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## Quasi BPS Wilson loops, localization of loop equation by homology and exact beta function in the large N limit of SU(N) Yang-Mills theory

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Relevância na Pesquisa

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We localize the loop equation of large-N YM theory in the ASD variables on a
critical equation for an effective action by means of homological methods as
opposed to the cohomological localization of equivariantly closed forms in
local field theory. Our localization occurs for some special simple quasi BPS
Wilson loops, that have no perimeter divergence and no cusp anomaly for
backtracking cusps, in a partial Eguchi-Kawai reduction from four to two
dimensions of the non-commutative theory in the limit of infinite
non-commutativity and in a lattice regularization in which the ASD integration
variables live at the points of the lattice, thus implying an embedding of
parabolic Higgs bundles in the YM functional integral. We find that the beta
function of the effective action is saturated by the non-commutative ASD
vortices of the EK reduction. An exact canonical beta function of NSVZ type
that reproduces the universal first and second perturbative coefficients
follows by the localization on vortices. Finally we argue that a scheme can be
found in which the canonical coupling coincides with the physical charge
between static quark sources in the large-N limit and we compare our
theoretical calculation with some numerical lattice result.; Comment: 52 pages...

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## Weak solutions of the cohomological equation on R^2 for regular vector fields

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 20/01/2014

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In a recent article, we studied the global solvability of the so-called
cohomological equation L_X f=g in C^\infty(\Rt), where X is a regular vector
field on the plane and L_X the corresponding Lie derivative. In a joint article
with T. Gramchev and A. Kirilov, we studied the existence of global L^1_{loc}
weak solutions of the cohomological equation for vector fields depending only
on one coordinate. Here we generalize the results of both articles by providing
explicit conditions for the existence of global weak solutions to the
cohomological equation when X is intrinsically Hamiltonian or of finite type.; Comment: 22 pages, 6 figures

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## Cohomological equation and cocycle rigidity of parabolic actions in $SL(n,\RR)$

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Relevância na Pesquisa

46.6%

For any unitary representation $(\pi,\mathcal{H})$ of $G=SL(n,\RR)$, $n\geq
3$ without non-trivial $G$-invariant vectors, we study smooth solutions of the
cohomological equation $\mathfrak{u}f=g$ where $\mathfrak{u}$ is a vector in
the root space of $\mathfrak{sl}(n,\RR)$ and $g$ is a given vector in
$\mathcal{H}$. We characterize the obstructions to solving the cohomological
equation, construct smooth solutions of the cohomological equation and obtain
tame Sobolev estimates for $f$.
We also study common solutions to (the infinitesimal version of) the cocycle
equation $\mathfrak{u}h=\mathfrak{v}g$, where $\mathfrak{u}$ and $\mathfrak{v}$
are commutative vectors in different root spaces of $\mathfrak{sl}(n,\RR)$ and
$g$ and $h$ are given vectors in $\mathcal{H}$. We give precisely the condition
under which the cocycle equation has common solutions: $(*)$ if $\mathfrak{u}$
and $\mathfrak{v}$ embed in $\mathfrak{sl}(2,\RR)\times \RR$, then the common
solution exists. Otherwise, we show counter examples in each $SL(n,\RR)$,
$n\geq 3$. As an application, we obtain smooth cocycle rigidity for higher rank
parabolic actions over $SL(n,\RR)/\Gamma$, $n\geq 4$ if the Lie algebra of the
acting parabolic subgroup contains a pair $\mathfrak{u}$ and $\mathfrak{v}$
satisfying property $(*)$ and prove that the cocycle rigidity fails otherwise.
Especially...

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## On the cohomological equation for interval exchange maps

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 28/04/2003

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46.45%

We exhibit an explicit full measure class of minimal interval exchange maps T
for which the cohomological equation $\Psi -\Psi\circ T=\Phi$ has a bounded
solution $\Psi$ provided that the datum $\Phi$ belongs to a finite codimension
subspace of the space of functions having on each interval a derivative of
bounded variation.
The class of interval exchange maps is characterized in terms of a
diophantine condition of ``Roth type'' imposed to an acceleration of the
Rauzy--Veech--Zorich continued fraction expansion associated to T.
Contents
0. French abridged version
1. Interval exchange maps and the cohomological equation. Main Theorem
2. Rauzy--Veech--Zorich continued fraction algorithm and its acceleration
3. Special Birkhoff sums
4. The Diophantine condition
5. Sketch of the proof of the theorem; Comment: 11 pages, french abstract and abridged version

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## On the cohomological equation of magnetic flows

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 29/07/2008

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46.27%

We consider a magnetic flow without conjugate points on a closed manifold $M$
with generating vector field $\G$. Let $h\in C^{\infty}(M)$ and let $\theta$ be
a smooth 1-form on $M$. We show that the cohomological equation \[\G(u)=h\circ
\pi+\theta\] has a solution $u\in C^{\infty}(SM)$ only if $h=0$ and $\theta$ is
closed. This result was proved in \cite{DP2} under the assumption that the flow
of $\G$ is Anosov.

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## Asymptotics for the wave equation on differential forms on Kerr-de Sitter space

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 10/02/2015

Relevância na Pesquisa

26.16%

#Mathematics - Analysis of PDEs#General Relativity and Quantum Cosmology#Mathematics - Spectral Theory#Primary 35P25, Secondary 35L05, 35Q61, 83C57

We study asymptotics for solutions of Maxwell's equations, in fact of the
Hodge-de Rham equation $(d+\delta)u=0$ without restriction on the form degree,
on a geometric class of stationary spacetimes with a warped product type
structure (without any symmetry assumptions), which in particular include
Schwarzschild-de Sitter spaces of all spacetime dimensions $n\geq 4$. We prove
that solutions decay exponentially to $0$ or to stationary states in every form
degree, and give an interpretation of the stationary states in terms of
cohomological information of the spacetime. We also study the wave equation on
differential forms and in particular prove analogous results on
Schwarzschild-de Sitter spacetimes. We demonstrate the stability of our
analysis and deduce asymptotics and decay for solutions of Maxwell's equations,
the Hodge-de Rham equation and the wave equation on differential forms on
Kerr-de Sitter spacetimes with small angular momentum.; Comment: 43 pages

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## Cohomological construction of relative twists

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Relevância na Pesquisa

26.15%

Let g be a complex, semi-simple Lie algebra, h a Cartan subalgebra of g and D
a subdiagram of the Dynkin diagram of g. Let g_D and l_D be the corresponding
semi-simple and Levi subalgebras and consider two invariant solutions Phi,
Phi_D of the pentagon equation for g and g_D respectively. Motivated by the
theory of quasi-Coxeter quasitriangular quasibialgebras \cite{TL3}, we study in
this paper the existence of a relative twist, that is an element F invariant
under l_D such that the twist of Phi by F is Phi_D. Adapting the method of
Donin and Shnider, who treated the case of an empty D, so that l_D=h and
Phi_D=1, we give a cohomological construction of such an F under the assumption
that Phi_D is the image of Phi under the generalised Harish-Chandra
homomorphism. We also show that F is unique up to a gauge transformation if l_D
is of corank 1 or F satisfies F^\Theta= F^{21} where \Theta is an involution of
g acting as -1 on h.; Comment: minor touch-ups, to appear in Advances in Math

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## Cohomological equations for suspension flows over Vershik automorphisms

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Relevância na Pesquisa

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In this paper we give sufficient conditions for existence of a solution of
cohomological equation for suspension flows over automorphisms of Markov
compacta, which were introduced by Vershik and Ito. The main result (Theorem 1)
can be regarded as a symbolic analogue of results due to Forni and Marmi,
Moussa and Yoccoz for translation flows and interval exchange transformations.; Comment: 10 pp. Comments are welcome! ver.2: small corrections are made

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## The cohomological equation for Roth type interval exchange maps

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 30/03/2004

Relevância na Pesquisa

46.56%

We exhibit an explicit full measure class of minimal interval exchange maps T
for which the cohomological equation $\Psi -\Psi\circ T=\Phi$ has a bounded
solution $\Psi$ provided that the datum $\Phi$ belongs to a finite codimension
subspace of the space of functions having on each interval a derivative of
bounded variation. The class of interval exchange maps is characterized in
terms of a diophantine condition of ``Roth type'' imposed to an acceleration of
the Rauzy--Veech--Zorich continued fraction expansion associated to T.
CONTENTS 0. Introduction 1. The continued fraction algorithm for interval
exchange maps 1.1 Interval exchnge maps 1.2 The continued fraction algorithm
1.3 Roth type interval exchange maps 2. The cohomological equation 2.1 The
theorem of Gottschalk and Hedlund 2.2 Special Birkhoff sums 2.3 Estimates for
functions of bounded variation 2.4 Primitives of functions of bounded variation
3. Suspensions of interval exchange maps 3.1 Suspension data 3.2 Construction
of a Riemann surface 3.3 Compactification of $M_\zeta^*$ 3.4 The cohomological
equation for higher smoothness 4. Proof of full measure for Roth type 4.1 The
basic operation of the algorithm for suspensions 4.2 The Teichm\"uller flow 4.3
The absolutely continuous invariant measure 4.4 Integrability of $\log\Vert
Z_{(1)}\Vert$ 4.5 Conditions (b) and (c) have full measure 4.6 The main step
4.7 Condition (a) has full measure 4.8 Proof of the Proposition Appendix A
Roth--type conditions in a concrete family of i.e.m. Appendix B A non--uniquely
ergodic i.e.m. satsfying condition (a) References; Comment: 64 pages...

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## BCOV theory via Givental group action on cohomological field theories

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 03/10/2008

Relevância na Pesquisa

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In a previous paper (arXiv:0704.1001), Losev, me, and Shneiberg constructed a
full descendant potential associated to an arbitrary cyclic Hodge dGBV algebra.
This contruction extended the construction of Barannikov and Kontsevich of
solution of the WDVV equation, based on the earlier paper of Bershadsky,
Cecotti, Ooguri, and Vafa. In the present paper, we give an interpretation of
this full descendant potential in terms of Givental group action on
cohomological field theories. In particular, the fact that it satisfies all
tautological equations becomes a trivial observation.; Comment: 22 pages

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## Four-dimensional couplings among BF and massless Rarita-Schwinger theories: a BRST cohomological approach

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 19/12/2008

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The local and manifestly covariant Lagrangian interactions in four spacetime
dimensions that can be added to a free model that describes a massless
Rarita-Schwinger theory and an Abelian BF theory are constructed by means of
deforming the solution to the master equation on behalf of specific
cohomological techniques.; Comment: 59 pages

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## H\"older regularity of the solutions of the cohomological equation for Roth type interval exchange maps

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 07/07/2014

Relevância na Pesquisa

46.27%

We prove that the solutions of the cohomological equation for Roth type
interval exchange maps are H\"older continuous provided that the datum is of
class $C^r$ with $r>1$ and belongs to a finite-codimension linear subspace.; Comment: 22 pages

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## The Cohomological Equation and Invariant Distributions for Horocycle Maps

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Relevância na Pesquisa

46.27%

We study the invariant distributions for the horocycle map on
$\Gamma\backslash SL(2, \mathbb{R})$ and prove Sobolev estimates for the
cohomological equation of the horocycle map. As an application, we obtain an
estimate for the rate of equidistribution for horocycle maps on compact
manifolds.; Comment: 44 pages

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## Heterotic horizons, Monge-Ampere equation and del Pezzo surfaces

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Relevância na Pesquisa

26.27%

Heterotic horizons preserving 4 supersymmetries have sections which are T^2
fibrations over 6-dimensional conformally balanced Hermitian manifolds. We give
new examples of horizons with sections S^3 X S^3 X T^2 and SU(3). We then
examine the heterotic horizons which are T^4 fibrations over a Kahler
4-dimensional manifold. We prove that the solutions depend on 6 functions which
are determined by a non-linear differential system of 6 equations that include
the Monge-Ampere equation. We show that this system has an explicit solution
for the Kahler manifold S^2 X S^2. We also demonstrate that there is an
associated cohomological system which has solutions on del Pezzo surfaces. We
raise the question of whether for every solution of the cohomological problem
there is a solution of the differential system, and so a new heterotic horizon.
The horizon sections have topologies which include ((k-1) S^2 X S^4 # k (S^3 X
S^3)) X T^2$ indicating the existence of exotic black holes. We also find an
example of a horizon section which gives rise to two different near horizon
geometries.; Comment: 33 pages, latex. Reference added

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## Epsilon-neighborhoods of orbits of parabolic diffeomorphisms and cohomological equations

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Relevância na Pesquisa

36.37%

In this article, we study analyticity properties of (directed) areas of
epsilon-neighborhoods of orbits of parabolic germs. The article is motivated by
the question of analytic classification using epsilon-neighborhoods of orbits
in the simplest formal class. We show that the coefficient in front of
epsilon^2 term in the asymptotic expansion in epsilon, which we call the
principal part of the area, is a sectorially analytic function of initial point
of the orbit. It satisfies a cohomological equation similar to the standard
trivialization equation for parabolic diffeomorphisms. We give necessary and
sufficient conditions on a diffeomorphism f for the existence of globally
analytic solution of this equation. Furthermore, we introduce new
classification type for diffeomorphisms implied by this new equation and
investigate the relative position of its classes with respect to the analytic
classes.; Comment: 41 pages, 1 figures

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## Quasianalytic monogenic solutions of a cohomological equation

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 18/01/2001

Relevância na Pesquisa

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We prove that the solutions of a cohomological equation of complex dimension
one and in the analytic category have a monogenic dependence on the parameter,
and we investigate the question of their quasianalyticity. This equation is the
standard linearized conjugacy equation for germs of holomorphic maps in a
neighborhood of a fixed point. The parameter is the eigenvalue $q$ of the
linear part. This problem has been first investigated by Arnol'd and Herman.
Herman raised the question whether the solutions of the cohomological equation
had a quasianalytic dependence on the parameter. Indeed they are analytic
outside $\S^1$ which is a natural boundary but the solutions are still defined
at points of $\S^1$ which lie ``far enough from resonances''. We adapt to our
case Herman's construction of an increasing sequence of compacts which avoid
resonances and prove that the solutions belong to the associated space of
monogenic functions. The solutions admit asymptotic expansions at the points of
$\S^1$ which satisfy some arithmetical condition, and Carleman's Theorem allows
us to answer negatively to the question of quasianalyticity at these points.
But resonances (roots of unity) lead to asymptotic expansions, for which
quasianalyticity is obtained as a particular case of \'Ecalle's theory of
resurgent functions. At constant-type points one can still recover the
solutions from their asymptotic expansions and obtain a special kind of
quasianalyticity. Our results are obtained by reducing the problem to the study
of a fundamental solution (which is the ``quantum logarithm''). We deduce as a
corollary of our work the proof of a conjecture of Gammel on the monogenic and
quasianalytic properties of a certain number-theoretical Borel-Wolff-Denjoy
series.; Comment: 76 pages...

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## Sobolev regularity of solutions of the cohomological equation

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Relevância na Pesquisa

46.27%

We refine the theory of the cohomological equation for translation flows on
higher genus surfaces with the goal of proving optimal results on the Sobolev
regularity of solutions and of distributional obstructions. For typical
translation surfaces our results are sharp and we find the expected relation
between the regularity of the distributional obstructions and the Lyapunov
exponents of the Kontsevich-Zorich renormalization cocycle. As a consequence we
exactly determine the dimension of the space of obstructions in each Sobolev
regularity class in terms of the Kontsevich-Zorich exponents. For a fixed
arbitrary translation surface and a typical direction, our results are probably
not optimal but are the best which can be achieved with the available harmonic
analysis techniques we have introduced in an earlier paper.; Comment: 119 pages

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## The cohomological equation for partially hyperbolic diffeomorphisms

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 29/09/2008

Relevância na Pesquisa

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We establish a theory for the existence and regularity of solutions to the
cohomological equation over an accessible, partially hyperbolic diffeomorphism.
As a by-product of our techniques, we show that for $r>1$, any $C^r$
homogeneous, locally compact submanifold of a $C^r$ manifold is in fact a $C^r$
submanifold.; Comment: 94 pages, 8 figures

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## Interacting mixed-symmetry type tensor gauge fields of degrees two and three: a four-dimensional cohomological approach

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 10/03/2003

Relevância na Pesquisa

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A special class of mixed-symmetry type tensor gauge fields of degrees two and
three in four dimensions is investigated from the perspective of the Lagrangian
deformation procedure based on cohomological BRST techniques. It is shown that
the deformed solution to the master equation can be taken to be nonvanishing
only at the first order in the coupling constant. As a consequence, we deduce
an interacting model with deformed gauge transformations, an open gauge algebra
and undeformed reducibility functions. The resulting coupled Lagrangian action
contains a quartic vertex and some ``mass'' terms involving only the tensor of
degree two. We discuss in what sense the results of the deformation procedure
derived here are complementary to recent others.; Comment: LaTeX 2.e, 10 pages

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## Equidistribution for higher-rank Abelian actions on Heisenberg nilmanifolds

Fonte: American Institute of Mathematical Sciences
Publicador: American Institute of Mathematical Sciences

Tipo: Artigo de Revista Científica

Publicado em /11/2015
ENG

Relevância na Pesquisa

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2010 Mathematics Subject Classification: Primary: 37C85, 37A17, 37A45; Secondary: 11K36, 11L07.; We prove quantitative equidistribution results for actions of Abelian subgroups of the (2g + 1)-dimensional Heisenberg group acting on compact (2g + 1)-dimensional homogeneous nilmanifolds. The results are based on the study of the C∞-cohomology of the action of such groups, on tame estimates of the associated cohomological equations and on a renormalization method initially applied by Forni to surface flows and by Forni and the second author to other parabolic flows. As an application we obtain bounds for finite Theta sums defined by real quadratic forms in g variables, generalizing the classical results of Hardy and Littlewood [25, 26] and the optimal result of Fiedler, Jurkat, and Körner [17] to higher dimension.

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