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Rinat Kedem

Publications and source records attributed to Rinat Kedem.

31 records · Page 2Linked to original sources
arXiv2006

Fusion products of Kirillov-Reshetikhin modules and fermionic multiplicity formulas

We give a complete description of the graded multiplicity space which appears in the Feigin-Loktev fusion product [FL99] of graded Kirillov-Reshetikhin modules for all simple Lie algebras. This construction is used to obtain an upper bound formula for the fusion coefficients in these cases. The formula generalizes the case of g=A_r [AKS06], where the multiplicities are generalized Kostka polynomials [SW99,KS02]. In the case of other Lie algebras, the formula is the the fermionic side of the X=M conjecture [HKO+99]. In the cases where the Kirillov-Reshetikhin conjecture, regarding the decomposition formula for tensor products of KR-modules, has been been proven in its original, restricted form, our result provides a proof of the conjectures of Feigin and Loktev regarding the fusion product multiplicites.

math.RT↗
arXiv2005

Fermionic characters of arbitrary highest-weight integrable sl_{r+1}-modules

We give a formula for the q-characters of arbitrary highest-weight integrable modules of sl_{r+1} as a linear combination of the fermionic q-characters of special fusion products of integrable modules. The coefficients in the sum are entries of the inverse matrix of generalized Kostka polynomials, which are in Z[q^{-1}]. In this paper we prove the relation between the character of the Feigin-Loktev graded tensor product and the generalized Kostka polynomial. We also prove the fermionic formula for the q-characters of the (unrestricted) fusion products of rectangular highest-weight integrable g-modules.

math.RT↗
arXiv2005

Fusion products, Kostka polynomials, and fermionic characters of su(r+1)_k

Using a form factor approach, we define and compute the character of the fusion product of rectangular representations of \hat{su}(r+1). This character decomposes into a sum of characters of irreducible representations, but with q-dependent coefficients. We identify these coefficients as (generalized) Kostka polynomials. Using this result, we obtain a formula for the characters of arbitrary integrable highest-weight representations of \hat{su}(r+1) in terms of the fermionic characters of the rectangular highest weight representations.

math-ph↗
arXiv2004

Filling the Bose sea: symmetric quantum Hall edge states and affine characters

We explore the structure of the bosonic analogues of the $k$-clustered ``parafermion'' quantum Hall states. We show how the many-boson wave functions of $k$-clustered quantum Hall droplets appear naturally as matrix elements of ladder operators in integrable representations of the affine Lie algebra $\hat{su}(2)_k$. Using results of Feigin and Stoyanovsky, we count the dimensions of spaces of symmetric polynomials with given $k$-clustering properties and show that as the droplet size grows the partition function of its edge excitations evolves into the character of the representation. This confirms that the Hilbert space of edge states coincides with the representation space of the $\hat{su}(2)_k$ edge-current algebra. We also show that a spin-singlet, two-component $k$-clustered boson fluid is similarly related to integrable representations of $\hat{su}(3)$. Parafermions are not necessary for these constructions.

cond-mat.mes-hall↗
arXiv2003

Fusion products, cohomology of GL(N) flag manifolds and Kostka polynomials

This paper explains the relation between the fusion product of symmetric power sl(n) evaluation modules, as defined by Feigin and Loktev, and the graded coordinate ring R(mu), which describes the cohomology ring of the flag variety Fl(mu) of GL(N). The graded multiplicity spaces appearing in the decomposition of the fusion product into irreducible sl(n)-modules are identified with the multiplicity spaces of the Specht modules in R(mu). This proves that the Kostka polynomial gives the character of the fusion product in this case. In the case of the product of fundamental evaluation modules, we give the precise correspondence with the reduced wedge product, and thus the usual wedge space construction of irreducible level-1 ^sl(n)-modules in the limit N->infty. The multiplicity spaces are W(sl(n))-algebra modules in this limit.

math.RT↗
arXiv1996

Singular R-matrices and Drinfeld's comultiplication

We compute the R-matrix which intertwines two dimensional evaluation representations with Drinfeld comultiplication for U_q(\widehat{sl}_2). This R-matrix contains terms proportional to the delta-function. We construct the algebra A(R) generated by the elements of the matrices L^\pm(z) with relations determined by R. In the category of highest weight representations, there is a Hopf algebra isomorphism between A(R) and an extension \overline{U}_q(\widehat{sl}_2)} of Drinfeld's algebra.

q-alg↗
arXiv1995

Difference Equations in Spin Chains with a Boundary

Correlation functions and form factors in vertex models or spin chains are known to satisfy certain difference equations called the quantum Knizhnik-Zamolodchikov equations. We find similar difference equations for the case of semi-infinite spin chain systems with integrable boundary conditions. We derive these equations using the properties of the vertex operators and the boundary vacuum state, or alternatively through corner transfer matrix arguments for the 8-vertex model with a boundary. The spontaneous boundary magnetization is found by solving such difference equations. The boundary $S$-matrix is also proposed and compared, in the sine-Gordon limit, with Ghoshal--Zamolodchikov's result. The axioms satisfied by the form factors in the boundary theory are formulated.

hep-th↗
arXiv1994

Quasi-particles in the chiral Potts model

We study the quasi-particle spectrum of the integrable three-state chiral Potts chain in the massive phase by combining a numerical study of the zeroes of associated transfer matrix eigenvalues with the exact results of the ferromagnetic three-state Potts chain and the three-state superintegrable chiral Potts model. We find that the spectrum is described in terms of quasi-particles with momenta restricted only to segments of the Brillouin zone $0\leq P \leq 2π$ where the boundaries of the segments depend on the chiral angles of the model.

hep-th↗
arXiv1993

The sums of Rogers, Schur and Ramanujan and the Bose-Fermi correspondence in $1+1$-dimensional quantum field theory

We discuss the relation of the two types of sums in the Rogers-Schur-Ramanujan identities with the Bose-Fermi correspondence of massless quantum field theory in $1+1$ dimensions. One type, which generalizes to sums which appear in the Weyl-Kac character formula for representations of affine Lie algebras and in expressions for their branching functions, is related to bosonic descriptions of the spectrum of the field theory (associated with the Feigin-Fuchs construction in conformal field theory). Fermionic descriptions of the same spectrum are obtained via generalizations of the other type of sums. We here summarize recent results for such fermionic sum representations of characters and branching functions. (To appear in C.N. Yang's 70th birthday Festschrift.)

hep-th↗
arXiv1992

Thermodynamics of the 3-State Potts Spin Chain

We demonstrate the relation of the infrared anomaly of conformal field theory with entropy considerations of finite temperature thermodynamics for the 3-state Potts chain. We compute the free energy and compute the low temperature specific heat for both the ferromagnetic and anti-ferromagnetic spin chains, and find the central charges for both.

hep-th↗
arXiv1992

Construction of Modular Branching Functions from Bethe's Equations in the 3-State Potts Chain

We use the single particle excitation energies and the completeness rules of the 3-state anti-ferromagnetic Potts chain, which have been obtained from Bethe's equation, to compute the modular invariant partition function. This provides a fermionic construction for the branching functions of the $D_4$ representation of $Z_4$ parafermions which complements the previous bosonic constructions. It is found that there are oscillations in some of the correlations and a new connection with the field theory of the Lee-Yang edge is presented.

hep-th↗
arXiv1992

Physics beyond quasi-particles: Spectrum and completeness of the 3 state superintegrable chiral Potts model

We find the rules which count the energy levels of the 3 state superintegrable chiral Potts model and demonstrate that these rules are complete. We then derive the complete spectrum of excitations in the thermodynamic limit in the massive phase and demonstrate the existence of excitations which do not have a quasi-particle form. The physics of these excitations is compared with the BCS superconductivity spectrum and the counting rules are compared with the closely related $S=1$ XXZ spin chain.

cond-mat↗
arXiv1992

Physics beyond quasi-particles: Spectrum and completeness of the 3 state superintegrable chiral Potts model

We find the rules which count the energy levels of the 3 state superintegrable chiral Potts model and demonstrate that these rules are complete. We then derive the complete spectrum of excitations in the thermodynamic limit in the massive phase and demonstrate the existence of excitations which do not have a quasi-particle form. The physics of these excitations is compared with the BCS superconductivity spectrum and the counting rules are compared with the closely related $S=1$ XXZ spin chain.

hep-th↗