Construction Of Field Theory Research Articles

Z(3)-symmetric effective theory for pure gauge QCD at high temperature

We review the construction and basic properties of a three-dimensional effective field theory for high-temperature SU(3) Yang-Mills theory, which respects its center symmetry and was introduced in Ref. [A. Vuorinen, L.G. Yaffe, Z(3)-symmetric effective theory for SU(3) Yang-Mills theory at high, Phys. Rev. D 74 (2006) 025011, hep-ph/0604100 ]. We explain why the phase diagram of the new theory is expected to closely resemble the one of the full theory and argue that this implies that it is applicable down to considerably lower temperatures than the usual non-Z(3)-symmetric 3d effective theory EQCD.

Critical Exponents from the Resummed Effective Potential of the $$({\frac{g}{4}}\phi^{4}-J\phi)_{1+1}$$ Scalar Field Theory

We establish a unified way for the calculation of the critical exponents, without the use of epsilon expansion, through the improvement of the perturbative effective potential of the 1+1 dimensional \(({\frac{g}{4}}\phi^{4}-J\phi)\) scalar field theory. First, we obtain the perturbation series for the effective potential up to g3. We improved the perturbative effective potential by establishing a parameter-free resummation algorithm, originally due to Kleinert, Thoms and Janke, which has the privilege of using the strong coupling as well as the large coupling behaviors rather than the conventional resummation techniques which use only the large order behavior. Accordingly, although the perturbation series available is up to g3 order, we found a complete agreement between our resummed effective potential and the well known features from constructive field theory. We prove that the 1-PI correlation functions and the effective potential ought to have the same large order as well as strong coupling behaviors. We computed the critical exponents and our results show a good agreement with the exact Ising model values.

Analysis and Topology in Interaction

The MFO workshop “Analysis and Topology in Interaction”, organised by Ulrich Bunke (Göttingen), Sebastian Goette (Regensburg), Kyoshi Igusa (Brandeis) and Thomas Schick (Göttingen), was held from March 12th through March 18th, 2006. The aim of this meeting was to reflect on the current state of the interaction between mathematical fields of analysis and topology. It brought together representatives of various mathematical communities ranging from homotopy theory, index theory, global analysis up to mathematical physics. The program of the conference has been a mixture of research and overview talks, sometimes with introductory elements. Topics of the latter kind were the connection of Quantum field theory constructions with twisted K-theory, and motivic groups. There were two more informal introductory lectures on stacks in topological and smooth categories, and the insights given by the study of the Brownian motions into Hodge theory. The research talks reflected recent developments in the corresponding fields and covered a broad area between topology, geometry, and analysis. The intention of the program was to communicate these developments across the borders of mathematical communities. They stimulated discussions and hopefully future collaborations. More specifically, the subjects covered by research talks included in particular Altogether, there were 23 “official” talks and a number of informal presentations (partly extended discussions), and ample free interaction between the participants. One of the evenings was devoted to a problem session; the open problems presented are included in this report. The conference was attended by 52 participants coming mainly from all over Europe, Northern America and East Asia. Among them, a couple of very young pre-doctoral mathematicians, supported by the EU, had the unique opportunity to participate in such a stimulating event at a very early stage of their career. It is a pleasure to thank the institute for providing a pleasant and stimulating atmosphere.

Quantum Field Theory on Curved Backgrounds. I. The Euclidean Functional Integral

We give a mathematical construction of Euclidean quantum field theory on certain curved backgrounds. We focus on generalizing Osterwalder Schrader quantization, as these methods have proved useful to establish estimates for interacting fields on flat space-times. In this picture, a static Killing vector generates translations in Euclidean time, and the role of physical positivity is played by positivity under reflection of Euclidean time. We discuss the quantization of flows which correspond to classical space-time symmetries, and give a general set of conditions which imply that broad classes of operators in the classical picture give rise to well-defined operators on the quantum-field Hilbert space. In particular, Killing fields on spatial sections give rise to unitary groups on the quantum-field Hilbert space, and corresponding densely-defined self-adjoint generators. We construct the Schrödinger representation using a method which involves localizing certain integrals over the full manifold to integrals over a codimension-one submanifold. This method is called sharp-time localization, and implies reflection positivity.

Generalised N=2 permutation branes

Generalised permutation branes in products of N=2 minimal models play an important role in accounting for all RR charges of Gepner models. In this paper an explicit conformal field theory construction of these generalised permutation branes for one simple class of examples is given. We also comment on how this may be generalised to the other cases.

Dual realizations of dynamical symmetry breaking

We show the infrared equivalence between a recently proposed model containing a six dimensional scalar field with a four-dimensional localized Higgs type potential and the four-dimensional Nambu-Jona-Lasinio (NJL) model. In the dual NJL description, the fermions are localized at the origin of a large two-dimensional compact space. Due to a classical running effect above the compactification scale, the four-fermion coupling of the NJL model increases from the cutoff scale down to the compactification scale, providing the large Fermi coupling needed for the dynamical symmetry breaking. We also present a string theory embedding of our field-theory construction. On more general grounds, our results suggest that 4d models with dynamical symmetry breaking can be given a higher dimensional description in terms of field theories with nontrivial boundary conditions in the internal space.

Transformations among large c conformal field theories

We show that there is a set of transformations that relates all of the 24 dimensional even self-dual (Niemeier) lattices, and also leads to non-lattice objects some of which can perhaps be interpreted as a basis for the construction of holomorphic conformal field theory. In the second part of this paper, we extend our observations to higher-dimensional conformal field theories build on extremal partition functions, where we generate c = 24 k theories. We argue that there exists generalizations of the c = 24 models based on Niemeier lattices and of the non-Niemeier spin-1 theories. The extremal cases have spectra decomposable into the irreducible representations of the Fischer–Griess Monster. This additional symmetry leads us to conjecture that these extremal theories, as well as the higher-dimensional analogs of the group lattice bases Niemeiers, will eventually yield to a full construction of their associated CFTs. We observe interesting periodicities in the coefficients of extremal partition functions and characters of the extremal vertex operator algebras.

Relativistically Covariant Quantum Field Theory of the Maslov Complex Germ

We consider an explicitly covariant formulation of the quantum field theory of the Maslov complex germ (semiclassical field theory) in the example of a scalar field. The main object in the theory is the “semiclassical bundle” whose base is the set of classical states and whose fibers are the spaces of states of the quantum theory in an external field. The respective semiclassical states occurring in the Maslov complex germ theory at a point and in the theory of Lagrangian manifolds with a complex germ are represented by points and surfaces in the semiclassical bundle space. We formulate semiclassical analogues of quantum field theory axioms and establish a relation between the covariant semiclassical theory and both the Hamiltonian formulation previously constructed and the axiomatic field theory constructions Schwinger sources, the Bogoliubov S-matrix, and the Lehmann-Symanzik-Zimmermann R-functions. We propose a new covariant formulation of classical field theory and a scheme of semiclassical quantization of fields that does not involve a postulated replacement of Poisson brackets with commutators.

Solitons in Affine and Permutation Orbifolds

We consider properties of solitons in general orbifolds in the algebraic quantum field theory framework and constructions of solitons in affine and permutation orbifolds. Under general conditions we show that our construction gives all the twisted representations of the fixed point subnet. This allows us to prove a number of conjectures: in the affine orbifold case we clarify the issue of ``fixed point resolutions''; in the permutation orbifold case we determine all irreducible representations of the orbifold, and we also determine the fusion rules in a nontrivial case, which imply an integral property of chiral data for any completely rational conformal net.

Field Theory on Kappa-spacetime

A general formalism is developed that allows the construction of field theory on quantum spaces which are deformations of ordinary spacetime. The symmetry group of spacetime is replaced by a quantum group. This formalism is demonstrated for the kappa-deformed Poincare algebra and its quantum space. The algebraic setting is mapped to the algebra of functions of commuting variables with a suitable star-product. Fields are elements of this function algebra. As an example, the Klein-Gordon equation is defined and derived from an action.

Superfield algorithm for higher order gauge field theories

We propose an algorithm for the construction of higher order gauge field theories from a superfield formulation within the Batalin-Vilkovisky formalism. This is a generalization of the superfield algorithm recently considered by Batalin and Marnelius. This generalization seems to allow for non-topological gauge field theories as well as alternative representations of topological ones. A five dimensional non-abelian Chern-Simons theory and a topological Yang-Mills theory are treated as examples.

LITTLE GROUPS OF PREON BRANES

Little groups for preon branes (i.e. configurations of branes with maximal (n-1)/n fraction of survived supersymmetry) for dimensions d=2,3,…,11 are calculated for all massless, and partially for massive orbits. For massless orbits little groups are semidirect product of d-2 translational group Td-2 on a subgroup of ( SO (d-2) × R-invariance) group. E.g. at d=9 the subgroup is exceptional G2 group. It is also argued, that 11D Majorana spinor invariants, which distinguish orbits, are actually invariant under d=2+10 Lorentz group. Possible applications of these results include construction of field theories in generalized spacetimes with brane charges coordinates, different problems of group's representations decompositions, spin-statistics issues.

Pauli–Villars regularization and Born–Infeld kinematics

Dynamical symmetries of Born–Infeld theory can be absorbed into the spacetime geometry, giving rise to relativistic kinematics with an additional invariant acceleration scale. The standard Poincaré group \U0001d4ab is thereby enhanced to its pseudo-complexified version, which is isomorphic to \U0001d4ab × \U0001d4ab. We construct the irreducible representations of this group, which yields the particle spectrum of a relativistic quantum theory that respects a maximal acceleration. It is found that each standard relativistic particle is associated with a ‘pseudo’-partner of equal spin but generically different mass. These pseudo-partners act as Pauli–Villars regulators for the other member of the doublet, as is found from the explicit construction of quantum field theory on pseudo-complex spacetime. Conversely, a Pauli–Villars regularized quantum field theory on real spacetime possesses a field phase space with integrable pseudo-complex structure, which gives rise to a quantum field theory on pseudo-complex spacetime. This equivalence between maximal acceleration kinematics, pseudo-complex quantum field theory and Pauli–Villars regularization rigorously establishes a conjecture on the regularizing property of the maximal acceleration principle in quantum field theory, by Nesterenko, Feoli, Lambiase and Scarpetta.

A review of non-commutative gauge theories

Construction of quantum field theory based on operators that are functions of noncommutative space-time operators is reviewed. Examples of Φ4 theory and QED are then discussed. Problems of extending the theories toSU(N) gauge theories and arbitrary charges in QED are considered. Construction of standard model on non-commutative space is then briefly discussed. The phenomenological implications are then considered. Limits on non-commutativity from atomic physics as well as accelerator experiments are presented.

The Partition Function of the Two-Dimensional Black Hole Conformal Field Theory

We compute the partition function of the conformal field theory on the two-dimensional euclidean black hole background using path-integral techniques. We show that the resulting spectrum is consistent with the algebraic expectations for the SL(2,R)/U(1) coset conformal field theory construction. In particular, we find confirmation for the bound on the spin of the discrete representations and we determine the density of the continuous representations. We point out the relevance of the partition function to all string theory backgrounds that include an SL(2,R)/U(1) coset factor.

A Note on a Proposal for the Tachyon State in Vacuum String Field Theory

We discuss the proposal of Hata and Kawano for the tachyon fluctuation around a solution of vacuum string field theory representing a D25 brane. We give a conformal field theory construction of their state -- a local insertion of a tachyon vertex operator on the sliver surface state, and explain why the on-shell momentum condition emerges correctly. We also show that a naive computation of the D25-brane tension using data for the three point coupling of this state gives an answer that is $(\pi^2/3)(16/27\ln 2)^3 \simeq 2.0558$ times the expected answer. We demonstrate that this problem arises because the HK state does not satisfy the equations of motion in a strong sense required for the computation of D-brane tension from the on-shell 3-tachyon coupling.

Building light nuclei from neutrons, protons, and pions

In these lectures I first explain, in a rather basic fashion, the construction of effective field theories. I then discuss some recent developments in the application of such theories to two- and three-nucleon systems.

Algebra of Deformed Differential Operators and Induced Integrable Toda Field Theory

We build in this paper the algebra of q-deformed pseudo-differential operators, shown to be an essential step toward setting a q-deformed integrability program. In fact, using the results of this q-deformed algebra, we derive the q-analogues of the generalized KdV hierarchy. We focus in particular on the first leading orders of this q-deformed hierarchy, namely the q-KdV and q-Boussinesq integrable systems. We also present the q-generalization of the conformal transformations of the currents un,n ≥ 2, and discuss the primary condition of the fields Wn, n ≥ 2, by using the Volterra gauge group transformations for the q-covariant Lax operators. An induced su(n)-Toda(su(2)-Liouville) field theory construction is discussed and other important features are presented.

Ghost structure and closed strings in vacuum string field theory

We complete the construction of vacuum string field theory by proposing a canonical choice of ghost kinetic term – a local insertion of the ghost field at the string midpoint with an infinite normalization. This choice, supported by level expansion studies in the Siegel gauge,

Superconformal field theories for compact manifolds with Spin(7) holonomy

We present a construction of superconformal field theories for manifolds with Spin(7) holonomy. Geometrically these models correspond to the realization of Spin(7) manifolds as anti-holomorphic quotients of Calabi-Yau fourfolds. Describing the fourfolds as Gepner models and requiring anomaly cancellation we determine the resulting Betti numbers of the Spin(7) superconformal field theory. As in the G_2 case, we find that the Gepner model and the geometric result disagree.