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Mirror Symmetry on Kummer Type K3 Surfaces

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We investigate both geometric and conformal field theoretic aspects of mirror symmetry on N=(4,4) superconformal field theories with central charge c=6. Our approach enables us to determine the action of mirror symmetry on (non-stable) singular fibers in elliptic fibrations of ℤ N orbifold limits of K3. The resulting map gives an automorphism of order 4,8, or 12, respectively, on the smooth universal covering space of the moduli space. We explicitly derive the geometric counterparts of the twist fields in our orbifold conformal field theories. The classical McKay correspondence allows for a natural interpretation of our results.

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References

  1. Andreas, B., Curio, G., Ruiperez, D.H., Yau, S.-T.: Fourier-Mukai transform and mirror symmetry for D-branes on elliptic Calabi-Yau. math.ag/0012196

  2. Andreas, B., Curio, G., Ruiperez, D.H., Yau, S.-T.: Fibrewise T-duality for D-branes on elliptic Calabi-Yau. JHEP 0103:020 (2001), hep-th/0101129

  3. Aspinwall, P.S., Greene, B.R., Morrison, D.R.: Calabi–Yau moduli space, mirror manifolds and spacetime topology change in string theory. Nucl. Phys. B416, 414–480 (1994), hep-th/9309097

    Google Scholar 

  4. Aspinwall, P.S., Greene, B.R., Morrison, D.R.: Measuring small distances in N=2 sigma models. Nucl. Phys. B420, 184–242 (1994), hep-th/9311042

    Google Scholar 

  5. Aspinwall, P.S., Morrison, D.R.: String theory on K3 surfaces. In: Greene, B., Yau, S.-T. (eds.), Mirror symmetry, Vol. II, Cambridge, MA: International Press, 1994, pp. 703–716, hep-th/9404151

  6. Aspinwall, P.S.: Enhanced gauge symmetries and K3 surfaces. Phys. Lett. B357, 329–334 (1995), hep-th/9507012

  7. Bartocci, C., Bruzzo, U., Ruiperez, D.H., Munoz Porras, J.M.: Mirror symmetry on elliptic K3 surfaces via Fourier-Mukai transform. Commun. Math. Phys. 195, 79–93 (1998), alg-geom/9704023

    Article  MathSciNet  MATH  Google Scholar 

  8. Batyrev, V.: Dual polyhedra and mirror symmetry for Calabi–Yau hypersurfaces in toric varieties. J. Algebr. Geom. 3, 493–535 (1994), alg-geom/9310003

    MathSciNet  MATH  Google Scholar 

  9. Batyrev, V.: Birational Calabi-Yau n-folds have equal Betti numbers. In: New trends in algebraic geometry (Warwick, 1996), London Math. Soc. Lecture Note Ser. 264, 1996. Cambridge, UK: Cambridge Univ. Press, 1999, pp. 1–11, alg-geom/9710020

    Google Scholar 

  10. Bergman, O., Gimon, E., Kol, B.: Strings on orbifold lines. JHEP 0105:19 (2001), hep-th/0102095

  11. Braam, P.J., van Baal, P.: Nahm’s transformation for instantons. Commun. Math. Phys. 122, 267–280 (1989)

    MathSciNet  MATH  Google Scholar 

  12. Bridgeland, T., King, A., Reid, M.: Mukai implies McKay: The McKay correspondence as an equivalence of derived categories. J. Am. Math. Soc. 14(3), 535–554 (2001), math.AG/9908027

    Article  MATH  Google Scholar 

  13. Brunner, I., Entin, R., Römelsberger, Ch.: D-branes on T 4/ℤ2 and T-Duality. JHEP 9906:16, (1999), hep-th/9905078

  14. Bruzzo, U., Sanguinetti, G.: Mirror symmetry on K3 surfaces as a hyper-Kaehler rotation. Lett. Math. Phys. 45, 295–301 (1998), physics/9802044

    Article  MathSciNet  MATH  Google Scholar 

  15. Candelas, P., De~La Ossa, X.C., Green, P.S., Parkes, L.: A pair of Calabi-Yau manifolds as an exactly soluble superconformal theory. Nucl. Phys. B359, 21–74 (1991)

  16. Cecotti, S.: N=2 supergravity, type IIB superstrings and algebraic geometry. Commun. Math. Phys. 131, 517–536 (1990)

    MathSciNet  MATH  Google Scholar 

  17. Chen, W., Ruan, Y.: Orbifold quantum cohomology. math.AG/0005198

  18. Cox, D.A., Katz, S.: Mirror symmetry and algebraic geometry. Providence, RI: Am. Math. Soc., 1999

  19. de~Boer, J., Dijkgraaf, R., Hori, K., Keurentjes, A., Morgan, J., Morrison, D.R., Sethi, S.: Triples, fluxes, and strings. Adv. Theor. Math. Phys. 4(5), 995–1186 (2000), hep-th/0103170

    Google Scholar 

  20. Denef, J., Loeser, F.: Germs of arcs on singular algebraic varieties and motivic integration. Invent. Math. 135(1), 201–232 (1999), math.AG/9803039

    Article  MATH  Google Scholar 

  21. Diaconescu, D.-E., Gomis, J.: Fractional branes and boundary states in orbifold theories. JHEP 0110:001 (2000), hep-th/9906242

  22. Dijkgraaf, R.: Instanton strings and hyperkaehler geometry. Nucl. Phys. B543, 545–571 (1999), hep-th/9810210

  23. Distler, J., Greene, B.: Some exact results on the superpotential from Calabi-Yau compactifications. Nucl. Phys. B309, 295–316 (1988)

    Google Scholar 

  24. Dixon, L.J.: Some world-sheet properties of superstring compactifications, on orbifolds and otherwise. Lectures given at the 1987 ICTP Summer Workshop in High Energy Physics and Cosmology, Trieste, June 29 - August 7

  25. Dixon, L.J., Friedan, D., Martinec, E., Shenker, S.: The conformal field theory of orbifolds. Nucl. Phys. B282, 13–73 (1987)

    Google Scholar 

  26. Eguchi, T., Taormina, A.: Extended superconformal algebras and string compactifications. Trieste School 1988: Superstrings, pp. 167–188

  27. Fantechi, B., Göttsche, L.: Orbifold cohomology for global quotients. math.AG/ 0104207

  28. Garcia-Compean, H.: D-branes in orbifold singularities and equivariant K- theory. Nucl. Phys. B557, 480–504 (1999), hep-th/9812226

  29. Gomez, C., Hernandez, R., Lopez, E.: S-Duality and the Calabi-Yau interpretation of the N = 4 to N = 2 flow. Phys. Lett. B386, 115–122 (1996), hep-th/9512017

    Google Scholar 

  30. Gonzalez-Sprinberg, G., Verdier, J.-L.: Construction géométrique de la correspondance de McKay. Ann. Sci. École Norm. Sup. 16(3), 409–449 (1983)

    MATH  Google Scholar 

  31. Greene, B.R., Plesser, M.R.: Duality in Calabi-Yau moduli space. Nucl. Phys. B338, 15–37 (1990)

    Google Scholar 

  32. Gross, M., Wilson, P.M.H.: Mirror symmetry via 3-tori for a class of Calabi-Yau threefolds. Math. Ann. 309(3), 505–531 (1997), alg-geom/9608004

    Article  MATH  Google Scholar 

  33. Gross, M., Wilson, P.M.H.: Large Complex Structure Limits of K3 Surfaces. J. Diff. Geom. 55(3), 475–546 (2000), math.DG/0008018

    MATH  Google Scholar 

  34. Hamidi, S., Vafa, C.: Interactions on orbifolds. Nucl. Phys. B279, 465–513 (1987)

    Google Scholar 

  35. Harvey, R., Lawson Jr., H.B.: Calibrated geometries. Acta Math. 148, 47–157 (1982)

    MathSciNet  MATH  Google Scholar 

  36. Hirzebruch, F.: The topology of normal singularities of an algebraic surface (after D. Mumford). In: Séminaire Bourbaki, vol. 8. Paris, France: Soc. Math. 1995, Exp. No. 250, pp. 129–137

  37. Huybrechts, D.: The Kaehler cone of a compact hyperkaehler manifold. math.AG/9909109

  38. Katz, S., Morrison, D.R., Plesser, M.R.: Enhanced Gauge Symmetry in Type II String Theory. Nucl. Phys. B477, 105–140 (1996), hep-th/9601108

    Google Scholar 

  39. Kiritsis, E., Obers, N., Pioline, B.: Heterotic/type II triality and instantons on K(3). JHEP 0001:019 (2000), hep-th/0001083

  40. Knörrer, H.: Group representations and the resolution of rational double points. In: Finite groups – coming of age (Montreal, Que., 1982). Providence, R.I.: Am. Math. Soc., 1985, pp. 175–222

  41. Kodaira, K.: On compact analytic surfaces: II. Ann. Math. 77, 563–626 (1963)

    MATH  Google Scholar 

  42. Kontsevich, M.: Homological algebra of mirror symmetry. In: Proceedings of the International Congress of Mathematicians (Zürich, 1994), Vol. 1,2, Basel: Birkhäuser, 1995, pp. 120–139, alg-geom/9411018

  43. Kontsevich, M.: Lecture at Orsay, December 7, 1995

  44. Kontsevich, M., Soibelman, Y.: Homological mirror symmetry and torus fibrations. math.SG/0011041

  45. Kutasov, D.: Geometry on the space of conformal field theories and contact terms. Phys. Lett. B220, 153–164, (1989)

    Google Scholar 

  46. Lepowsky, J.: Calculus of twisted vertex operators. Proc. Nat. Acad. Sci. U.S.A. 82(24), 8295–8299 (1985)

    MATH  Google Scholar 

  47. Lerche, W., Vafa, C., Warner, N.P.: Chiral rings in N=2 superconformal theories. Nucl. Phys. B324, 427–474 (1989)

    Google Scholar 

  48. McKay, J.: Graphs, singularities, and finite groups. In: The Santa Cruz Conference on Finite Groups (Univ. California, Santa Cruz, Calif., 1979). Providence, R.I.: Am. Math. Soc., 1980, pp. 183–186

  49. McKay, J.: Cartan matrices, finite groups of quaternions, and Kleinian singularities. Proc. Am. Math. Soc. 81(1), 153–154 (1981)

    MATH  Google Scholar 

  50. Morrison, D.R.: Mirror symmetry and rational curves on quintic threefolds: A guide for mathematicians. J. Am. Math. Soc. 6(1), 223–247 (1993), alg-geom/9202004

    MATH  Google Scholar 

  51. Morrison, D.R.: The Geometry Underlying Mirror Symmetry. In: New trends in algebraic geometry (Warwick, 1996). Cambridge, UK: Cambridge Univ. Press, 1999, pp. 283–310, alg-geom/9608006

  52. Mumford, D.: The topology of normal singularities of an algebraic surface and a criterion for simplicity. Inst. Hautes Études Sci. Publ. Math. 9, 5–22 (1961)

    MATH  Google Scholar 

  53. Nahm, W.: The construction of all self–dual monopoles by the ADHM method. In: Craigie, N.S., Goddard, P., Nahm, W. (eds.), Monopoles in quantum field theory (Trieste, 1981). Singapore: World Scientific, 1982, pp. 87–94

  54. Nahm, W.: Self–dual monopoles and calorons. In: Denardo, G., et~al. (eds.), Lecture Notes in Physics, Vol. 201. Berlin-Heidelberg-New York: Springer, 1984, pp. 189–200

  55. Nahm, W., Wendland, K.: A hiker’s guide to K3 – Aspects of N=(4,4) superconformal field theory with central charge c=6. Commun. Math. Phys. 216, 85–138 (2001), hep-th/9912067

    MathSciNet  MATH  Google Scholar 

  56. Nahm, W.: Strings on K3. In: Proceedings of the ICMP 2000. London, 2000

  57. Narain, K.S.: New heterotic string theories in uncompactified dimensions < 10. Phys. Lett. 169B, 41–46 (1986)

    Article  MathSciNet  Google Scholar 

  58. Narain, K.S., Sarmadi, M.H., Vafa, C.: Asymmetric orbifolds. Nucl. Phys. B288, 551–557 (1987)

    Google Scholar 

  59. Nikulin, V.V.: On Kummer Surfaces. Math. USSR Isv. 9, 261–275 (1975)

    MATH  Google Scholar 

  60. Obers, N.A., Pioline, B.: Exact thresholds and instanton effects in string theory. Fortsch. Phys. 49, 359–375 (2001), hep-th/0101122

    Article  MathSciNet  MATH  Google Scholar 

  61. Ruan, Y.: Stringy geometry and topology of orbifolds. math.AG/0011149

  62. Ruan, Y.: Cohomolgy ring of crepant resolutions. math.AG/0108195

  63. Schenk, H.: On a generalized Fourier transform of instantons over flat tori. Commun. Math. Phys. 116, 177–183 (1988)

    MathSciNet  MATH  Google Scholar 

  64. Seiberg, N.: Observations on the moduli space of superconformal field theories. Nucl. Phys. B303, 286–304 (1988)

    Google Scholar 

  65. Strominger, A., Yau, S.-T., Zaslow, E.: Mirror symmetry is T-duality. Nucl. Phys. B479, 243–259 (1996), hep-th/9606040

    Google Scholar 

  66. Thomas, R.P.: Mirror symmetry and actions of braid groups on derived categories. In: Proceedings of the Harvard Winter School on Mirror Symmetry, Vector Bundles and Lagrangian Submanifolds, January 1999. Cambridge MA: International Press, 2001, pp. 187–201

  67. Thurston, W.P.: Three-dimensional geometry and topology. Vol. 1. Levy, S. (ed.). Princeton, NJ: Princeton University Press, 1997

  68. Vafa, C., Witten, E.: On orbifolds with discrete torsion. J. Geom. Phys. 15, 189–214 (1995), hep-th/9409188

    Article  MathSciNet  MATH  Google Scholar 

  69. van Enckevort, Ch.: Mirror Symmetry and T–Duality. Ph.D. thesis, Universiteit Utrecht, 2000

  70. Wendland, K.: Consistency of orbifold conformal field theories on K3. Adv. Theor. Math. Phys. 5(3), (2001), hep-th/0010281

  71. Witten, E.: String theory dynamics in various dimensions. Nucl. Phys. B443, 85–126 (1995), hep-th/9503124

  72. Witten, E.: D-branes and K-theory. JHEP 9812:019 (1998), hep-th/9810188

  73. Wolf, J.A.: Complex homogeneous contact manifolds and quaternionic symmetric spaces. J. Math. Mech. 14, 1033–1047 (1965)

    MATH  Google Scholar 

  74. Zamolodchikov, A.B.: Irreversibility of the flux of the renormalization group in a 2-D field theory. JETP Lett. 43, 730–732 (1986)

    MathSciNet  Google Scholar 

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  1. Physikalisches Institut, Universität Bonn, Nußallee 12, 53115, Bonn, Germany

    Werner Nahm

  2. Dept. of Physics and Astronomy, UNC at Chapel Hill, 141 Phillips Hall, CB #3255, Chapel Hill, NC 27599, USA

    Katrin Wendland

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  1. Werner Nahm
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Communicated By R.H. Dijkgraaf

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Nahm, W., Wendland, K. Mirror Symmetry on Kummer Type K3 Surfaces. Commun. Math. Phys. 243, 557–582 (2003). https://doi.org/10.1007/s00220-003-0985-3

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