Abstract
I review how recent results in quantum field theory confirm two general predictions of the mirror symmetry program in the special case of elliptic curves: (1) counting functions of holomorphic curves on a Calabi-Yau space (Gromov-Witten invariants) are ‘quasimodular forms’ for the mirror family; (2) they can be computed by a summation over trivalent Feynman graphs.
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References
AFMO] H. Awata, M. Fukuma, Y. Matsuo, S. Odake, Representation theory of the W 1+∞ algebra, hep-th/9408158.
BCOV] M. Bershadsky, S. Cecotti, H. Ooguri, C. Vafa, Holomorphic anomalies in topological field theories, Nucl. Phys. B405 (1993) 279–304, hep-th/9302103; Kodaira-Spencer theory of gravity and exact results for quantum string amplitude, Commun. Math. Phys. 165 (1994) 311-428, hep-th/9309140.
CMR] S. Cordes, G. Moore, S. Rangoolam, Large N 2-D Yang-Mills theory and topological string theory, hep-th/9402107; Lectures on 2-D Yang-Mills theory, equivariant cohomology and topological field theories, hep-th/9411210 G. Moore, 2-D Yang-Mills theory and topological field theory, ICM 1994 con¬tribution, hep-th/9409044.
Dou] M.R. Douglas, Conformal field theory techniques in large N Yang-Mills theory, hep-th/9311130.
Dij] R. Dijkgraaf, Chiral deformations of conformal field theories on a torus, to appear.
R. Dijkgraaf, E. Verlinde and H. Verlinde, On moduli spaces of conformal field theories with c ≥ 1, in Perspectives in String Theory, P. Di Vecchia and J.L. Petersen Eds. (World Scientific, 1988 ).
R. Dijkgraaf and E. Witten, Topological gauge theories and group cohomology, Commun. Math. Phys. 129 (1990) 393.
A. Floer, Symplectic fixed points and holomorphic spheres, Commun. Math. Phys. 120 (1989), 575.
F.G. Frobenius, Über die Charaktere der symmetrischen Gruppe, Sitz. König. Preuß. Akad. Wissen. (1900) 516–534 = Ges. Abh., Band III ( Springer-Verlag, Berlin, 1968 ), 148–166
D. Freed and F. Quinn, Chern-Simons gauge theory with a finite gauge group, Commun. Math. Phys. 156 (1993) 435–472.
GPR] A. Giveon, M. Porrati, E. Rabinovoco, Target space duality in string theory,Phys. Rep. 244 (1994) 77-202, hep-th/9401139.
M. Gromov, Pseudo-holomorphic curves on almost complex manifolds, Invent. Math. 82 (1985) 307.
GT] D.J. Gross and W. Taylor IV, Two dimensional QCD is a string theory, Nucl. Phys. B400 (1993) 181–210, hep-th/9301068.
Hur] A. Hurwitz, Ueber die Anzahl der Riemannschen Flächen mit gegebener Verzweigungspunkten, Math. Ann. 55 (1902) 53.
tH] G.’t Hooft, A planar diagram theory for strong interactions, Nucl. Phys. B72 (1974) 461.
Kon] M. Kontsevich, Enumeration of rational curves via torus actions, in this volume, hep-th/9405035.
KZ] M. Kaneko and D. Zagier, A generalized Jacobi theta function and quasimodular forms, in this volume, 165–172.
W. Lerche, C. Vafa, and N.P. Warner, Chiral rings in N = 2 superconformal theories, Nucl. Phys. B324 (1989) 427.
Ru] R. Rudd, The String Partition Function for QCD on the Torus, hep-th/9407176.
E. Verlinde, Fusion rules and modular transformations in 2d conformal field theory, Nucl. Phys. B300 (1988) 360.
281; Two dimensional gravity and intersection theory on moduli space, Surveys In Diff. Geom. 1 (1991) 243.
Yau] Essays on Mirror manifolds, Ed. S-T Yau (International Press, Hong Kong, 1992).
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Dijkgraaf, R. (1995). Mirror Symmetry and Elliptic Curves. In: Dijkgraaf, R.H., Faber, C.F., van der Geer, G.B.M. (eds) The Moduli Space of Curves. Progress in Mathematics, vol 129. Birkhäuser Boston. https://doi.org/10.1007/978-1-4612-4264-2_5
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DOI: https://doi.org/10.1007/978-1-4612-4264-2_5
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