Quantum field theories and critical phenomena on defects

We derive a nonlinear integral equation (NLIE) for some bulk excited states of the sine-Gordon model on a finite interval with general integrable boundary interactions, including boundary terms proportional to the first time derivative of the field. We use this NLIE to compute numerically the dimensions of these states as a function of scale, and check the UV and IR limits analytically. We also find further support for the ground-state NLIE by comparison with boundary conformal perturbation theory (BCPT), boundary truncated conformal space approach (BTCSA) and the boundary analogue of the Lüscher formula.     

Conformal field theories and modular invariant partition functions for parafermion field theories

Parafermion conformal field theories with D_2N discrete symmetry are examined in detail. The structure of field space of parafermion field theories is studied with the help of a projection operator G. Characters of the representations of the twist sector of parafermion algebra and projected characters are given. A new class of modular invariant partition functions, therefore conformal field theories, for parafermion theories are found. We argue that the principal theories correspond to the generic critical SOS models of Andrew, Baxter and Forrest.     

Conformal field theories via Hamiltonian reduction

Constraining theSL(3) WZW-model we construct a reduced theory which is invariant with respect to the new chiral algebraW ₃ ² . This symmetry is generated by the stress-energy tensor, two bosonic currents with spins 3/2 and theU(1) current. We conjecture a Kac formula that describes the highly reducible representation for this algebra. We also discuss the quantum Hamiltonian reduction for the general type of constraints that leads to the new extended conformal algebras.Address after September 1990: Lyman Laboratory, Harvard University, Cambridge, MA 02138, USA     

Conformal field theories,representations and lattice constructions

An account is given of the structure and representations of chiral bosonic meromorphic conformal field theories (CFT's), and, in particular, the conditions under which such a CFT may be extended by a representation to form a new theory. This general approach is illustrated by considering the untwisted andZ ₂-twisted theories, () and respectively, which may be constructed from a suitable even Euclidean lattice . Similarly, one may construct lattices and by analogous constructions from a doubly-even binary code . In the case when is self-dual, the corresponding lattices are also. Similarly, () and are self-dual if and only if is. We show that has a natural triality structure, which induces an isomorphism and also a triality structure on . For the Golay code, is the Leech lattice, and the triality on is the symmetry which extends the natural action of (an extension of) Conway's group on this theory to the Monster, so setting triality and Frenkel, Lepowsky and Meurman's construction of the natural Monster module in a more general context. The results also serve to shed some light on the classification of self-dual CFT's. We find that of the 48 theories () and with central charge 24 that there are 39 distinct ones, and further that all 9 coincidences are accounted for by the isomorphism detailed above, induced by the existence of a doubly-even self-dual binary code.     

Conformal field theories in six-dimensional twistor space

This article gives a study of the higher-dimensional Penrose transform between conformally invariant massless fields on space–time and cohomology classes on twistor space, where twistor space is defined to be the space of projective pure spinors of the conformal group. We focus on the six-dimensional case in which twistor space is the 6-quadric Q$Q$ in CP⁷${\mathbb{C}\mathbb{P}}^7$ with a view to applications to the self-dual (0,2)$(0,2)$-theory. We show how spinor-helicity momentum eigenstates have canonically defined distributional representatives on twistor space (a story that we extend to arbitrary dimension). These yield an elementary proof of the surjectivity of the Penrose transform. We give a direct construction of the twistor transform between the two different representations of massless fields on twistor space (H²$H^2$ and H³$H^3$) in which the H³$H^3$s arise as obstructions to extending the H²$H^2$s off Q$Q$ into CP⁷${\mathbb{C}\mathbb{P}}^7$.     

Conformal field theories near a boundary in general dimensions

The implications of restricted conformal invariance under conformal transformations preserving a plane boundary are discussed for general dimensions d. Calculations of the universal function of a conformal invariant ξ which appears in the two-point function of scalar operators in conformally invariant theories with a plane boundary are undertaken to first order in the ge = 4 − d expansion for the operator φ² in φ⁴ theory. The form for the associated functions of ξ for the two-point functions for the basic field φ^α and the auxiliary field λ in the N → ∞ limit of the O(N) nonlinear sigma model for any d in the range 2 < d < 4 are also rederived. These results are obtained by integrating the two-point functions over planes parallel to the boundary, defining a restricted two-point function which may be obtained more simply. Assuming conformal invariance this transformation can be inverted to recover the full two-point function. Consistency of the results is checked by considering the limit d → 4 and also by analysis of the operator product expansions for φ^αφ^β and λλ. Using this method the form of the two-point function for the energy-momentum tensor in the conformal O(N) model with a plane boundary is also found. General results for the sum of the contributions of all derivative operators appearing in the operator product expansion, and also in a corresponding boundary operator expansion, to the two-point functions are also derived making essential use of conformal invariance.     

Mean field theory of self-organized critical phenomena

A mean field theory is presented for the recently discovered self-organized critical phenomena. The critical exponents are calculated and found to be the same as the mean field values for percolation. The power spectrum has 1/f behavior with exponentg4=1.     

Conformal invariance and scalar-tensor theories

A new generalisation of Einstein’s theory is proposed which is invariant under conformal mappings. Two scalar fields are introduced in addition to the metric tensor field, so that two special choices of gauge are available for physical interpretation, the ‘Einstein gauge’ and the ‘atomic gauge’. The theory is not unique but contains two adjustable parameters ζ anda. Witha=1 the theory viewed from the atomic gauge is Brans-Dicke theory (ω=−3/2+ζ/4). Any other choice ofa leads to a creation-field theory. In particular the theory given by the choicea=−3 possesses a cosmological solution satisfying Dirac’s ‘large numbers’ hypothesis.     

Conformal theories and punctured surfaces

We define conformal theories as realizations of certain operations involving punctured Riemann surfaces (with coordinates chosen at the punctures) in a Hilbert space. We describe the connections of our formalism with other formulations of conformal theories.     

Random field effects in classical and quantum critical phenomena

The effect of quenched random fields on classical and quantum critical behaviour is studied by means of the ?-analysis for a number of systems. The investigation is performed in terms of a generalized random-field correlation function. The interplay of short-range correlations as well as of a parameter-dependent variety of long-range correlations with thermal and quantum fluctuations is revealed.     

Conformal transformations in scalar-tensor gravitation theories

Conformal transformations in scalar-tensor gravitation theories are investigated in the present paper. It is demonstrated that the field equations of these theories can be expressed in the Vaidya form. A scalar field equation is derived based on the compatibility condition for the field equations. The conformal factor, dilaton, and restrictions on the metric are determined for the diagonal metric of type I in the Bianchi classification. __________ Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 2, pp. 10–14, February, 2006.     

Conformal invariance and surface critical behavior

Conformal invariance constrains the form of correlation functions near a free surface. In two dimensions, for a wide class of models, it completely determines the correlation functions at the critical point, and yields the exact values of the surface critical exponents. They are related to the bulk exponents in a non-trivial way. For the Q-state Potts model (0 Q 4) we find η_<|; = 2/(3v − 1), and for the O(N) model (−2 N 2), η_<|; = (2v − 1)/(4v − 1).     

Conformal transformations and string field redefinitions

It is shown that conformal transformations of interaction vertices in Witten-type string field theories are realized to lowest order as nonlinear string field redefinitions. However, these redefinitions induce higher-order interactions. Unlike the redefinitions commonly done in local field theory, the S-matrix beyond tree level is affected. In particular, the change of variables determinant contributes integrals over different regions of moduli space.     

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.     

Unconventional field theories

Two relativistic, local, renormalizable field theories are defined. The first one is characterized by not having free particles and could provide a model for the permanent binding of quarks. The second one is a four fermion interaction which is constructed in a way similar to QED. It gives exactly the V-A type of coupling in first order, but higher orders are finite.Invited talk at the Symposium on Mathematical Methods in the Theory of Elementary Particles, Liblice castle, Czechoslovakia, June 18–23, 1978.