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, L (4) SD are encoded in a ranktwo 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.

Using the replica method, we analyze the mass dependence of the QCD 3 partition function in a parameter range where the leading contribution is from the zero momentum Goldstone fields. Three complementary approaches are considered in this article. First, we derive exact relations between the QCD 3 partition function and the QCD 4 partition function continued to half-integer topological charge. The replica limit of these formulas results in exact relations between the corresponding microscopic spectral densities of QCD 3 and QCD 4 . Replica calculations, which are exact for QCD 4 at half-integer topological charge, thus result in exact expressions for the microscopic spectral density of the QCD 3 Dirac operator. Second, we derive Virasoro constraints for the QCD 3 partition function. They uniquely determine the small-mass expansion of the partition function and the corresponding sum rules for inverse Dirac eigenvalues. Due to de Wit-'t Hooft poles, the replica limit only reproduces the small mass expansion of the resolvent up to a finite number of terms. Third, the large mass expansion of the resolvent is obtained from the replica limit of a loop expansion of the QCD 3 partition function. Because of Duistermaat-Heckman localization exact results are obtained for the microscopic spectral density in this way.

The direct sum of a couple of Maxwell-Chern-Simons (MCS) 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 two analogues of MCS 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 we end up after soldering exactly with the new massive gravity theory of Bergshoeff, Hohm and Townsend in its linearized approximation.

We start this work by revisiting the problem of the soldering of two chiral Schwinger models of opposite chiralities. We verify that, different from what one can conclude from the current literature, the usual sum of these models is, in fact, gauge invariant and corresponds to a composite model, where the component models are the vector and axial Schwinger models. As a consequence, we reinterpret this formalism as a kind of degree of freedom reduction mechanism. This result has led us to discover a second soldering possibility giving rise to the axial Schwinger model. This new result is seemingly rather general. We explore it here in the soldering of two Maxwell-Chern-Simons theories with different masses.

We investigate the replica trick for the microscopic spectral density, ρ s (x), of the Euclidean QCD Dirac operator. Our starting point is the low-energy limit of the QCD partition function for n fermionic flavors (or replicas) in the sector of topological charge ν. In the domain of the smallest eigenvalues, this partition function is simply given by a U(n) unitary matrix integral. We show that the asymptotic behavior of ρ s (x) for x → ∞ is obtained from the n → 0 limit of this integral. The smooth contributions to this series are obtained from an expansion about the replica symmetric saddle-point, whereas the oscillatory terms follow from an expansion about a saddle-point that breaks the replica symmetry. For ν = 0 we recover the small-x logarithmic singularity of the resolvent by means of the replica trick. For half integer ν, when the saddle point expansion of the U(n) integral terminates, the replica trick reproduces the exact analytical result. In all other cases only an asymptotic series that does not uniquely determine the microscopic spectral density is obtained. We argue that bosonic replicas fail to reproduce the microscopic spectral density. In all cases, the exact answer is obtained naturally by means of the supersymmetric method.

By means of a triple master action we deduce here a linearized version of the "New Massive Gravity" (NMG) in arbitrary dimensions. The theory contains a 4th-order and a 2nd-order term in derivatives. The 4th-order term is invariant under a generalized Weyl symmetry. The action is formulated in terms of a traceless η µν Ω µνρ = 0 mixed symmetry tensor Ω µνρ = −Ω µρν and corresponds to the massive Fierz-Pauli action with the replacement e µν = ∂ ρ Ω µνρ . The linearized 3D and 4D NMG theories are recovered via the invertible maps Ω µνρ = ǫ

In ferromagnetic spin models above the critical temperature ͑T Ͼ T cr ͒ the partition function zeros accumulate at complex values of the magnetic field ͑H E ͒ with a universal behavior for the density of zeros ͑H͒ ϳ͉H − H E ͉ . The critical exponent is believed to be universal at each space dimension and it is related to the magnetic scaling exponent y h via = ͑d − y h ͒ / y h . In two dimensions we have y h =12/ 5 ͑ =−1/ 6͒ while y h =2 ͑ =−1/ 2͒ in d = 1. For the one-dimensional Blume-Capel and Blume-Emery-Griffiths models we show here, for different temperatures, that a value y h =3 ͑ =−2/ 3͒ can emerge if we have a triple degeneracy of the transfer matrix eigenvalues.

We derive Virasoro constraints for the zero momentum part of the QCD-like partition functions in the sector of topological charge . The constraints depend on the topological charge only through the combination N f ϩ␤/2 where the value of the Dyson index ␤ is determined by the reality type of the fermions. This duality between flavor and topology is inherited by the small-mass expansion of the partition function and all spectral sum rules of inverse powers of the eigenvalues of the Dirac operator. For the special case ␤ϭ2 but arbitrary topological charge the Virasoro constraints are solved uniquely by a generalized Kontsevich model with the potential V(X)ϭ1/X. DOI: 10.1103/PhysRevD.64.054002 PACS number͑s͒: 12.39.Fe, 11.30.Rd, 12.38.Lg I. FLAVOR-TOPOLOGY DUALITYThrough the work of 't Hooft we know that the lowenergy limit of QCD is dominated by light flavors and topology ͓1͔. We expect that the same will be the case for the low-lying eigenvalues of the Dirac operator. Indeed, for massless flavors, the fermion determinant results in the repulsion of eigenvalues away from ϭ0. It is perhaps less known that the presence of exactly zero eigenvalues has the same effect. The reason is the repulsion of eigenvalues which occurs in all interacting systems and has probably best been understood in the context of random matrix theory where the eigenvalues obey the Wigner repulsion law ͓2͔.In QCD, the fluctuations of the low-lying eigenvalues of the Dirac operator are described by chiral random matrix theory ͑chRMT͒ ͓3-5͔. This is a random matrix theory with the global symmetries of the QCD partition function. It is characterized by the Dyson index ͓6͔ ␤ which is equal to the number of independent variables per matrix element. For QCD with fundamental fermions we have ␤ϭ1 for N c ϭ2 and ␤ϭ2 for N c Ͼ2. For QCD with adjoint fermions and N c у2 the Dirac matrix can be represented in terms of selfdual quaternions with ␤ϭ4. The main ingredient of the chRMT partition function is the integration measure which includes the Vandermonde determinant. In terms of Dirac eigenvalues i k it is given byTherefore, the presence of N f massless flavors, with the fermion determinant given by ͟ k k 2N f , has the same effect on eigenvalue correlations as ϭ2N f /␤ zero eigenvalues. More precisely, the joint eigenvalue distribution only depends on the combination 2N f ϩ␤ ͓4,7͔. Based on the conjecture ͓3,4͔ that the zero-momentum part of the QCD partition function, Z N f , (m 1 , . . . ,m N f ), is a chiral random matrix theory, it was suggested ͓8͔ that its mass dependence obeys the duality relation ͑for ␤ϭ2͒This relation, which is now known as flavor topologyduality, is a trivial consequence of the flavor dependence of the chRMT joint eigenvalue distribution. The mass dependence of the chRMT partition function can be reduced to a unitary matrix integral which is known from the zero momentum limit of chiral perturbation theory ͓9,10͔. Starting from this representation of the low energy limit of the QCD partition function, also known as the finite...