D. Dalmazi

A new spin-2 self-dual model in D = 2+1

There are three self-dual models of massive particles of helicity +2 (or -2) in D = 2+1. Each model is of first, second, and third-order in derivatives. Here we derive a new self-dual model of fourth-order, SD(4), which follows from the third-order model (linearized topologically massive gravity) via Noether embedment of the linearized Weyl symmetry. In fact, each self-dual model can be obtained from the previous one SD(i) → SD(i+1) , i = 1,2,3 by the Noether embedment of an appropriate gauge symmetry, culminating in SD(4). The new model may be identified with the linearized version of HDTMG = μνρΓμγ[∂νΓργ+(2/3)ΓνδγΓρδ]/8m + (−g)1/2[RμνRνμ−3 R2/8]/2m2. We also construct a master action relating the third-order self-dual model to SD(4) by means of a mixing term with no particle content which assures spectrum equivalence of SD(4) to other lower-order self-dual models despite its pure higher derivative nature and the absence of the Einstein-Hilbert action. The relevant degrees of freedom of SD(4) are encoded in a rank-two tensor which is symmetric, traceless and transverse due to trivial (non-dynamic) identities, contrary to other spin-2 self-dual models. We also show that the Noether embedment of the Fierz-Pauli theory leads to the new massive gravity of Bergshoeff, Hohm and Townsend.

Generalized soldering of +/- 2 helicity states in D=2+1

The direct sum of a couple of Maxwell-Chern-Simons gauge theories of opposite helicities {+-}1 does not lead to a Proca theory in D=2+1, although both theories share the same spectrum. However, it is known that by adding an interference term between both helicities we can join the complementary pieces together and obtain the physically expected result. A generalized soldering procedure can be defined to generate the missing interference term. Here, we show that the same procedure can be applied to join together {+-}2 helicity states in a full off-shell manner. In particular, by using second-order (in derivatives) self-dual models of helicities {+-}2 (spin-2 analogues of Maxwell-Chern-Simmons models) the Fierz-Pauli theory is obtained after soldering. Remarkably, if we replace the second-order models by third-order self-dual models (linearized topologically massive gravity) of opposite helicities, after soldering, we end up exactly with the new massive gravity theory of Bergshoeff, Hohm, and Townsend in its linearized approximation.

Dual descriptions of spin two massive particles in D = 2 + 1 via master actions

In the first part of this work we show the decoupling (up to contact terms) of redundant degrees of freedom which appear in the covariant description of spin-two massive particles in D=2+1. We make use of a master action which interpolates, without solving any constraints, between a first-, second-, and third-order (in derivatives) self-dual model. An explicit dual map between those models is derived. In our approach the absence of ghosts in the third-order self-dual model, which corresponds to a quadratic truncation of topologically massive gravity, is due to the triviality (no particle content) of the Einstein-Hilbert action in D=2+1. In the second part of the work, also in D=2+1, we prove the quantum equivalence of the gauge invariant sector of a couple of self-dual models of opposite helicities (+2 and -2) and masses m{sub +} and m{sub -} to a generalized self-dual model which contains a quadratic Einstein-Hilbert action, a Chern-Simons term of first order, and a Fierz-Pauli mass term. The use of a first-order Chern-Simons term instead of a third-order one avoids conflicts with the sign of the Einstein-Hilbert action.

Duality of parity doublets of helicity {+-}2 in D=2+1 dimensions

In D = 2+ 1 dimensions there are two dual descriptions of parity singlets of helicity ±1, namely the self-dual model of first-order (in derivatives) and the Maxwell-ChernSimons theory of second-order. Correspondingly, for helicity ±2 there are four models S (r) SD± describing parity singlets of helicities ±2. They are of first-, second-,thirdand fourth-order (r = 1, 2, 3, 4) respectively. Here we show that the generalized soldering of the opposite helicity models S (4) SD+ and S (4) SD− leads to the linearized form of the new massive gravity suggested by Bergshoeff, Hohm and Townsend (BHT) similarly to the soldering of S (3) SD+ and S (3) SD−. We argue why in both cases we have the same result. We also find out a triple master action which interpolates between the three dual models: linearized BHT theory, S (3) SD++S (3) SD− and S (4) SD++S (4) SD−. By comparing gauge invariant correlation functions we deduce dual maps between those models. In particular, we learn how to decompose the field of the linearized BHT theory in helicity eigenstates of the dual models up to gauge transformations.

Ghost free dual vector theories in 2+1 dimensions

We explore here the issue of duality versus spectrum equivalence in dual theories generated through the master action approach. Specifically we examine a generalized self-dual (GSD) model where a Maxwell term is added to the self-dual model. A gauge embedding procedure applied to the GSD model leads to a Maxwell-Chern-Simons (MCS) theory with higher derivatives. We show here that the latter contains a ghost mode contrary to the original GSD model. By figuring out the origin of the ghost we are able to suggest a new master action which interpolates between the local GSD model and a nonlocal MCS model. Those models share the same spectrum and are ghost free. Furthermore, there is a dual map between both theories at classical level which survives quantum correlation functions up to contact terms. The remarks made here may be relevant for other applications of the master action approach.

Generalized duality between local vector theories in D = 2 + 1

The existence of an interpolating master action does not guarantee the same spectrum for the interpolated dual theories. In the specific case of a generalized self-dual (GSD) model defined as the addition of the Maxwell term to the self-dual model in D = 2+1, previous master actions have furnished a dual gauge theory which is either nonlocal or contains a ghost mode. Here we show that by reducing the Maxwell term to first order by means of an auxiliary field we are able to define a master action which interpolates between the GSD model and a couple of non-interacting Maxwell-Chern-Simons theories of opposite helicities. The presence of an auxiliary field explains the doubling of fields in the dual gauge theory. A generalized duality transformation is defined and both models can be interpreted as self-dual models. Furthermore, it is shown how to obtain the gauge invariant correlators of the non-interacting MCS theories from the correlators of the self-dual field in the GSD model and vice-versa. The derivation of the non-interacting MCS theories from the GSD model, as presented here, works in the opposite direction of the soldering approach.

The Yang?Lee edge singularity in spin models on connected and non-connected rings

Renormalization group arguments based on a ?3 field theory lead us to expect a certain universal behavior for the density of partition function zeros in spin models with short-range interaction. Such universality has been tested analytically and numerically in different d = 1 and higher dimensional spin models. In d = 1, one finds usually the critical exponent ? = ?1/2. Recently, we have shown in the d = 1 Blume?Emery?Griffiths (BEG) model on a periodic static lattice (one ring) that a new critical behavior with ? = ?2/3 can arise if we have a triple degeneracy of the transfer matrix eigenvalues. Here we define the d = 1 BEG model on a dynamic lattice consisting of connected and non-connected rings (non-periodic lattice) and check numerically that also in this case we have mostly ? = ?1/2 while the new value ? = ?2/3 can arise under the same conditions of the static lattice (triple degeneracy) which is a strong check of universality of the new value of ?. We also show that although such conditions are necessary, they are not sufficient to guarantee the new critical behavior.

Massive spin-2 particles via embedment of the Fierz-Pauli equations of motion

Here we obtain alternative descriptions of massive spin-2 particles by an embedding procedure of the Fierz–Pauli equations of motion. All models are free of ghosts at the quadratic level, although most of them are of a higher order in derivatives. The models that we obtain can be nonlinearly completed in terms of a dynamic and a fixed metric. They include some f(R) massive gravities recently considered in the literature. In some cases there is an infrared (no derivative) modification of the Fierz–Pauli mass term altogether, with higher order terms in derivatives. The analytic structure of the propagator of the corresponding free theories is not affected by the extra terms in the action, as compared to the usual second order Fierz–Pauli theory.

Massive ``spin-2{''} theories in arbitrary D >= 3 dimensions

Abstract Here we show that in arbitrary dimensions D ≥ 3 there are two families of linearized second order Lagrangians describing massive “spin-2” particles via a nonsymmetric rank-2 tensor. They differ from the usual Fierz–Pauli theory in general. At zero mass one of the families is Weyl invariant. Such massless theory has no particle content in D = 3 and gives rise, via master action, to a dual higher order (in derivatives) description of massive spin-2 particles in D = 3 where both the second and the fourth order terms are Weyl invariant, contrary to the linearized New Massive Gravity. However, only the fourth order term is invariant under arbitrary antisymmetric shifts. Consequently, the antisymmetric part of the tensor e [ μ ν ] propagates at large momentum as 1 / p 2 instead of 1 / p 4 . So, the same kind of obstacle for the renormalizability of the New Massive Gravity reappears in this nonsymmetric higher order description of massive spin-2 particles.

Generalized partition function zeros of 1D spin models and their critical behavior at edge singularities

Here we study the partition function zeros of the one-dimensional Blume–Emery–Griffiths model close to their edge singularities. The model contains four couplings (H, J, Δ, K) including the magnetic field H and the Ising coupling J. We assume that only one of the three couplings (J, Δ, K) is complex and the magnetic field is real. The generalized zeros zi tend to form continuous curves on the complex z-plane in the thermodynamic limit. The linear density at the edges zE diverges usually with ρ(z) ~ |z − zE|σ and σ = −1/2. However, as in the case of complex magnetic fields (Yang–Lee edge singularity), if we have a triple degeneracy of the transfer matrix eigenvalues a new critical behavior with σ = −2/3 can appear as we prove here explicitly for the cases where either Δ or K is complex. Our proof applies for a general three-state spin model with short-range interactions. The Fisher zeros (complex J) are more involved; in practice, we have not been able to find an explicit example with σ = −2/3 as far as the other couplings (H, Δ, K) are kept as real numbers. Our results are supported by numerical computations of zeros. We show that it is absolutely necessary to have a non-vanishing magnetic field for a new critical behavior. The appearance of σ = −2/3 at the edge closest to the positive real axis indicates its possible relevance for tricritical phenomena in higher-dimensional spin models.