Strings on orbifolds (II)

A general framework for analyzing string propagation on the quotient of a manifold by a discrete group is presented.The conditions necessary for modular invariance and absence of tachyons are discussed. Models with four or fewer generations which correspond to the standard embedding of the spin connection in the gauge group are analyzed. We all discuss some generalizations which may lead to more phenomenologically promising models.     

Modular invariance,chiral anomalies and contact terms

The chiral anomaly in heterotic strings with full and partial modular invariance in D = 2n + 2 dimensions is calculated. The boundary terms which were present in previous calculations are shown to be cancelled in the modular-invariant case by contact terms, which can be obtained by an appropriate analytic continuation. The relation to the low-energy field theory is explained. In theories with partial modular invariance, an expression for the anomaly iis obtained and shown to be non-zero in general.     

Generalized lattice fermion actions and applications

We derive the general form of lattice fermion action consistent with the requirements of gauge invariance, translation invariance, reflection positivity and invariance under 90° rotations, and involving only bilinear, nearest neighbour couplings. The meaning of the parameters occuring in the action is discussed analyzing the spectrum, the symmetries and the axial Ward identities of the theory, and their renormalization is studied within the Migdal-Kadanoff approximation. In particular we give the relation between the dependence of the vacuum energy density on the CP phases appearing in the action and the mean topological density and susceptibility.     

Classification of symmetry breaking by Wilson lines on the Z orbifold

We give a complete classification of gauge symmetry breaking by Wilson lines on the standard Z orbifold by deriving the general formula of the conditions of modular invariance and group invariance in the presence of background gauge fields. All possible E₆×SU(3) breaking in terms of one Wilson line is given. The symmetries of the electroweak and grand unification are obtained by combining two Wilson lines.     

Remarks on Wilson lines,modular invariance and possible string relics in Calabi-Yau compactifications

When one mods out a (2,2) conformal field theory by the action of a discrete group, it is possible to include Wilson lines to break the gauge symmetry. We simplify and generalize an earlier analysis by Witten of the constraints that modular invariance places on the allowed symmetry breaking patterns. The analysis does not depend on the details of the original conformal field theory. We then consider the fractionally charged states in such theories, first discussed by Wen and Witten. We note that these are rather generic, and consider the possibilities for their detection. We also note that, while in general they are expected to be massive (∼M_Planck), in models based on free fields, such as orbifold compactifications, there are likely to be massless (very light) fractionally charged states.     

Operator expression of Witten's superstring vertex

The explicit operator expressions of both the 3-Neveu-Schwarz (NS) and Neveu-Schwarz-Ramond-Ramond vertices in Witten's superstring field theory are given by using the method recently developed by Kunitomo and the author. The BRST invariance of these vertices is proved at the critical dimension d=10. Further, we clarify the relation between the prefactor of the 3-NS vertex required by the BRST invariance and the picture changing operator.     

Modular invariance and the spectrum of two dimensional conformal field theories

Modular invariance has recently emerged as a powerful tool in conformal field theory. In conjunction with the representation theory of infinite dimensional Lie algebras, the study of modular invariance gave the spectrum of several families of theories. These include the minimal conformal models (Cardy and others), WZW theories which describe string propagation on group manifolds (Gepner and Witten) and parafermionic field theories (Gepner and Qiu). The minimal conformal models models were shown to be a product of two SU(2) WZW theories (Gepner). These results represent a step towards a complete classification of conformal field theories, an important goal both for the study of critical phenomena and string theory.     

Orbifold models and modular transformation

The orbifold models of the heterotic string are constructed on the quotient spaces of generalized tori by translational and rotational discrete symmetries. In order to obtain the consistent orbifold models, the conditions of the modular invariance are derived from a one-loop vacuum amplitude. Z₃ orbifold models satisfying such conditions are searched systematically. It is shown that there are infinite possible models with N = 2 supersymmetry. Among these models, two examples having E₆ and E₇ gauge groups are discussed. The orbifold models with N = 1 supersymmetry are also discussed in detail. It is shown that there are only five consistent models in the class of these models based on E₈ ⊗ E′₈ heterotic string in which the extra six-dimensional torus and the E₈ ⊗ E′₈ maximal torus are modded out by the rotational and the translational Z₃ symmetries respectively.     

4d fermionic superstrings with arbitrary twists

We present the rules for systematically constructing all consistent four-dimensional string theories, using free world-sheet fermions which pick up arbitrary phases when parallel transported around the string. These rules are necessary and sufficient for multi-loop modular invariance. They lead to theories with general Z_N (GSO-type) projections, whose merits for model-building we discuss. We classify all boundary conditions yielding massless space-time spinors. We show that, in contrast to the case of only real 2d fermions, all possible realizations of world-sheet supersymmetry are now allowed. This opens the way for the construction of a new class of supersymmetric string models.     

Conformal invariance and conserved quantities of general holonomic systems in phase space

This paper studies the conformal invariance and conserved quantities of general holonomic systems in phase space. The definition and the determining equation of conformal invariance for general holonomic systems in phase space are provided. The conformal factor expression is deduced from conformal invariance and Lie symmetry. The relationship between the conformal invariance and the Lie symmetry is discussed, and the necessary and sufficient condition that the conformal invariance would be the Lie symmetry of the system under the infinitesimal single-parameter transformation group is deduced. The conserved quantities of the system are given. An example is given to illustrate the application of the result.     

Gaugino masses in modular invariant supergravity

We calculate gaugino masses in string-derived models with hidden-sector gaugino condensation. The linear multiplet formulation for the dilaton superfield is used to implement perturbative modular invariance. The contribution arising from quantum effects in the observable sector includes the term recently found in generic supergravity models. A much larger contribution is present if matter fields with Standard Model gauge couplings also couple to the Green–Schwarz counter term. We comment on the relation of our Kähler U(1) superspace formalism to other calculations.     

Conformal invariance and conserved quantities of a general holonomic system with variable mass

Conformal invariance and conserved quantities of a general holonomic system with variable mass are studied. The definition and the determining equation of conformal invariance for a general holonomic system with variable mass are provided. The conformal factor expression is deduced from conformal invariance and Lie symmetry. The relationship between the conformal invariance and the Lie symmetry is discussed, and the necessary and sufficient condition under which the conformal invariance would be the Lie symmetry of the system under an infinitesimal one-parameter transformation group is deduced. The conserved quantities of the system are given. An example is given to illustrate the application of the result.     

Two-dimensional conformal invariant theories on a torus

The study of unitary conformal invariant theories on a torus reveals two important properties: the partition function and correlation functions may be expressed in terms of free (gaussian) field modes, and the modular invariance dictates the operator content of the theory: for a generic value of the central charge c = 1−/m(m + 1), there exist at least two distinct models depending whether m = 0,3 mod or m = 1,2 mod 4. The case of non-unitary c < 1 theories is also briefly discussed.     

Heterotic string theory: (II). The interacting heterotic string

The theory of interacting heterotic strings is presented. Vertex operators are derived in both the bosonic and fermionic formulations of the theory and are shown to be consistent with gauge invariance, Lorentz invariance, and supersymmetry. Three- and four-point amplitudes for the scattering of massless string states are calculated and used to derive the low-energy field theory limit of the heterotic string. Divergences in string theories are discussed and it is shown that one-loop heterotic string amplitudes are finite and modular invariant only for gauge group E₈×E₈ or spin (32)/Z₂.     

Violation of CP invariance in the two-doublet higgs sector of the MSSM

Problems of substantiation and investigation of CP violation in models with the extended Higgs sector are discussed. The general form of the effective two-doublet Higgs potential with complex parameters whose CP invariance is violated both explicitly and spontaneously is considered. For the special case of the two-doublet Higgs sector of the minimum supersymmetric model in which the CP invariance of the effective potential is violated by the interaction of Higgs fields with third generation scalar quarks, the physical states of Higgs bosons, their masses and interaction constants are obtained. The basic phenomenological scenarios and predictions for investigation of the properties of the Higgs sector are considered.     

Modular bootstrap in Liouville field theory

The modular matrix for the generic 1-point conformal blocks on the torus is expressed in terms of the fusion matrix for the 4-point blocks on the sphere. The modular invariance of the toric 1-point functions in the Liouville field theory with DOZZ structure constants is proved.     

s-channel discontinuities of reggeon diagrams with planar vertices

The mutual relations between s-channel discontinuities of reggeon diagrams with planar triple-Regge vertices are discussed. It is shown that, in general, contributions coming from different cuttings are not simply related to each other as certain vertices acquire non-trivial phase factors depending on the nature of the cutting. For diagrams with only triple-Regge vertices this phase may be related to the triple-Regge phase. The complete four-reggeon vertices are not known in general. An approximation is considered which amounts to preserving only the absorptive part of triple- or four-reggeon amplitudes in certain parts of the diagrams. This approximation turns out to be equivalent to a simple absorbed multiperipheral model. The problem of exchange degeneracy breaking within the dual unitarisation Scheme is briefly discussed as an example.     

Conformal geometry and string field theory

The covariant perturbation theory rules that should arise from any gauge invariant string field theory, such as those proposed on the basis of BRST formalism, are set forth. The resulting path integral expressions naturally produce coordinate invariant densities on the moduli space of Riemann surfaces; these include the Koba-Nielsen amplitudes. The connection between string field theory and modular invariance is discussed, and it is proposed that future explorations in string field theory focus on coordinate invariant quantities on moduli space.     

Topological weights and combinatorical unitarity in first and second quantization

We give a unified and coherent account on particle and string theories with a particular emphasis on the comparison of first and second quantization. The formalism described is applicable to other (higher dimensional) propagating objects. The weights of topologically different graphs determine the possible interactions. Restrictions on them, imposed by combinatorical aspects of unitarity are discussed. These restrictions are particularly strong for string theories. Reparametrization and modular invariance are also considered in this light.