arXiv subjects
Publications and source records attributed to W. Lerche.
After briefly reviewing basic concepts of perturbative string theory, we explain in simple terms some of the new findings that created excitement among the string physicists. These developments include non-perturbative dualities and a unified picture that embraces the so-far known theories. (An overview for non-experts)
We study limits of four-dimensional type II Calabi-Yau compactifications with vanishing four-cycle singularities, which are dual to $\IT^2$ compactifications of the six-dimensional non-critical string with $E_8$ symmetry. We define proper subsectors of the full string theory, which can be consistently decoupled. In this way we obtain rigid effective theories that have an intrinsically stringy BPS spectrum. Geometrically the moduli spaces correspond to special geometry of certain non-compact Calabi-Yau spaces of an intriguing form. An equivalent description can be given in terms of Seiberg-Witten curves, given by the elliptic simple singularities together with a peculiar choice of meromorphic differentials. We speculate that the moduli spaces describe non-perturbative non-critical string theories.
We give an elementary introduction to the recent solution of $N=2$ supersymmetric Yang-Mills theory. In addition, we review how it can be re-derived from string duality.
We show that the genus 34 Seiberg-Witten curve underlying $N=2$ Yang-Mills theory with gauge group $E_6$ yields physically equivalent results to the manifold obtained by fibration of the $E_6$ ALE singularity. This reconciles a puzzle raised by $N=2$ string duality.
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 {\it stable} BPS states in field theory. Moreover, we identify the six-dimensional space $\IR^4\times Σ$ as the world-volume of a five-brane and show that BPS states correspond to two-branes ending on this five-brane.
Using heterotic/type II string duality, we obtain exact nonperturbative results for the point particle limit (alpha' -> 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.
We review some simple group theoretical properties of BPS states, in relation with the singular homology of level surfaces. Primary focus is on classical and quantum N=2 supersymmetric Yang-Mills theory, though the considerations can be applied to string theory as well.
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.
We elaborate on our previous work on N=2 supersymmetric Yang-Mills theory. In particular, we show how to explicitly determine the low energy quantum effective action for $G=SU(3)$ from the underlying hyperelliptic Riemann surface, and calculate the leading instanton corrections. This is done by solving Picard-Fuchs equations and asymptotically evaluating period integrals. We find that the dynamics of the $SU(3)$ theory is governed by an Appell system of type $F_4$, and compute the exact quantum gauge coupling explicitly in terms of Appell functions.
We show that the BRST structure of the topological string is encoded in the ``small'' $N=4$ superconformal algebra, enabling us to obtain, in a non-trivial way, the string theory from hamiltonian reduction of $A(1|1)$. This leads to the important conclusion that not only ordinary string theories, but topological strings as well, can be obtained, or even defined, by hamiltonian reduction from WZW models. Using two different gradations, we find either the standard $N=2$ minimal models coupled to topological gravity, or an embedding of the bosonic string into the topological string. We also comment briefly on the generalization to super Lie algebras $A(n|n)$.
We review the generalization of the work of Seiberg and Witten on N=2 supersymmetric SU(2) Yang-Mills theory to SU(n) gauge groups. The quantum moduli spaces of the effective low energy theory parametrize a special family of hyperelliptic genus n-1 Riemann surfaces. We discuss the massless spectrum and the monodromies.
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 investigate the algebraic structure of integrable hierarchies that, we propose, underlie models of $W$-gravity coupled to matter. More precisely, we concentrate on the dispersionless limit of the topological subclass of such theories, by making use of a correspondence between Drinfeld-Sokolov systems, principal $s\ell(2)$ embeddings and certain chiral rings. We find that the integrable hierarchies can be viewed as generalizations of the usual matrix Drinfeld-Sokolov systems to higher fundamental representations of $s\ell(n)$. The underlying Heisenberg algebras have an intimate connection with the quantum cohomology of grassmannians. The Lax operators are directly given in terms of multi-field superpotentials of the associated topological LG theories. We view our construction as a prototype for a multi-variable system and suspect that it might be useful also for a class of related problems.
We investigate how specific free-field realizations of twisted N=2 supersymmetric coset models give rise to natural extensions of the ``matter'' Hilbert spaces in such a manner as to incorporate the various gravitational excitations. In particular, we show that adopting a particular screening prescription is equivalent to imposing the requisite equivariance condition on cohomology. We find a simple algebraic characterization of the $W_n$-gravitational ground ring spectra of these theories in terms of affine-$SU(n)$ highest weights..
In any string theory there is a hidden, twisted superconformal symmetry algebra, part of which is made up by the BRST current and the anti-ghost. We investigate how this algebra can be systematically constructed for strings with $N\!-\!2$ supersymmetries, via quantum Hamiltonian reduction of the Lie superalgebras $osp(N|2)$. The motivation is to understand how one could systematically construct generalized string theories from superalgebras. We also briefly discuss the BRST algebra of the topological string, which is a doubly twisted $N\!=\!4$ superconformal algebra.
We study the equivariant cohomology of a class of multi-field topological LG models, and find that such systems carry intrinsic information about $W$-gravity. As a result, we can construct the gravitational chiral ring in terms of LG polynomials. We find, in particular, that the spectrum of such theories seems to be richer than so far expected. We also briefly discuss the BRST operator for non-linear topological $W$-gravity.
(Talk given at Strings '93, Berkeley, and at XXVII. Internationales Symposium über Elementarteilchentheorie, Wendisch-Rietz, 1993) We review the superconformal properties of matter coupled to $2d$ gravity, and $W$-extensions thereof. We show in particular how the \nex2 structure provides a direct link between certain matter-gravity systems and matrix models. We also show that much, probably all, of this can be generalized to $W$-gravity, and this leads to an infinite class of new exactly solvable systems. These systems are governed by certain integrable hierarchies, which are generalizations of the usual KdV hierarchy and whose algebraic structure is given in terms of quantum cohomology rings of grassmannians.
Talk given at the 26th Workshop: ``From Superstrings to Supergravity", Erice - Sicily, 5-12 December 1992. We review the superconformal properties of 2d matter coupled to gravity, and extensions thereof. Focusing on topological strings, we recall how the superconformal structure helps to provide a direct link between Liouville theory coupled to matter, and matrix models. We also construct an infinite class of new theories based on $W$-gravity.