scholarly journals Hidden relations of central charges and OPEs in holographic CFT

2021 ◽  
Vol 2021 (11) ◽  
Author(s):  
Yue-Zhou Li ◽  
H. Lü ◽  
Liang Ma
Keyword(s):  
Differential Operators ◽  
Riemann Tensor ◽  
Point Function ◽  
Linear Combinations ◽  
Tensor Invariants ◽  
Higher Dimensional ◽  
Holographic Approach ◽  
Higher Order Corrections ◽  
Radial Variable ◽  
Field Redefinition

Abstract It is known that the (a, c) central charges in four-dimensional CFTs are linear combinations of the three independent OPE coefficients of the stress-tensor three-point function. In this paper, we adopt the holographic approach using AdS gravity as an effect field theory and consider higher-order corrections up to and including the cubic Riemann tensor invariants. We derive the holographic central charges and OPE coefficients and show that they are invariant under the metric field redefinition. We further discover a hidden relation among the OPE coefficients that two of them can be expressed in terms of the third using differential operators, which are the unit radial vector and the Laplacian of a four-dimensional hyperbolic space whose radial variable is an appropriate length parameter that is invariant under the field redefinition. Furthermore, we prove that the consequential relation c = 1/3ℓeff∂a/∂ℓeff and its higher-dimensional generalization are valid for massless AdS gravity constructed from the most general Riemann tensor invariants.

scholarly journals Symmetric Determinants and the Cayley and Capelli Operator

1948 ◽  
Vol 8 (2) ◽  
pp. 76-86 ◽  
Author(s):  
H. W. Turnbull
Keyword(s):  
Invariant Theory ◽  
Differential Operators ◽  
Matrix Operator ◽  
Linear Combinations ◽  
Classical Invariant Theory ◽  
Independent Variables ◽  
The Matrix ◽  
Classical Invariant ◽  
Initial Discovery ◽  
Special Case

The result obtained by Lars Gårding, who uses the Cayley operator upon a symmetric matrix, is of considerable interest. The operator Ω = |∂/∂xij|, which is obtained on replacing the n2 elements of a determinant |xij by their corresponding differential operators and forming the corresponding n-rowed determinant, is fundamental in the classical invariant theory. After the initial discovery in 1845 by Cayley further progress was made forty years later by Capelli who considered the minors and linear combinations (polarized forms) of minors of the same order belonging to the whole determinant Ω: but in all this investigation the n2 elements xij were regarded as independent variables. The apparently special case, undertaken by Gårding when xij = xji and the matrix [xij] is symmetric, is essentially a new departure: and it is significant to have learnt from Professor A. C. Aitken in March this year 1946, that he too was finding the symmetrical matrix operator [∂/∂xij] of importance and has already written on the matter.

scholarly journals DIFFERENTIAL STRUCTURE OF ABELIAN FUNCTIONS

2008 ◽  
Vol 19 (02) ◽  
pp. 145-171 ◽  
Author(s):  
KOJI CHO ◽  
ATSUSHI NAKAYASHIKI
Keyword(s):  
Abelian Variety ◽  
Differential Operators ◽  
Free Resolution ◽  
Holomorphic Differential ◽  
Differential Structure ◽  
Linear Basis ◽  
Higher Dimensional ◽  
Abelian Functions ◽  
Derivatives Of ◽  
Logarithmic Derivatives

The space of Abelian functions of a principally polarized abelian variety (J,Θ) is studied as a module over the ring [Formula: see text] of global holomorphic differential operators on J. We construct a [Formula: see text] free resolution in case Θ is non-singular. As an application, in the case of dimensions 2 and 3, we construct a new linear basis of the space of abelian functions which are singular only on Θ in terms of logarithmic derivatives of the higher-dimensional σ-function.

scholarly journals The Effect of Higher Dimensional QCD Operators on the Spectroscopy of Bottom-Up Holographic Models

Universe ◽  
2021 ◽  
Vol 7 (4) ◽  
pp. 102
Author(s):  
Sergey Afonin
Keyword(s):  
Excited States ◽  
Normal Modes ◽  
Mass Range ◽  
Scalar Case ◽  
Holographic Model ◽  
Bottom Up ◽  
Higher Dimensional ◽  
Soft Wall ◽  
Holographic Approach ◽  
Meson States

Within the bottom-up holographic approach to QCD, the highly excited hadrons are identified with the bulk normal modes in the fifth “holographic” dimension. We show that additional states in the same mass range can appear also from taking into consideration the 5D fields dual to higher dimensional QCD operators. The possible effects of these operators have not been taken into account in virtually any phenomenological applications. Using the scalar case as the simplest example, we demonstrate that the additional higher dimensional operators lead to a large degeneracy of highly excited states in the soft wall holographic model, and in the hard wall holographic model, they result in a proliferation of excited states. The considered model can be viewed as the first analytical toy model predicting a one-to-one mapping of the excited meson states to definite QCD operators, to which they prefer to couple.

scholarly journals Efficiently transforming from values of a function on a sparse grid to basis coefficients

2021 ◽  
Author(s):  
Tucker CARRINGTON ◽  
Robert Wodraszka
Keyword(s):  
Uncertainty Quantification ◽  
Efficient Method ◽  
Basis Function ◽  
Piecewise Linear ◽  
Basis Functions ◽  
Sparse Grid ◽  
Point Function ◽  
Linear Combinations ◽  
Basis Expansion ◽  
The Cost

In many contexts it is necessary to determine coefficients of a basis expansion of a function ${f}\left(x_1, \ldots, x_D\right) $ from values of the function at points on a sparse grid. Knowing the coefficients, one has an interpolant or a surrogate. For example, such coefficients are used in uncertainty quantification. In this chapter, we present an efficient method for computing the coefficients. It uses basis functions that, like the familiar piecewise linear hierarchical functions, are zero at points in previous levels. They are linear combinations of any, e.g. global, nested basis functions $\varphi_{i_k}^{\left(k\right)}\left(x_k\right)$. Most importantly, the transformation from function values to basis coefficients is done, exploiting the nesting, by evaluating sums sequentially. When the number of functions in level $\ell_k$ equals $\ell_k$ (i.e. when the level index is increased by one, only one point (function) is added) and the basis function indices satisfy ${\left\lVert\mathbf{i}-\mathbf{1}\right\lVert_1 \le b}$, the cost of the transformation scales as $\mathcal{O}\left(D \left[\frac{b}{D+1} + 1\right] N_\mathrm{sparse}\right)$, where $N_\mathrm{sparse}$ is the number of points on the sparse grid. We compare the cost of doing the transformation with sequential sums to the cost of other methods in the literature.

scholarly journals Gaudin models and multipoint conformal blocks. Part II. Comb channel vertices in 3D and 4D

2021 ◽  
Vol 2021 (11) ◽  
Author(s):  
Ilija Burić ◽  
Sylvain Lacroix ◽  
Jeremy Mann ◽  
Lorenzo Quintavalle ◽  
Volker Schomerus
Keyword(s):  
Differential Operators ◽  
Arbitrary Dimension ◽  
Tensor Fields ◽  
Conformal Blocks ◽  
Conformal Field ◽  
Higher Dimensional ◽  
Sutherland Model ◽  
Joint Eigenfunctions ◽  
Space Formalism ◽  
Gaudin Models

Abstract It was recently shown that multi-point conformal blocks in higher dimensional conformal field theory can be considered as joint eigenfunctions for a system of commuting differential operators. The latter arise as Hamiltonians of a Gaudin integrable system. In this work we address the reduced fourth order differential operators that measure the choice of 3-point tensor structures for all vertices of 3- and 4-dimensional comb channel conformal blocks. These vertices come associated with a single cross ratio. Remarkably, we identify the vertex operators as Hamiltonians of a crystallographic elliptic Calogero-Moser-Sutherland model that was discovered originally by Etingof, Felder, Ma and Veselov. Our construction is based on a further development of the embedding space formalism for mixed-symmetry tensor fields. The results thereby also apply to comb channel vertices of 5- and 6-point functions in arbitrary dimension.

A cyclic non-singular universe from Gauss–Bonnet and superstring corrections

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Rami Ahmad El-Nabulsi
Keyword(s):  
Scalar Field ◽  
Time Derivative ◽  
Low Energy ◽  
Exponential Potential ◽  
Field Equations ◽  
Superstring Theory ◽  
Higher Dimensional ◽  
Free Parameters ◽  
Higher Order Corrections ◽  
The Universe

Abstract In this study, we have constructed a viable cosmological model characterized by the presence of the Gauss–Bonnet four-dimensional invariant, higher-order corrections to the low energy effective action motivated from heterotic superstring theory and a general exponential potential comparable to those obtained in higher dimensional supergravities. The field equations were studied by assuming a particular relation between the Hubble parameter and the time derivative of the scalar field. It was observed that, for specific relations between the free parameters in the theory, the universe is cyclic, expands and contracts alternately without singularity with an equation of state oscillating around −1. The model is found to fit the recent astrophysical data.

The saturation class for linear combinations of Szász–Mirakjan operators

2020 ◽  
Vol 18 (06) ◽  
pp. 2050044
Author(s):  
Linsen Xie ◽  
Shuli Wang ◽  
Kaifei Zhao
Keyword(s):  
Differential Operators ◽  
Approximation Order ◽  
Linear Combinations ◽  
Saturation Class

This paper investigates the regularity of a class of differential operators generated by Szász–Mirakjan operators. With the aid of the regularity, the relation between the uniform saturated approximation order of linear combinations of Szász–Mirakjan operators and the smoothness of the approximated functions is studied. The saturation class for linear combinations of Szász–Mirakjan operators is characterized.

scholarly journals A calculus for conformal hypersurfaces and new higher Willmore energy functionals

2020 ◽  
Vol 20 (1) ◽  
pp. 29-60 ◽  
Author(s):  
A. Rod Gover ◽  
Andrew Waldron
Keyword(s):  
Differential Operators ◽  
Yamabe Problem ◽  
Conformal Class ◽  
Energy Functions ◽  
Energy Functionals ◽  
Conformally Compact ◽  
Higher Dimensional ◽  
Invariant Differential ◽  
The Right ◽  
Willmore Energy

AbstractThe invariant theory for conformal hypersurfaces is studied by treating these as the conformal infinity of a conformally compact manifold. Recently it has been shown how, given a conformal hypersurface embedding, a distinguished ambient metric is found (within its conformal class) by solving a singular version of the Yamabe problem [21]. This enables a route to proliferating conformal hypersurface invariants. The aim of this work is to give a self contained and explicit treatment of the calculus and identities required to use this machinery in practice. In addition we show how to compute the solution’s asymptotics. We also develop the calculus for explicitly constructing the conformal hypersurface invariant differential operators discovered in [21] and in particular how to compute extrinsically coupled analogues of conformal Laplacian powers. Our methods also enable the study of integrated conformal hypersurface invariants and their functional variations. As a main application we prove that a class of energy functions proposed in a recent work have the right properties to be deemed higher-dimensional analogues of the Willmore energy. This complements recent progress on the existence and construction of different functionals in [22] and [20].

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