Deformation Quantization Research Articles

On deformation quantizations of hypertoric varieties

Based on a construction by Kashiwara and Rouquier, we present an analogue of the Beilinson- Bernstein localization theorem for hypertoric varieties. In this case, sheaves of differential operators are replaced by sheaves of W-algebras. As a special case, our result gives a localization theorem for rational Cherednik algebras associated to cyclic groups.

Fusion symmetric spaces and subfactors

We construct analogs of the embedding of orthogonal and symplectic groups into unitary groups in the context of fusion categories. At least some of the resulting module categories also appear in boundary conformal field theory. We determine when these categories are unitarizable, and explicitly calculate the index and principal graph of the resulting subfactors. This paper is a sequel of our previous paper [W4], where we introduced a q-deformation of Brauer’s centralizer algebra for orthogonal and symplectic groups; this algebra had already appeared more or less before in [Mo], see also discussion in [W4]. It is motivated by finding a deformation of orthogonal or symplectic subgroups of a unitary group which is compatible with the standard quantum deformation of the big group. This has been done before on the level of coideal subalgebras of Hopf algebras by Letzter. However, our categorical approach also allows us to extend this to the level of fusion tensor categories, where we find finite analogs of symmetric spaces related to the already mentioned groups. Moreover, we can establish C∗ structures, necessary for the construction of subfactors, in this categorical setting; this is not so obvious to see in the setting of co-ideal algebras. It is well-known how one can use a subgroup H of a (for simplicity here) finite group G to construct a module category of the representation category Rep G of G. This module category also appears in the context of subfactors of II1 von Neumann factors as follows: Let R be the hyperfinite II1 factor, and let N = RG ⊂M = RH be the fixed points under outer actions of G and H. Then the category of N − N bimodules is equivalent to Rep G, and the module category is given via the M−N bimodules of the inclusion N ⊂ M; its simple objects are labeled by the irreducible representations of H. In particular, an important invariant called the principal graph of the subfactor is determined by the restriction rules for representations from G to H. Important examples of subfactors were constructed from fusion categories whose Grothendieck semirings are quotients of the ones of semisimple Lie groups. So a natural question to ask is whether one can perform a similar construction in this context. More precisely, can we find restriction rules for type A fusion categories which describe a subfactor as before, and which will approach in the classical limit the usual restriction rules from U(N) to O(N). We answer this question in the positive in this paper via a fairly elementary construction. We show that certain semisimple quotients of the q-Brauer algebras have a C∗ structure and contain C∗-quotients of Hecke algebras of type A. The subfactor is then obtained as the closure of inductive limits of such algebras. Due to its close connection to Lie groups, we can

The quantum deformed mirror TBA I

We derive the ground state thermodynamic Bethe ansatz equations for the quantum deformation of the AdS5 × S5 mirror model, taking the deformation parameter to be a root of unity. By virtue of the deformation, the resulting equations show an interesting structure between a finite number of Y-functions.

Quantum mechanics in phase space: the Schrödinger and the Moyal representations

We present a phase space formulation of quantum mechanics in the Schr\"odinger representation and derive the associated Weyl pseudo-differential calculus. We prove that the resulting theory is unitarily equivalent to the standard "configuration space" formulation and show that it allows for a uniform treatment of both pure and mixed quantum states. In the second part of the paper we determine the unitary transformation (and its infinitesimal generator) that maps the phase space Schr\"odinger representation into another (called Moyal) representation, where the wave function is the cross-Wigner function familiar from deformation quantization. Some features of this representation are studied, namely the associated pseudo-differential calculus and the main spectral and dynamical results. Finally, the relation with deformation quantization is discussed.

Membrane sigma-models and quantization of non-geometric flux backgrounds

We develop quantization techniques for describing the nonassociative geometry probed by closed strings in flat non-geometric R-flux backgrounds M. Starting from a suitable Courant sigma-model on an open membrane with target space M, regarded as a topological sector of closed string dynamics in R-space, we derive a twisted Poisson sigma-model on the boundary of the membrane whose target space is the cotangent bundle T^*M and whose quasi-Poisson structure coincides with those previously proposed. We argue that from the membrane perspective the path integral over multivalued closed string fields in Q-space is equivalent to integrating over open strings in R-space. The corresponding boundary correlation functions reproduce Kontsevich's deformation quantization formula for the twisted Poisson manifolds. For constant R-flux, we derive closed formulas for the corresponding nonassociative star product and its associator, and compare them with previous proposals for a 3-product of fields on R-space. We develop various versions of the Seiberg-Witten map which relate our nonassociative star products to associative ones and add fluctuations to the R-flux background. We show that the Kontsevich formula coincides with the star product obtained by quantizing the dual of a Lie 2-algebra via convolution in an integrating Lie 2-group associated to the T-dual doubled geometry, and hence clarify the relation to the twisted convolution products for topological nonassociative torus bundles. We further demonstrate how our approach leads to a consistent quantization of Nambu-Poisson 3-brackets.

GENERALIZED COHERENT STATES APPROACH TO DEFORMATION QUANTIZATION

Generalized (f)-coherent state approach in deformation quantization framework is investigated by using a *-eigenvalue equation. For this purpose we introduce a new Moyal star product called f-star product, so that by using this *f-eigenvalue equation one can obtain exactly the spectrum of a general Hamiltonian of a deformed system. Eventually the method is supported with some examples.

Shape invariance in phase space

AbstractShape invariance is a powerful solvability condition, that allows for complete knowledge of the energy spectrum, and eigenfunctions of a system. After a short introduction into the deformation quantization formalism, this paper explores the implications of the supersymmetric quantum mechanics and shape invariance techniques to the phase space formalism. We show that shape invariance induces a new set of relations between the Wigner functions of the system, that allows for their direct calculation, once we know one of them. The simple harmonic oscillator and the Morse potential are solved as examples.

Explicit formulas for noncommutative deformations of ${\mathbb C}{P^N}$CPN and ${\mathbb C}{H^N}$CHN

We give explicit expressions of a deformation quantization with separation of variables for \documentclass[12pt]{minimal}\begin{document}${\mathbb C}P^N$\end{document}CPN and \documentclass[12pt]{minimal}\begin{document}${\mathbb C}H^N$\end{document}CHN. This quantization method is one of the ways to perform a deformation quantization of Kähler manifolds, which is introduced by Karabegov. Star products are obtained as explicit formulas in all order in the noncommutative parameter. We also give the Fock representations of the noncommutative \documentclass[12pt]{minimal}\begin{document}${\mathbb C}P^N$\end{document}CPN and \documentclass[12pt]{minimal}\begin{document}${\mathbb C}H^N$\end{document}CHN.

Bulk Deformations of Open Topological String Theory

We present a general method to construct bulk-deformed open topological string theories from Landau-Ginzburg models. To this end we obtain a weak version of deformation quantisation, and we show how this together with the technique of homological perturbation allows to explicitly compute all bulk-deformed open topological string amplitudes at tree-level before tadpole-cancellation. Our approach is based on a coherent treatment of the problem in terms of the fundamental A ∞- and L ∞-structures involved.

Nonassociative Strict Deformation Quantization of C*-Algebras and Nonassociative Torus Bundles

In this paper, we initiate the study of nonassociative strict deformation quantization of C*-algebras with a torus action. We shall also present a definition of nonassociative principal torus bundles, and give a classification of these as nonassociative strict deformation quantization of ordinary principal torus bundles. We then relate this to T-duality of principal torus bundles with $H$-flux. We also show that the Octonions fit nicely into our theory.

The Lefschetz–Lunts formula for deformation quantization modules

We adapt to the case of deformation quantization modules a formula of V. Lunts who calculates the trace of a kernel acting on Hochschild homology.

Hitchin’s connection, Toeplitz operators, and symmetry invariant deformation quantization

We introduce the notion of a rigid family of Kähler structures on a symplectic manifold. We then prove that a Hitchin connection exists for any rigid holomorphic family of Kähler structures on any compact pre-quantizable symplectic manifold which satisfies certain simple topological constraints. Using Toeplitz operators we prove that the Hitchin connection induces a unique formal connection on smooth functions on the symplectic manifold. Parallel transport of this formal connection produces equivalences between the corresponding Berezin–Toeplitz deformation quantizations. In the cases where the Hitchin connection is projectively flat, the formal connections will be flat and we get a symmetry-invariant formal quantization. If a certain cohomological condition is satisfied a global trivialization of this algebra bundle is constructed. As a corollary we get a symmetry-invariant deformation quantization. Finally, these results are applied to the moduli space situation in which Hitchin originally constructed his connection. First we get a proof that the Hitchin connection in this case is the same as the connection constructed by Axelrod, Della Pietra, and Witten. Second we obtain in this way a mapping class group invariant formal quantization of the smooth symplectic leaves of the moduli space of flat SU(n)-connections on any compact surface.

Fréchet algebraic deformation quantization of the Poincaré disk

Abstract. Starting from formal deformation quantization we use an explicit formula for a star product on the Poincaré disk 𝔻 n $\mathbb {D}_n$ to introduce a Fréchet topology making the star product continuous. To this end a general construction of locally convex topologies on algebras with countable vector space basis is introduced and applied. Several examples of independent interest are investigated as, e.g., group algebras over finitely generated groups and infinite matrices. In the case of the star product on 𝔻 n $\mathbb {D}_n$ the resulting Fréchet algebra is shown to have many nice features: it is a strongly nuclear Köthe space, the symmetry group SU ( 1 , n ) $\mathrm {SU}(1, n)$ acts smoothly by continuous automorphisms with an inner infinitesimal action, and evaluation functionals at all points of 𝔻 n $\mathbb {D}_n$ are continuous positive functionals.

3d gravity and quantum deformations: a Drinfel'd double approach

The constant curvature spacetimes of 3d gravity and their associated symmetry algebras are shown to arise from the 6d Drinfel'd double that underlies the two-parametric 'hybrid' quantum deformation of the \U0001d530\U0001d529(2, ℛ) algebra. Moreover, the quantum deformation supplies the additional structures (star structure and pairing) that enter in the Chern-Simons formulation of the theory, thus establishing a direct link between quantum \U0001d530\U0001d529(2, ℛ) algebras and 3d gravity models. In this approach the flat spacetimes and Newtonian models arise as Lie algebra contractions that are governed by two dimensionful \U0001d530\U0001d529(2, ℛ) deformation parameters, which are directly related to the cosmological constant and to the speed of light.

The eigenvalue equation for a 1-D Hamilton function in deformation quantization

The eigenvalue equation has been found for a Hamilton function in a form independent of the choice of a potential. This Letter proposes a modified Fedosov construction on a flat symplectic manifold. Necessary and sufficient conditions for solutions of an eigenvalue equation to be Wigner functions of pure states are presented. The 1-D harmonic oscillator eigenvalue equation in the coordinates time and energy is solved. A perturbation theory based on the variables time and energy is elaborated.

Deformation Quantization by Moyal Star-Product and Stratonovich Chaos

We make a deformation quantization by Moyal star-product on a space of functions endowed with the normalized Wick product and where Stratonovich chaos are well defined.

Quantized Algebras of Functions on Homogeneous Spaces with Poisson Stabilizers

Let G be a simply connected semisimple compact Lie group with standard Poisson structure, K a closed Poisson-Lie subgroup, 0<q<1. We study a quantization C(G_q/K_q) of the algebra of continuous functions on G/K. Using results of Soibelman and Dijkhuizen-Stokman we classify the irreducible representations of C(G_q/K_q) and obtain a composition series for C(G_q/K_q). We describe closures of the symplectic leaves of G/K refining the well-known description in the case of flag manifolds in terms of the Bruhat order. We then show that the same rules describe the topology on the spectrum of C(G_q/K_q). Next we show that the family of C*-algebras C(G_q/K_q), 0<q\le1, has a canonical structure of a continuous field of C*-algebras and provides a strict deformation quantization of the Poisson algebra \C[G/K]. Finally, extending a result of Nagy, we show that C(G_q/K_q) is canonically KK-equivalent to C(G/K).

Exotic ℝ4 and quantum field theory

Recent work on exotic smooth ℝ4,s, i.e. topological ℝ4 with exotic differential structure, shows the connection of 4-exotics with the codimension-1 foliations of S3, SU(2) WZW models and twisted K-theory KH(S3), H ∊ H3(S3,ℤ). These results made it possible to explicate some physical effects of exotic 4-smoothness. Here we present a relation between exotic smooth ℝ4 and operator algebras. The correspondence uses the leaf space of the codimension-1 foliation of S3 inducing a von Neumann algebra W(S3) as description. This algebra is a type III1 factor lying at the heart of any observable algebra of QFT. By using the relation to factor II, we showed that the algebra W(S3) can be interpreted as Drinfeld-Turaev deformation quantization of the space of flat SL(2, ℂ) connections (or holonomies). Thus, we obtain a natural relation to quantum field theory. Finally we discuss the appearance of concrete action functionals for fermions or gauge fields and its connection to quantum-field-theoretical models like the Tree QFT of Rivasseau.

The Chevalley–Eilenberg Complex and Deformation Quantization in Presence of Two Branes

In this note, we prove that, for a finite-dimensional Lie algebra $\mathfrak g$ over a field $\mathbb K$ of characteristic 0 which contains $\mathbb C$ , the Chevalley–Eilenberg complex $\mathrm U(\mathfrak g)\otimes \wedge(\mathfrak g)$ , which is in a natural way a deformation quantization of the Koszul complex of $\mathrm S(\mathfrak g)$ , is A ∞-quasi-isomorphic to the deformation quantization of the A ∞-bimodule $K=\mathbb K$ provided by the Formality Theorem in presence of two branes (Calaque et al., Comput Math 147(01):105–160, 2011).

Quantum deformation of two four-dimensional spin foam models

We construct the q-deformed version of two four-dimensional spin foam models, the Euclidean and Lorentzian versions of the Engle, Pereira, Rovelli and Livine (EPRL) model. The q-deformed models are based on the representation theory of two copies of \documentclass[12pt]{minimal}\begin{document}$U_q(\mathfrak {su}(2))$\end{document}Uq(su(2)) at a root of unity and on the quantum Lorentz group with a real deformation parameter. For both models, we give a definition of the quantum EPRL intertwiners, study their convergence and braiding properties, and construct an amplitude for the four-simplexes. We find that both of the resulting models are convergent.