Gravity and Yang-Mills theory

Three of the four forces of Nature are described by quantum Yang-Mills theories with remarkable precision. The fourth force, gravity, is described classically by the Einstein-Hilbert theory. There appears to be an inherent incompatibility between quantum mechanics and the Einstein-Hilbert theory which prevents us from developing a consistent quantum theory of gravity. The Einstein-Hilbert theory is therefore believed to differ greatly from Yang-Mills theory (which does have a sensible quantum mechanical description). It is therefore very surprising that these two theories actually share close perturbative ties. This article focuses on these ties between Yang-Mills theory and the Einstein-Hilbert theory. We discuss the origin of these ties and their implications for a quantum theory of gravity.Comment: 6 pages, based on contribution to GRF 2010, to appear in a special edition of IJMP

Dualities in integrable systems and N=2 theories

We discuss dualities of the integrable dynamics behind the exact solution to the N=2 SUSY YM theory. It is shown that T duality in the string theory is related to the separation of variables procedure in dynamical system. We argue that there are analogues of S duality as well as 3d mirror symmetry in the many-body systems of Hitchin type governing low-energy effective actions.Comment: 16 pages, Latex, Talk given at QFTHEP-99, Moscow, May 27-June

Fractional quantum Hall effect on the two-sphere: a matrix model proposal

We present a Chern-Simons matrix model describing the fractional quantum Hall effect on the two-sphere. We demonstrate the equivalence of our proposal to particular restrictions of the Calogero-Sutherland model, reproduce the quantum states and filling fraction and show the compatibility of our result with the Haldane spherical wavefunctions.Comment: 26 pages, LaTeX, no figures, references adde

Derived traces of Soergel categories

We study two kinds of categorical traces of (monoidal) dg categories, with particular interest in categories of Soergel bimodules. First, we explicitly compute the usual Hochschild homology, or derived vertical trace, of the category of Soergel bimodules in arbitrary types. Secondly, we introduce the notion of derived horizontal trace of a monoidal dg category and compute the derived horizontal trace of Soergel bimodules in type A. As an application we obtain a derived annular Khovanov-Rozansky link invariant with an action of full twist insertion, and thus a categorification of the HOMFLY-PT skein module of the solid torus

Combining historic records and multi-criteria habitat suitability analysis for the potential reintroduction of Lake Sturgeon (Acipenser fulvescens Rafinesque) into tributaries of Lake Erie

Predicting the location and quality of habitat for imperiled species is an increasingly important application of modeling technology. The Lake Sturgeon (Acipenser fulvescens) is a widely-extirpated fish of the Laurentian Great Lakes whose recovery is dependent on the availability and connectivity of suitable stream habitat today. This is especially true in Lake Erie, where the largest Lake Sturgeon fishery was once found. I predicted that modern habitat suitability would be dependent on land use legacies from the past 200 years, with western Lake Erie tributaries having less suitable habitat compared to the eastern Lake Erie tributaries. I developed a multi-criteria habitat suitability analysis framework that was applied to two different spatial scales (watershed and stream segment) to predict the location and quality of habitat for spawning adults and juveniles in historically-used U.S. tributaries of Lake Erie. I also tested the transferability of the model framework by applying it to a stream where extant Lake Sturgeon spawn currently: the Black River in northern Michigan. My results suggest that a broad range of habitat qualities exist across the study region, with predictions aligning with several smaller-scale habitat suitability projects in the past in several of the watersheds analyzed here. Most low-scoring watersheds were located to the west, while the highest-scoring watersheds were located to the east, as predicted. The model found a high degree of agreement between the watershed scale and reach scale, suggesting that the framework could be applied at either scale accurately depending on input data availability. The model predicted that the Black River watershed is fairly suitable (40-50% suitable) for Lake Sturgeon, which warrants further investigation and ground-truthing of the model’s real-world accuracy given that the Black River is known to be highly suitable for the species. Additional spatially-explicit analysis of these results in the future will aim to reveal patterns in habitat connectivity from river mouth upstream for each watershed. My results can be used on the fine scale by managers seeking to develop local Lake Sturgeon reintroduction and restoration projects but also at the large scale for the purpose of communication and habitat connectivity for the benefit of multiple populations of Lake Sturgeon

Coadjoint orbits of the Virasoro algebra and the global Liouville equation

The classification of the coadjoint orbits of the Virasoro algebra is reviewed and is then applied to analyze the so-called global Liouville equation. The review is self-contained, elementary and is tailor-made for the application. It is well-known that the Liouville equation for a smooth, real field $\phi$ under periodic boundary condition is a reduction of the SL(2,R) WZNW model on the cylinder, where the WZNW field g in SL(2,R) is restricted to be Gauss decomposable. If one drops this restriction, the Hamiltonian reduction yields, for the field $Q=\kappa g_{22}$ where $\kappa\neq 0$ is a constant, what we call the global Liouville equation. Corresponding to the winding number of the SL(2,R) WZNW model there is a topological invariant in the reduced theory, given by the number of zeros of Q over a period. By the substitution $Q=\pm\exp(- \phi/2)$, the Liouville theory for a smooth $\phi$ is recovered in the trivial topological sector. The nontrivial topological sectors can be viewed as singular sectors of the Liouville theory that contain blowing-up solutions in terms of $\phi$. Since the global Liouville equation is conformally invariant, its solutions can be described by explicitly listing those solutions for which the stress-energy tensor belongs to a set of representatives of the Virasoro coadjoint orbits chosen by convention. This direct method permits to study the `coadjoint orbit content' of the topological sectors as well as the behaviour of the energy in the sectors. The analysis confirms that the trivial topological sector contains special orbits with hyperbolic monodromy and shows that the energy is bounded from below in this sector only.Comment: Plain TEX, 48 pages, final version to appear in IJMP

Anomalous Zero Sound

We show that the anomalous term in the current, recently suggested by Son and Yamamoto, modifies the structure of the zero sound mode in the Fermi liquid in a magnetic field.Comment: 14 pages, 2 figure

More on the Tensor Response of the QCD Vacuum to an External Magnetic Field

In this Letter we discuss a few issues concerning the magnetic susceptibility of the quark condensate and the Son-Yamamoto (SY) anomaly matching equation. It is shown that the SY relation in the IR implies a nontrivial interplay between the kinetic and WZW terms in the chiral Lagrangian. It is also demonstrated that in a holographic framework an external magnetic field triggers mixing between scalar and tensor fields. Accounting for this, one may calculate the magnetic susceptibility of the quark condensate to all orders in the magnetic field.Comment: 20 pages, 2 figure