arXiv subjects
Publications and source records attributed to L. Griguolo.
In these notes Yang-Mills theories in 1+1 dimensions are reviewed. Instantons on a sphere prove to be -in the decompactification limit- the key issue to clarify an old controversy between equal-time and light-front quantization.
The exact expression for Wilson loop averages winding n times on a closed contour is obtained in two dimensions for pure U(N) Yang-Mills theory and, rather surprisingly, it displays an interesting duality in the exchange $n \leftrightarrow N$. The large-N limit of our result is consistent with previous computations. Moreover we discuss the limit of small loop area ${\cal A}$, keeping $n^2 {\cal A}$ fixed, and find it coincides with the zero-instanton approximation. We deduce that small loops, both at finite and infinite "volume", are blind to instantons. Next we check the non-perturbative result by resumming 't Hooft-CPV and Wu-Mandelstam-Leibbrandt (WML)-prescribed perturbative series, the former being consistent with the exact result, the latter reproducing the zero-instanton contribution. A curious interplay between geometry and algebraic invariants is observed. Finally we compute the spectral density of the Wilson loop operator, at large $N$, via its Fourier representation, both for 't Hooft and WML: for small area they exhibit a gap and coincide when the theory is considered on the sphere $S^2$.
The instanton contributions to the partition function and to homologically trivial Wilson loops for a U(N) Yang-Mills theory on a torus $T^2$ are analyzed. An exact expression for the partition function is obtained as a sum of contributions localized around the classical solutions of Yang-Mills equations, that appear according to the general classification of Atiyah and Bott. Explicit expressions for the exact Wilson loop averages are obtained when N=2, N=3. For general $N$ the contribution of the zero-instanton sector has been carefully derived in the decompactification limit, reproducing the sum of the perturbative series on the plane, in which the light-cone gauge Yang-Mills propagator is prescribed according to Wu-Mandelstam-Leibbrandt (WML). Agreement with the results coming from $S^2$ is therefore obtained, confirming the truly perturbative nature of the WML computations.
The exact Wilson loop expression for the pure Yang-Mills U(N) theory on a sphere $S^2$ of radius $R$ exhibits, in the decompactification limit $R\to \infty$, the expected pure area exponentiation. This behaviour can be understood as due to the sum over all instanton sectors. If only the zero instanton sector is considered, in the decompactification limit one exactly recovers the sum of the perturbative series in which the light-cone gauge Yang-Mills propagator is prescribed according to Wu-Mandelstam-Leibbrandt. When instantons are disregarded, no pure area exponentiation occurs, the string tension is different and, in the large-N limit, confinement is lost.
Two different families of abelian chiral gauge theories on the torus are investigated: the aim is to test the consistency of two-dimensional anomalous gauge theories in the presence of global degrees of freedom for the gauge field. An explicit computation of the partition functions shows that unitarity is recovered in particular regions of parameter space and that the effective dynamics is described in terms of fermionic interacting models. For the first family, this connection with fermionic models uncovers an exact duality which is conjectured to hold in the nonabelian case as well.
We point out that the claims made in the paper ``Non-thermalizability of a Quantum Field Theory'' (hep-th/9802008) by C. R. Hagen are irrelevant to our recent results concerning large gauge invariance of the effective action in thermal QED.
We present two compact derivations of the correct definition of the Chern-Simons term in the topologically non trivial context of thermal $QED_3$. One is based on a transgression descent from a D=4 background connection, the other on embedding the abelian model in SU(2). The results agree with earlier cohomology conclusions and can be also used to justify a recent simple heuristic approach. The correction to the naive Chern-Simons term, and its behavior under large gauge transformations are displayed.
We analyze and give explicit representations for the effective abelian vector gauge field actions generated by charged fermions with particular attention to the thermal regime in odd dimensions, where spectral asymmetry can be present. We show, through $ζ-$function regularization, that both small and large gauge invariances are preserved at any temperature and for any number of fermions at the usual price of anomalies: helicity/parity invariance will be lost in even/odd dimensions, and in the latter even at zero mass. Gauge invariance dictates a very general ``Fourier'' representation of the action in terms of the holonomies that carry the novel, large gauge invariant, information. We show that large (unlike small) transformations and hence their Ward identities, are not perturbative order-preserving, and clarify the role of (properly redefined) Chern-Simons terms in this context. From a powerful representation of the action in terms of massless heat kernels, we are able to obtain rigorous gauge invariant expansions, for both small and large fermion masses, of its separate parity even and odd parts in arbitrary dimension. The representation also displays both the nonperturbative origin of a finite renormalization ambiguity, and its physical resolution by requiring decoupling at infinite mass. Finally, we illustrate these general results by explicit computation of the effective action for some physical examples of field configurations in the three dimensional case, where our conclusions on finite temperature effects may have physical relevance. Nonabelian results will be presented separately.
The soliton structure of a gauge theory recently proposed to describe chiral excitations in the Fractional Quantum Hall Effect is investigated. A new type of non-linear derivative Schrödinger equation emerges as an effective description of the system that supports novel chiral solitons. We discuss the classical properties of solutions with vanishing and non-vanishing boundary conditions (dark solitons) and we explain their relation to integrable systems. The quantum analysis is also addressed in the framework of a semiclassical approximation improved by Renormalization Group arguments.
The effective gauge field actions generated by charged fermions in $QED_3$ and $QCD_3$ can be made invariant under both small and large gauge transformations at any temperature by suitable regularization of the Dirac operator determinant, at the price of parity anomalies. We resolve the paradox that the perturbative expansion is not invariant, as manifested by the temperature dependence of the induced Chern-Simons term, by showing that large (unlike small) transformations and hence their Ward identities, are not perturbative order-preserving. Our results are illustrated through concrete examples of field configurations.
We investigate the quantum effects of the non-minimal matter-gravity couplings derived by Cangemi and Jackiw in the realm of a specific fermionic theory, namely the abelian Thirring model on a Riemann surface of genus zero and one. The structure and the strength of the new interactions are seen to be highly constrained, when the topology of the underlying manifold is taken into account. As a matter of fact, by requiring to have a well-defined action, we are led both to quantization rules for the coupling constants and to selection rules for the correlation functions. Explicit quantum computations are carried out in genus one (torus). In particular the two-point function and the chiral condensate are carefully derived for this case. Finally the effective gravitational action, coming from integrating out the fermionic degrees of freedoom, is presented. It is different from the standard Liouville one: a new non-local functional of the conformal factor arises and the central charge is improved, depending also on the Thirring coupling constant. This last feature opens the possibility of giving a new explicit representation of the minimal series in terms of a fermionic interacting model.
We show that excitations in a recently proposed gauge theory for anyons on a line in fact do not obey anomalous statistics. On the other hand, the theory supports novel chiral solitons. Also we construct a field-theoretic description of lineal anyons, but gauge fields play no role.
We show that the replacement of the ``instantaneous'' 't Hooft's potential with the causal form suggested by equal time canonical quantization in light-cone gauge, which entails the occurrence of negative probability states, does not change the bound state spectrum when the difference is treated as a single insertion in the kernel.
The radiation from the black holes of a 1+1-dimensional chiral quantum gravity model is studied. Most notably, a non-trivial dependence on a renormalization parameter that characterizes the anomaly relations is uncovered in an improved semiclassical approximation scheme; this dependence is not present in the naive semiclassical approximation.
The propagation of a localized wave packet in the conical space-time created by a pointlike massive source in 2+1 dimensional gravity is analyzed. The scattering amplitude is determined and shown to be finite along the classical scattering directions due to interference between the scattered and the transmitted wave functions. The analogy with diffraction theory is emphasized.
Pure Yang-Mills theories on the $S_1\times R$ cylinder are quantized in light-cone gauge $A_-=0$ by means of ${\bf equal-time}$ commutation relations. Positive and negative frequency components are excluded from the ``physical" Hilbert space by imposing Gauss' law in a weak sense. Zero modes, related to the winding on the cylinder, provide non trivial topological variables of the theory. A Wilson loop with light-like sides is studied: in the abelian case it can be exactly computed obtaining the expected area result, whereas difficulties are pointed out in non abelian cases.
We study a scalar field theory coupled to gravity on a flat background, below Planck's energy. Einstein's theory is treated as an effective field theory. Within the context of Wilson's renormalization group, we compute gravitational corrections to the beta functions and the anomalous dimension of the scalar field, taking into account threshold effects.
A family of theories which interpolate between vector and chiral Schwinger models is studied on the two--sphere $S^{2}$. The conflict between the loss of gauge invariance and global geometrical properties is solved by introducing a fixed background connection. In this way the generalized Dirac--Weyl operator can be globally defined on $S^{2}$. The generating functional of the Green functions is obtained by taking carefully into account the contribution of gauge fields with non--trivial topological charge and of the related zero--modes of the Dirac determinant. In the decompactification limit, the Green functions of the flat case are recovered; in particular the fermionic condensate in the vacuum vanishes, at variance with its behaviour in the vector Schwinger model.