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arXiv:0911.5393 · Journal of Physics A Mathematical and General

Analytic Bethe ansatz and functional relations related to tensor-like representations of type-II Lie superalgebras B ( r | s ) and D ( r | s )

Zengo Tsuboi

Published Sep 28, 1999• 20 citations
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Abstract

An analytic Bethe ansatz is carried out related to tensor-like representations of the type-II Lie superalgebras B ( r | s ) = osp (2 r +1|2 s ) ( r 0 , s 1 ) and D ( r|s ) = osp (2 r|2s ) ( r 2 , s 1 ). We present eigenvalue formulae of transfer matrices in dressed vacuum forms (DVFs) labelled by Young (super) diagrams. A class of transfer matrix functional relations ( T -system) is discussed. In particular for the B (0| s ) = osp (1|2 s ) ( s 1 ) case, a complete set of functional relations is proposed by using duality among DVFs.

DOI
10.1088/0305-4470/32/41/311

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TTT-system and thermodynamic Bethe ansatz equations for solvable lattice models associated with superalgebras (Bilinear Method in the Study of Integrable Systems and Related Topics)
An analytic Bethe ansatz is carried out related to the Lie superal- gebra osp(1∣2s)osp(1|2s)osp(1∣2s) .We present an eigenvahie formula of atransfer matrix in dressed vacuum form (DVF) labeled by aYoung (super) diagram.Remarkable duality among DVFs is found.Acomplete set of transfer matrix functional relations ( TTT -systern)is proposed as areduction of a Hirota-Miwa equation.We also derive athermodynamic Bethe ansatz (TBA) equation from this TTT -system and the quantum transfer matrix method.This TBA equation is identical to the one from the string hypothesis.1Introduction Solvable lattice models related to Lie superalgebras [1] he˙ιv(\\mathrm{h}_{\dot{\mathrm{e}}}\iota \mathrm{v}(^{\backslash }he˙​ιv(\ rpceived Inι1(jh\mathrm{I}\mathrm{n}\iota 1(j\mathrm{h}Inι1(jh attentions [2, 3,4,5,6,7,8,9].To construct cigellval\iota \iota e formulae of transfer matrices for such models is an irnportant problem irl\mathrm{i}\mathrm{r}\mathrm{l}irl mathernatical pbysics.To achieve this pr()gl⋅alll,\mathrm{p}\mathrm{r}()\mathrm{g}\mathrm{l}\cdot \mathrm{a}\mathrm{l}\mathrm{l}\mathrm{l},pr()gl⋅alll, t11(Y\mathrm{t}11(^{\mathrm{Y}}t11(Y Betbe ϵ˙ιr1ιs1atlz\dot{\epsilon}\iota \mathrm{r}1_{\iota}\mathrm{s}^{1}\mathrm{a}\mathrm{t}_{l}\mathrm{z}ϵ˙ιr1ι​s1atl​z has been often t1.bτt\r1\mathrm{t}1.\mathrm{b}^{\tau}\mathrm{t}^{\backslash}\mathrm{r}1t1.bτt\r1 .Nowadays, there is nmch 1itI(Y,1⋅c˙tfu1⋅t11\mathrm{i}\mathrm{t}\mathrm{I}(^{\mathrm{Y}},1^{\cdot}\dot{c}\mathrm{t}\mathrm{f}\mathrm{u}1^{\cdot}\mathrm{t}^{1}1itI(Y,1⋅c˙tfu1⋅t1 (see for (\X⊂˙llIr11)1t1,.′[4.1t],(^{\backslash }\mathrm{X}\dot{\subset}\mathrm{l}_{l}\mathrm{I}\mathrm{r}11^{)}1\mathrm{t}^{1},.'[4.1\mathrm{t}],(\X⊂˙​ll​Ir11)1t1,.′[4.1t],11,12,1311,12,1311,12,13 , 14, 15, 16, 17, 18, 19, 2tI,2\mathrm{t}\mathrm{I},2tI, 21,22,23,9‾421,22,23,\underline{9}421,22,23,9​4 ] allcl r(1f\\mathrm{r}(^{1\mathrm{f}\backslash }r(1f\ ( Γt\11Ces\Gamma \mathrm{t}^{\backslash }11\mathrm{C}\mathrm{e}\mathrm{s}Γt\11Ces therein.)()I1B(\t11t1()\mathrm{I}1\mathrm{B}(^{\backslash }\mathrm{t}1_{1\mathrm{t}^{1}}()I1B(\t11t1​
Extending the method proposed in [1], we derive QQ-relations (functional relations among Baxter Q-functions) and T-functions (eigenvalues of transfer matrices) for fusion vertex models associated with the twisted quantum affine superalgebras Uq(gl(2r+1|2s)(2)), Uq(gl(2r|2s+1)(2)), Uq(gl(2r|2s)(2)), Uq(osp(2r|2s)(2)) and the untwisted quantum affine orthosymplectic superalgebras Uq(osp(2r+1|2s)(1)) and Uq(osp(2r|2s)(1)) (and their Yangian counterparts, Y(osp(2r+1|2s)) and Y(osp(2r|2s))) as reductions (a kind of folding) of those associated with Uq(gl(M|N)(1)). In particular, we reproduce previously proposed generating functions (difference operators) of the T-functions for the symmetric or anti-symmetric representations, and tableau sum expressions for more general representations for orthosymplectic superalgebras [2], [3], and obtain Wronskian-type expressions (analogues of Weyl-type character formulas) for them. T-functions for spinorial representations are related to reductions of those for asymptotic limits of typical representations of Uq(gl(M|N)(1)).