A bstract We describe an algebra of observables for a static patch in de Sitter space, with operators gravitationally dressed to the worldline of an observer. The algebra is a von Neumann algebra of Type II 1 . There is a natural notion of entropy for a state of such an algebra. There is a maximum entropy state, which corresponds to empty de Sitter space, and the entropy of any semiclassical state of the Type II 1 algebras agrees, up to an additive constant independent of the state, with the expected generalized entropy S gen = ( A/ 4 G N ) + S out . An arbitrary additive constant is present because of the renormalization that is involved in defining entropy for a Type II 1 algebra.
It is shown that in compactification of superstrings on manifolds of SU(3) holonomy, the superpotential receives no string theoretic corrections from the form it takes in the field theoretic limit, at least to all finite orders in sigma model perturbation theory. Modulo nonperturbative effects, this implies that those manifolds do indeed obey the exact classical equations of super-string theory, as has been argued on other grounds. Also, it is pointed out that the superpotential – even in the field theory limit – contains terms coupling charged fields to E6 singlets as well as self-couplings of the charged fields. A slightly tentative argument is given that on certain manifolds of SU(3) holonomy – though not all – it is possible to find conformally invariant sigma models that, while keeping unbroken supersymmetry, break E8 to SO(10) or SU(5) rather than E6. Including the effects of Wilson lines this would mean that E8 could be broken precisely to SU(3) × SU(2) × U(1) while keeping unbroken supersymmetry. These facts may open avenues for solving the problems associated with neutrino masses, proton decay, and renormalization group calculations of coupling constants. They also may lead to models with fewer unknown parameters than have been present in previous quasi-realistic models.
We discuss high energy properties of states for (possibly interacting) quantum fields in curved spacetimes. In particular, if the spacetime is real analytic, we show that an analogue of the timelike tube theorem and the Reeh–Schlieder property hold with respect to states satisfying a weak form of microlocal analyticity condition. The former means the von Neumann algebra of observables of a spacelike tube equals the von Neumann algebra of observables of a significantly bigger region that is obtained by deforming the boundary of the tube in a timelike manner. This generalizes theorems by Araki (Helv Phys Acta 36:132–139, 1963) and Borchers (Nuovo Cim (10) 19:787–793, 1961) to curved spacetimes.
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In these notes, I will sketch a new approach to Khovanov homology of knots and links based on counting the solutions of certain elliptic partial differential equations in four and five dimensions.The equations are formulated on four and five-dimensional manifolds with boundary, with a rather subtle boundary condition that encodes the knots and links.The construction is formally analogous to Floer and Donaldson theory in three and four dimensions.It was discovered using quantum field theory arguments but can be described and understood purely in terms of classical gauge theory.
No abstract is provided for this article.
In these lectures, I will describe some aspects of an approach to the Jones polynomial of knots, and its generalizations, that is based on a three-dimensional quantum Yang-Mills theory in which the usual Yang-Mills Lagrangian is replaced by a Chern-Simons action. This approach gives a manifestly three-dimensional approach to the subject, but some of the key aspects of the story, involving the Feynman path integral, are somewhat beyond the reach of present rigorous understanding. In the first two lectures, I will describe aspects of the subject that can be developed rigorously at present. This basically consists of a gauge theory approach to the Jones representations of the braid group and their generalizations. In the last lecture, I will describe the more ambitious Feynman path integral approach, which is an essential part of the way that physicists actually think about problems such as this one, and which gives the most far-reaching results.
The theoretical physics of the first quarter of the twentieth century —centering around relativity theory and nonrelativistic quantum mechanics—has had a broad influence mathematically. The main achievement of theoretical physics in the following half-century was the development of quantum field theory or QFT. Yet the mathematical influence of QFT still belongs largely to the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="21 Superscript s t"> <mml:semantics> <mml:msup> <mml:mn>21</mml:mn> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>s</mml:mi> <mml:mi>t</mml:mi> </mml:mrow> </mml:msup> <mml:annotation encoding="application/x-tex">21^{st}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> century, because its mathematical foundations are still not well-understood.
In this paper, we explore supersymmetric and 2d analogs of the SYK model. We begin by working out a basis of (super)conformal eigenfunctions appropriate for expanding a four-point function. We use this to clarify some details of the 1d supersymmetric SYK model. We then introduce new bosonic and supersymmetric analogs of SYK in two dimensions. These theories consist of $N$ fields interacting with random $q$-field interactions. Although models built entirely from bosons appear to be problematic, we find a supersymmetric model that flows to a large $N$ CFT with interaction strength of order one. We derive an integral formula for the four-point function at order $1/N$, and use it to compute the central charge, chaos exponent and some anomalous dimensions. We describe a problem that arises if one tries to find a 2d SYK-like CFT with a continuous global symmetry.
The correspondence between string theory in Anti-de Sitter space and super Yang Mills theory is an example of the Holographic principle according to which a quantum theory with gravity must be describable by a boundary theory. However, arguments given so far are incomplete because, while the bulk theory has been related to a boundary theory, the holographic bound saying that the boundary theory has only one bit of information per Planck area has not been justified. We show here that this bound is the physical interpretation of one of the unusual aspects of the correspondence between Anti-de Sitter space and the boundary conformal field theory, which is that infrared effects in the bulk theory are reflected as ultraviolet effects in the boundary theory.
Perturbative fermion anomalies in spacetime dimension $d$ have a well-known relation to Chern-Simons functions in dimension $D=d+1$. This relationship is manifested in a beautiful way in "anomaly inflow" from the bulk of a system to its boundary. Along with perturbative anomalies, fermions also have global or nonperturbative anomalies, which can be incorporated by using the $η$-invariant of Atiyah, Patodi, and Singer instead of the Chern-Simons function. Here we give a nonperturbative description of anomaly inflow, involving the $η$-invariant. This formula has been expected in the past based on the Dai-Freed theorem, but has not been fully justified. It leads to a general description of perturbative and nonperturbative fermion anomalies in $d$ dimensions in terms of an $η$-invariant in $D$ dimensions. This $η$-invariant is a cobordism invariant whenever perturbative anomalies cancel.
We show that three dimensional Chern-Simons gauge theories with a compact gauge groupG (not necessarily connected or simply connected) can be classified by
The Feynman path integral of ordinary quantum mechanics is complexified and it is shown that possible integration cycles for this complexified integral are associated with branes in a two-dimensional A-model.This provides a fairly direct explanation of the relationship of the A-model to quantum mechanics; such a relationship has been explored from several points of view in the last few years.These phenomena have an analog for Chern-Simons gauge theory in three dimensions: integration cycles in the path integral of this theory can be derived from N = 4 super Yang-Mills theory in four dimensions.Hence, under certain conditions, a Chern-Simons path integral in three dimensions is equivalent to an N = 4 path integral in four dimensions.
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No abstract is provided for this article.
Symmetry protected topological (SPT) states have boundary 't Hooft anomalies that obstruct an effective boundary theory realized in its own dimension with UV completion and an on-site $G$-symmetry. In this work, yet we show that a certain anomalous non-on-site $G$ symmetry along the boundary becomes on-site when viewed as an extended $H$ symmetry, via a suitable group extension $1\to K\to H\to G\to1$. Namely, a non-perturbative global (gauge/gravitational) anomaly in $G$ becomes anomaly-free in $H$. This guides us to construct exactly soluble lattice path integral and Hamiltonian of symmetric gapped boundaries, always existent for any SPT state in any spacetime dimension $d \geq 2$ of any finite symmetry group, including on-site unitary and anti-unitary time-reversal symmetries. The resulting symmetric gapped boundary can be described either by an $H$-symmetry extended boundary of bulk $d \geq 2$, or more naturally by a topological emergent $K$-gauge theory with a global symmetry $G$ on a 3+1D bulk or above. The excitations on such a symmetric topologically ordered boundary can carry fractional quantum numbers of the symmetry $G$, described by representations of $H$. (Apply our approach to a 1+1D boundary of 2+1D bulk, we find that a deconfined gauge boundary indeed has spontaneous symmetry breaking with long-range order. The deconfined symmetry-breaking phase crosses over smoothly to a confined phase without a phase transition.) In contrast to known gapped interfaces obtained via symmetry breaking (either global symmetry breaking or Anderson-Higgs mechanism for gauge theory), our approach is based on symmetry extension. More generally, applying our approach to SPT, topologically ordered gauge theories and symmetry enriched topologically ordered (SET) states, leads to generic boundaries/interfaces constructed with a mixture of symmetry breaking, symmetry extension, and dynamical gauging.
We analyze the u-plane contribution to Donaldson invariants of a four-manifold X. For $b_2^+(X)>1$, this contribution vanishes, but for $b_2^+=1$, the Donaldson invariants must be written as the sum of a u-plane integral and an SW contribution. The u-plane integrals are quite intricate, but can be analyzed in great detail and even calculated. By analyzing the u-plane integrals, the relation of Donaldson theory to N=2 supersymmetric Yang-Mills theory can be described much more fully, the relation of Donaldson invariants to SW theory can be generalized to four-manifolds not of simple type, and interesting formulas can be obtained for the class numbers of imaginary quadratic fields. We also show how the results generalize to extensions of Donaldson theory obtained by including hypermultiplet matter fields.
DOE Final Report “Strings 2014” PI: Edward Witten, Institute for Advanced Study, Princeton, NJ 08540 CO-PI: Igor Klebanov, Princeton University, Princeton, NJ 08540 DOE Grant Number: DE-SC0011919 The Strings 2014 meeting was held at Princeton University in June 2014, co-sponsored by Princeton University and the Institute for Advanced Study. Plenary lectures at Strings 2014 were held in Richardson Auditorium of Princeton University. This comfortable and spacious facility easily accommodated the 616 participants registered participants at Strings 2014. The rental fee for the auditorium was $11,000. This grant provided $5,500 from the Department of Energy to pay for one-half of the cost of the facility rental and videotaping. Speakers were supported with funds from the National Science Foundation Clay Mathematics Institute, the Institute for Advanced Study and Princeton University. The organization of Strings 2014 consisted of an International Organizing Committee of 60 prominent scientists around the world, and a Local Advisory Committee consisting of an additional 15 distinguished scientists from neighboring institutions. Additionally, the Local Organizing Committee assisted them with about 15 members (mostly faculty at Princeton University and the Institute for Advanced Study). These groups (which are listed at the end of this narrative) offered important input concerning the selectionmore » of speakers and helped to ensure that the speakers were selected from the broadest possible pool. The conference was held on June 23-7 at Princeton University and the Institute for Advanced Study. The 616 registered participants included 272 participants from the United States and 344 from 32 institutions outside of the U.S. We believe that we were successful at providing a stimulating and up-to-date overview of research in string theory and its relations to other areas of physics and mathematics, ranging from geometry to quantum field theory, condensed matter physics, and more. There were a total of 45 plenary speakers and 27 speakers at parallel sessions. (Parallel sessions were held at the Institute for Advanced Study.) Overall the speakers did an excellent job of presenting their topics and some presented surprising and novel results. The talks at Strings 2014 were videotaped and are available on the conference website: http://physics.princeton.edustrings2014/Talk_titles.shtml. One important facet of Strings 2014 and one of the reasons it was so well-attended was that it had a strong educational component. The week before the meeting, there was a summer school, Prospects in Theoretical Physics (PiTP), held at the Institute for Advanced Study on the subject of string theory. 260 graduate students attended both PiTP and Strings 2014. The group consisted of 25 females and 235 males; 208 graduate students and 52 postdocs. 129 participants were from the United States, and 131 participants came from institutions in 25 countries outside of the U.S. The Institute for Advanced Study substantially subsidized the summer school for students. Over two dozen students had the chance to give short (six minute) talks at the “gong shows” that were held at PiTP and Strings 2014, and nearly 60 students and postdocs made poster presentations at Strings 2014.« less
