scholarly journals THE LOGARITHMIC RESIDUE DENSITY OF A GENERALIZED LAPLACIAN

2011 ◽  
Vol 90 (1) ◽  
pp. 53-80 ◽  
Author(s):  
JOUKO MICKELSSON ◽  
SYLVIE PAYCHA
Keyword(s):  
Dirac Operator ◽  
Differential Form ◽  
Flat Space ◽  
Invariant Polynomial ◽  
Four Dimensions ◽  
Special Cases ◽  
Twisted Dirac Operator ◽  
Finite Sum ◽  
Logarithmic Residue ◽  
Generalized Laplacian

AbstractWe show that the residue density of the logarithm of a generalized Laplacian on a closed manifold defines an invariant polynomial-valued differential form. We express it in terms of a finite sum of residues of classical pseudodifferential symbols. In the case of the square of a Dirac operator, these formulas provide a pedestrian proof of the Atiyah–Singer formula for a pure Dirac operator in four dimensions and for a twisted Dirac operator on a flat space of any dimension. These correspond to special cases of a more general formula by Scott and Zagier. In our approach, which is of perturbative nature, we use either a Campbell–Hausdorff formula derived by Okikiolu or a noncommutative Taylor-type formula.

scholarly journals Chiral correlators in $$ \mathcal{N} $$ = 2 superconformal quivers

2021 ◽  
Vol 2021 (5) ◽  
Author(s):  
Francesco Galvagno ◽  
Michelangelo Preti
Keyword(s):  
Field Theory ◽  
Perturbation Theory ◽  
Flat Space ◽  
Field Theories ◽  
Superconformal Field Theories ◽  
Four Dimensions ◽  
Yang Mills ◽  
The Matrix ◽  
Superspace Formalism ◽  
Model Approach

Abstract We consider a family of $$ \mathcal{N} $$ N = 2 superconformal field theories in four dimensions, defined as ℤq orbifolds of $$ \mathcal{N} $$ N = 4 Super Yang-Mills theory. We compute the chiral/anti-chiral correlation functions at a perturbative level, using both the matrix model approach arising from supersymmetric localisation on the four-sphere and explicit field theory calculations on the flat space using the $$ \mathcal{N} $$ N = 1 superspace formalism. We implement a highly efficient algorithm to produce a large number of results for finite values of N , exploiting the symmetries of the quiver to reduce the complexity of the mixing between the operators. Finally the interplay with the field theory calculations allows to isolate special observables which deviate from $$ \mathcal{N} $$ N = 4 only at high orders in perturbation theory.

Integrable and solvable systems, and relations among them

1985 ◽  
Vol 315 (1533) ◽  
pp. 451-457 ◽  
Keyword(s):  
Gordon Equation ◽  
Soliton Solutions ◽  
The Self ◽  
Sine Gordon Equation ◽  
Field Equations ◽  
Four Dimensions ◽  
Painlevé Property ◽  
All Solutions ◽  
Special Cases ◽  
Integrable Dynamical Systems

There are several different classes of differential equations that may be described as ‘integrable’ or ‘solvable’. For example, there are completely integrable dynamical systems; equations such as the sine—Gordon equation, which admit soliton solutions; and the self-dual gauge-field equations in four dimensions (with generalizations in arbitrarily large dimension). This lecture discusses two ideas that link all of these together: one is the Painlevé property, which says (roughly speaking) that all solutions to the equations are meromorphic; the other is that many of the equations are special cases (i.e. reductions) of others.

scholarly journals THE FLUCTUATION SPECTRA AROUND A GAUSSIAN CLASSICAL SOLUTION OF A TENSOR MODEL AND THE GENERAL RELATIVITY

2008 ◽  
Vol 23 (05) ◽  
pp. 693-718 ◽  
Author(s):  
NAOKI SASAKURA
Keyword(s):  
General Relativity ◽  
Classical Solution ◽  
Flat Space ◽  
Effective Field ◽  
Tensor Model ◽  
Metric Tensor ◽  
Four Dimensions ◽  
Tensor Models ◽  
Momentum Spectra ◽  
General Coordinate Transformation

Tensor models can be interpreted as theory of dynamical fuzzy spaces. In this paper, I study numerically the fluctuation spectra around a Gaussian classical solution of a tensor model, which represents a fuzzy flat space in arbitrary dimensions. It is found that the momentum distribution of the low-lying low-momentum spectra is in agreement with that of the metric tensor modulo the general coordinate transformation in the general relativity at least in the dimensions studied numerically, i.e. one to four dimensions. This result suggests that the effective field theory around the solution is described in a similar manner as the general relativity.

The Ambient Metric (AM-178)

2011 ◽  
Author(s):  
Charles Fefferman ◽  
C. Robin Graham
Keyword(s):  
Power Series ◽  
Formal Power Series ◽  
Conformal Geometry ◽  
Four Dimensions ◽  
Conformal Curvature ◽  
Special Cases ◽  
Einstein Spaces ◽  
Explicit Analysis ◽  
Curvature Tensors ◽  
Formal Power

This book develops and applies a theory of the ambient metric in conformal geometry. This is a Lorentz metric in n+2 dimensions that encodes a conformal class of metrics in n dimensions. The ambient metric has an alternate incarnation as the Poincaré metric, a metric in n+1 dimensions having the conformal manifold as its conformal infinity. In this realization, the construction has played a central role in the AdS/CFT correspondence in physics. The existence and uniqueness of the ambient metric at the formal power series level is treated in detail. This includes the derivation of the ambient obstruction tensor and an explicit analysis of the special cases of conformally flat and conformally Einstein spaces. Poincaré metrics are introduced and shown to be equivalent to the ambient formulation. Self-dual Poincaré metrics in four dimensions are considered as a special case, leading to a formal power series proof of LeBrun's collar neighborhood theorem proved originally using twistor methods. Conformal curvature tensors are introduced and their fundamental properties are established. A jet isomorphism theorem is established for conformal geometry, resulting in a representation of the space of jets of conformal structures at a point in terms of conformal curvature tensors. The book concludes with a construction and characterization of scalar conformal invariants in terms of ambient curvature, applying results in parabolic invariant theory.

DETERMINANTS, DIRAC OPERATORS, AND ONE-LOOP PHYSICS

1989 ◽  
Vol 04 (06) ◽  
pp. 1467-1484 ◽  
Author(s):  
STEVEN K. BLAU ◽  
MATT VISSER ◽  
ANDREAS WIPF
Keyword(s):  
Dirac Operator ◽  
Dirac Operators ◽  
World Sheet ◽  
Four Dimensions ◽  
Arbitrary Genus ◽  
String Theories

We consider the Dirac operator. Its determinant is examined and in two Euclidean dimensions is explicitly evaluated in terms of geometrical quantities. This leads us to consider a generalization of the Wess-Zumino action that is applicable to arbitrary genus. Our analysis is relevant to a number of interesting systems: Schwinger models on curved two-manifolds; string theories with world-sheet vectors; and as an exploration of possible directions in evaluating determinants in four dimensions.

scholarly journals A study of practical implementations of the overlap Dirac operator in four dimensions

1999 ◽  
Vol 540 (1-2) ◽  
pp. 457-471 ◽  
Author(s):  
Robert G. Edwards ◽  
Urs M. Heller ◽  
Rajamani Narayanan
Keyword(s):  
Dirac Operator ◽  
Four Dimensions

A super-twisted Dirac operator and Novikov inequalities

2000 ◽  
Vol 43 (5) ◽  
pp. 470-480 ◽  
Author(s):  
Huitao Feng ◽  
Enli Guo
Keyword(s):  
Dirac Operator ◽  
Twisted Dirac Operator

scholarly journals Local index theory and the Riemann–Roch–Grothendieck theorem for complex flat vector bundles

2018 ◽  
Vol 12 (04) ◽  
pp. 941-987
Author(s):  
Man-Ho Ho
Keyword(s):  
Vector Bundle ◽  
Dirac Operator ◽  
Differential Form ◽  
Vector Bundles ◽  
Variational Formula ◽  
Index Theory ◽  
Index Theorem ◽  
Local Index ◽  
Chern Simons ◽  
Local Family

The purpose of this paper is to give a proof of the real part of the Riemann–Roch–Grothendieck theorem for complex flat vector bundles at the differential form level in the even dimensional fiber case. The proof is, roughly speaking, an application of the local family index theorem for a perturbed twisted spin Dirac operator, a variational formula of the Bismut–Cheeger eta form without the kernel bundle assumption in the even dimensional fiber case, and some properties of the Cheeger–Chern–Simons class of complex flat vector bundle.

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