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.     

Non-linear sigma models on arbitrary genus Riemann surfaces

A Ward-Takahashi type identity is obtained for two insertions of the energy-momentum tensor of the non-linear sigma model on an arbitrary Riemann surface. The identity shows explicitly how the Virasoro algebra is violated by spurious terms generated by the trace anomaly. Requiring these terms to vanish leads to a set of constraints on the graviton and dilaton background fields, which are necessary for the algebra to be restored. Although the modular parameters play an important role in the computation, the background field equations turn out to be genus independent up to order α′.     

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.     

Anomaly cancelling terms from the elliptic genus

We calculate the heterotic string one-loop diagram in 2n + 2 dimensions with one external B_μν and n external gravitons and/or gauge bosons. The result is a modular integral over the weight zero terms of the character valued partition function (or elliptic genus) of the theory, and can be directly expressed in terms of the factor which multiplies TrF² − TrR² in the field theory anomaly. The integrands have a non-trivial dependence on the modular parameter τ, reflecting contributions not only from the physical massless states but also from an infinity of “unphysical” modes. Some of them are identical to integrands which have been discussed recently in relation with Atkin-Lehner symmetry and the cosmological constant. As a corollary we find a method to compute these integrals without using Atkin-Lehner transformations.     

New heterotic string theories in uncompactified dimensions < 10

It is shown that infinitely many heterotic string theories exist in uncompactified dimensions less than 10, that are one-loop finite (for massless external legs). Tachyons are removed by compactifying into tori (10 − d) and (26 −d) dimensions of the right-moving superstring and left-moving bosonic string sectors, respectively. The condition for modular invariance is shown to be equivalent to self-duality condition on even lorentzian lattices with (10 −d) and (26 −d) timelike and spacelike directions, respectively. The construction results in a (10 −d)(26 −d) parameter family of one-loop finite string theories. The zero mass sector of these theories for d = 4 and 6 correspond to N = 4 and 2 supergravity coupled to super Yang-Mills with many possible groups, some of which cannot be obtained by compactifying d = 10 heterotic string theory.     

On the absence of supersymmetry anomalies in gauge theories

It is shown that N = 2 non-abelian gauge theory can be renormalized consistent with gauge invariance, supersymmetry and unitarity. There is no perturbative supersymmetry anomaly: the proof is independent of any specific regulator.     

TFT construction of RCFT correlators I: partition functions

We formulate rational conformal field theory in terms of a symmetric special Frobenius algebra A and its representations. A is an algebra in the modular tensor category of Moore–Seiberg data of the underlying chiral CFT. The multiplication on A corresponds to the OPE of boundary fields for a single boundary condition. General boundary conditions are A-modules, and (generalised) defect lines are A–A-bimodules.The relation with three-dimensional TFT is used to express CFT data, like structure constants or torus and annulus coefficients, as invariants of links in three-manifolds. We compute explicitly the ordinary and twisted partition functions on the torus and the annulus partition functions. We prove that they satisfy consistency conditions, like modular invariance and NIM-rep properties.We suggest that our results can be interpreted in terms of non-commutative geometry over the modular tensor category of Moore–Seiberg data.     

Conformal anomaly in 4D gravity-matter theories non-minimally coupled with a dilaton

The conformal anomaly for 4D gravity-matter theories, which are non-minimally coupled with the dilaton, is systematically studied. Special care is taken of rescaling of fields, treatment of total derivatives, hermiticity of the system operator and the choice of measure. Scalar, spinor and vector fields are taken as the matter quantum fields and their explicit conformal anomalies in the gravity-dilaton background are found. The cohomology analysis is carried out and some new conformal invariants and trivial terms, involving the dilaton, are obtained. The symmetry of the constant shift of the dilaton field plays an important role. The general structure of the conformal anomaly is examined. It is shown that the dilaton affects the conformal anomaly characteristically for each case: (1) [Scalar] The dilaton changes the conformal anomaly only by a new conformal invariant, I₄; (2) [Spinor] The dilaton does not change the conformal anomaly; (3) [Vector] The dilaton changes the conformal anomaly by three new (generalized) conformal invariants, I₄, I₂, I₁. We present some new anomaly formulae which are useful for practical calculations. Finally, the anomaly induced action is calculated for the dilatonic Wess-Zumino model. We point out that the coefficient of the total derivative term in the conformal anomaly for the 2D scalar coupled to a dilaton is ambiguous. This resolves the disagreement between earlier calculations and the result of Hawking and Bousso.     

Renormalization of Lorentz non-invariant actions and manifest T-duality

We study general two-dimensional σ-models which do not possess manifest Lorentz invariance. We show how demanding that Lorentz invariance is recovered as an emergent on-shell symmetry constrains these σ-models. The resulting actions have an underlying group-theoretic structure and resemble Poisson–Lie T-duality invariant actions. We consider the one-loop renormalization of these models and show that the quantum Lorentz anomaly is absent. We calculate the running of the couplings in general and show, with certain non-trivial examples, that this agrees with that of the T-dual models obtained classically from the duality invariant action. Hence, in these cases solving constraints before and after quantization are commuting operations.     

Chiral deformations of conformal field theories

We study general perturbations of two-dimensional conformal field theories by holomorphic fields. It is shown that the genus one partition function is controlled by a contact term (pre-Lie) algebra given in terms of the operator product expansion. These models have applications to vertex operator algebras, two-dimensional QCD, topological strings, holomorphic anomaly equations and modular properties of generalized characters of chiral algebras such as the W_1+∞ algebra, that is treated in detail.     

Trace anomalies and λφ4 theory in curved space

It is shown for a conformally invariant λφ⁴ theory in a weakly curved background, how to extend previous results to obtain full information about the trace anomaly in perturbation theory, including the “topological” term in the gravitational part of the anomaly. There is a strong connection among renormalisability of the curved space theory, finiteness of the energy-momentum tensor, and the role of normal products. Combined with a renormalisation-group analysis this provides an efficient means of calculating some terms in the anomaly to high orders of perturbation theory. In particular, the first λ-dependent coefficient of the topological part of the anomaly appears at O(λ⁴) and can be deduced from simple flat-space results without the calculation of any further Feynman diagrams. Some techniques based on an absorptive-part argument are developed in order to compute other anomalous coefficients, and a direct 5-loop calculation confirms the indirect renormalisation-group derivation of a non-vanishing R² anomaly at O(λ⁵). All the essential information can be obtained from the massless theory. The underlying ideas are applicable to other theories, and similar results for massless QED are obtained in a subsequent paper.     

Cancellation of the chiral anomaly in a model with spontaneous symmetry breaking

A perturbatively renormalized Abelian Higgs-Kibble model with a chirally coupled fermion is considered. The Slavnov identity is fulfilled to all orders of perturbation theory, which is crucial for renormalizability in models with vector bosons. BRS invariance, i.e. the validity of the identity, forces the chiral anomaly to be cancelled by Wess-Zumino counterterms. This procedure preserves the renormalizability in the one-loop approximation but it violates the Froissart bounds for partial wave amplitudes above some energy and destroys renormalizability from the second order in? onwards due to the counterterms.     

Off-shell conformal methods for the superstring

Superconformal field theory is used to compute off-shell amplitudes for the superstring. Off-shell three- and four-point functions in Witten's superstring field theory are obtained. The problem of gauge invariance due to the associativity anomaly in the bosonic sector is resolved to order g².     

Scalar mesons and glueball in a quark model allowing for gluon anomalies

This paper is a sequel of a previous one (Scalar mesons in a chiral quark model with glueball, Eur. Phys. J. A 8, 567 (2000)) where an attempt to construct an effective U(3)×U(3)-symmetric meson Lagrangian with a scalar glueball was made. The glueball was introduced by using the dilaton model on the base of scale invariance. The scale invariance breaking because of current quark masses and the scale anomaly of QCD, reproduced by the dilaton potential, was taken into account. However, in the previous paper, the scale invariance breaking because of the terms like h _φφ² and h _σ , where φ and are the pseudoscalar and scalar isosinglets, was not taken into account. These terms are produced by the part of the 't Hooft interaction that is connected with gluon anomalies. Allowing for the scale invariance breaking by these terms has a decisive effect on the quarkonium-glueball mixing and noticeably changes the widths of glueball strong decays. Taking account of this additional source of the scale invariance breaking and its implications are the subject of the present work. It is also shown that in the decay of a glueball into four pions, the channel with two ρ-resonances dominates. Received: 11 January 2001 / Accepted: 25 January 2001     

Universal volume of groups and anomaly of Vogel’s symmetry

We show that integral representation of universal volume function of compact simple Lie groups gives rise to six analytic functions on \({ CP }^2\), which transform as two triplets under group of permutations of Vogel’s projective parameters. This substitutes expected invariance under permutations of universal parameters by more complicated covariance. We provide an analytic continuation of these functions and calculate their change (anomaly) under permutations of parameters (Vogel’s symmetry). This last relation is universal generalization, for an arbitrary simple Lie group and, moreover, to an arbitrary point in Vogel’s plane, of the Kinkelin’s reflection relation on Barnes’ \(G(1+N)\) function. Kinkelin’s relation gives asymmetry of the \(G(1+N)\) function (which is essentially reciprocal of the volume function for \({ SU }(N)\) groups) under \(N\leftrightarrow -N\) transformation (which is equivalent of the permutation of Vogel’s parameters for \({ SU }(N)\) groups), and coincides with above-mentioned anomaly of permutations at the \({ SU }(N)\) line on Vogel’s plane. Our results also give an anomaly of Vogel’s symmetry of the universal partition function of Chern–Simons theory on three-dimensional sphere. This effect is analogous to modular covariance, instead of invariance, of partition functions of appropriate gauge theories under modular transformation of couplings.     

Superconvergent functional output for time-dependent problems using finite differences on summation-by-parts form

Finite difference operators satisfying the summation-by-parts (SBP) rules can be used to obtain high order accurate, energy stable schemes for time-dependent partial differential equations, when the boundary conditions are imposed weakly by the simultaneous approximation term (SAT).In general, an SBP-SAT discretization is accurate of order p + 1 with an internal accuracy of 2p and a boundary accuracy of p. Despite this, it is shown in this paper that any linear functional computed from the time-dependent solution, will be accurate of order 2p when the boundary terms are imposed in a stable and dual consistent way.The method does not involve the solution of the dual equations, and superconvergent functionals are obtained at no extra computational cost. Four representative model problems are analyzed in terms of convergence and errors, and it is shown in a systematic way how to derive schemes which gives superconvergent functional outputs.     

Quasielastic rho-meson knockout from nucleons by high-energy electrons

The electroproduction of charged rho mesons on a nucleon under conditions of quasielastic kinematics [Q ² > 2 (GeV/c)², W > 3 GeV, and ω _ρ ? q ⁰ in the laboratory frame] is considered. The main diagrams involved in this process are taken into account, and it is shown that the case of rho-meson knockout by a longitudinal photon is dominated by the t-channel pole diagram featuring a rho meson, so that direct rho-meson electroknockout makes a leading contribution to the process being considered. Questions concerning the gauge invariance of the electroproduction process are considered in detail, along with contact terms that inevitably arise in employing form factors in ρNN vertices and which are necessary for restoring gauge invariance. It is shown that the structure of these contact terms is not determined exclusively by the requirement that the electroproduction amplitude be gauge-invariant and may be such that the corresponding correction to the dominant diagram due to these terms proves to be suppressed in the quasielastic region (with increasing Q ²), with the result that the t-pole process featuring a rho meson remains dominant; at the same time, the contribution of this correction is significant at low Q ² or in the case of photoproduction, modifying substantially the electroproduction cross section.     

Lorentz anomaly in arbitrary dimensions

It is shown that the Lorentz invariance is broken in gauge theories of chiral Weyl fermions in flat space-time via one-loop quantum corrections. Abelian gauge fields contribute to this anomaly in even dimensions larger than or equal to four and non-Abelian gauge fields do in even dimensions larger than or equal to six. The anomaly is proportional toD/2–1 power to the charge, whereD is a number of space-time dimensions.     

The one-loop N = 2 supercurrent anomaly

A one-loop superspace calculation of the N = 2 supercurrent anomaly is performed. The method preserves explicitly both N = 2 supersymmetry and gauge invariance.