Functional tetrahedron equation

Kashaev, Rinat; Korepanov, I. G.; Sergeev, S. M.

doi:10.1007/bf02557179

Cited by 67 publications

(131 citation statements)

References 5 publications

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“…This algebra is, obviously, a quantum counterpart of the Poisson algebra (24). In the previous Section we have already mentioned the result of [21] that (i) the map (22) is an automorphism of the tensor cube of the Poisson algebra (24) (remind that the relation (20) should be taken into account in (22)). …”

Section: Tetrahedron Equationmentioning

confidence: 98%

“…Then, following the arguments of [19], one can show that the map (3) satisfies the functional tetrahedron equation [20] …”

Section: Consider Four Pointsmentioning

confidence: 99%

“…The formulae (22) for the quantum map stay exactly the same, but the relation (20) should be replaced by either of the two relations on the second line of (53), for instance, k 2 = q (1 − a * a). In particular, (23) should be replaced with (k…”

Section: Tetrahedron Equationmentioning

confidence: 99%

“…The analog of the Yang-Baxter equation for integrable quantum systems in 3D is called the tetrahedron equation. It was introduced by Zamolodchikov in [13,14] (see also [15][16][17][18][19][20][21] for further results in this field, used in this paper). Similarly to the Yang-Baxter equation the tetrahedron equation provides local integrability conditions which are not related to the size of the lattice.…”

Section: Introductionmentioning

confidence: 99%

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Quantum geometry of three-dimensional lattices

Bazhanov¹,

Mangazeev²,

Sergeev³

2008

J. Stat. Mech.

100

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We study geometric consistency relations between angles on 3-dimensional (3D) circular quadrilateral lattices -lattices whose faces are planar quadrilaterals inscribable into a circle. We show that these relations generate canonical transformations of a remarkable "ultra-local" Poisson bracket algebra defined on discrete 2D surfaces consisting of circular quadrilaterals. Quantization of this structure leads to new solutions of the tetrahedron equation (the 3D analog of the Yang-Baxter equation). These solutions generate an infinite number of non-trivial solutions of the Yang-Baxter equation and also define integrable 3D models of statistical mechanics and quantum field theory. The latter can be thought of as describing quantum fluctuations of lattice geometry. The classical geometry of the 3D circular lattices arises as a stationary configuration giving the leading contribution to the partition function in the quasi-classical limit.

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Section: Tetrahedron Equationmentioning

confidence: 98%

“…Then, following the arguments of [19], one can show that the map (3) satisfies the functional tetrahedron equation [20] …”

Section: Consider Four Pointsmentioning

confidence: 99%

Section: Tetrahedron Equationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Quantum geometry of three-dimensional lattices

Bazhanov¹,

Mangazeev²,

Sergeev³

2008

J. Stat. Mech.

100

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show abstract

“…The relation (3.5) is equivalent to quantum Korepanov equation, it can be also seen as the tetrahedral Zamolodchikov algebra/local Yang-Baxter equation for the adjoint action of R. See the long story of [12,13,14,15,4,11] for details. We fix the normalization of R by…”

Section: Theorem 31 There Is a Unique (Up To A Constant Multiple) Imentioning

confidence: 99%

Tetrahedron Equation and Quantum R Matrices for Spin Representations of $${B^{(1)}_n}$$ B n ( 1 ) , $${D^{(1)}_n}$$ D n ( 1 ) and $${D^{(2)}_{n+1}}$$ D n + 1 ( 2 )

Kuniba

Sergeev

2013

Commun. Math. Phys.

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(2) n+1 ATSUO KUNIBA AND SERGEY SERGEEV Dedicated to Professor Vladimir Bazhanov on the occasion of his sixtieth birthday. AbstractIt is known that a solution of the tetrahedron equation generates infinitely many solutions of the Yang-Baxter equation via suitable reductions. In this paper this scheme is applied to an oscillator solution of the tetrahedron equation involving bosons and fermions by using special 3d boundary conditions. The resulting solutions of the Yang-Baxter equation are identified with the quantum R matrices for the spin representations of B

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Multidimensional analogues of the geometrics⇌t duality

Korepanov

2000

Theor Math Phys

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The usual propetry of s ↔ t duality for scattering amplitudes, e.g. for Veneziano amplitude, is deeply connected with the 2-dimensional geometry. In particular, a simple geometric construction of such amplitudes was proposed in a joint work by this author and S. Saito (solv-int/9812016). Here we propose analogs of one of those amplitudes associated with multidimensional euclidean spaces, paying most attention to the 3-dimensional case. Our results can be regarded as a variant of "Regge calculus" intimately connected with ideas of the theory of integrable models.

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Functional tetrahedron equation

Cited by 67 publications

References 5 publications

Quantum geometry of three-dimensional lattices

Quantum geometry of three-dimensional lattices

Tetrahedron Equation and Quantum R Matrices for Spin Representations of $${B^{(1)}_n}$$ B n ( 1 ) , $${D^{(1)}_n}$$ D n ( 1 ) and $${D^{(2)}_{n+1}}$$ D n + 1 ( 2 )

Multidimensional analogues of the geometrics⇌t duality

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