Action Of SU Research Articles

Harmonic forms on asymptotically ADS metrics

In this paper we study the rotationally invariant harmonic cohomology of a two-parameter family of Einstein metrics g which admits a cohomogeneity one action of SU(2) × U(1) and has AdS asymptotics. Depending on the values of the parameters, g is either of NUT type, if the fixed-point locus of the U(1) action is zero-dimensional, or of bolt type, if it is two-dimensional. We find that if g is of NUT type then the space of SU(2)-invariant harmonic two-forms is three-dimensional and consists entirely of self-dual forms; if g is of bolt type it is four-dimensional. In both cases we explicitly determine a basis. The pair (g, F) for F a self-dual harmonic two-form is also a solution of the bosonic sector of 4D supergravity. We determine for which choices it is a supersymmetric solution and the amount of preserved supersymmetry.

The chiral algebra of genus two class mathcal{S} theory

We construct the chiral algebra associated with the A1-type class mathcal{S} theory for the genus two Riemann surface without punctures. By solving the BRST cohomology problem corresponding to a marginal gauging in four dimensions, we find a set of chiral algebra generators that form closed OPEs. Given the fact that they reproduce the spectrum of chiral algebra operators up to large dimensions, we conjecture that they are the complete set of generators. Remarkably, their OPEs are invariant under an action of SU(2) which is not associated with any conserved one-form current in four dimensions. We find that this novel SU(2) strongly constrains the OPEs of non-scalar Schur operators. For completeness, we also check the equivalence of Schur indices computed in two S-dual descriptions with a non-vanishing flavor fugacity turned on.

Abelian decomposition and Weyl symmetric effective action of SU(3) QCD

We show how to calculate the effective potential of SU(3) QCD which tells that the true minimum is given by the monopole condensation. To do this we make the gauge independent Weyl symmetric Abelian decomposition of the SU(3) QCD which decomposes the gluons to the color neutral neurons and the colored chromons. In the perturbative regime this decomposes the Feynman diagram in such a way that the conservation of color is explicit. Moreover, this shows the existence of two gluon jets, the neuron jet and chromon jet, which can be verified by experiment. In the non-perturbative regime, the decomposition puts QCD to the background field formalism and reduces the non-Abelian gauge symmetry to a discrete color reflection symmetry, and provides us an ideal platform to calculate the one-loop effective action of QCD. Integrating out the chromons from the Weyl symmetric Abelian decomposition of QCD gauge invariantly imposing the color reflection invariance, we obtain the SU(3) QCD effective potential which generates the stable monopole condensation and the mass gap. We discuss the physical implications of our result, in particular the possible existence of the vacuum fluctuation mode of the monopole condensation in QCD.

Principal fibrations over noncommutative spheres

We present examples of noncommutative four-spheres that are base spaces of SU(2)-principal bundles with noncommutative seven-spheres as total spaces. The noncommutative coordinate algebras of the four-spheres are generated by the entries of a projection which is invariant under the action of SU(2). We give conditions for the components of the Connes–Chern character of the projection to vanish but the second (the top) one. The latter is then a non-zero Hochschild cycle that plays the role of the volume form for the noncommutative four-spheres.

Finding the effective Polyakov line action for SU(3) gauge theories at finite chemical potential

Motivated by the sign problem, we calculate the effective Polyakov line action corresponding to certain SU(3) lattice gauge theories on a $1{6}^{3}\ifmmode\times\else\texttimes\fi{}6$ lattice via the ``relative weights'' method introduced in our previous papers. The calculation is carried out at $\ensuremath{\beta}=5.6$, 5.7 for the pure gauge theory and at $\ensuremath{\beta}=5.6$ for the gauge field coupled to a relatively light scalar particle. In the latter example we determine the effective theory also at finite chemical potential and show how observables relevant to phase structure can be computed in the effective theory via mean field methods. In all cases a comparison of Polyakov line correlators in the effective theory and the underlying lattice gauge theory, computed numerically at zero chemical potential, shows accurate agreement down to correlator magnitudes of order $1{0}^{\ensuremath{-}5}$. We also derive the effective Polyakov line action corresponding to a gauge theory with heavy quarks and large chemical potential and apply mean field methods to extract observables.

Gribov-Zwanziger action in SU(2) maximally Abelian gauge withU(1)3Landau gauge

We construct the local Gribov-Zwanziger action for SU(2) Euclidean Yang-Mills theories in the maximally Abelian (MA) gauge with $\mathrm{U}(1{)}_{3}$ Landau gauge fixing based on Zwanziger's work in the Landau gauge. By restricting the functional integral region to the Gribov region in the MA gauge, we give the nonlocal action. We localize the action with new fields and obtain the action with the shift of the new scalar fields, which has the terms corresponding to the localized action of the horizon function in the MA gauge. The diagonal gluon propagator in the MA gauge at tree level behaves like the propagator from the Gribov-Zwanziger action in the Landau gauge, and shows the violation of the Kallen-Lehmann representation. From our result, the validity of the Gribov-Zwanziger scenario in the MA gauge can be studied by comparing with the lattice result in two dimensions, which is qualitatively consistent with the result of the original Gribov-Zwanziger action in the Landau gauge.

On Holomorphic Differential Operators Equivariant for the Inclusion of Sp(n,R) in U(n,n)

We construct differential operators which act on holomorphic functions defined on the Hermitian half space of degree n and are equivariant under the action of Sp(n,R), thereby increasing the weight by ν. For this, we construct and use as a tool the νth Rankin–Cohen bracket for holomorphic functions on the Hermitian half space of degree n, which is equivariant under the action of SU(n,n). The explicit combinatorial description of the differential operators is given for n=2, an arbitrary ν; and for an arbitrary n, ν=1. We also treat the vector-valued case, where a twisted automorphy factor shows up naturally for equivariance under the group U(n,n). Furthermore, our differential operators are unique upto scalars after restriction of variables to the Siegel half space.

Toric varieties and spherical embeddings over an arbitrary field

We are interested in two classes of varieties with group action, namely toric varieties and spherical embeddings. They are classified by combinatorial objects, called fans in the toric setting, and colored fans in the spherical setting. We characterize those combinatorial objects corresponding to varieties defined over an arbitrary field k. Then we provide some situations where toric varieties over k are classified by Galois-stable fans, and spherical embeddings over k by Galois-stable colored fans. Moreover, we construct an example of a smooth toric variety under a 3-dimensional nonsplit torus over k whose fan is Galois-stable but which admits no k-form. In the spherical setting, we offer an example of a spherical homogeneous space X 0 over R of rank 2 under the action of SU ( 2 , 1 ) and a smooth embedding of X 0 whose fan is Galois-stable but which admits no R -form.

Constraints on SU(2) ⊗ SU(2) invariant polynomials for a pair of entangled qubits

We discuss the entanglement properties of two qubits in terms of polynomial invariants of the adjoint action of SU(2) ⊕ SU(2) group on the space of density matrices \(\mathfrak{P}_ +\) . Since elements of \(\mathfrak{P}_ +\) are Hermitian, non-negative fourth-order matrices with unit trace, the space of density matrices represents a semi-algebraic subset, \(\mathfrak{P}_ + \in \mathbb{R}^{15}\) . We define \(\mathfrak{P}_ +\) explicitly with the aid of polynomial inequalities in the Casimir operators of the enveloping algebra of SU(4) group. Using this result the optimal integrity basis for polynomial SU(2) ⊕ SU(2) invariants is proposed and the well-known Peres-Horodecki separability criterion for 2-qubit density matrices is given in the form of polynomial inequalities in three SU(4) Casimir invariants and two SU(2) ⊕ SU(2) scalars; namely, determinants of the so-called correlation and the Schlienz-Mahler entanglement matrices.

Finite and infinite-dimensional symmetries of pure 𝒩 = 2 supergravity inD= 4

We study the symmetries of pure = 2 supergravity in D = 4. As is known, this theory reduced on one Killing vector is characterised by a non-linearly realised symmetry SU(2,1) which is a non-split real form of SL(3,). We consider the BPS brane solutions of the theory preserving half of the supersymmetry and the action of SU(2,1) on them. Furthermore we provide evidence that the theory exhibits an underlying algebraic structure described by the Lorentzian Kac-Moody group SU(2,1)+++. This evidence arises both from the correspondence between the bosonic space-time fields of = 2 supergravity in D = 4 and a one-parameter sigma-model based on the hyperbolic group SU(2,1)++, as well as from the fact that the structure of BPS brane solutions is neatly encoded in SU(2,1)+++. As a nice by-product of our analysis, we obtain a regular embedding of the Kac-Moody algebra (2,1)+++ in 11 based on brane physics.

Locally smooth SU(n + 1)-actions on SU(n + 1)/S(U(n - 1) × U(2)) are unique

If the Lie group SU(n + 1) acts nontrivially and in a locally smooth way on the Grassmannian Gn+1,2, then the action is conjugate to the standard action by left translation. We prove this theorem using a former result of the author on the nonexistence of fixed points for actions of SU(n + 1) on generalized flag manifolds.

Soliton solutions in an effective action for SU(2) Yang—Mills theory: including effects of higher-derivative term

The Skyrme-Faddeev-Niemi (SFN) model, which is an O(3) σ model in three dimensional space, upto fourth-order in the first derivative is regarded as a low-energy effective theory of SU(2) Yang-Mills theory. One can show from the Wilsonian renormalization group argument that the effective action of Yang-Mills theory recovers the SFN in the infrared region. However, the theory contains an additional fourth-order term which destabilizes the soliton solution. In this paper, we derive the second-derivative term perturbatively and show that the SFN model with the second-derivative term possesses soliton solutions.

Uncertainty for spin systems

A modified definition of quantum mechanical uncertainty Δ for spin systems, which is invariant under the action of SU(2), is suggested. Its range is shown to be ℏ2j⩽Δ⩽ℏ2j(j+1) within any irreducible representation j of SU(2) and its mean value in Hilbert space computed using the Fubini–Study metric is determined to be mean(Δ)=ℏ2j(j+1/2). The most used sets of coherent states in spin systems coincide with the set of minimum Δ uncertainty states.

Bilinear Generating Functions for Orthogonal Polynomials

Using realizations of the positive discrete series representations of the Lie algebra su(1,1) in terms of Meixner—Pollaczek polynomials, the action of su(1,1) on Poisson kernels of these polynomials is considered. In the tensor product of two such representations, two sets of eigenfunctions of a certain operator can be considered and they are shown to be related through continuous Hahn polynomials. As a result, a bilinear generating function for continuous Hahn polynomials is obtained involving the Poisson kernel of Meixner—Pollaczek polynomials; this result is also known as the Burchnall—Chaundy formula. For the positive discrete series representations of the quantized universal enveloping algebra U q (su(1,1)) a similar analysis is performed and leads to a bilinear generating function for Askey—Wilson polynomials involving the Poisson kernel of Al-Salam and Chihara polynomials.

Action and topological density carried by Abelian monopoles in finite temperature pure SU(2) gauge theory: An analysis using RG smoothing

We test a new parametrization of a suitably truncated classically perfect action for SU(2) pure gauge theory with respect to self-consistency and locate the deconfinement transition on a 12^3X4 lattice. Using the technique of smoothing (blocking followed by inverse blocking) we demonstrate clustering of action and topological charge density. Concentrating on the Abelian monopoles found in smoothed configurations after Abelian projection from the maximally Abelian gauge, we present evidence for their role as carriers of non-Abelian action and topological charge.

FROM PRINCIPAL CHIRAL MODEL TO SELF-DUAL GRAVITY

It is demonstrated that the action of SU(N) principal chiral model leads in the limit N →∞ to the action for the Park-Husain (P-H) heavenly equation. The principal chiral model in the Hilbert space L 2(ℝ1) is considered and it is shown, that in this case the chiral equation is equivalent to the Moyal deformation of the P-H heavenly equation. A new method of searching for solutions to this latter equation, via Lie algebra representations in L 2(ℝ1) is given.

Conformal Lorentz geometry revisited

The group U(2,2) and its subgroup SU(2,2) were considered by Penrose in his study of the conformal compactification ℳ of the Minkowski space M [R. Penrose and W. Rindler, Spinors and Space-Time (Cambridge University, Cambridge, 1986) and R. O. Wells, Jr., Bull. Am. Math. Soc. I, 2 (1979)]. The standard representation of SU(2,2) in C4 and in ℳ are the corner stones of twistor theory, which was created by Penrose to the double purpose of obtaining new solutions of Einstein equations and new insights on gravitational radiation. We think that other representations of SU(2,2) or U(2,2) could also bring some information in relativity [see also, Barut O. Asjim, in Noncompact Lie Groups and some of their Applications, edited by E. A. Tanner and R. Wilson (Kluwer Academic, Dordrecht, 1994), p. 103] and, accordingly, we propose an extension of Penrose twistor program. In this paper we deal with a new U(2,2)-space, which we denote by W. We show first that the SU(2,2)-space ℳ introduced by Penrose is isomorphic to U(2), endowed with an action of SU(2,2) given by non-Abelian homographic transformations. These transformations keep invariant the equation det(u−v)=0, characterizing the pairs (u,v)∈U(2)×U(2) such that ‘‘u lies on the light-cone of v.’’ By definition, our space W consists of all pairs (u,v)∈U(2)×U(2) satisfying the condition det(u−v)≠0. The starting point of this article is the observation that W carries an SU(2,2)-invariant pseudo-Riemannian metric L:=Tr[(u−v)−1u̇ ×(u−v)−1v̇], of signature (4,4). (W,L) is in fact an irreducible symmetric space in Cartan’s sense, which is isomorphic to the quotient SO(2,4)/S[O(1,1)×O(1,3)]. As an irreducible symmetric space, it is an 8-dimensional Einstein space, whose Ricci tensor is proportional to the metric tensor. We study the geodesic paths of this space giving the general solutions in terms of initial data and studying the constants of motion. In particular we determine the geodesic paths which exhibit two periods. We also show that Mach’s principle on inertial motions receives an explanation in our theory by considering the particular geodesic paths, for which one of the partners of an interacting pair is fixed and sent to infinity. In fact we study a dynamical system (W,L) which presents some formal and topological similarities with a system of two particles interacting gravitationally. (W,L) is the only conformally invariant relativistic two-point dynamical system. At the end we show that W can be naturally regarded as the base of a principal GL(2,C)-bundle which comes with a natural connection. We study this bundle from differential geometric point of view. Physical interpretations will be discussed in a future paper. This text is an improvement of a previous version, which was submitted under the title ‘‘Hypertwistor Geometry.’’ [See, K. Teleman, ‘‘Hypertwistor Geometry (abstract),’’ 14th International Conference on General Relativity and Gravitation, Florence, Italy, 1995.] The change of the title and many other improvements are due to the valuable comments of the referee, who also suggested the author to avoid hazardous interpretations.

Bicovariant DeRham cohomology Suq (2)

Bicovariant differential forms over the quantum group Su q (2) are considered. Their multiplet structure is determined under the action of SU q (2). This gives a basis of the forms. The exterior derivative of forms and elements of the Hopf algebra A , which is generated by the matrix elements M j i , is explicitly calculated. Then the closedand exact forms are compared. This gives the bicovariant DeRham Cohomology of SU q (2).

Analysis of the static spherically symmetricSU(n)-Einstein-Yang-Mills equations

The singular boundary value problem that arises for the static spherically symmetricSU(n)-Einstein-Yang-Mills equations in the so-called magnetic case is analyzed. Among the possible actions ofSU(2) on aSU(n)-principal bundles over space-time there is one which appears to be the most natural. If one assumes that no electrostatic type component is present in the Yang-Mills fields and the gauge is suitably fixed a set ofn-1 second order and two first order differential equations is obtained forn-1 gauge potentials and two metric components as functions of the radial distance. This system generalizes the one for the casen=2 that leads to the discrete series of the Bartnick-Mckinnon and the corresponding black hole solutions. It is highly nonlinear and singular atr=∞ and atr=0 or at the black hole horizon but it is known to admit at least one series of discrete solutions which are scaled versions of then=2 case. In this paper local existence and uniqueness of solutions near these singular points is established which turns out to be a nontrivial problem for generaln. Moreover, a number of new numerical soliton (i.e. globally regular) numerical solutions of theSU(3)-EYM equations are found that are not scaledn=2 solutions.

Kähler–Einstein metrics with SU(2) action

The aim of this paper is to analyse Riemannian Kähler–Einstein metrics g in real dimension four admitting an isometric action of SU(2) with generically three-dimensional orbits. The Kähler condition means that there is a complex structure I, with respect to which the metric is hermitian, such that the two-form Ωdefined byis closed. It is well-known that if this condition holds then Ω is in fact covariant constant.