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We discuss some aspects of symmetry breakings in string theories. We mainly focus on the algebra of bosonic strings, but suspect similar ideas may be used for superstring theories. We point out a curious relation between the partition function of certain bosonic string theories and superstring theories.
Revised 4 July 1986 The connection of recently constructed lower dimensional heterotic strings with conventional toroidal compactification is clarified.
We generalize the recently discovered relationship between JT gravity and double-scaled random matrix theory to the case that the boundary theory may have time-reversal symmetry and may have fermions with or without supersymmetry. The matching between variants of JT gravity and matrix ensembles depends on the assumed symmetries. Time-reversal symmetry in the boundary theory means that unorientable spacetimes must be considered in the bulk. In such a case, the partition function of JT gravity is still related to the volume of the moduli space of conformal structures, but this volume has a quantum correction and has to be computed using Reidemeister-Ray-Singer "torsion." Presence of fermions in the boundary theory (and thus a symmetry $(-1)^F$) means that the bulk has a spin or pin structure. Supersymmetry in the boundary means that the bulk theory is associated to JT supergravity and is related to the volume of the moduli space of super Riemann surfaces rather than of ordinary Riemann surfaces. In all cases we match JT gravity or supergravity with an appropriate random matrix ensemble. All ten standard random matrix ensembles make an appearance -- the three Dyson ensembles and the seven Altland-Zirnbauer ensembles. To facilitate the analysis, we extend to the other ensembles techniques that are most familiar in the case of the original Wigner-Dyson ensemble of hermitian matrices. We also generalize Mirzakhani's recursion for the volumes of ordinary moduli space to the case of super Riemann surfaces.
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For various reasons, it seems necessary to include complex saddle points in the Euclidean path integral of General Relativity. But some sort of restriction on the allowed complex saddle points is needed to avoid various unphysical examples. In this article, a speculative proposal is made concerning a possible restriction on the allowed saddle points in the gravitational path integral. The proposal is motivated by recent work of Kontsevich and Segal on complex metrics in quantum field theory, and earlier work of Louko and Sorkin on topology change from a real time point of view.
We derive a holomorphic anomaly equation for the Vafa-Witten partition function for twisted four-dimensional N = 4 super Yang-Mills theory on CP 2 for the gauge group SO(3) from the path integral of the effective theory on the Coulomb branch.The holomorphic kernel of this equation, which receives contributions only from the instantons, is not modular but 'mock modular'.The partition function has correct modular properties expected from S-duality only after including the anomalous nonholomorphic boundary contributions from anti-instantons.Using M-theory duality, we relate this phenomenon to the holomorphic anomaly of the elliptic genus of a two-dimensional noncompact sigma model and compute it independently in two dimensions.The anomaly both in four and in two dimensions can be traced to a topological term in the effective action of six-dimensional (2, 0) theory on the tensor branch.We consider generalizations to other manifolds and other gauge groups to show that mock modularity is generic and essential for exhibiting duality when the relevant field space is noncompact.
K-theory provides a framework for classifying Ramond-Ramond (RR) charges and fields. K-theory of manifolds has a natural extension to K-theory of noncommutative algebras, such as the algebras considered in noncommutative Yang-Mills theory or in open string field theory. In a number of concrete problems, the K-theory analysis proceeds most naturally if one starts out with an infinite set of D-branes, reduced by tachyon condensation to a finite set. This suggests that string field theory should be reconsidered for N=∞.
Chromodynamics with $n$ flavors of massless quarks is invariant under chiral $\mathrm{U}(n)\ensuremath{\bigotimes}\mathrm{U}(n)$. It is shown that in the limit of a large number of colors, under reasonable assumptions, this symmetry group must spontaneously break down to diagonal $\mathrm{U}(n)$.
The gauge theory approach to the geometric Langlands program is extended to the case of wild ramification. The new ingredients that are required, relative to the tamely ramified case, are differential operators with irregular singularities, Stokes phenomena, isomonodromic deformation, and, from a physical point of view, new surface operators associated with higher order singularities.
A brane configuration is described that is relevant to understanding the dynamics of N = 1 supersymmetric Yang-Mills theory. Confinement and spontaneous breaking of a discrete chiral symmetry can be understood as consequences of the topology of the brane. Because of the symmetry breaking, there can be domain walls separating different vacua; the QCD string can end on such a domain wall. The model in which these properties can be understood semiclassically does not coincide with supersymmetric Yang-Mills theory but is evidently in the same universality class.
A proposal for a gauge invariant nonlinear classical field theory of open superstrings is explored. The gauge fixing that gives rise to the standard structure in the Ramond sector is described, answering questions that have been raised in recent literature. The model is shown to possess space-time supersymmetry as well as gauge invariance. The proof of space-time supersymmetry uses the covariant form of the fermion emission vertex and the associated picture changing operation, in accord with suggestions by several authors. Also discussed are the canonical formulation of the theory, and some issues connected with the extension of the work to closed strings.
We analyze global anomalies for elementary Type II strings in the presence of D-branes. Global anomaly cancellation gives a restriction on the D-brane topology. This restriction makes possible the interpretation of D-brane charge as an element of K-theory.
It is shown that world-sheets violate the Peccie-Quinn symmetries associated with some of the axions that arise in superstring compactification. In the case of unbroken low-energy supersymmetry, the Peccie-Quinn violating operators that arise are D terms of dimension six. One remaining axion is unaffected.
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