Monodromy properties of energy momentum tensor on general algebraic curves

A new approach to analyze the properties of the energy momentum tensor T (z) of conformal field theories on generic Riemann surfaces (RS) is proposed. T (z) is decomposed into N components with different monodromy properties, where N is the number of branches in the realization of RS as branch covering over the complex sphere. This decomposition gives rise to new infinite dimensional Lie algebra which can be viewed as a generalization of Virasoro algebra containing information about the global properties of the underlying RS. In the simplest case of hyperelliptic curves the structure of the algebra is calculated in two ways and its central extension is explicitly given. The algebra possess an interesting symmetry with a clear interpretation in the framework of the radial quantization of CFTs with multivalued fields on the complex sphere.     

Remarks on the quantization of conformal fields

The quantization of a general (b, c) system in two dimensions is formulated in terms of an infinite hierarchy of modules for the Virasoro algebra that interpolate between the space of classical conformal fields of weight j and the Dirac sea of semi-infinite forms. This provides a natural framework in which to study the relation between algebraic geometry and representations of the Virasoro algebra with central charge c_j=−2(6j²−6j+1). The importance of the construction is discussed in the context of string theory.     

Central charges in the canonical realization of asymptotic symmetries: An example from three dimensional gravity

It is shown that the global charges of a gauge theory may yield a nontrivial central extension of the asymptotic symmetry algebra already at the classical level. This is done by studying three dimensional gravity with a negative cosmological constant. The asymptotic symmetry group in that case is eitherR×SO(2) or the pseudo-conformal group in two dimensions, depending on the boundary conditions adopted at spatial infinity. In the latter situation, a nontrivial central charge appears in the algebra of the canonical generators, which turns out to be just the Virasoro central charge.     

Kac-Moody current algebras of D = 2 massless gauge theories,their representations and applications

We give a classification of the Kac-Moody current algebras of all the possible massless fermion-gauge theories in two dimensions. It is shown that only Kac-Moody algebras based on A_N, B_N, C_N, and D_N in the Cartan classification with all possible central charge occur. The representation of local fermion fields and simply laced Kac-Moody algebras with minimal central charge in terms of free boson fields on a compactified space is discussed in detail, where stress is laid on the role played by the boundary conditions on the various collective modes. Fractional solitons and the possible soliton representation of certain nonsimply laced algebras is also analysed. We briefly discuss the relationship between the massless bound state sector of these two-dimensioned gauge theories and the critically coupled two-dimensional nonlinear sigma model, which share the same current algebra. Finally we briefly discuss the relevance of Sp(n) Kac-Moody algebras to the physics of monopole-fermion systems.     

Spinorial charges and their role in the fusion of internal and space-time symmetries

The advent of supersymmetry immediately led to speculations that a non-trivial mixing of internal and space-time symmetries could be achieved within its framework. In fact, the well-known no-go theorems do not apply to the supersymmetry algebra due to the presence, in the latter, of (anticommuting) spinorial charges. However, not until the recent work of Haag, Lopuszanski and Sohnius did a clearcut picture emerge as to how the aforementioned nontrivial mixing can take place. Most notably, the presence of the conformal algebra within the supersymmetry algebra turns out to be vital. We solidify the findings of Haag et al. through an explicit construction which uses as underlying space the pseudo-Euclidean space E(4,2), i.e. the space for which the conformal group is the group of rotations, and which employs as main tools the spinors associated with the space E(4,2). We follow the algebro-geometric approach of Cartan in order to understand both the introduction and the properties of these spinors. In this manner, we gain many insights regarding the mathematical foundations of supersymmetry. Thus, we fully understand the emergence of the anticommutator, rather than the commutator, among spinor charges as a natural algebraic consequence and not as an a priori given fact. In addition, we clearly see how an (internal) unitary symmetry group can make its appearance within the supersymmetry scheme and verify, via our explicit construction, the results of Haag et al.     

General charges and conformal supergravity

It is shown how to incorporate multiplets with central charges in superconformal gravity. The linear and scalar N=2 multiplets are presented, and the full superconformal action of scalar multiplets with gauged central charge is given.     

N=2 extended superconformal algebra with “central” charges

We show how central charges may be incorporated in a superconformal (D = 4) algebra for N = 2. The charges are no longer truly central and so are at variance with the well-known theorems on (super-) symmetries of the S-matrix. We discuss the possible relevance of the algebra and justify our interest in it.     

Non-abelian D = 11 supermembrane

We obtain a U(M) action for super membranes with central charges in the Light Cone Gauge (LCG). The theory realizes all of the symmetries and constraints of the supermembrane together with the invariance under a U(M) gauge group with M arbitrary. The worldvolume action has (LCG) N = 8 supersymmetry arid it corresponds to M parallel supermembranes minimally immersed on the target M g × T ² (MIM2). In order to ensure the invariance under the symmetries and to close the corresponding algebra, a star-product determined by the central charge condition is introduced. It is constructed with a nonconstant symplectic two-form where curvature terms are also present. The theory is in the strongly coupled gauge-gravity regime. At low energies, the theory enters in a decoupling limit and it is described by an ordinary N = 8 SYM in the IR phase for any number of M2-branes.     

On the topological charge concentrated at a point

For SU(2) gauge fields over the 4-dimensional sphere with a finite number of points x₁, x₂, ..., and x_N removed, there are gauge transformations which modify the topological charge concentrated at x_j by adding n_j, where n₁, n₂, …, and n_N. are integers such that Σ^N_{j = 1}n_j = 0. However, the reduction modulo Z of the topological charge at a point is well defined, being given in terms of the secondary characteristic classes of Chern and Simons, except when the topological charge is indeterminate.     

Fundamental monopoles and multimonopole solutions for arbitrary simple gauge groups

Magnetic monopole solutions for an arbitrary compact simple gauge group are considered in the Prasad-Sommerfield limit. For each group and choice of symmetry breaking there is a set of fundamental monopoles with minimal topological charges and possessing no internal degrees of freedom; the number of these is less than or equal to the rank of the gauge group. It is shown that if the unbroken gauge group is abelian, all solutions with higher topological charges belong to p-parameter families, where p is the number of position and group orientation parameters needed to describe a set of non-interacting fundamental monopoles with the given topological charge. It is argued that these solutions, some examples of which are given, should therefore be interpreted as multimonopole configurations. An extension of these results to the case of a non-albelian unbroken gauge symmetry is conjecture and shown to be valid for a number of examples.     

Modular invariance in N=2 superconformal field theories

The modular invariance properties of two-dimensional N=2 superconformal field theories are studied. It is shown that the character formulae of the central charge c<3 unitary highest weight representation for the untwisted algebras can be written in terms of the string functions and the theta functions of the affine su(2) Kac-Moody algebra. Deriving the modular transformation of the characters we construct the modular invariant partition functions on a torus. The character relation corresponding to the coset space construction of the unitary discrete series in the N=2 algebra is also obtained.     

AdS superalgebras with brane charges

We consider the inclusion of brane charges in AdS₅ superalgebras that contain the maximal central extension of the super-Poincaré algebra on ∂AdS₅. For theories with N supersymmetries on the boundary, the maximal extension is OSp(1/8N,R), which contains the group Sp(8N,R)⊃U(2N,2N)⊃SU(2,2)×U(N) as extension of the conformal group. An “intermediate” extension to U(2N,2N/1) is also discussed, as well as the inclusion of brane charges in AdS₇ and AdS₄ superalgebras. BPS conditions in the presence of brane charges are studied in some details.     

Central Charge Extended Supersymmetric Structures for Fundamental Fermions Around Non-Abelian Vortices

Fermionic zero modes around non-abelian vortices are shown that they constitute two N = 2, d = 1 supersymmetric quantum mechanics algebras. These two algebras can be combined under certain circumstances to form a central charge extended N = 4 supersymmetric quantum algebra. We thoroughly discuss the implications of the existence of supersymmetric quantum mechanics algebras, in the quantum Hilbert space of the fermionic zero modes.     

The soliton supermultiplet in 4 + 1 dimensions

We consider the SO(4) = SU(2) ? USp(2) Clifford algebra, obtained by the supersymmetry algebra for the N = 2 supersymmetric Yang-Mills theory in 4+1 dimensions, which, in the phase of unbroken gauge symmetry, has a topological charge as central charge. We find that, even if the Higgs mechanism is absent, the massive soliton supermultiplet contains the same number of states as the massless supermultiplet of elementary particles.     

Atomic charges in Mg2SiO4 (forsterite), fitted to thermoelastic and structural properties

First and second derivatives of cohesion energy of forsterite with respect to cell edges (generalized Born-Mayer model) are related to thermoelastic tensors; first derivatives with respect to five structural parameters (rotation angle of the SiO₄ tetrahedron, translations of Si and Mg(2) atoms) are set equal to zero by the zero-force principle. All differentiations are performed by keeping constant the internal geometry of the SiO₄ group. Fourteen equations are then obtained and solved numerically in the magnesium and oxygen atomic charges, and in three repulsive parameters, considered as unknowns. The best fitting charge distribution is: z_Mg = 1.38, z_o = ?1.05, z_si = l.44 e. Elastic constants are reproduced with an average relative deviation of 4%, and calculated atomic positions show an average shift of 0.014 A from experimental values. Results are discussed and compared with atomic charges determined from X-ray electron density measurements and vibrational spectroseopy data.     

Current algebra of quarks confined to a cavity

Equal-time commutators of fields with charges are calculated in a cavity approximation to the MIT bag model, with N flavours of non-interacting quarks confined to a rigid spherical cavity and SU(N) symmetry arbitrarily broken by mass terms. It is proved that inside the cavity the algebra is identical with that of free field theory, whilst on the boundary quark fields commute with axial charges. Vector divergences and sigma commutators belong to a $\text{(N,}\text{N}\text{) + (}\text{N}\text{, N)}$ multiplet of chiral SU(N) × SU(N). Axial divergences contain additional surface terms which do not contribute to sigma commutators. A non-strange quark mass in the range 20–44 MeV is required to give a value 30–70 MeV for the nucleon matrix element of the sigma commutator relevant to pion-nucleon scattering.     

All possible generators of supersymmetries of the S-matrix

A generator of a symmetry or supersymmetry of the S-matrix has to have three simple properties (see sect. 2). Starting from these properties one can give a complete analysis of the possible structure of the pseudo Lie algebra of these generators. In a theory with non-vanishing masses one finds that the only extension of previously known relations is the possible appearance of “central charges” as anticommutators of Fermi charges. In the massless case (disregarding infrared problems and symmetry breaking) the Fermi charges may generate the conformal group together with a unitary internal symmetry group.     

Yang-Baxter algebras of monodromy matrices in integrable quantum field theories

We consider a large class of two-dimensional integrable quantum field theories with non-abelian internal symmetry and classical scale invariance. We present a general procedure to determine explicitly the conserved quantum monodromy operator generating infinitely many non-local charges. The main features of our method are a factorization principle and the use of $\text{P}$, $\text{T}$, and internal symmetries. The monodromy operator is shown to satisfy a Yang-Baxter algebra, the structure constants (i.e. the quantum R-matrix) of which are determined by two-particle S-matrix of the theory. We apply the method to the chiral SU(N) and the O(2N) Gross-Neveu models.     

Finite-N fluctuation formulas for random matrices

For the Gaussian and Laguerre random matrix ensembles, the probability density function (p.d.f.) for the linear statistic Σ _j ^N =1 (x _j ? 〈x〉) is computed exactly and shown to satisfy a central limit theorem asN → ∞. For the circular random matrix ensemble the p.d.f.’s for the statistics ½Σ _j ^N =1 (θ _j ?π) and ? Σ _j ^N =1 log 2 |sinθ _j/2| are calculated exactly by using a constant term identity from the theory of the Selberg integral, and are also shown to satisfy a central limit theorem asN → ∞.