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Ohta S.,Kyoto University | Ohta S.,Max Planck Institute For Mathematik | Sturm K.-T.,University of Bonn
Archive for Rational Mechanics and Analysis

We study contractivity properties of gradient flows for functions on normed spaces or, more generally, on Finsler manifolds. Contractivity of the flows turns out to be equivalent to a new notion of convexity for the functions. This is different from the usual convexity along geodesics in non-Riemannian Finsler manifolds. As an application, we show that the heat flow on Minkowski normed spaces other than inner product spaces is not contractive with respect to the quadratic Wasserstein distance. © 2012 Springer-Verlag. Source

Gukov S.,California Institute of Technology | Gukov S.,Max Planck Institute For Mathematik
Journal of High Energy Physics

Abstract: Interpreting renormalization group flows as solitons interpolating between different fixed points, we ask various questions that are normally asked in soliton physics but not in renormalization theory. Can one count RG flows? Are there different “topological sectors” for RG flows? What is the moduli space of an RG flow, and how does it compare to familiar moduli spaces of (supersymmetric) dowain walls? Analyzing these questions in a wide variety of contexts — from counting RG walls to AdS/CFT correspondence — will not only provide favorable answers, but will also lead us to a unified general framework that is powerful enough to account for peculiar RG flows and predict new physical phenomena. Namely, using Bott’s version of Morse theory we relate the topology of conformal manifolds to certain properties of RG flows that can be used as precise diagnostics and “topological obstructions” for the strong form of the C-theorem in any dimension. Moreover, this framework suggests a precise mechanism for how the violation of the strong C-theorem happens and predicts “phase transitions” along the RG flow when the topological obstruction is non-trivial. Along the way, we also find new conformal manifolds in well-known 4d CFT’s and point out connections with the superconformal index and classifying spaces of global symmetry groups. © 2016, The Author(s). Source

Dimofte T.,Institute for Advanced Study | Dimofte T.,Cambridge College | Gukov S.,California Institute of Technology | Gukov S.,Max Planck Institute For Mathematik
Journal of High Energy Physics

We study S-dualities in analytically continued SL(2) Chern-Simons theory on a 3-manifold M. By realizing Chern-Simons theory via a compactification of a 6d five-brane theory on M, various objects and symmetries in Chern-Simons theory become related to objects and operations in dual 2d, 3d, and 4d theories. For example, the space of flat SL(2, ℂ) connections on M is identified with the space of supersymmetric vacua in a dual 3d gauge theory. The hidden symmetry [InlineMediaObject not available: see fulltext.] of SL(2) Chern-Simons theory can be identified as the S-duality transformation of N =4 super-Yang-Mills theory (obtained by compactifying the five-brane theory on a torus); whereas the mapping class group action in Chern-Simons theory on a three-manifold M with boundary C is realized as S-duality in 4d N =2 super-Yang-Mills theory associated with the Riemann surface C. We illustrate these symmetries by considering simple examples of 3-manifolds that include knot complements and punctured torus bundles, on the one hand, and mapping cylinders associated with mapping class group transformations, on the other. A generalization of mapping class group actions further allows us to study the transformations between several distinguished coordinate systems on the phase space of Chern-Simons theory, the SL(2) Hitchin moduli space. © 2013 SISSA, Trieste, Italy. Source

D'Adderio M.,Max Planck Institute For Mathematik | Moci L.,TU Berlin
European Journal of Combinatorics

We prove that the Ehrhart polynomial of a zonotope is a specialization of the arithmetic Tutte polynomial introduced by Moci (2012) [16]. We derive some formulae for the volume and the number of integer points of the zonotope. © 2012 Elsevier Ltd. Source

Mautner C.,Max Planck Institute For Mathematik
Selecta Mathematica, New Series

We give geometric descriptions of the category {Mathematical expression} of rational polynomial representations of {Mathematical expression} over a field {Mathematical expression} of degree {Mathematical expression} for {Mathematical expression}, the Schur functor and Schur-Weyl duality. The descriptions and proofs use a modular version of Springer theory and relationships between the equivariant geometry of the affine Grassmannian and the nilpotent cone for the general linear groups. Motivated by this description, we propose generalizations for an arbitrary connected complex reductive group of the category {Mathematical expression} and the Schur functor. © 2014 Springer Basel. Source

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