Quantum Volterra model

Volkov, А. Yu.

doi:10.1016/0375-9601(92)90270-v

Cited by 86 publications

(53 citation statements)

References 8 publications

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“…The deformation parameter b plays the role of the quantum parameter of the theory. The result (8) coincides with the partition function of the Faddeev-Volkov model [16,17], which describes a discrete Virasoro symmetry.…”

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confidence: 73%

Emergent 3-Manifolds from Four Dimensional Superconformal Indices

Terashima

Yamazaki

2012

Phys. Rev. Lett.

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We show that the smooth geometry of a hyperbolic 3-manifold emerges from a classical spin system defined on a 2d discrete lattice, and moreover show that the process of this "dimensional oxidation" is equivalent with the dimensional reduction of a supersymmetric gauge theory from 4d to 3d. More concretely, we propose an equality between (1) the 4d superconformal index of a 4d N = 1 superconformal quiver gauge theory described by a bipartite graph on T 2 and (2) the partition function of a classical integrable spin chain on T 2 . The 2d spin system is lifted to a hyperbolic 3-manifold after the dimensional reduction and the Higgsing of the 4d gauge theory.Introduction.-The concept of spacetime has been of crucial importance in our understanding of Nature. However, in the theory of quantum gravity, it is widely believed that even the notion of classical spacetime is of secondary nature, and emerges from a more fundamental structure. One proposal for such a structure is the spin network [1], a spin system defined on a discrete lattice.In a different line of development, more recently there have been important developments in supersymmetric gauge theories suggesting that the spacetime geometry could be traded for another "internal" geometry. This has been discussed for a class of supersymmetric gauge theories compactified on a compact curved manifold, which is thought of as the Euclidean version of the spacetime for the theory. The idea is simple; we begin with a D-dimensional field theory and compactify the theory on a class of d 1 -dimensional manifolds C. The resulting d 2 -dimensional theory is defined on a fixed d 2 -dimensional compact manifold S, where d 1 + d 2 = D. We could instead first compactify on S, and then we have a d 1 -dimensional theory on C. Thus we have a correspondence between the d 2 -dimensional field theory on S and the d 1 -dimensional field theory on C.While the idea itself is rather general, in practice it is a rather difficult problem to make a precise identification between the observables of the two theories, since a quantity on one side could take a rather different form on the other. A successful example of such a quantitative identification is the relation between the S 4 partition function of 4d N = 2 superconformal field theories (SCFT) arising from a compactification of 6d (2, 0) theory on a Riemann surface C [2] and a correlation function of 2d Liouville theory on C [3].The goal of this Letter is to unify these two apparently unrelated ideas in supersymmetric gauge theories and gravity. This gives new perspectives on the emergence of classical geometry, and surprisingly the process has a counterpart in the supersymmetric gauge theory.We analyze the 4d superconformal index for quiver gauge theories dual to toric Calabi-Yau 3-folds, and find that the 4d index is equivalent to the partition function of an integrable spin system in 2d. We then discuss dimensional reduction from the 4d index to the 3d partition

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confidence: 73%

Emergent 3-Manifolds from Four Dimensional Superconformal Indices

Terashima

Yamazaki

2012

Phys. Rev. Lett.

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“…This matrix relation contains exactly nine scalar equations where the LHS only depends on the front angles (8), while the RHS only depends on the back angles (10). Solving these equations one can obtain explicit form of the map (3).…”

Section: Consider Four Pointsmentioning

confidence: 99%

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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“…We consider a periodic lattice with N + 1 sites, and on each lattice site we attach a dynamical variable a i on which we impose the Faddeev-Takhtadjan Poisson bracket [7]: {a i , a j } = a i a j (4 − a i − a j )(δ i,j+1 − δ j,i+1 ) − a j+1 δ i,j+2 + a i+1 δ j,i+2 (5) This bracket 4 is interesting because taking the continuum limit as a i ≃ 1 + ∆ 2 u(x), ∆ = 1 (N + 1) it becomes the Virasoro Poisson bracket eq.(4). For precisely this reason, and in this perspective, the lattice model has been extensively studied both at the classical level [7,8] and at the quantum level [9,10,11,13,12]. The present paper is one more contribution this series of works.…”

Section: The Volterra Modelmentioning

confidence: 93%

Continuum Limit of the Volterra Model, Separation of Variables and Non-Standard Realizations of the Virasoro Poisson Bracket

Babelon

2006

Commun. Math. Phys.

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The classical Volterra model, equipped with the Faddeev-Takhtadjan Poisson bracket provides a lattice version of the Virasoro algebra. The Volterra model being integrable, we can express the dynamical variables in terms of the so called separated variables. Taking the continuum limit of these formulae, we obtain the Virasoro generators written as determinants of infinite matrices, the elements of which are constructed with a set of points lying on an infinite genus Riemann surface. The coordinates of these points are separated variables for an infinite set of Poisson commuting quantities including L 0 . The scaling limit of the eigenvector can also be calculated explicitly, so that the associated Schroedinger equation is in fact exactly solvable.

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Quantum Volterra model

Cited by 86 publications

References 8 publications

Emergent 3-Manifolds from Four Dimensional Superconformal Indices

Emergent 3-Manifolds from Four Dimensional Superconformal Indices

Quantum geometry of three-dimensional lattices

Continuum Limit of the Volterra Model, Separation of Variables and Non-Standard Realizations of the Virasoro Poisson Bracket

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