Gauge invariant and gauge fixed actions for various higher-spin fields from string field theory

We propose a systematic procedure for extracting gauge invariant and gauge fixed actions for various higher-spin gauge field theories from covariant bosonic open string field theory. By identifying minimal gauge invariant part for the original free string field theory action, we explicitly construct a class of covariantly gauge fixed actions with BRST and anti-BRST invariance. By expanding the actions with respect to the level N   of string states, the actions for various massive fields including higher-spin fields are systematically obtained. As illustrating examples, we explicitly investigate the level N?3$N?3$ part and obtain the consistent actions for massive graviton field, massive 3rd rank symmetric tensor field, or anti-symmetric field. We also investigate the tensionless limit of the actions and explicitly derive the gauge invariant and gauge fixed actions for general rank n symmetric and anti-symmetric tensor fields.     

Supersymmetry anomalies and some aspects of renormalization

We show how the $\text{L}$-matrix elements avoid the problem of supersymmetry breaking by the gauge fixing and ghost terms for renormalization in the Wess-Zumino gauge. Possible origins of supersymmetry anomalies are discussed. Gauge and gravitational anomalies induce a supersymmetry anomaly which has two distinct terms, one of which is gauge invariant. We give the expression for the noninvariant term for 2n-dimensional spacetime and for the invariant part in four dimensions. This anomaly, although cohomologically nontrivial, is still consistent with result that in superspace no supersymmetry anomaly is generated.     

General theory of renormalization of gauge invariant operators

We study the question of renormalization of gauge invariant operators in the gauge theories. Our discussion applies to gauge invariant operators of arbitrary dimensions and tensor structure. We show that the gauge noninvariant (and ghost) operators that mix with a given set of gauge invariant operators form a complete set of local solutions of a functional differential equation. We show that this set of gauge noninvariant operators together with the gauge invariant operators close under renormalization to all orders. We obtain a complete set of local solutions of the differential equation. The form of these solutions has recently been conjectured by Kluberg Stern and Zuber. With the help of our solutions, we show that there exists a basis of operators in which the gauge noninvariant operators “decouple” from the gauge invariant operators to all orders in the sense that eigenvalues corresponding to the eigenstates containing gauge invariant operators can be computed without having to compute the full renormalization metrix. We further discuss the substructure of the renormalization matrix.     

The off-shell infrared problem in N = 1 supersymmetric yang-mills theories (II). General massless models

We prove that in a general massless N = I SYM theory off-shell Green functions exist such that Green functions of gauge invariant operators are supersymmetrically covariant. The off-shell infrared problem present in the superfield treatment of these theories is thus shown to remain a gauge artefact. The N = 2, 4 pure SYM theories are covered by this result and thus exist as N = 1 SYM theories.     

U(1)-invariant current forms

Let λ : U(1) → GL (V) be a linear representation. Lagrangians defined on J¹(T*M × V)whose universal current form is gauge invariant, are described in terms of gauge invariant Lagrangians and invariant functions on V. A characterization of gauge invariant Lagrangians depending on matter fields only, that is, L : J¹V → ?, is also obtained.     

Theory of nonlinear optical response of gauge invariant currents to linear and nonlinear couplings

When a gauge field interacts with a quantum condensed matter system, at first order of the gauge field it couples to the current operator of the electrons. Higher orders of the gauge field couple to electrons through other operators such as the stress tensor, etc. On the other hand, when one performs a measurement on a quantum system, not only the current operator, but also stress tensor operator of the electrons, etc. are hidden in the measurement, as they contribute to the gauge invariant current. We formulate a general problem of nonlinear optical response of the gauge invariant currents in presence of nonlinear couplings. We show that the new couplings along with new responses arising from field current have a very simple structure which can be formulated as time ordered multi-particle correlation functions. We also obtain their Lehman representation and thereby show that one need not use non-equilibrium formulations to deal with them. These new correlation functions suggest that in nonlinear optical response many new processes are possible. The experimental detection of the new terms in the current operator, and application corresponding multi-photon processes needs further theoretical and experimental investigations.     

Review of AdS/CFT Integrability, Chapter I.1: Spin Chains in {\mathcal{N}=4} Super Yang-Mills

In this chapter of Review of AdS/CFT Integrability we introduce ${\mathcal{N}=4}$ Super Yang-Mills. We discuss the global superalagebra PSU(2, 2|4) and its action on gauge invariant operators. We then discuss the computation of the correlators of certain gauge invariant operators, the so-called single trace operators in the large N limit. We show that interactions in the gauge theory lead to mixing of the operators. We compute this mixing at the one-loop level and show that the problem maps to a one-dimensional spin chain with nearest neighbor interactions. For operators in the SU(2) sector we show that the spin chain is the ferromagnetic Heisenberg spin chain whose eigenvalues are determined by the Bethe equations.     

Generalized non-linear σ-models with gauge invariance

Generalizations of the non-linear σ-model and of the CP^N model are considered. They are invariant under a global group G and under a local gauge subgroup H. Whenever G/H defines a symmetric space, we can show that these models are asymptotically free. Furthermore, in two dimensions the local gauge invariance leads to confinement.     

Photoproduction ofN = (1236) by polarized γ-Quanta

The one-pion-exchange model with gauge invariant extension is used to calculate cross sections and decay distributions for the reactionγp→N⁼⁺⁺ π ^? where theγ-quanta are linearly polarized. We show that measurements with polarizedγ-quanta would provide a sensitive check of the model. In particular, the relevance of the terms added to the one-pion-exchange part to guarantee gauge invariance could be tested.     

Non-abelian bosonization in higher genus Riemann surfaces

We propose a generalization of the character formulas of the SU (2) Kac-Moody algebra to higher genus Riemann surfaces. With this construction, we show that the modular invariant partition function of the SO(4) k=1 Wess-Zumino model is equivalent, in arbitrary genus Riemann surfaces, to that of free fermion theory.     

Higher gauge theory and a non-Abelian generalization of 2-form electrodynamics

In conventional gauge theory, a charged point particle is described by a representation of the gauge group. If we propagate the particle along some path, the parallel transport of the gauge connection acts on this representation. The Lagrangian density of the gauge field depends on the curvature of the connection which can be calculated from the holonomy around (infinitesimal) loops. For Abelian symmetry groups, say G=U(1), there exists a generalization, known as p-form electrodynamics, in which (p−1)-dimensional charged objects can be propagated along p-surfaces and in which the Lagrangian depends on a generalized curvature associated with (infinitesimal) closed p-surfaces. In this article, we use Lie 2-groups and ideas from higher category theory in order to formulate a discrete gauge theory which generalizes these models at the level p=2 to possibly non-Abelian symmetry groups. An important feature of our model is that it involves both parallel transports along paths and generalized transports along surfaces with a non-trivial interplay of these two types of variables. Our main result is the geometric picture, namely the assignment of non-Abelian quantities to geometrical objects in a coordinate free way. We construct the precise assignment of variables to the curves and surfaces, the generalized local symmetries and gauge invariant actions and we clarify which structures can be non-Abelian and which others are always Abelian. A discrete version of connections on non-Abelian gerbes is a special case of our construction. Even though the motivation sketched so far suggests applications mainly in string theory, the model presented here is also related to spin foam models of quantum gravity and may in addition provide some insight into the role of centre monopoles and vortices in lattice QCD.     

Infrared behaviour of Yang-Mills theory at high temperatures

We employ Monte Carlo methods to probe the infrared structure of SU(2) Yang-Mills theory at high temperatures. In particular, we study the free energy of the gauge invariant non-abelian magnetic flux. The Monte Carlo support the conjecture that the magnetic flux is screened. The inverse magnetic screening length provides a non-perturbative infrared cut-off for the quantum statistics of the SU(2) gauge theory and is estimated to be about 0.24 g²T.     

Self-intersection numbers and random surfaces on the lattice

String theory in four dimensions has the unique feature that a topological term, the oriented self-intersection number, can be added to the usual action. It has been suggested that the corresponding theory of random surfaces would be free from the problem encountered in the scaling of the string tension. Unfortunately, in the usual dynamical triangulation it is not clear how to write such a term. We show that for random surfaces on a hypercubic lattice however, the analogue of the oriented self-intersection number I[σ] can be defined and computed in a straightforward way. Furthermore, I[σ] has a genuine topological meaning in the sense that it is invariant under the discrete analogue of continuous deformations. The resulting random surface model is no longer free and may lead to a non-trivial continuum limit.     

States and automorphism groups of operator algebras

Suppose that a group of automorphisms of a von Neumann algebraM, fixes the center elementwise. We show that if this group commutes with the modular (KMS) automorphism group associated with a normal faithful state onM, then this state is left invariant by the group of automorphisms. As a result we obtain a “noncommutative” ergodic theorem. The discrete spectrum of an abelian unitary group acting as automorphisms ofM is completely characterized by elements inM. We discuss the KMS condition on the CAR algebra with respect to quasi-free automorphisms and gauge invariant generalized free states. We also obtain a necessary and sufficient condition for the CAR algebra and a quasi-free automorphism group to be η-abelian.     

Finite matrix models with continuous local gauge invariance

We construct a hamiltonian lattice gauge theory which possesses local SU (2) gauge invariance and yet is defined on a Hilbert space of 5-dimensional real vectors for every link. This construction does not allow for generalization to arbitrary SU(N), but a small variation of it can be generalized to an SU(N) × U(1) local gauge invariant model. The latter is solvable in simple gauge sectors leading to trivial spectra. We display these by studying a U(1) local gauge invariant model with similar characteristics.     

Pseudointegrable systems in classical and quantum mechanics

We study particles moving in planar polygonal enclosures with rational angles, and show by several methods that trajectories in the classical phase space explore two-dimensional invariant surfaces which are generically not tori as in integrable systems but instead have the topology of multiply-handled spheres. The quantum mechanics of one such ‘pseudointegrable system’ is studied in detail by computing energy levels using an exact formalism. This system consists of motion on a unit coordinate torus containing a square reflecting obstacle with side L. We find that neighbouring levels avoid degeneracies as L varies, and that the probability distribution for the spacing S of adjacent levels vanishes linearly as S→0 (‘level repulsion’). The Weyl area rule plus edge and corner corrections gives a very accurate approximation for the mean level density. Oscillatory corrections to the mean level density are given as a sum over closed classical paths; for pseudointegrable systems these closed paths form families covering part of the phase-space invariant surfaces.     

Gauge parameter dependence in the background field gauge and the construction of an invariant charge

By using the enlarged BRS transformations we control the gauge parameter dependence of Green functions in the background field gauge. We show that it is unavoidable — also if we consider the local Ward identity — to introduce the normalization value ξ₀ of the gauge parameter ξ. The dependence of Green functions on ξ₀ is governed by a partial differential equation in a similar manner as the dependence on the normalization point κ is governed by the RG equation. By modifying the Ward identity we are able to construct in 1-loop order a gauge parameter independent combination of 2-point vector and background vector functions. By explicit construction of the next orders we show that this combination can be used to construct a gauge parameter independent RG-invariant charge. However, it is seen that this RG-invariant charge does not satisfy the differential equation of the normalization value ξ₀ of the gauge parameter, and, hence, is not ξ₀-independent as required.     

Chern–Simons theory with vector fermion matter

We study three-dimensional conformal field theories described by U(N) Chern?CSimons theory at level k coupled to massless fermions in the fundamental representation. By solving a Schwinger?CDyson equation in light-cone gauge, we compute the exact planar free energy of the theory at finite temperature on ?² as a function of the ??t?Hooft coupling ??=N/k. Employing a dimensional reduction regularization scheme, we find that the free energy vanishes at |??|=1; the conformal theory does not exist for |??|>1. We analyze the operator spectrum via the anomalous conservation relation for higher spin currents, and in particular show that the higher spin currents do not develop anomalous dimensions at leading order in 1/N. We present an integral equation whose solution in principle determines all correlators of these currents at leading order in 1/N and present explicit perturbative results for all three-point functions up to two loops. We also discuss a light-cone Hamiltonian formulation of this theory where a W _?? algebra arises. The maximally supersymmetric version of our theory is ABJ model with one gauge group taken to be U(1), demonstrating that a pure higher spin gauge theory arises as a limit of string theory.