Global Conformal Invariance Research Articles

Complete Contraction of Tensor and Its Application

This article presents a mathematical model for tensor complete contraction utilizing permutation group theory. The model helps in identifying the index class that dictates the complete contraction of even-order tensors. Additionally, the concept of a complete contraction closed loop is introduced along with a method for calculating the number of completely contracted images of any even-order tensor under any λ type permutation. The model is applied to investigate the algebraic structure of global conformal invariants on three-dimensional hypersurfaces. Moreover, it is shown that the generalized Willmore functional is the sole global conformal invariant in three-dimensional hypersurfaces when the difference is a constant multiple.

Extrinsic Paneitz operators and Q-curvatures for hypersurfaces

For any hypersurface M of a Riemannian manifold X, recent works introduced the notions of extrinsic conformal Laplacians and extrinsic Q-curvatures. Here we derive explicit formulas for the extrinsic version P4 of the Paneitz operator and the corresponding extrinsic fourth-order Q-curvature Q4 in general dimensions. In the critical dimension n=4, this result yields a closed formula for the global conformal invariant ∫MQ4dvol (for closed M) and various decompositions of Q4, which are analogs of the Alexakis/Deser-Schwimmer type decompositions of global conformal invariants. These results involve a series of obvious local conformal invariants of the embedding M4↪X5 (defined in terms of the Weyl tensor and the trace-free second fundamental form) and a non-trivial local conformal invariant C. In turn, we identify C as a linear combination of two local conformal invariants J1 and J2. We also observe that these are special cases of local conformal invariants for hypersurfaces in backgrounds of general dimension. Moreover, in the critical dimension n=4, a linear combination of J1 and J2 can be expressed in terms of obvious local conformal invariants of the embedding M↪X. This finally reduces the non-trivial part of the structure of Q4 to the non-trivial invariant J1. For totally umbilic M, the invariants Ji vanish, and the formula for P4 substantially simplifies. For closed M4↪R5, we relate the integrals of Ji to functionals of Guven and Graham-Reichert. Moreover, we establish a Deser-Schwimmer type decomposition of the Graham-Reichert functional of a hypersurface M4↪X5 in general backgrounds. In this context, we find one further local conformal invariant J3. Finally, we derive an explicit formula for the singular Yamabe energy of a closed M. The resulting explicit formulas show that it is proportional to the total extrinsic fourth-order Q-curvature. This observation confirms a special case of a general fact and serves as an additional cross-check of our main result. We carefully discuss the relations of our results to the recent literature, in particular to the work of Blitz, Gover and Waldron.

Fresh perspective on gauging the conformal group

We consider the construction of gauge theories of gravity that are invariant under local conformal transformations. We first clarify the geometric nature of global conformal transformations, in both their infinitesimal and finite forms, and the consequences of global conformal invariance for field theories, before reconsidering existing approaches for gauging the conformal group, namely auxiliary conformal gauge theory and biconformal gauge theory, neither of which is generally accepted as a complete solution. We then demonstrate that, provided any matter fields belong to an irreducible representation of the Lorentz group, the recently proposed extended Weyl gauge theory (eWGT) may be considered as an alternative method for gauging the conformal group, since eWGT is invariant under the full set of local conformal transformations, including inversions, as well as possessing conservation laws that provide a natural local generalisation of those satisfied by field theories with global conformal invariance, and also having an `ungauged' limit that corresponds to global conformal transformations. By contrast, although standard Weyl gauge theory also enjoys the first of these properties, it does not share the other two, and so cannot be considered a valid gauge theory of the conformal group.

Conformal equations that are not Virasoro or Weyl invariant

While the argument by Zamolodchikov and Polchinski suggests global conformal invariance implies Virasoro invariance in two-dimensional unitary conformal field theories with discrete dilatation spectrum, it is not the case in more general situations without these assumptions. We indeed show that almost all the globally conformal invariant differential equations in two dimensions are neither Virasoro invariant nor Weyl invariant. The only exceptions are the higher spin conservation laws, conformal Killing tensor equations and the Laplace equation of a conformal scalar.

A classification of global conformal invariants**Dedicated to the memory of our colleague Christian Duval 1947–2018.

We provide the full classification, in arbitrary even and odd dimensions, of global conformal invariants, i.e. scalar densities in the spacetime metric and its derivatives that are invariant, possibly up to a total derivative, under local Weyl rescalings of the metric. We use cohomological techniques that have already proved instrumental in the classification of Weyl anomalies in arbitrary dimensions. The approach we follow is purely algebraic and borrows techniques originating from perturbative quantum field theory for which locality is crucial.

Hyperunified field theory and Taiji program in space for GWD

In this paper, I present the recently established hyperunified field theory (HUFT)[Formula: see text] for all basic forces and elementary particles within the framework of gravitational quantum field theory (GQFT)[Formula: see text] in hyper-space–time. GQFT treats gravity as a gauge theory in the framework of quantum field theory to avoid the long term obstacle between general relativity and quantum mechanics. HUFT is built based on the guiding principle: the dimension of hyper-space–time correlates to intrinsic quantum numbers of basic building blocks of nature, and the action describing the laws of nature obeys the gauge invariance and coordinate independence, which is more fundamental than that proposed by Einstein for general relativity. The basic gravitational field is defined in biframe hyper-space–time as a bicovariant vector field, it is a gauge-type hyper-gravifield rather than a metric field. HUFT is characterized by a bimaximal Poincaré and hyper-spin gauge symmetry [Formula: see text] with a global and local conformal scaling invariance in biframe hyper-space–time. The gravitational origin of gauge symmetry is revealed through the hyper-gravifield that plays an essential role as a Goldstone-like field, which enables us to demonstrate the gauge-gravity and gravity-geometry correspondences and to corroborate the gravitational gauge-geometry duality with an emergent hidden general linear group symmetry [Formula: see text]. The Taiji Program in Space for the gravitational wave detection in China[Formula: see text] is briefly outlined.

Energy transport in an integrable parafermionic chain via generalized hydrodynamics

We study energy transport in the integrable $\mathbb Z_3$ parafermionic chain using the partitioning protocol. By exploiting the Bethe-ansatz solution for the thermodynamics of the system, we develop a generalized hydrodynamic description of the non-equilibrium steady states, which we benchmark using numerical simulations based on matrix product states. The model features a low-energy conformal limit with central charge $c=4/5$, which affects the low-temperature energy current, as we explicitly show. Moreover, we exploit that, for energies close to the maximally excited state, the system is also critical and described by a conformal field theory with $c=1$. By considering the two halves prepared at two temperatures both low in value but opposite in sign, we are able to investigate in an exact and controlled way the junction between two conformal field theories with different central charges. Notwithstanding the absence of global conformal invariance, we find results that approximate to a high degree those of out-of-equilibrium conformal field theories. Our study extends the generalized hydrodynamics to a novel framework, where it can be profitably used for exploring new physical phenomena.

Global Conformal Invariants of Submanifolds

The goal of the present paper is to investigate the algebraic structure of global conformal invariants of submanifolds. These are defined to be conformally invariant integrals of geometric scalars of the tangent and normal bundle. A famous example of a global conformal invariant is the Willmore energy of a surface. In codimension one we classify such invariants, showing that under a structural hypothesis (more precisely we assume the integrand depends separately on the intrinsic and extrinsic curvatures, and not on their derivatives) the integrand can only consist of an intrinsic scalar conformal invariant, an extrinsic scalar conformal invariant and the Chern-Gauss-Bonnet integrand. In particular, for codimension one surfaces, we show that the Willmore energy is the unique global conformal invariant, up to the addition of a topological term (the Gauss curvature, giving the Euler Characteristic by the Gauss Bonnet Theorem). A similar statement holds also for codimension two surfaces, once taking into account an additional topological term given by the Chern-Gauss-Bonnet integrand of the normal bundle. We also discuss existence and properties of natural higher dimensional (and codimensional) generalizations of the Willmore energy.

Minimal area surfaces in AdS3 through integrability

Minimal area surfaces in ending on a given curve at the boundary are dual to planar Wilson loops in SYM. In previous work it was shown that the problem of finding such surfaces can be recast as the one of finding an appropriate parameterization of the boundary contour that corresponds to conformal gauge. Dekel was able to find such reparameterization in a perturbative expansion around a circular contour. In this work we show that for more general contours such reparameterization can be found using a numerical procedure that does not rely on a perturbative expansion. This provides further checks and applications of the integrability method. An interesting property of the method is that it uses as data the Schwarzian derivative of the contour and therefore it has manifest global conformal invariance. Finally, we apply Shanks transformation to extend the near circular expansion to larger deformations. The results are in agreement with the new method.

DS/CFT and the operator product expansion

Global conformal invariance determines the form of two- and three-point functions of quasiprimary operators in a conformal field theory and generates nontrivial relations between terms in the operator product expansion. These ideas are generalized to the principal and complementary series representations, which play an important role in the conjectured dS/CFT correspondence. The conformal partial wave expansions are constructed for these representations, which in turn determine the operator product expansion. This leads us to conclude that conformal field theories containing such representations have essential singularities, so they cannot be realized as conventional field theories.

Conformal geometry of timelike curves in the [formula omitted]-Einstein universe

We study the conformal geometry of timelike curves in the (1+2)-Einstein universe, the conformal compactification of Minkowski 3-space defined as the quotient of the null cone of R2,3 by the action by positive scalar multiplications. The purpose is to describe local and global conformal invariants of timelike curves and to address the question of existence and properties of closed trajectories for the conformal strain functional. Some relations between the conformal geometry of timelike curves and the geometry of knots and links in the 3-sphere are discussed.

The decomposition of global conformal invariants: Some technical proofs II

This paper forms part of a larger work where we prove a conjecture of Deser and Schwimmer regarding the algebraic structure of global conformal invariants; these are defined to be conformally invariant integrals of geometric scalars. The conjecture asserts that the integrand of any such integral can be expressed as a linear combination of a local conformal invariant, a divergence and of the Chern-Gauss-Bonnet integrand.

The Decomposition of Global Conformal Invariants: Some Technical Proofs. I

This paper forms part of a larger work where we prove a conjecture of Deser and Schwimmer regarding the algebraic structure of "global conformal invariants"; these are defined to be conformally invariant integrals of geometric scalars. The conjecture asserts that the integrand of any such integral can be expressed as a linear combination of a local conformal invariant, a divergence and of the Chern-Gauss-Bonnet integrand.

The decomposition of global conformal invariants IV: A proposition on local Riemannian invariants

This is the fourth in a series of papers where we prove a conjecture of Deser and Schwimmer regarding the algebraic structure of “global conformal invariants”; these are defined to be conformally invariant integrals of geometric scalars. The conjecture asserts that the integrand of any such integral can be expressed as a linear combination of a local conformal invariant, a divergence and of the Chern–Gauss–Bonnet integrand. The present paper lays out the second half of this entire work: The second half proves certain purely algebraic statements regarding local Riemannian invariants; these were used extensively in the first two papers in this series, see Alexakis (2007, 2009) [2,3]. These results may be of independent interest, applicable to related problems.

Deviation from Alday–Maldacena duality for wavy circle

Alday–Maldacena conjecture is stated in this paper that the area A Π of the minimal surface in AdS 5 space with a boundary Π, located in Euclidean space at infinity of AdS 5 , coincides with a double integral D Π along Π, the Abelian Wilson average in an auxiliary dual model. This comes from Alday and Maldacena's original proposal and the BDS conjecture on the extrapolation of the MHV amplitudes. The boundary Π is a polygon formed by momenta of n external light-like particles in N = 4 SYM theory, and in a certain n = ∞ limit it can be substituted by an arbitrary smooth curve (wavy circle). The Alday–Maldacena conjecture is known to be violated for n > 5 , when it fails to be supported by the peculiar global dual conformal invariance, however, the structure of deviations remains obscure. The case of wavy lines can appear more convenient for analysis of these deviations due to the systematic method developed in [H. Itoyama, A. Mironov, A. Morozov, Anomaly in n = ∞ Alday–Maldacena duality for wavy circle, JHEP 0807 (2008) 024, arXiv:0803.1547] for (perturbative) evaluation of minimal areas, which is not yet available in the presence of angles at finite n. We correct a mistake in that paper and explicitly evaluate the h 2 h ¯ 2 terms, where the first deviation from the Alday–Maldacena duality arises for the wavy circle.

Topological foundations of electromagnetism

Electromagnetic Phenomena Not Explained by Maxwell's Equations Sagnac Effect: A Consequence of Conservation of Action Due to Gauge Field Global Conformal Invariance in a Multiply Joined Topology of Coherent Fields Topological Approaches to Electromagnetism.

Infinite-dimensional Lie algebras in 4D conformal quantum field theory**Lecture at the workshops ‘Lie Theory and Its Applications in Physics’, 18–24 June 2007, Varna, Bulgaria; ‘Infinite- Dimensional Algebras and Quantum Integrable Systems’, 23–27 July, 2007, Faro, Portugal; and ‘Supersymmetries and Quantum Symmetries’, 30 July–4 August, 2007, Dubna, Russia.

The concept of global conformal invariance (GCI) opens the way of applying algebraic techniques, developed in the context of two-dimensional chiral conformal field theory, to a higher (even) dimensional spacetime. In particular, a system of GCI scalar fields of conformal dimension two gives rise to a Lie algebra of harmonic bilocal fields, VM(x, y), where the M span a finite dimensional real matrix algebra closed under transposition. The associative algebra is irreducible iff its commutant coincides with one of the three real division rings. The Lie algebra of (the modes of) the bilocal fields is in each case an infinite-dimensional Lie algebra: a central extension of corresponding to the field of reals, of u(∞, ∞) associated with the field of complex numbers, and of so*(4∞) related to the algebra of quaternions. They give rise to quantum field theory models with superselection sectors governed by the (global) gauge groups O(N), U(N) and , respectively.

Logarithmic [formula omitted] minimal models, their logarithmic couplings, and duality

A natural construction of the logarithmic extension of the M ( 2 , p ) (chiral) minimal models is presented, which generalises our previous model of percolation ( p = 3 ). Its key aspect is the replacement of the minimal model irreducible modules by reducible ones obtained by requiring that only one of the two principal singular vectors of each module vanish. The resulting theory is then constructed systematically by repeatedly fusing these building block representations. This generates indecomposable representations of the type which signify the presence of logarithmic partner fields in the theory. The basic data characterising these indecomposable modules, the logarithmic couplings, are computed for many special cases and given a new structural interpretation. Quite remarkably, a number of them are presented in closed analytic form (for general p). These are the prime examples of “gauge-invariant” data—quantities independent of the ambiguities present in defining the logarithmic partner fields. Finally, mere global conformal invariance is shown to enforce strong constraints on the allowed spectrum: It is not possible to include modules other than those generated by the fusion of the model's building blocks. This generalises the statement that there cannot exist two effective central charges in a c = 0 model. It also suggests the existence of a second “dual” logarithmic theory for each p. Such dual models are briefly discussed.

PONTRJAGIN FORMS AND INVARIANT OBJECTS RELATED TO THE Q-CURVATURE

It was shown by Chern and Simons that the Pontrjagin forms are conformally invariant. We show them to be the Pontrjagin forms of the conformally invariant tractor connection. The Q-curvature is intimately related to the Pfaffian. Working on even-dimensional manifolds, we show how the k-form operators Qkof [12], which generalize the Q-curvature, retain a key aspect of the Q-curvature's relation to the Pfaffian, by obstructing certain representations of natural operators on closed forms. In a closely related direction, we show that the Qkgive rise to conformally invariant quadratic forms Θkon cohomology that interpolate, in a suitable sense, between the integrated metric pairing (at k = n/2) and the Pfaffian (at k = 0). Using a different construction, we show that the Qkoperators yield a map from conformal structures to Lagrangian subspaces of the direct sum Hk⊕ Hk(where Hkis the dual of the de Rham cohomology space Hk); in an appropriate sense this generalizes the period map. We couple the Qkoperators with the Pontrjagin forms to construct new natural densities that have many properties in common with the original Q-curvature; in particular these integrate to global conformal invariants. We also work out a relevant example, and show that the proof of the invariance of the (nonlinear) action functional whose critical metrics have constant Q-curvature extends to the action functionals for these new Q-like objects. Finally we set up eigenvalue problems that generalize to Qk-operators the Q-curvature prescription problem.

Unitary Positive-Energy Representations of Scalar Bilocal Quantum Fields

The superselection sectors of two classes of scalar bilocal quantum fields in D ≥ 4 dimensions are explicitly determined by working out the constraints imposed by unitarity. The resulting classification in terms of the dual of the respective gauge groups U(N) and O(N) confirms the expectations based on general results obtained in the framework of local nets in algebraic quantum field theory, but the approach using standard Lie algebra methods rather than abstract duality theory is complementary. The result indicates that one does not lose interesting models if one postulates the absence of scalar fields of dimension D−2 in models with global conformal invariance. Another remarkable outcome is the observation that, with an appropriate choice of the Hamiltonian, a Lie algebra embedded into the associative algebra of observables completely fixes the representation theory.