Generating functions for noncanonical maps

A differential representation, analogous to the generating function representation for canonical maps, is obtained for arbitrary smooth maps in even-dimensional spaces. A few two-dimensional examples are worked out.     

Quantum fields presentation and generating functions of symplectic Schur functions and symplectic universal characters

This paper is concerned with construction of quantum fields presentation and generating functions of symplectic Schur functions and symplectic universal characters. The boson-fermion correspondence for these symmetric functions have been presented. In virtue of quantum fields, we derive a series of infinite order nonlinear integrable equations, namely, universal character hierarchy, symplectic KP hierarchy and symplectic universal character hierarchy, respectively. In addition, the solutions of these integrable systems have been discussed.     

On the universal covering group of the real symplectic group

A model for the universal covering group of the symplectic group as a Lie group, and some calculations based on the model, as well as defining a similar model for the Lagrangian Grassmannian and relating our construction to the Maslov Index.     

The generation problem and the symplectic group

The generation structure of quarks and leptons is studied in a series of extended electroweak models. They are gauge models constructed from subgroups of left-right symmetric SU(4) × S_P^L(6) × S_P^R(6). In particular, SU_C(3) × S_P^L(6) × U_Y(1), as well as SU_C(3) × SU_L(3) × SU_R(3) × U_X(1), are investigated in greater detail. It is shown that models based on such subgroups yield massless first generation fermions at the tree level. They acquire masses from those of the third generation via two-loop radiative corrections. Moreover, since such corrections are induced by charged scalar fields, it is found that there is a natural correlation between the two apparently conflicting inequalities m_u < m_d and m_t > m_b.     

Generating functions for Hermite polynomials of arbitrary order

We propose a systematic method for the construction of generating functions for Hermite polynomials of arbitrary order. The procedure is based on a suitable formula for the Hermite polynomials and our results contain ones obtained earlier by Nieto and Truax [Phys. Lett. A 208 (1995) 8] as particular cases.     

Generating functions and sum rules for quantum oscillator

Generating functions and sum rules are discussed for transition probabilities between quantum oscillator eigenstates with time-dependent parameters.     

Braid group action and quantum affine algebras

We lift the lattice of translations in the extended affine Weyl group to a braid group action on the quantum affine algebra. This action fixes the Heisenberg subalgebra pointwise. Loop-like generators of the algebra are obtained which satisfy the relations of Drinfel'd's new realization. Coproduct formulas are given and a PBW type basis is constructed.     

Calculating frequency response functions for uncertain systems using complex affine analysis

This paper is concerned with the effect of parametric uncertainty upon the response of mechanical systems. Two uncertainty propagation techniques are employed and compared against a benchmark based upon randomly scanning input parameter ranges. The first technique, complex interval analysis, is computationally very efficient but suffers from the issue of dependency which may result in large overestimation of the system response. The second technique, namely complex affine analysis, is presented here for the first time and it keeps track of variable dependencies throughout the calculation thus limiting overestimation, albeit at some computational expense. The methods are demonstrated on the problem of calculation of the frequency response functions of a simple lumped mass system with uncertain parameters for purposes of transparency of the presented ideas, although these ideas are equally applicable to more complex systems. It transpired that the complex affine analysis performed significantly better than its interval counterpart.     

Generating functions in two-dimensional quantum gravity

We solve general 1-matrix models without taking the double scaling limit. A method of computing generating functions is presented. We calculate the generating functions for a simple and a double torus. Our method is also applicable to higher genus. Each generating function can be expressed by a “specific heat” function for the sphere. Universal terms, which survived in the double scaling limit can be easily picked out from our exact solutions. We also find that the regular part of the spherical generating function is at most bilinear in the coupling constants of the source terms.     

On the solvability of the transvection group of extrinsic symplectic symmetric spaces

Let M$M$ be a symplectic symmetric space, and let ?:M→V$?:M\to V$ be an extrinsic symplectic symmetric immersion in the sense of Krantz and Schwachhöfer (2010) [7], i.e., (V,Ω)$(V,\Omega )$ is a symplectic vector space and ?$?$ is an injective symplectic immersion such that for each point p∈M$p\in M$, the geodesic symmetry in p$p$ is compatible with the reflection in the affine normal space at ?(p)$?\left(p\right)$.     

Renormalization group functions for the radiative symmetry breaking scheme

We obtain the renormalization group (RG) functions for the massless scalar field theory where symmetry breaking occurs radiatively. After obtaining the effective potential for the radiative symmetry breaking scheme by finite transformations for the classical field and coupling constant, we obtain the corresponding RG functions from that of the minimal subtraction (MS) scheme.     

A locally compact quantum group arising from quantization of the affine group of a local field

Letters in Mathematical Physics - Using methods coming from non-formal equivariant quantization, we construct in this short note a unitary dual 2-cocycle on a discrete family of quotient groups of...     

Generating functions for effective hamiltonians. Symmetry,topology, combinatorics

Recent developments associated with old technique of generating functions and invariant theory which I have started to apply to molecular problems due to my collaboration with Yu.F. Smirnov about 25 years ago are discussed. Three aspects are presented: the construction of diagonal in polyad quantum number effective resonant vibrational Hamiltonians using the symmetrized Hadamard product; the decomposition of the number of state generating function into regular and oscillatory contributions and its relation with Todd polynomials and topological characterization of energy bands; qualitative aspects of resonant oscillators and fractional monodromy as one of new generalizations of Hamiltonian monodromy.     

Torus structure on graphs and twisted partition functions for minimal and affine models

Using the Ocneanu quantum geometry of ADE diagrams (and of other diagrams belonging to higher Coxeter–Dynkin systems), we discuss the classification of twisted partition functions for affine and minimal models in conformal field theory and study several examples associated with the WZW, Virasoro and cases.     

Complex extension of the representation of the symplectic group associated with the canonical commutation relations

The aim of the investigation is to extend the representation of the real symplectic group associated with the canonical commutation relations into the complex symplectic group. It is shown that an extension exists to a semigroup S such that Sp(2n, R) ? S ? Sp(2n, C). The construction of the extension is achieved by extensive use of Bargmann's reproducing kernel space. We are able to give a simple geometric model for the semigroup S: it is exactly the semigroup of all complex symplectic transformations which increase the U(n, n) “length”.     

A family of affine quantum group invariant integrable extensions of the Hubbard Hamiltonian

We construct a family of spin chain Hamiltonians, which have the affine quantum group symmetry . Their eigenvalues coincide with the eigenvalues of the usual spin chain Hamiltonians, but have the degeneracy of levels, corresponding to the affine . The space of states of these spin chains is formed by the tensor product of the fully reducible representations of the quantum group.

The fermionic representations of the constructed spin chain Hamiltonians show that we have obtained new extensions of the Hubbard Hamiltonians. All of them are integrable and have the affine quantum group symmetry. The exact ground state of such type of model is presented, exhibiting superconducting behavior via the η-pairing mechanism.     

A renormalization group analysis of correlation functions for the dipole gas

We develop a renormalization group method for analyzing the generating functional for charge correlations of a dilute classical dipole gas. It is based on and extends the renormalization group analysis introduced by Brydges and Yau for the dipole gas partition function. Our method leads to systematic formulas for the large-distance behavior of correlation functions of all orders. We prove that in any dimensiond2, at any value>0 of the inverse temperature, and at sufficiently small activityz, the correlation functions exhibit at large distances the same behavior as for a vacuum (z=0), but with a new dielectric constant 1+ over which we have good control. The results proved here extend existing results on the two-point correlations to all higher correlations, and constitute a general confirmation of the fact that dipoles do not screen.     

Generating functions for characters and weight multiplicities of irreducible (4)-modules

Generating functions for the characters of the irreducible representations of simple Lie algebras are rational functions where both the numerator and denominator can be expressed as polynomials in the characters corresponding to the fundamental weights. They encode much information on the representation theory of the algebra, but their explicit expressions are in general very complicated. In fact, it seems that rank three is the highest rank tractable. In this paper, we use a method based on the quantum Calogero-Sutherland model to compute the full generating function for the characters of irreducible modules over the complex Lie algebra (4), and exploit this result to obtain also generating functions giving the multiplicities of some low order weights in all representations. We have applied the same method to compute the generating function for the characters of the modules the other rank three simple Lie algebras, but in these cases the full expressions are very long and appear only in the arXiv version of the paper (arXiv:1705.03711 [math-ph]). Nevertheless, when the generating functions are limited to some particular subsets of characters, the results are quite simple and we present them here.