arXiv subjects
Publications and source records attributed to Edward Witten.
We consider the interpretation in classical geometry of conformal field theories constructed from orbifolds with discrete torsion. In examples we can analyze, these spacetimes contain ``stringy regions'' that from a classical point of view are singularities that are to be neither resolved nor blown up. Some of these models also give particularly simple and clear examples of mirror symmetry.
In 2 + 1 dimensions, in the presence of gravity, supersymmetry can ensure the vanishing of the cosmological constant without requiring the equality of bose and fermi masses.
By studying the partition function of $N=4$ topologically twisted supersymmetric Yang-Mills on four-manifolds, we make an exact strong coupling test of the Montonen-Olive strong-weak duality conjecture. Unexpected and exciting links are found with two-dimensional rational conformal field theory.
It is shown that in string theory mirror duality is a gauge symmetry (a Weyl transformation) in the moduli space of $N=2$ backgrounds on group manifolds, and we conjecture on the possible generalization to other backgrounds, such as Calabi-Yau manifolds.
This article is a collection of letters solicited by the editors of the Bulletin in response to a previous article by Jaffe and Quinn [math.HO/9307227]. The authors discuss the role of rigor in mathematics and the relation between mathematics and theoretical physics.
By exploiting standard facts about $N=1$ and $N=2$ supersymmetric Yang-Mills theory, the Donaldson invariants of four-manifolds that admit a Kahler metric can be computed. The results are in agreement with available mathematical computations, and provide a powerful check on the standard claims about supersymmetric Yang-Mills theory.
We derive a condition under which (0,2) linear sigma models possess a ``left-moving'' conformal stress tensor in $\bq$ cohomology (i.e. which leaves invariant the ``right-moving'' ground states) even away from their critical points. At the classical level this enforces quasihomogeneity of the superpotential terms. The persistence of this structure at the quantum level on the worldsheet is obstructed by an anomaly unless the charges and superpotential degrees satisfy a condition which is equivalent to the condition for the cancellation of the anomaly in a particular ``right-moving'' U(1) R-symmetry.
The article is devoted to a quantum field theory explanation of the relationship (noticed some years ago by Gepner) between the Verlinde algebra of the group $U(k)$ at level $N-k$ and the cohomology of the Grassmannian. The argument proceeds by starting with the two dimensional sigma model whose target space is the Grassmannian and integrating out some fields in a standard way. It has long been known that the resulting low energy effective action describes a theory with a mass gap; the novelty here is that this theory in fact is equivalent at long distances to a gauged WZW model of $U(k)/U(k)$, and hence is related to the Verlinde algebra.
We develop techniques to compute the complete the massless spectrum in heterotic string compactification on N=2 supersymmetric Landau-Ginzburg orbifolds. This includes not just the familiar charged fields, but also the gauge singlets. The number of gauge singlets can vary in the moduli space of a given compactification and can differ from what it would be in the large radius limit of the corresponding Calabi-Yau. Comparison with exactly soluble Gepner models provides a confirmation of our results at Gepner points. Our methods carry over straightforwardly to $(0,2)$ Landau-Ginzburg models.
The conjecture that $N=2$ minimal models in two dimensions are critical points of a super-renormalizable Landau-Ginzburg model can be tested by computing the path integral of the Landau-Ginzburg model with certain twisted boundary conditions. This leads to simple expressions for certain characters of the $N=2$ models which can be verified at least at low levels. An $N=2$ superconformal algebra can in fact be found directly in the {\it noncritical} Landau-Ginzburg system, giving further support for the conjecture.
Not only in physical string theories, but also in some highly simplified situations, background independence has been difficult to understand. It is argued that the ``holomorphic anomaly'' of Bershadsky, Cecotti, Ooguri, and Vafa gives a fundamental explanation of some of the problems. Moreover, their anomaly equation can be interpreted in terms of a rather peculiar quantum version of background independence: in systems afflicted by the anomaly, background independence does not hold order by order in perturbation theory, but the exact partition function as a function of the coupling constants has a background independent interpretation as a state in an auxiliary quantum Hilbert space. The significance of this auxiliary space is otherwise unknown.
Simple mean field methods can be used to describe transitions between different space-time models in string theory. These include transitions between different Calabi-Yau manifolds, and more exotic things such as the Calabi-Yau/Landau-Ginzberg correspondence.
A recent proposal for a background independent open string field theory is studied in detail for a class of backgrounds that correspond to general quadratic boundary interactions on the world-sheet. A short-distance cut-off is introduced to formulate the theory with a finite number of local and potentially unrenormalizable boundary couplings. It is shown that renormalization of the boundary couplings makes both the world-sheet partition function and the string field action finite and cut-off independent, although the resulting string field action has an unpalatable dependence on the leading unrenormalizable coupling.
This is a study of the Landau-Ginzburg/Calabi-Yau correspondence, and related matters, using linear sigma models.
Recently, background independent open-string field theory has been formally defined in the space of all two-dimensional world-sheet theories. In this paper, to make the construction more concrete, I compute the action for an off-shell tachyon field of a certain simple type. From the computation it emerges that, although the string field action does not coincide with the world-sheet (matter) partition function in general, these functions do coincide on shell. This can be demonstrated in general, as long as matter and ghosts are decoupled.
A framework for background independent open-string field theory is proposed. The approach involves using the BV formalism -- in a way suggested by recent developments in closed-string field theory -- to implicitly define a gauge invariant Lagrangian in a hypothetical ``space of all open-string world-sheet theories.'' It is built into the formalism that classical solutions of the string field theory are BRST invariant open-string world-sheet theories and that, when expanding around a classical solution, the infinitesimal gauge transformations are generated by the world-sheet BRST operator.
We present a conformal field theory -- obtained from a gauged WZW model -- that describes a closed, inhomogeneous expanding and recollapsing universe in $3+1$ dimensions. A possible violation of cosmic censorship is avoided because the universe recollapses just when a naked singularity was about to form. The model has been chosen to have $c=4$ (or $\widehat c=4$ in the supersymmetric case), just like four dimensional Minkowski space.
This lecture surveys a few loosely related topics, ranging from the scarcity of quantum field theories -- and the role that this has played, and still plays, in physics -- to paradoxes involving black holes in soluble two dimensional string theory and the question of whether naked singularities might be of even greater interest to string theorists than black holes.