Non-fundamental Representations Research Articles

Onsager symmetries in -invariant clock models

We show how the Onsager algebra, used in the original solution of the two-dimensional Ising model, arises as an infinite-dimensional symmetry of certain self-dual models that also have a symmetry. We describe in detail the example of nearest-neighbour n-state clock chains whose symmetry is enhanced to . As a consequence of the Onsager-algebra symmetry, the spectrum of these models possesses degeneracies with multiplicities 2N for positive integer N. We construct the elements of the algebra explicitly from transfer matrices built from non-fundamental representations of the quantum-group algebra . We analyse the spectra further by using both the coordinate Bethe ansatz and a functional approach, and show that the degeneracies result from special exact n-string solutions of the Bethe equations. We also find a family of commuting chiral Hamiltonians that break the degeneracies and allow an integrable interpolation between ferro- and antiferromagnets.

Schrödinger functional boundary conditions and improvement for N >3

The standard method to calculate non-perturbatively the evolution of the running coupling of a SU(N) gauge theory is based on the Schr\"odinger functional (SF). In this paper we construct a family of boundary fields for general values of N which enter the standard definition of the SF coupling. We provide spatial boundary conditions for fermions in several representations which reduce the condition number of the squared Dirac operator. In addition, we calculate the O(a) improvement coefficients for N>3 needed to remove boundary cutoff effects from the gauge action. After this, residual cutoff effects on the step scaling function are shown to be very small even when considering non-fundamental representations. We also calculate the ratio of Lambda parameters between the MS-bar and SF schemes.

Fiscal Foresight and Information Flows

News�or foresight�about future economic fundamentals can create rational expectations equilibria with non-fundamental representations that pose substantial challenges to econometric efforts to recover the structural shocks to which economic agents react. Using tax policies as a leading example of foresight, simple theory makes transparent the economic behavior and information structures that generate non-fundamental equilibria. Econometric analyses that fail to model foresight will obtain biased estimates of output multipliers for taxes; biases are quantitatively important when two canonical theoretical models are taken as data generating processes. Both the nature of equilibria and the inferences about the effects of anticipated tax changes hinge critically on hypothesized information flows. Different methods for extracting or hypothesizing the information flows are discussed and shown to be alternative techniques for resolving a non-uniqueness problem endemic to moving average representations.

Identifying News Shocks from SVARs

This paper investigates the reliability of SVARs in identifying the dynamic effects of news shocks. Using a simple but insightful model with a non-fundamental representation, we show analytically under which conditions SVARs are likely to be successful at identifying news shocks. We find that the dynamic responses to news shocks identified using a short-run restriction are biased. However, this bias is smaller if news shocks account for most of the variability of the endogenous variable and the economy exhibits strong forward-looking behavior. Our simulation experiments confirm this finding and further suggest that the number of lags in the VAR and the anticipation horizon are key ingredients for the success of the VAR setup.

CHARACTER EXPANSION FOR HOMFLY POLYNOMIALS III: ALL 3-STRAND BRAIDS IN THE FIRST SYMMETRIC REPRESENTATION

We continue the program of systematic study of extended HOMFLY polynomials, suggested in [A. Mironov, A. Morozov and And. Morozov, arXiv:1112.5754] and [A. Mironov, A. Morozov and And. Morozov, J. High Energy Phys. 03, 034 (2012), arXiv:1112.2654]. Extended polynomials depend on infinitely many time-variables, are close relatives of integrable τ-functions, and depend on the choice of the braid representation of the knot. They possess natural character decompositions, with coefficients which can be defined by exhaustively general formula for any particular number m of strands in the braid and any particular representation R of the Lie algebra GL(∞). Being restricted to "the topological locus" in the space of time-variables, the extended HOMFLY polynomials reproduce the ordinary knot invariants. We derive such a general formula, for m = 3, when the braid is parametrized by a sequence of integers (a1, b1, a2, b2, …) and for the first nonfundamental representation R = [2]. Instead of calculating the mixing matrices directly, as suggested [A. Mironov, A. Morozov and And. Morozov, J. High Energy Phys. 03, 034 (2012), arXiv:1112.2654], we deduce them from comparison with the known answers for torus and composite knots. A simple reflection symmetry converts the answer for the symmetric representation [2] into that for the antisymmetric one [1, 1]. The result applies, in particular, to the figure eight knot 41, and was further extended to superpolynomials in arbitrary symmetric and antisymmetric representations in H. Itoyama, A. Mironov, A. Morozov and And. Morozov, arXiv:1203.5978.

Fiscal Foresight and Information Flows

News - or foresight - about future economic fundamentals can create rational expectations equilibria with non-fundamental representations that pose substantial challenges to econometric efforts to recover the structural shocks to which economic agents react. Using tax policies as a leading example of foresight, simple theory makes transparent the economic behavior and information structures that generate non-fundamental equilibria. Econometric analyses that fail to model foresight will obtain biased estimates of output multipliers for taxes; biases are quantitatively important when two canonical theoretical models are taken as data generating processes. Both the nature of equilibria and the inferences about the effects of anticipated tax changes hinge critically on hypothesized information flows. Different methods for extracting or hypothesizing the information flows are discussed and shown to be alternative techniques for resolving a non-uniqueness problem endemic to moving average representations.

Fiscal Foresight and Information Flows

News - or foresight - about future economic fundamentals can create rational expectations equilibria with non-fundamental representations that pose substantial challenges to econometric efforts to recover the structural shocks to which economic agents react. Using tax policies as a leading example of foresight, simple theory makes transparent the economic behavior and information structures that generate non-fundamental equilibria. Econometric analyses that fail to model foresight will obtain biased estimates of output multipliers for taxes; biases are quantitatively important when two canonical theoretical models are taken as data generating processes. Both the nature of equilibria and the inferences about the effects of anticipated tax changes hinge critically on hypothesized information flows. Different methods for extracting or hypothesizing the information flows are discussed and shown to be alternative techniques for resolving a non-uniqueness problem endemic to moving average representations.

Monetary Policy Shocks - a Nonfundamental Look at the Data

VAR studies of the effects of monetary policy on output suggest that a contractionary impulse results in a drawn-out, hump-shaped response of output. Standard structural economic models are generally not able to reproduce such a response. In this paper I look at nonfundamental representations that are observationally equivalent to a VAR. I find that the quantitative effect of a monetary policy shock on output might be much smaller and much more short-lived than the VAR studies suggest. I conclude that the apparent discrepancy between the VAR findings and standard structural models may be spurious and that the general tendency to append non-structural, ad hoc features to structural models should be questioned.

Quantization ambiguities in isotropic quantum geometry

Some typical quantization ambiguities of quantum geometry are studied within isotropic models. Since this allows explicit computations of operators and their spectra, one can investigate the effects of ambiguities in a quantitative manner. It is shown that these ambiguities do not affect the fate of the classical singularity, demonstrating that the absence of a singularity in loop quantum cosmology is a robust implication of the general quantization scheme. The calculations also allow conclusions about modified operators in the full theory. In particular, using holonomies in a non-fundamental representation of SU(2) to quantize connection components turns out to lead to significant corrections to classical behaviour at macroscopic volume for large values of the spin of the chosen representation.

QUANTIZING SL(N) SOLITONS AND THE HECKE ALGEBRA

The problem of quantizing a class of two-dimensional integrable quantum field theories is considered. The classical equations of the theory are the complex sl(n) affine Toda equations which admit soliton solutions with real masses. The classical scattering theory of the solitons is developed using Hirota’s solution techniques. A form for the soliton S matrix is proposed based on the constraints of S matrix theory, integrability and the requirement that the semiclassical limit is consistent with the semiclassical WKB quantization of the classical scattering theory. The proposed S matrix is an intertwiner of the quantum group associated to sl(n), where the deformation parameter is a function of the coupling constant. It is further shown that the S matrix describes a nonunitary theory, which reflects the fact that the classical Hamiltonian is complex. The spectrum of the theory is found to consist of the basic solitons, excited (or ‘breathing’) solitons, scalar states (or breathers) and solitons transforming in nonfundamental representations. For some region of coupling constant space only the original solitons are in the spectrum and so the S matrix is complete, in addition arguments are presented which indicate that in a more restricted region the theory is actually unitary. It is also noted that the construction of the S matrix is valid for any representation of the Hecke algebra, allowing the definition of restricted S matrices, which lie in the unitary and complete region.

The braid group representations associated with some non-fundamental representations of Lie algebras

The braid group representations associated with the eight-dimensional representation of B3 and six-dimensional representation of A2, including the quantum Lie algebra connected with the latter, are explicitly calculated.