Mirror Symmetry and Elliptic Curves
01 Jan 1995-Iss: 129, pp 149-163
TL;DR: In this article, the authors review how recent results in quantum field theory confirm two general predictions of the mirror symmetry program in the special case of elliptic curves: counting functions of holomorphic curves on a Calabi-Yau space (Gromov-Witten invariants) are quasimodular forms for the mirror family; they can be computed by a summation over trivalent Feynman graphs.
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.
Citations
More filters
26 Jun 2003
TL;DR: In this paper, the authors investigated various representations for this partition function: a statistical sum over random partitions, a partition function of the ensemble of random curves, and a free fermion correlator.
Abstract: We study \( \mathcal{N} = 2 \) supersymmetric four-dimensional gauge theories, in a certain 525-02 = 2 supergravity background, called theΩ-background. The partition function of the theory in the Ω-background can be calculated explicitly. We investigate various representations for this partition function: a statistical sum over random partitions, a partition function of the ensemble of random curves, and a free fermion correlator.
1,350 citations
TL;DR: The Schur measure on partitions as discussed by the authors is a generalization of the Plancherel measure, and it has been shown that correlation functions for random partitions are determinantal functions for the Toda lattice hierarchy.
Abstract: We use representation theory to obtain a number of exact results for random partitions. In particular, we prove a simple determinantal formula for correlation functions of what we call the Schur measure on partitions (which is a far reaching generalization of the Plancherel measure; see [3], [8]) and also observe that these correlations functions are
$ \tau $
-functions for the Toda lattice hierarchy. We also give a new proof of the formula due to Bloch and the author [5] for the so-called n-point functions of the uniform measure on partitions and comment on the local structure of a typical partition.
370 citations
TL;DR: In this article, an explicit equivalence between the stationary sector of the Gromov-Witten theory of a target curve X and the enumeration of Hurwitz coverings of X in the basis of completed cycles was established.
Abstract: We establish an explicit equivalence between the stationary sector of the Gromov-Witten theory of a target curve X and the enumeration of Hurwitz coverings of X in the basis of completed cycles. The stationary sector is formed, by definition, by the descendents of the point class. Completed cycles arise naturally in the theory of shifted symmetric functions. Using this equivalence, we give a complete description of the stationary Gromov-Witten theory of the projective line and elliptic curve. Toda equations for the relative stationary theory of the projective line are derived.
360 citations
01 Jan 1995
TL;DR: In this article, a direct proof using the theory of modular forms of a beautiful fact explained in the preceding paper by Robbert Dijkgraaf was given, and the proof was used to give a proof for the existence of quasi-modular forms on the full modular group.
Abstract: In this note we give a direct proof using the theory of modular forms of a beautiful fact explained in the preceding paper by Robbert Dijkgraaf [1, Theorem 2 and Corollary]. Let \( {\tilde M_*}({\Gamma _1}) \) denote the graded ring of quasi-modular forms on the full modular group Γ= PSL(2, ℤ). This is the ring generated by G2, G4, G6, and graded by assigning to each G k the weight where \( {G_k} = - \frac{{{B_k}}}{{2k}} + \sum\limits_{n = 1}^\infty {\left( {{{\sum\limits_{d|n} d }^{k - 1}}} \right)} {q^n}\left( {k \geqslant 2,{B_k} = kth Bernoulli number} \right) \) are the classical Eisenstein series, all of which except G 2 are modular.
346 citations
TL;DR: In this paper, it was shown that the generating function for the numbers of such coverings is a tau-function for the Toda lattice hierarchy of Ueno and Takasaki.
Abstract: We consider ramified coverings of P^1 with arbitrary ramification type over 0 and infinity and simple ramifications elsewhere and prove that the generating function for the numbers of such coverings is a tau-function for the Toda lattice hierarchy of Ueno and Takasaki.
308 citations
References
More filters
CERN1
TL;DR: In this article, it was shown that only planar diagrams with the quarks at the edges dominate; the topological structure of the perturbation series in 1/N is identical to that of the dual models, such that the number 1/n corresponds to the dual coupling constant.
4,449 citations
"Mirror Symmetry and Elliptic Curves..." refers background in this paper
...The idea to treat the classical Lie groups in perturbation theory around innite rank is a very productive idea in physics conceived of by ’t Hooft [ 13 ]....
[...]
TL;DR: In this article, the authors define a parametrized (pseudo holomorphic) J-curve in an almost complex manifold (IS, J) is a holomorphic map of a Riemann surface into Is, say f : (S, J3 ~(V, J).
Abstract: Definitions. A parametrized (pseudo holomorphic) J-curve in an almost complex manifold (IS, J) is a holomorphic map of a Riemann surface into Is, say f : (S, J3 ~(V, J). The image C=f(S)C V is called a (non-parametrized) J-curve in V. A curve C C V is called closed if it can be (holomorphically !) parametrized by a closed surface S. We call C regular if there is a parametrization f : S ~ V which is a smooth proper embedding. A curve is called rational if one can choose S diffeomorphic to the sphere S 2.
2,482 citations
TL;DR: In this paper, the authors developed techniques to compute higher loop string amplitudes for twisted N = 2 theories with ε = 3 (i.e. the critical case) by exploiting the discovery of an anomaly at every genus in decoupling of BRST trivial states, captured by a master anomaly equation.
Abstract: We develop techniques to compute higher loop string amplitudes for twistedN=2 theories withĉ=3 (i.e. the critical case). An important ingredient is the discovery of an anomaly at every genus in decoupling of BRST trivial states, captured to all orders by a master anomaly equation. In a particular realization of theN=2 theories, the resulting string field theory is equivalent to a topological theory in six dimensions, the Kodaira-Spencer theory, which may be viewed as the closed string analog of the Chern-Simons theory. Using the mirror map this leads to computation of the ‘number’ of holomorphic curves of higher genus curves in Calabi-Yau manifolds. It is shown that topological amplitudes can also be reinterpreted as computing corrections to superpotential terms appearing in the effective 4d theory resulting from compactification of standard 10d superstrings on the correspondingN=2 theory. Relations withc=1 strings are also pointed out.
1,633 citations
TL;DR: In this paper, the authors studied conformal field theories with a finite number of primary fields with respect to some chiral algebra and showed that the fusion rules are completely determined by the behavior of the characters under the modular group.
1,506 citations
Additional excerpts
...The proof follows essentially the argument of the Verlinde formula [ 21 ]....
[...]
TL;DR: A variant of the usual supersymmetric nonlinear sigma model is described in this article, governing maps from a Riemann surface to an arbitrary almost complex manifold, which possesses a fermionic BRST-like symmetry, conserved for arbitrary Σ, and obeying Q 2 = 0.
Abstract: A variant of the usual supersymmetric nonlinear sigma model is described, governing maps from a Riemann surfaceΣ to an arbitrary almost complex manifoldM. It possesses a fermionic BRST-like symmetry, conserved for arbitraryΣ, and obeyingQ
2=0. In a suitable version, the quantum ground states are the 1+1 dimensional Floer groups. The correlation functions of the BRST-invariant operators are invariants (depending only on the homotopy type of the almost complex structure ofM) similar to those that have entered in recent work of Gromov on symplectic geometry. The model can be coupled to dynamical gravitational or gauge fields while preserving the fermionic symmetry; some observations by Atiyah suggest that the latter coupling may be related to the Jones polynomial of knot theory. From the point of view of string theory, the main novelty of this type of sigma model is that the graviton vertex operator is a BRST commutator. Thus, models of this type may correspond to a realization at the level of string theory of an unbroken phase of quantum gravity.
1,173 citations
"Mirror Symmetry and Elliptic Curves..." refers background in this paper
...They appeared in Gromov’s fundamental work on pseudo-holomorphic curves in symplectic geometry [2] and Witten’s equally fundamental study of topological sigma models [ 3 ]....
[...]