arXiv subjects

C. Vafa

Publications and source records attributed to C. Vafa.

46 records · Page 3Linked to original sources

arXiv1993

Kodaira-Spencer Theory of Gravity and Exact Results for Quantum String Amplitudes

We develop techniques to compute higher loop string amplitudes for twisted $N=2$ theories with $\hat c=3$ (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 the $N=2$ 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--Simon 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 corresponding $N=2$ theory. Relations with $c=1$ strings are also pointed out.

hep-th↗

arXiv1993

Holomorphic Anomalies in Topological Field Theories

We study the stringy genus one partition function of $N=2$ SCFT's. It is shown how to compute this using an anomaly in decoupling of BRST trivial states from the partition function. A particular limit of this partition function yields the partition function of topological theory coupled to topological gravity. As an application we compute the number of holomorphic elliptic curves over certain Calabi-Yau manifolds including the quintic threefold. This may be viewed as the first application of mirror symmetry at the string quantum level.

hep-th↗

arXiv1993

Two dimensional black-hole as a topological coset model of c=1 string theory

We show that a special superconformal coset (with $\hat c =3$) is equivalent to $c=1$ matter coupled to two dimensional gravity. This identification allows a direct computation of the correlation functions of the $c=1$ non-critical string to all genus, and at nonzero cosmological constant, directly from the continuum approach. The results agree with those of the matrix model. Moreover we connect our coset with a twisted version of a Euclidean two dimensional black hole, in which the ghost and matter systems are mixed.

hep-th↗

arXiv1992

On Classification of N=2 Supersymmetric Theories, (e-mail uncorrupted version)

We find a relation between the spectrum of solitons of massive $N=2$ quantum field theories in $d=2$ and the scaling dimensions of chiral fields at the conformal point. The condition that the scaling dimensions be real imposes restrictions on the soliton numbers and leads to a classification program for symmetric $N=2$ conformal theories and their massive deformations in terms of a suitable generalization of Dynkin diagrams (which coincides with the A--D--E Dynkin diagrams for minimal models). The Landau-Ginzburg theories are a proper subset of this classification. In the particular case of LG theories we relate the soliton numbers with intersection of vanishing cycles of the corresponding singularity; the relation between soliton numbers and the scaling dimensions in this particular case is a well known application of Picard-Lefschetz theory.

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arXiv1992

Quantum Affine Symmetry as Generalized Supersymmetry

The quantum affine $\CU_q (\hat{sl(2)}) $ symmetry is studied when $q^2$ is an even root of unity. The structure of this algebra allows a natural generalization of N=2 supersymmetry algebra. In particular it is found that the momentum operators $P ,\bar{P}$, and thus the Hamiltonian, can be written as generalized multi-commutators, and can be viewed as new central elements of the algebra $\CU_q (\hat{sl(2)})$. We show that massive particles in (deformations of) integer spin representions of $sl(2)$ are not allowed in such theories. Generalizations of Witten's index and Bogomolnyi bounds are presented and a preliminary attempt in constructing manifestly $\CU_q (\hat{sl(2)})$ invariant actions as generalized supersymmetric Landau-Ginzburg theories is made.

hep-th↗

arXiv1992

Ising Model and $N=2$ Supersymmetric Theories

We establish a direct link between massive Ising model and arbitrary massive $N=2$ supersymmetric QFT's in two dimensions. This explains why the equations which appear in the computation of spin-correlations in the non-critical Ising model are the same as those describing the geometry of vacua in $N=2$ theories. The tau-function appearing in the Ising model (i.e., the spin correlation function) is reinterpreted in the $N=2$ context as a new `index'. In special cases this new index is related to Ray-Singer analytic torsion, and can be viewed as a generalization of that to loop space of Kähler manifolds.

hep-th↗

arXiv1992

Massive Orbifold

We study some aspects of 2d supersymmetric sigma models on orbifolds. It turns out that independently of whether the 2d QFT is conformal the operator products of twist operators are non-singular, suggesting that massive (non-conformal) orbifolds also `resolve singularities' just as in the conformal case. Moreover we recover the OPE of twist operators for conformal theories by considering the UV limit of the massive orbifold correlation functions. Alternatively, we can use the OPE of twist fields at the conformal point to derive conditions for the existence of non-singular solutions to special non-linear differential equations (such as Painleve III).

hep-th↗

arXiv1991

Topological Mirrors and Quantum Rings

Aspects of duality and mirror symmetry in string theory are discussed. We emphasize, through examples, the importance of loop spaces for a deeper understanding of the geometrical origin of dualities in string theory. Moreover we show that mirror symmetry can be reformulated in very simple terms as the statement of equivalence of two classes of topological theories: Topological sigma models and topological Landau-Ginzburg models. Some suggestions are made for generalization of the notion of mirror symmetry.

hep-th↗

arXiv1991

Exact Results for Supersymmetric Sigma Models

We show that the metric and Berry's curvature for the ground states of $N=2$ supersymmetric sigma models can be computed exactly as one varies the Kahler structure. For the case of $CP^n$ these are related to special solutions of affine toda equations. This allows us to extract exact results (including exact instanton corrections). We find that the ground state metric is non-singular as the size of the manifold shrinks to zero thus suggesting that 2d QFT makes sense even beyond zero radius. In other words it seems that manifolds with zero size are non-singular as target spaces for string theory (even when they are not conformal). The cases of $CP^1$ and $CP^2$ are discussed in more detail.

hep-th↗

arXiv1991

Elements of String Cosmology

Aspects of string cosmology for critical and non-critical strings are discussed emphasizing the necessity to account for the dilaton dynamics for a proper incorporation of ``large - small" duality. This drastically modifies the intuition one has with Einstein's gravity. For example winding modes, even though contribute to energy density, oppose expansion and if not annihilated will stop the expansion. Moreover we find that the radiation dominated era of the standard cosmology emerges quite naturally in string cosmology. Our analysis of non-critical string cosmology provides a reinterpretation of the (universal cover of the) recently studied two dimensional black hole solution as a conformal realization of cosmological solutions found previously by Mueller.

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