We investigate the properties of chiral operators in N = 2 superconformal theories. In particular, we study the spectral flow of such models under a one-parameter family of twists generated by the U(1) current, and use this to deduce various properties of the ring of chiral primary fields. We furthermore investigate under what conditions a given superconformal theory can be represented as the fixed point of an N = 2 Landau-Ginzburg theory and show how to determine the superpotential. We also investigate the coset models of Kazama and Suzuki and find a simple cohomological characterization for the elements of the chiral primary ring. Moreover we show how some of them can be represented as LG models. * This is the analog of the Hodge decomposition for differential forms. See also the discussion in sect. 3. * This can be seen by considering the contour integral of G+(z) about a pair of chiral fields, and noting that the definition of chiral state demands that the contour integral of G+(z) encircling a chiral field vanishes. ** The ring is commutative up to +_ signs, due to the fact that the ring is defined before the GSO projection. **'* This is not to be confused with the commutative ring which appears for rational conformal field theories [9]. In particular the ring we obtain here is nilpotent, which is not the case for RCFT operator algebra. * For the behavior of Tr(1) F under asymmetric spectral flow see ref. [12]. ** These formulas generalize to relations between the full N = 2 characters for all levels [11,13]. * Toroidal compactification [25] and orbifolds based on them [26] could also be viewed as orbifolds of Landau-Ginzburg models. ** Our conventions in this section are opposite to that of sect. 3, where the (c, c) corresponded to Kghler deformations.

We show how the Riemann surface Σ of N = 2 Yang-Mills field theory arises in type II string compactifications on Calabi-Yau threefolds. The relevant local geometry is given by fibrations of ALE spaces. The 3-branes that give rise to BPS multiplets in the string descend to self-dual strings on the Riemann surface, with tension determined by a canonically fixed Seiberg-Witten differential λ. This gives, effectively, a dual formulation of Yang-Mills theory in which gauge bosons and monopoles are treated on equal footing, and represents the rigid analog of type II-heterotic string duality. The existence of BPS states is essentially reduced to a geodesic problem on the Riemann surface with metric |λ| 2 . This allows us, in particular, to easily determine the spectrum of stable BPS states in field theory. Moreover, we identify the six-dimensional space IR 4 × Σ as the world-volume of a five-brane and show that BPS states correspond to two-branes ending on this five-brane.

We present a first step towards generalizing the work of Seiberg and Witten on N = 2 supersymmetric Yang-Mills theory to arbitrary gauge groups. Specifically, we propose a particular sequence of hyperelliptic genus n−1 Riemann surfaces to underly the quantum moduli space of SU (n) N = 2 supersymmetric gauge theory. These curves have an obvious generalization to arbitrary simply laced gauge groups, which involves the A-D-E type simple singularities. To support our proposal, we argue that the monodromy in the semiclassical regime is correctly reproduced. We also give some remarks on a possible relation to string theory.

We analyze the map between heterotic and type II N=2 supersymmetric string theories for certain two and three moduli examples found by Kachru and Vafa. The appearance of elliptic j-functions can be traced back to specializations of the Picard-Fuchs equations to systems for K 3 surfaces. For the three-moduli example we write the mirror maps and Yukawa couplings in the weak coupling limit in terms of j-functions; the expressions agree with those obtained in perturbative calculations in the heterotic string in an impressive way. We also discuss symmetries of the world-sheet instanton numbers in the type II theory, and interpret them in terms of S-duality of the non-perturbative heterotic string.

Using heterotic/type II string duality, we obtain exact nonperturbative results for the point particle limit (α ′ → 0) of some particular four dimensional, N = 2 supersymmetric compactifications of heterotic strings. This allows us to recover recent exact nonperturbative results on N = 2 gauge theory directly from tree-level type II string theory, which provides a highly non-trivial, quantitative check on the proposed string duality. We also investigate to what extent the relevant singular limits of Calabi-Yau manifolds are related to the Riemann surfaces that underlie rigid N = 2 gauge theory.