flops and F-theory: ${\rm SU}(5)$ gauge theories, J. High Energy Phys. 2013 (2013), no. 10, 046, 43 pages, arXiv:1304.1678.
  109. Heckman J.J., Morrison D.R., Rudelius T., Vafa C., Atomic classification of 6D SCFTs, Fortschr. Phys. 63 (2015), 468-530, arXiv:1502.05405.
  110. Heckman J.J., Morrison D.R., Vafa C., On the classification of 6D SCFTs and generalized ADE orbifolds, J. High Energy Phys. 2014 (2014), no. 5, 028, 49 pages, arXiv:1312.5746.
  111. Heller S., Schaposnik L.P., Branes through finite group actions, J. Geom. Phys. 129 (2018), 279-293, arXiv:1611.00391.
  112. Hitchin N., The self-duality equations on a Riemann surface, Proc. London Math. Soc. 55 (1987), 59-126.
  113. Hitchin N., Stable bundles and integrable systems, Duke Math. J. 54 (1987), 91-114.
  114. Hitchin N., Higgs bundles and characteristic classes, in Arbeitstagung Bonn 2013, Progr. Math., Vol. 319, Birkhäuser/Springer, Cham, 2016, 247-264, arXiv:1308.4603.
  115. Hitchin N., Spinors, Lagrangians and rank 2 Higgs bundles, Proc. London Math. Soc. 115 (2017), 33-54, arXiv:1605.06385.
  116. Hitchin N., Schaposnik L.P., Nonabelianization of Higgs bundles, J. Differential Geom. 97 (2014), 79-89, arXiv:1307.0960.
  117. Hoskins V., Schaffhauser F., Group actions on quiver varieties and applications, arXiv:1612.06593.
  118. Hoskins V., Schaffhauser F., Rational points of quiver moduli spaces, arXiv:1704.08624.
  119. Igusa J.-I., Fibre systems of Jacobian varieties, Amer. J. Math. 78 (1956), 171-199.
  120. Intriligator K., Morrison D.R., Seiberg N., Five-dimensional supersymmetric gauge theories and degenerations of Calabi-Yau spaces, Nuclear Phys. B 497 (1997), 56-100, hep-th/9702198.
  121. Kapustin A., Witten E., Electric-magnetic duality and the geometric Langlands program, Commun. Number Theory Phys. 1 (2007), 1-236, hep-th/0604151.
  122. Kausz I., A Gieseker type degeneration of moduli stacks of vector bundles on curves, Trans. Amer. Math. Soc. 357 (2005), 4897-4955, math.AG/0201197.
  123. Kimura T., Pestun V., Quiver elliptic W-algebras, arXiv:1608.04651.
  124. Klevers D., Taylor W., Three-index symmetric matter representations of ${\rm SU}(2)$ in F-theory from non-Tate form Weierstrass models, J. High Energy Phys. 2016 (2016), no. 6, 171, 32 pages, arXiv:1604.01030.
  125. Kodaira K., On compact analytic surfaces. II, Ann. of Math. 77 (1963), 563-626.
  126. Kodaira K., On compact analytic surfaces. III, Ann. of Math. 78 (1963), 1-40.
  127. Konno H., Construction of the moduli space of stable parabolic Higgs bundles on a Riemann surface, J. Math. Soc. Japan 45 (1993), 253-276.
  128. Kontsevich M., Soibelman Y., Stability structures, motivic Donaldson-Thomas invariants and cluster transformations, arXiv:0811.2435.
  129. Krause S., Mayrhofer C., Weigand T., $G_4$-flux, chiral matter and singularity resolution in F-theory compactifications, Nuclear Phys. B 858 (2012), 1-47, arXiv:1109.3454.
  130. Laumon G., Un analogue global du cône nilpotent, Duke Math. J. 57 (1988), 647-671.
  131. Lawrie C., Schäfer-Nameki S., The Tate form on steroids: resolution and higher codimension fibers, J. High Energy Phys. 2013 (2013), no. 4, 061, 66 pages, arXiv:1212.2949.
  132. Marchesano F., Savelli R., Schwieger S., Compact T-branes, J. High Energy Phys. 2017 (2017), no. 9, 132, 28 pages, arXiv:1707.03797.
  133. Marsano J., Schäfer-Nameki S., Yukawas, $G$-flux, and spectral covers from resolved Calabi-Yau's, J. High Energy Phys. 2011 (2011), no. 11, 098, 59 pages, arXiv:1108.1794.
  134. Matsuki K., Weyl groups and birational transformations among minimal models, Mem. Amer. Math. Soc. 116 (1995), vi+133 pages.
  135. Mayrhofer C., Morrison D.R., Till O., Weigand T., Mordell-Weil torsion and the global structure of gauge groups in F-theory, J. High Energy Phys. 2014 (2014), no. 10, 016, 47 pages, arXiv:1405.3656.
  136. Mazzeo R., Swoboda J., Weiss H., Witt F., Limiting configurations for solutions of Hitchin's equation, Semin. Theor. Spectr. Geom. 31 (2012-2014), 91-116, arXiv:1502.01692.
  137. Mazzeo R., Swoboda J., Weiss H., Witt F., Ends of the moduli space of Higgs bundles, Duke Math. J. 165 (2016), 2227-2271, arXiv:1405.5765.
  138. Mazzeo R., Swoboda J., Weiss H., Witt F., Asymptotic geometry of the Hitchin metric, arXiv:1709.03433.
  139. Melo M., Rapagnetta A., Vivani F., Fourier-Mukai and autoduality for compactified Jacobians, I, arXiv:1207.7233.
  140. Mestrano N., Simpson C., Moduli of sheaves, in Development of Moduli Theory - Kyoto 2013, Adv. Stud. Pure Math., Vol. 69, Math. Soc. Japan, Tokyo, 2016, 77-172.
  141. Miranda R., Smooth models for elliptic threefolds, in The Birational Geometry of Degenerations (Cambridge, Mass., 1981), Progr. Math., Vol. 29, Birkhäuser, Boston, Mass., 1983, 85-133.
  142. Mochizuki T., Asymptotic behavior of certain families of harmonic bundles on Riemann surfaces, J. Topology 9 (2016), 1021-1073, arXiv:1508.05997.
  143. Morrison D.R., Taylor W., Classifying bases for 6D F-theory models, Cent. Eur. J. Phys. 10 (2012), 1072-1088, arXiv:1201.1943.
  144. Morrison D.R., Taylor W., Matter and singularities, J. High Energy Phys. 2012 (2012), no. 1, 022, 54 pages, arXiv:1106.3563.
  145. Morrison D.R., Taylor W., Toric bases for 6D F-theory models, Fortschr. Phys. 60 (2012), 1187-1216, arXiv:1204.0283.
  146. Morrison D.R., Taylor W., Non-Higgsable clusters for 4D F-theory models, J. High Energy Phys. 2015 (2015), no. 5, 080, 37 pages, arXiv:1412.6112.
  147. Morrison D.R., Vafa C., Compactifications of $F$-theory on Calabi-Yau threefolds. II, Nuclear Phys. B 476 (1996), 437-469, hep-th/9603161.
  148. Nadler D., Perverse sheaves on real loop Grassmannians, Invent. Math. 159 (2005), 1-73, math.AG/0202150.
  149. Nahm W., Supersymmetries and their representations, Nuclear Phys. B 135 (1978), 149-166.
  150. Neitzke A., Notes on a new construction of hyperkähler metrics, arXiv:1308.2198.
  151. Néron A., Modèles minimaux des variétés abéliennes sur les corps locaux et globaux, Inst. Hautes Études Sci. Publ. Math. 21 (1964), 5-125.
  152. Pandharipande R., A compactification over $\overline {M}_g$ of the universal moduli space of slope-semistable vector bundles, J. Amer. Math. Soc. 9 (1996), 425-471, alg-geom/9502020.
  153. Peón-Nieto A., Higgs bundles, real forms and the Hitchin fibration, Ph.D. Thesis, Universidad Autónoma de Madrid, 2013.
  154. Peón-Nieto A., Cameral data for ${\rm SU}(p,p+1)$-Higgs bundles, arXiv:1506.01318.
  155. Radko O., A classification of topologically stable Poisson structures on a compact oriented surface, J. Symplectic Geom. 1 (2002), 523-542, math.SG/0110304.
  156. Schaposnik L.P., Monodromy of the ${\rm SL}_2$ Hitchin fibration, Internat. J. Math. 24 (2013), 1350013, 21 pages, arXiv:1111.2550.
  157. Schaposnik L.P., Spectral data for G-Higgs bundles, Ph.D. Thesis, University of Oxford, 2013, arXiv:1301.1981.
  158. Schaposnik L.P., Spectral data for $U(m,m)$-Higgs bundles, Int. Math. Res. Not. 2015 (2015), 3486-3498, arXiv:1307.4419.
  159. Schaposnik L.P., Higgs bundles and applications, Oberwolfach Reports, 2015, 31-33, arXiv:1603.06691.
  160. Schaposnik L.P., An introduction to spectral data for Higgs bundles, Lecture Note Series, Vol. 65, Institute for Mathematical Sciences, National University of Singapore, 2017, 38 pages, arXiv:1408.0333.
  161. Schaposnik L.P., A geometric approach to orthogonal Higgs bundles, Eur. J. Math., to appear, arXiv:1608.00300.
  162. Schaub D., Courbes spectrales et compactifications de jacobiennes, Math. Z. 227 (1998), 295-312.
  163. Schmitt A.H.W., Singular principal $G$-bundles on nodal curves, J. Eur. Math. Soc. 7 (2005), 215-251.
  164. Scognamillo R., An elementary approach to the abelianization of the Hitchin system for arbitrary reductive groups, Compositio Math. 110 (1998), 17-37, alg-geom/9412020.
  165. Seiberg N., Witten E., Comments on string dynamics in six dimensions, Nuclear Phys. B 471 (1996), 121-134, hep-th/9603003.
  166. Seshadri C., Degenerations of the moduli spaces of vector bundles on curves, Lecture given at the School on Algebraic Geometry, 1999.
  167. Seshadri C.S., Moduli spaces of torsion free sheaves on nodal curves and generalisations. I, in Moduli Spaces and Vector Bundles, London Math. Soc. Lecture Note Ser., Vol. 359, Cambridge University Press, Cambridge, 2009, 484-505.
  168. Simpson C.T., Constructing variations of Hodge structure using Yang-Mills theory and applications to uniformization, J. Amer. Math. Soc. 1 (1988), 867-918.
  169. Simpson C.T., Higgs bundles and local systems, Inst. Hautes Études Sci. Publ. Math. (1992), 5-95.
  170. Simpson C.T., Moduli of representations of the fundamental group of a smooth projective variety. I, Inst. Hautes Études Sci. Publ. Math. 79 (1994), 47-129.
  171. Simpson C.T., Moduli of representations of the fundamental group of a smooth projective variety. II, Inst. Hautes Études Sci. Publ. Math. 80 (1994), 5-79.
  172. Strominger A., Yau S.-T., Zaslow E., Mirror symmetry is $T$-duality, Nuclear Phys. B 479 (1996), 243-259, hep-th/9606040.
  173. Szydlo M.G., Flat regular models of elliptic schemes, Ph.D. Thesis, Harvard University, 1999.
  174. Tachikawa Y., ${\mathcal N}=2$ supersymmetric dynamics for pedestrians, Lecture Notes in Physics, Vol. 890, Springer, Cham, Hindustan Book Agency, New Delhi, 2015, arXiv:1312.2684.
  175. Tatar R., Walters W., GUT theories from Calabi-Yau 4-folds with ${\rm SO}(10)$ singularities, J. High Energy Phys. 2012 (2012), no. 12, 092, 24 pages, arXiv:1206.5090.
  176. Tate J., Algorithm for determining the type of a singular fiber in an elliptic pencil, in Modular Functions of One Variable, IV (Proc. Internat. Summer School, University Antwerp, Antwerp, 1972), Lecture Notes in Math., Vol. 476, Springer, Berlin, 1975, 33-52.
  177. Vafa C., Evidence for $F$-theory, Nuclear Phys. B 469 (1996), 403-415, hep-th/9602022.
  178. Witten E., Gauge theory and wild ramification, arXiv:0710.0631.
  179. Witten E., Conformal field theory in four and six dimensions, in Topology, Geometry and Quantum Field Theory, London Math. Soc. Lecture Note Ser., Vol. 308, Cambridge University Press, Cambridge, 2004, 405-419, arXiv:0712.0157.
  180. Xie D., General Argyres-Douglas theory, J. High Energy Phys. 2013 (2013), no. 1, 100, 52 pages, arXiv:1204.2270.

Previous article  Next article   Contents of Volume 14 (2018)