Two-Term Dilogarithm Identities Related to Conformal Field Theory

We study 2 × 2 matrices A such that the corresponding thermodynamic Bethe ansatz (TBA) equations yield in the form of the effective central charge of a minimal Virasoro model. Certain properties of such matrices and the corresponding solutions of the TBA equations are established. Several continuous families and a discrete set of admissible matrices A are found. The corresponding two-term dilogarithm identities (some of which appear to be new) are obtained. Most of them are proven or shown to be equivalent to previously known identities.     

The Relation Between Two Types of Characteristic Equations for Quantum Matrices

A definition (modification) of the power of quantum matrices using the -matrix has recently been proven useful to obtain generalizations of many well known theorems from linear algebra to the quantum case, among which are the Cayley–Hamilton theorem and the Newton identities. A separate effort has provided another generalization of the Cayley–Hamilton theorem for GL _q (n), which uses usual matrix powers but diagonal matrices as coefficients.We show that the latter generalization can be derived in the aforementioned more general framework and it is the expression of the modified quantum power in terms of the usual ones that accounts for the appearance of diagonal matrices.     

Anderson Localization for a Supersymmetric Sigma Model

We study a lattice sigma model which is expected to reflect the Anderson localization and delocalization transition for real symmetric band matrices in 3D. In this statistical mechanics model, the field takes values in a supermanifold based on the hyperbolic plane. The existence of a diffusive phase in 3 dimensions was proved in Disertori et al. (Commun. Math. Phys., doi:, 2009) [2] for low temperatures. Here we prove localization at high temperatures for any dimension d ≥ 1. Our analysis uses Ward identities coming from internal supersymmetry.     

One Way to Solve the Puzzle of γ5 in the Dimensional Regularization

By calculating the Dirac matrices in four dimensions and performing the dimensional regularization for loop momentum integration variables, various Ward identities can be obtained straightforwardly and the puzzle for extending γ5 to n dimensions can be solved.     

The Andrews–Gordon Identities and q-Multinomial Coefficients

We prove polynomial boson-fermion identities for the generating function of the number of partitions of n of the form , with , and . The bosonic side of the identities involves q-deformations of the coefficients of x ^a in the expansion of . A combinatorial interpretation for these q-multinomial coefficients is given using Durfee dissection partitions. The fermionic side of the polynomial identities arises as the partition function of a one-dimensional lattice-gas of fermionic particles. In the limit , our identities reproduce the analytic form of Gordon's generalization of the Rogers–Ramanujan identities, as found by Andrews. Using the duality, identities are obtained for branching functions corresponding to cosets of type of fractional level . Received: 22 January 1996 /Accepted: 4 September 1996     

On Gell-Mann's λ-matrices,d- andf-tensors,octets, and parametrizations ofS U (3)

The algebra ofS U (3) is developed on the basis of the matrices _i ofGell-Mann, and identities involving the tensorsd _{i j k} andf _{i j k} occurring in their multiplication law are derived. Octets and the tensor analysis of the adjoint groupS U (3)/Z(3) ofS U (3) are discussed. Various explicit parametrizations ofS U (3) are presented as generalizations of familiarS U (2) results.     

Drinfeld–Sokolov reduction in quantum algebras: canonical form of generating matrices

We define the second canonical forms for the generating matrices of the Reflection Equation algebras and the braided Yangians, associated with all even skew-invertible involutive and Hecke symmetries. By using the Cayley–Hamilton identities for these matrices, we show that they are similar to their canonical forms in the sense of Chervov and Talalaev (J Math Sci (NY) 158:904–911, 2008).     

Autocorrelation of Random Matrix Polynomials

 We calculate the autocorrelation functions (or shifted moments) of the characteristic polynomials of matrices drawn uniformly with respect to Haar measure from the groups U(N), O(2N) and USp(2N). In each case the result can be expressed in three equivalent forms: as a determinant sum (and hence in terms of symmetric polynomials), as a combinatorial sum, and as a multiple contour integral. These formulae are analogous to those previously obtained for the Gaussian ensembles of Random Matrix Theory, but in this case are identities for any size of matrix, rather than large-matrix asymptotic approximations. They also mirror exactly the autocorrelation formulae conjectured to hold for L-functions in a companion paper. This then provides further evidence in support of the connection between Random Matrix Theory and the theory of L-functions. Received: 1 August 2002 / Accepted: 25 December 2002 Published online: 7 May 2003 Communicated by P. Sarnak     

New Jacobi-like identities for Z K parafermion characters

We state and prove various new identities involving theZ _K parafermion characters (or level-K string functions)c _n ^l for the casesK=4,K=8, andK=16. These identities fall into three classes: identities in the first class are generalizations of the famous Jacobi -function identity (which is theK=2 special case), identities in another class relate the levelK>2 characters to the Dedekind -function, and identities in a third class relate theK>2 characters to the Jacobi -functions. These identities play a crucial role in the interpretation of fractional superstring spectra by indicating spacetime supersymmetry and aiding in the identification of the spacetime spin and statistics of fractional superstring states.     

Dilogarithm identities in conformal field theory and group homology

Recently, Rogers' dilogarithm identities have attracted much attention in the setting of conformal field theory as well as lattice model calculations. One of the connecting threads is an identity of Richmond-Szekeres that appeared in the computation of central charges in conformal field theory. We show that the Richmond-Szekeres identity and its extension by Kirillov-Reshetikhin (equivalent to an identity found earlier by Lewin) can be interpreted as a lift of a generator of the third integral homology of a finite cyclic subgroup sitting inside the projective special linear group of all 2×2 real matrices viewed as adiscrete group. This connection allows us to clarify a few of the assertions and conjectures stated in the work of Nahm-Recknagel-Terhoven concerning the role of algebraic K-theory and Thurston's program on hyperbolic 3-manifolds. Specifically, it is not related to hyperbolic 3-manifolds as suggested but is more appropriately related to the group manifold of the universal covering group of the projective special linear group of all 2×2 real matrices viewed as a topological group. This also resolves the weaker version of the conjecture as formulated by Kirillov. We end with a summary of a number of open conjectures on the mathematical side.To Professor C. N. Yang for his 70^th birthdayThis work was partially supported by grants from the Statens Naturvidenskabelige Forskningsraad and the Paul and Gabriella Rosenbaum Foundation     

On the validity of ward identities

Ward identities for matrix elements of covariant two-point time-ordered operators in the presence of an arbitrary number of subtractions are investigated. Neither the existence of naiveT-products nor the existence of equal-time commutators between current densities will be assumed. It is shown by means of the Jost-Lehmann-Dyson representation thatT*-products can always be defined such that normal Ward identities with respect to one current are valid. The simultaneous validity of normal Ward identities with respect to two currents requires a relation between equal-time charge-current commutators.Our results show that the usual realization of current algebra in the form of Ward identities is possible even if subtractions are necessary. Some examples are discussed in detail.     

Constrained Correlation Dynamics of QED in Canonical Form(Ⅱ)Equal Time Limit and Two-Body Correlation Dynamics

The equations of motion for multi-time correlation Green's functions are transformed into those for equal-time correlation Green's functions,which include the equations of motion for electron's and photon's density matrices as well as vertex functions.In two-body correlation truncation approximation,we present the explicit expressions for the equations of motion,Gauss law and Ward identities explicitly.     

The KT-BRST Complex of a Degenerate Lagrangian System

Quantization of a Lagrangian field system essentially depends on its degeneracy and implies its BRST extension defined by sets of non-trivial Noether and higher-stage Noether identities. However, one meets a problem how to select trivial and non-trivial higher-stage Noether identities. We show that, under certain conditions, one can associate to a degenerate Lagrangian L the KT-BRST complex of fields, antifields and ghosts whose boundary and coboundary operators provide all non-trivial Noether identities and gauge symmetries of L. In this case, L can be extended to a proper solution of the master equation.      

Local identities involving Jacobi elliptic functions

We derive a number of local identities involving Jacobi elliptic functions and use them to obtain several new results. First, we present an alternative, simpler derivation of the cyclic identities discovered by us recently, along with an extension to several new cyclic identities. Second, we obtain a generalization to cyclic identities in which successive terms have a multiplicative phase factor exp(2iπ/s), wheres is any integer. Third, we systematize the local identities by deriving four local ‘master identities’ analogous to the master identities for the cyclic sums discussed by us previously. Fourth, we point out that many of the local identities can be thought of as exact discretizations of standard non-linear differential equations satisfied by the Jacobi elliptic functions. Finally, we obtain explicit answers for a number of definite integrals and simpler forms for several indefinite integrals involving Jacobi elliptic functions.     

Bilinear identities for the constrained modified KP hierarchy

In this paper, we mainly investigate an equivalent form of the constrained modified KP hierarchy: the bilinear identities. By introducing two auxiliary functions ρ and σ, the corresponding identities are written into the Hirota forms. Also, we give the explicit solution forms of ρ and σ.     

A bijection which implies Melzer's polynomial identities: theχ 1,1 (p,p+1) case

We obtain a bijection between certain lattice paths and partitions. This implies a proof of polynomial identities conjectured by Melzer. In a limit, these identities reduce to Rogers-Ramanujantype identities for the _1,1 ^(p,p+1) (q) Virasoro characters, conjectured by the Stony Brook group.     

q-power function over q-commuting variables and deformed XXX and XXZ chains

We find certain functional identities for the Gauss q-power function of a sum of q-commuting variables. Then we use these identities to obtain two-parameter twists of the quantum affine algebra $U_q (\widehat{sl}_2 )$ and of the Yangian Y(sl ₂). We determine the corresponding deformed trigonometric and rational quantum R matrices, which then are used in the computation of deformed XXX and XXZ Hamiltonians.     

Generalized Cayley-Hamilton-Newton identities

The q-generalizations of the two fundamental statements of matrix algebra — the Cayley-Hamilton theorem and the Newton relations — to the cases of quantum matrix algebras of RTT and Reflection-equation types have been obtained. We construct a family of matrix identities which we call Cayley-Hamilton-Newton identities and which underlie the characteristic identity as well as the Newton relations for the RTT and Reflection equation algebras, in the sense that both the characteristic identity and the Newton relations are direct consequences of the Cayley-Hamilton-Newton identities.     

The most general L operator for the R-matrix of the XXX model

The problem of describing all the monodromy matrices for R matrices of the XXX and XXZ models is discussed. It is shown that the L operator of the lattice nonlinear Schrödinger model generates all possible monodromy matrices for the XXX R matrix.