Abstract
We study the geometric interpretation of two dimensional rational conformal field theories, corresponding to sigma models on Calabi-Yau manifolds. We perform a detailed study of RCFT’s corresponding to the T 2 target and identify the Cardy branes with geometric branes. The T 2’s leading to RCFT’s admit ‘‘complex multiplication’’ which characterizes Cardy branes as specific D0-branes. We propose a condition for the conformal sigma model to be RCFT for arbitrary Calabi-Yau n-folds, which agrees with the known cases. Together with recent conjectures by mathematicians it appears that rational conformal theories are not dense in the space of all conformal theories, and sometimes appear to be finite in number for Calabi-Yau n-folds for n>2. RCFT’s on K3 may be dense. We speculate about the meaning of these special points in the moduli spaces of Calabi-Yau n-folds in connection with freezing geometric moduli.
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References
Friedan, D., Qiu, Z., Shenker, S.: Conformal Invariance, Unitarity, and Two-Dimensional Critical Exponents. Phys. Rev. Lett. 52, 1575 (1984)
Belavin, A.A., Polyakov, A.M., Zamolodchikov, A.B.: Infinite Conformal Symmetry in Two-Dimensional Quantum Field Theory. Nucl. Phys. B241, 333 (1984)
Verlinde, E.: Fusion Rules and Modular Transformations in 2D Conformal Field Theory. Nucl. Phys. B300, 360 (1988)
Moore, G., Seiberg, N.: Polynomial Equations for Rational Conformal Field Theories. Phys. Lett. B212, 451 (1988); Classical and Quantum Conformal Field Theory. Commun. Math. Phys. 123, 177 (1989); Taming the Conformal Zoo. Phys. Lett. 220, 422 (1989)
Cardy, J.L.: Boundary Conditions, Fusion Rules and the Verlinde Formula. Nucl. Phys. B 324, 581 (1989)
Ishibashi, N.: The Boundary And Crosscap States In Conformal Field Theories. Mod. Phys. Lett. A4, 161 (1989)
Moore, G.: Arithmetic and attractors. arXiv:hep-th/9807087
Mumford, D.: A note on Shimura’s paper Discontinuous Groups and Abelian Varieties. Math. Ann. 181, 345 (1969)
Pjateckii-Shapiro, I., Shafarevich, I.R.: The Arithmetic of K3 Surfaces. Proc. Steklov Inst. Math. 132, 45 (1973)
Borcea, C.: Calabi-Yau Threefolds and Complex Multiplication. In: Essays on Mirror Manifolds, S.-T. Yau, (ed.), Cambridge, MA: International Press, 1992
Lian, B.H., Yau, S.-T.: Arithmetic Properties of Mirror Map and Quantum Coupling. Commun. Math. Phys. 176, 163 (1996)
Miller, S.D., Moore, G.: Landau-Siegel zeroes and black hole entropy. arXiv:hep-th/9903267
Candelas, P., de la Ossa, X., Rodriguez-Villegas, F.: Calabi-Yau manifolds over finite fields, I. arXiv:hep-th/0012233
Schimmrigk, R.: Arithmetic of Calabi-Yau varieties and rational conformal field theory. arXiv:hep-th/0111226
Kachru, S., Schulz, M., Trivedi, S.: Moduli Stabilization from Fluxes in a Simple IIB Orientifold. hep-th/0201028; see also a talk of S. Kachru at Strings 2002 Conference, http://www.damtp.cam.ac.uk/strings02/avt/kachru
Manin, Y., Marcolli, M.: Holography Principle and Arithmetic of Algebraic Curves. hep-th/0201036
Fuchs, J., Schweigert, C.: Branes: From free fields to general backgrounds. Nucl. Phys. B 530, 99 (1998) [arXiv:hep-th/9712257]
Maldacena, J., Moore, G., Seiberg, N.: Geometrical interpretation of D-branes in gauged WZW models. JHEP 0107, 046 (2001)
Cappelli, A., D’Appollonio, G.: Boundary States of c=1 and c=3/2 Rational Conformal Field Theories. JHEP 0202, 039 (2002) hep-th/0201173
Gaberdiel, M.R., Recknagel, A.: Conformal boundary states for free bosons and fermions. JHEP 0111, 016 (2001) [arXiv:hep-th/0108238]
Gannon, T.: Monstrous Moonshine and the Classification of CFT. math.QA/ 9909080
Ooguri, H., Oz, Y., Yin, Z.: D-Branes on Calabi-Yau Spaces and Their Mirrors. Nucl. Phys. B477, 407 (1996)
Dijkgraaf, R., Verlinde, E., Verlinde, H.: On Moduli Spaces of Conformal Field Theories with c ≥1. Proc. of 1987 Copenhagen Conference Perspectives in String Theory
Recknagel, A., Schomerus, V.: D-branes in Gepner models. Nucl. Phys. B 531, 185 (1998) [arXiv:hep-th/9712186]
Gutperle, M., Satoh, Y.: D-branes in Gepner models and supersymmetry. Nucl. Phys. B 543, 73 (1999) [arXiv:hep-th/9808080]
Vafa, C.: Quantum Symmetries of String Vacua. Mod. Phys. Lett. A4, 1615 (1989)
Mizoguchi, S., Tani, T.: Wound D-Branes in Gepner Models. Nucl. Phys. B611, 253 (2001)
Parshin, A.N., Shafarevich, I.R., (eds.): Number Theory II. Berlin-Heidelberg-New York: Springer-Verlag, 1992
Shimura, G., Taniyama, Y: Complex Multiplication of Abelian Varieties and its Applications to Number Theory. Japan Math. Soc. 1961
Lang, S., (ed.): Number Theory III. Berlin-Heidelberg-New York: Springer-Verlag, 1991
Borevic, Z., Shafarevich, I.: Number theory. 1985
Conway, J.H., Sloane, N.J.A.: Sphere Packings, Lattices, and Codes. Berlin-Heidelberg-New York: Springer-Verlag, 1993
Nikulin, V.: Integral symmetric bilinear forms and some of their Applications. Math. Izv. 14, 103 (1980)
Dolgachev, I.: Integral quadratic forms: Applications to algebraic geometry. Sem. Bourbaki 611, 251 (1982)
Wendland, W.: Moduli Spaces of Unitary Conformal Field Theories. PhD Thesis, September 2000
André, Y.: G-functions and Geometry. Aspects of Mathematics, Vol. E13, Braunshweig: Vieweg, 1989; André, Y.: Distribution des points CM sur les sous-variétés de modules de variétés abéliennes, 1997
Oort, F.: Canonical Lifts and Dense Sets of CM-points. Arithmetic Geometry. Proc. Cortona Symposium 1994, F. Catanese, (ed.), Symposia Math., Vol. XXXVII, Cambridge: Cambridge Univ. Press, 1997, p. 228
Shioda, T.: What is known about the Hodge conjecture? In: Algebraic Varieties and Analytic Varieties. Adv. Studies Pure Math 1, 55 (1983); Shioda, T.: Geometry of Fermat Varieties. In: Number Theory Related to Fermat’s Last Theorem, Progress in Math. 26, 45 (1982)
Deligne, P.: Local Behavior of Hodge Structures at Infinity. AMS/IP Studies Adv. Math. 1, 683 (1997)
Deligne, P. (Notes by Milne, J.): Hodge Cycles on Abelian Varieties. Lecture Notes in Math 900, 9 (1982)
Coleman, R.: Torsion Points on Curves. In Galois representations and arithmetic algebraic geometry, Y. Ihara, ed., Adv. Studies Pure Math. 12, 235 (1987)
de Jong, J., Noot, R.: Jacobians with Complex Multiplication. In: Arithmetic Algebraic Geometry, G. van der Geer, F. Oort, J. Steenbrink, (eds.), Basel-Boston: Birkhäuser, 1991
Hori, K., Iqbal, A., Vafa, C.: D-Branes and Mirror Symmetry. hep-th/0005247.
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Communicated by Y. Kawahigashi
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Gukov, S., Vafa, C. Rational Conformal Field Theories and Complex Multiplication. Commun. Math. Phys. 246, 181–210 (2004). https://doi.org/10.1007/s00220-003-1032-0
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DOI: https://doi.org/10.1007/s00220-003-1032-0