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## On a graph-theoretic formula of Gammelgaard for Berezin-Toeplitz quantization

Venue: | Alexander Karabegov) Department of Mathematics, Abilene Christian University |

Citations: | 4 - 1 self |

### Citations

885 | Deformation quantization of Poisson manifolds, - Kontsevich - 2003 |

344 |
Deformation theory and quantization
- Bayen, Flato, et al.
- 1978
(Show Context)
Citation Context ...maier’s identification theorem. We also identify the dual Karabegov-Bordemann-Waldmann star product. 1. Introduction Deformation quantization on a symplectic manifold M was introduced by Bayen et al. =-=[2]-=- as a deformation of the usual pointwise product of C∞(M) into a noncommutative associative ⋆-product of the formal series C∞(M)[[ν]]. The celebrated work of Kontsevich [25] completely solved existenc... |

305 |
A simple geometrical construction of deformation quantization,
- Fedosov
- 1994
(Show Context)
Citation Context ...orm 1νω−1 in Karabegov’s classification [18]. It was also independently constructed by Bordemann andWaldmann [6] by modifying Fedosov’s geometric construction of star products on symplectic manifolds =-=[15]-=-. The identification of Bordemann and Waldmann’s star product was due to Karabegov [20]. Neumaier [30] showed that each star product of separation of variables type can be obtained following Bordemann... |

144 |
Szego kernels and a theorem of
- Zelditch
- 1998
(Show Context)
Citation Context ...of their asymptotic expansions. Finally the Berezin-Toeplitz star product was identified by applying Schlichenmaier’s Theorem 2.1, and the Berezin star product was identified using Zelditch’s theorem =-=[38]-=-. We hope our approach through the graph theoretic formulae of Berezin transform and Bergman kernel could provide further insights to Karabegov-Schlichenmaier’s identification theorem. Theorem 5.1 ([2... |

134 | Toeplitz quantization of Kähler manifolds and gl
- Bordemann, Meinrenken, et al.
- 1994
(Show Context)
Citation Context ... 1/ν in I(m). Recall that the Toeplitz operator T (m) f for f ∈ C∞(M) is defined by (14) T (m) f := Π (m)(f ·) : H0(M,Lm)→ H0(M,Lm), where Π(m) : L2(M,Lm) → H0(M,Lm) is the orthogonal projection. See =-=[5]-=- for a detailed study of their semiclassical properties. The following celebrated theorem of Schlichenmaier [32] shows that Berezin-Toeplitz operator quantization and deformation quantization are clos... |

92 |
Quantization of Kähler manifolds
- Cahen, Gutt, et al.
- 1995
(Show Context)
Citation Context ...x) = ∞∑ k=0 Qkf(x)m −k, where Qk are linear differential operators. The Berezin star product was introduced by Berezin [3] through symbol calculus for linear operators on weighted Bergman spaces (cf. =-=[7, 14, 34]-=-). Karabegov [19] noted that any star product with separation of variables can be constructed from a unique formal Berezin transform. In our case, if we write (11) Qjf = ∑ α,β multiindices cjαβ∂ α∂̄βf... |

67 | A Fedosov Star Product of Wick Type for Kähler Manifolds
- Bordemann, Waldmann
- 1997
(Show Context)
Citation Context ...formation quantization on Kähler manifolds, i.e. it corresponds to the trivial Karabegov form 1νω−1 in Karabegov’s classification [18]. It was also independently constructed by Bordemann andWaldmann =-=[6]-=- by modifying Fedosov’s geometric construction of star products on symplectic manifolds [15]. The identification of Bordemann and Waldmann’s star product was due to Karabegov [20]. Neumaier [30] showe... |

67 |
Deformation quantization: genesis, developments and metamorphoses, pp. 9–54 in: Deformation quantization
- Dito, Sternheimer
- 2001
(Show Context)
Citation Context ...utative associative ⋆-product of the formal series C∞(M)[[ν]]. The celebrated work of Kontsevich [25] completely solved existence and classification of star-products on general Poisson manifolds. See =-=[10]-=- for a comprehensive survey of deformation quantization on symplectic and Poisson manifolds. This paper will restrict to study differentiable deformation quantization with separation of variables on a... |

64 |
Quantization in complex symmetric spaces
- Berezin
- 1975
(Show Context)
Citation Context ... f2 := ∞∑ j=0 νjCBTj (f1, f2), such that the following asymptotic expansion holds (16) T (m) f1 T (m) f2 ∼ ∞∑ j=0 m−jT (m) CBTj (f1,f2) , m→∞. 4 HAO XU The Berezin transform was introduced by Berezin =-=[4]-=- for symmetric domains in Cn and later extended by many authors (see e.g. [12, 14]). Karabegov and Schlichenmaier [24] proved the asymptotic expansion of the Berezin transform for compact Kähler mani... |

61 | Deformation Quantization with Separation of Variables on a Kähler
- Karabegov
- 1996
(Show Context)
Citation Context ...the C-bilinear operators Cj satisfy (4) C0(f1, f2) = f1f2, C1(f1, f2)− C1(f2, f1) = i{f1, f2}. A star product is called differentiable, if each Cj is a bidifferential operator. According to Karabegov =-=[18]-=-, a star product has the property of separation of variables (Wick type), if it satisfies f ⋆ h = f · h and h ⋆ g = h · g for any locally defined antiholomorphic function f , holomorpihc function g an... |

61 | Identification of Berezin-Toeplitz deformation quantization,
- Karabegov, Schlichenmaier
- 2001
(Show Context)
Citation Context ... obtained a remarkable universal formula for any star product with separation of variables corresponding to a given classifying Karabegov form. Using Karabegov-Schlichenmaier’s identification theorem =-=[24]-=- for Berezin-Toeplitz quantization (cf. Theorem 5.1), Gammelgaard’s formula specializes to a graph expansion for BerezinToeplitz quantizaion over acyclic graphs, which can be equivalently formulated i... |

50 | Deformation quantization of compact Kahler manifolds by Berezin-Toeplitz quantization, Conference Moshe Flato
- Schlichenmaier
- 1999
(Show Context)
Citation Context ...: H0(M,Lm)→ H0(M,Lm), where Π(m) : L2(M,Lm) → H0(M,Lm) is the orthogonal projection. See [5] for a detailed study of their semiclassical properties. The following celebrated theorem of Schlichenmaier =-=[32]-=- shows that Berezin-Toeplitz operator quantization and deformation quantization are closed related. Theorem 2.1 ([32]). The Berezin-Toeplitz star product (13) is the unique star product (15) f1 ⋆BT f2... |

47 | On the asymptotic expansion of Bergman kernel,
- Dai, Liu, et al.
- 2006
(Show Context)
Citation Context ... in the compact Kähler case, see [24, 38]. The asymptotic expansion of the Bergman kernel in the setting when Ω is a compact Kähler manifold was also extensively studied. For recent works, see e.g. =-=[9, 11, 17, 27]-=-. The Berezin transform is the integral operator (9) Imf(x) = ∫ Ω f(y) |Km(x, y)|2 Km(x, x) e−mΦ(y) wng (y) n! . At any point for which Km(x, x) invertible, the integral converges for each bounded mea... |

44 | Asymptotic faithfulness of the quantum SU(n) representations of the mapping class groups,
- Andersen
- 2006
(Show Context)
Citation Context ... with a vector bundle. Zelditch [39] studied quantizations of symplectic maps on compact Kähler manifolds and uncovered a connection between Berezin-Toeplitz quantization and quantum chaos. Andersen =-=[1]-=- proved asymptotic faithfulness of the mapping class groups action on Verlinde bundles by using Berezin-Toeplitz technique (cf. also [33]). 3. Differential operators encoded by graphs Throughout this ... |

43 | Berezin-Toeplitz operators, a semi-classical approach,
- Charles
- 2003
(Show Context)
Citation Context ... Definition 3.8). In [37], an explicit formula of Berezin transform in terms of strongly connected graphs was obtained, building on the works of Englǐs [13], Loi [26], and related results of Charles =-=[8]-=-. At a first look, acyclic graphs and strongly connected graphs are quite different. As noted by Schlichenmaier [35], it shall be interesting to clarify their relations. We will give a purely graph th... |

36 | Berezin-Toeplitz Quantization for Compact Kahler Manifolds. A Review of Results,
- Schlichenmaier
- 2010
(Show Context)
Citation Context ...ction, we briefly recall the asymptotic expansion of Berezin transform and related constructions of star products. More detailed expositions and historical remarks can be found in recent nice surveys =-=[34, 35]-=-. Let Φ(x, y) be an almost analytic extension of Φ(x) to a neighborhood of the diagonal, i.e. ∂̄xΦ and ∂yΦ vanish to infinite order for x = y. We can assume Φ(x, y) = Φ(y, x). ON A FORMULA OF GAMMELGA... |

25 | Weighted Bergman kernels and quantization
- Englǐs
(Show Context)
Citation Context ...x) = ∞∑ k=0 Qkf(x)m −k, where Qk are linear differential operators. The Berezin star product was introduced by Berezin [3] through symbol calculus for linear operators on weighted Bergman spaces (cf. =-=[7, 14, 34]-=-). Karabegov [19] noted that any star product with separation of variables can be constructed from a unique formal Berezin transform. In our case, if we write (11) Qjf = ∑ α,β multiindices cjαβ∂ α∂̄βf... |

23 |
The asymptotics of a Laplace integral on a Kähler manifold,
- Engliš
- 2000
(Show Context)
Citation Context ...d Γ(f1, f2) is the partition function of Γ (see Definition 3.8). In [37], an explicit formula of Berezin transform in terms of strongly connected graphs was obtained, building on the works of Englǐs =-=[13]-=-, Loi [26], and related results of Charles [8]. At a first look, acyclic graphs and strongly connected graphs are quite different. As noted by Schlichenmaier [35], it shall be interesting to clarify t... |

21 | Toeplitz operators on symplectic manifolds,
- Ma, Marinescu
- 2008
(Show Context)
Citation Context ...died in the literature and had found many applications. Karabegov [21] constructed an algebra of Toeplitz elements that is isomorphic to the algebra of Berezin-Toeplitz quantization. Ma and Marinescu =-=[29]-=- developed the theory of Toeplitz operators on symplectic manifolds twisted with a vector bundle. Zelditch [39] studied quantizations of symplectic maps on compact Kähler manifolds and uncovered a co... |

20 | On Fedosov’s approach to deformation quantization with separation of variables (in
- Karabegov
(Show Context)
Citation Context ...Bordemann andWaldmann [6] by modifying Fedosov’s geometric construction of star products on symplectic manifolds [15]. The identification of Bordemann and Waldmann’s star product was due to Karabegov =-=[20]-=-. Neumaier [30] showed that each star product of separation of variables type can be obtained following BordemannWaldmann’s construction. The following theorem is due to Gammelgaard [16]. It can also ... |

20 | Universality of Fedosov’s construction for star products of Wick type on pseudo-Kähler manifolds
- Neumaier
(Show Context)
Citation Context ...ldmann [6] by modifying Fedosov’s geometric construction of star products on symplectic manifolds [15]. The identification of Bordemann and Waldmann’s star product was due to Karabegov [20]. Neumaier =-=[30]-=- showed that each star product of separation of variables type can be obtained following BordemannWaldmann’s construction. The following theorem is due to Gammelgaard [16]. It can also be deduced from... |

19 | Berezin-Toeplitz quantization on Kähler manifolds
- Ma, Marinescu
(Show Context)
Citation Context ...C) denotes the length of C. We regard a single vertex as a 0-cycle and a loop as a 1-cycle. 4. Berezin-Toeplitz quantization The first few terms of Berezin-Toeplitz quantization have been computed in =-=[14, 24, 28, 37]-=-. Since Berezin-Toeplitz quantization corresponds to the inverse Berezin transform, the following theorem completely determines the structure of Berezin-Toeplitz quantization and implies (5) in Theore... |

16 | A closed formula for the asymptotic expansion of the Bergman kernel.
- Xu
- 2012
(Show Context)
Citation Context ... graph, G = (V,E) is defined to be a directed multi-graph, i.e. it has finite number of vertices and edges with multi-edges and loops allowed. Recall the definition of stable and semistable graphs in =-=[36, 37]-=-. These graphs were used to represent Weyl invariants, which encode the coefficients of the asymptotic expansion. Definition 3.1. We call a vertex v of a digraph G semistable if we have deg−(v) ≥ 1, d... |

15 |
Berezin quantization and reproducing kernels on complex domains
- Englǐs
- 1996
(Show Context)
Citation Context ...olds (16) T (m) f1 T (m) f2 ∼ ∞∑ j=0 m−jT (m) CBTj (f1,f2) , m→∞. 4 HAO XU The Berezin transform was introduced by Berezin [4] for symmetric domains in Cn and later extended by many authors (see e.g. =-=[12, 14]-=-). Karabegov and Schlichenmaier [24] proved the asymptotic expansion of the Berezin transform for compact Kähler manifolds. Berezin-Toeplitz quantization for compact Kähler manifolds was extensively... |

15 |
On The Canonical Normalization Of A Trace Density Of Deformation Quantization
- Karabegov
- 1998
(Show Context)
Citation Context ...where Qk are linear differential operators. The Berezin star product was introduced by Berezin [3] through symbol calculus for linear operators on weighted Bergman spaces (cf. [7, 14, 34]). Karabegov =-=[19]-=- noted that any star product with separation of variables can be constructed from a unique formal Berezin transform. In our case, if we write (11) Qjf = ∑ α,β multiindices cjαβ∂ α∂̄βf, then the coeffi... |

13 |
The Tian-Yau-Zelditch asymptotic expansion for real analytic Kähler metrics.
- Loi
- 2004
(Show Context)
Citation Context ...) is the partition function of Γ (see Definition 3.8). In [37], an explicit formula of Berezin transform in terms of strongly connected graphs was obtained, building on the works of Englǐs [13], Loi =-=[26]-=-, and related results of Charles [8]. At a first look, acyclic graphs and strongly connected graphs are quite different. As noted by Schlichenmaier [35], it shall be interesting to clarify their relat... |

10 | An Explicit Formula for the Berezin Star Product
- Xu
(Show Context)
Citation Context ...t(Γ)|Γ(f1, f2), where GssBT (see (27) for the definition) is a certain subset of strongly connected semistable one-pointed graphs and Γ(f1, f2) is the partition function of Γ (see Definition 3.8). In =-=[37]-=-, an explicit formula of Berezin transform in terms of strongly connected graphs was obtained, building on the works of Englǐs [13], Loi [26], and related results of Charles [8]. At a first look, acy... |

9 | A universal formula for deformation quantization on Kähler manifolds
- Gammelgaard
(Show Context)
Citation Context ...zin-Toeplitz star product is of Wick type and the Berezin star product is of anti-Wick type. They are dual opposite to each other. Motivated by the work of Reshetikhin and Takhtajan [31], Gammelgaard =-=[16]-=- obtained a remarkable universal formula for any star product with separation of variables corresponding to a given classifying Karabegov form. Using Karabegov-Schlichenmaier’s identification theorem ... |

9 |
Quantum maps and automorphisms. In The breadth of symplectic and Poisson geometry, volume 232
- Zelditch
(Show Context)
Citation Context ...nts that is isomorphic to the algebra of Berezin-Toeplitz quantization. Ma and Marinescu [29] developed the theory of Toeplitz operators on symplectic manifolds twisted with a vector bundle. Zelditch =-=[39]-=- studied quantizations of symplectic maps on compact Kähler manifolds and uncovered a connection between Berezin-Toeplitz quantization and quantum chaos. Andersen [1] proved asymptotic faithfulness o... |

8 | Asymptotics of spectral function of lower energy forms and Bergman kernel of semi-positive and big line bundles,
- Hsiao, Marinescu
- 2014
(Show Context)
Citation Context ... in the compact Kähler case, see [24, 38]. The asymptotic expansion of the Bergman kernel in the setting when Ω is a compact Kähler manifold was also extensively studied. For recent works, see e.g. =-=[9, 11, 17, 27]-=-. The Berezin transform is the integral operator (9) Imf(x) = ∫ Ω f(y) |Km(x, y)|2 Km(x, x) e−mΦ(y) wng (y) n! . At any point for which Km(x, x) invertible, the integral converges for each bounded mea... |

6 | An invariant formula for a star product with separation of variables arXiv:1107.5832
- Karabegov
(Show Context)
Citation Context ...uct of separation of variables type can be obtained following BordemannWaldmann’s construction. The following theorem is due to Gammelgaard [16]. It can also be deduced from recent works of Karabegov =-=[22, 23]-=-. Theorem 6.1 ([16]). On a Kähler manifold, the KBW star product ⋆S is given by (45) f1 ⋆S f2(x) = ∑ Γ=(V ∪{f},E)∈GS ν|E|−|V | (−1)|V | |Aut(Γ)|Γ(f1, f2) ∣∣∣ x , where GsconS is the set of strongly c... |

6 | Generalized asymptotic expansions of Tian–Yau–Zelditch, arXiv:0909.4591
- Liu, Lu
(Show Context)
Citation Context ... in the compact Kähler case, see [24, 38]. The asymptotic expansion of the Bergman kernel in the setting when Ω is a compact Kähler manifold was also extensively studied. For recent works, see e.g. =-=[9, 11, 17, 27]-=-. The Berezin transform is the integral operator (9) Imf(x) = ∫ Ω f(y) |Km(x, y)|2 Km(x, x) e−mΦ(y) wng (y) n! . At any point for which Km(x, x) invertible, the integral converges for each bounded mea... |

4 |
Bergman Kernel from Path
- Douglas, Klevtsov
(Show Context)
Citation Context |

3 | A formal model of Berezin-Toeplitz quantization
- Karabegov
(Show Context)
Citation Context ...f the Berezin transform for compact Kähler manifolds. Berezin-Toeplitz quantization for compact Kähler manifolds was extensively studied in the literature and had found many applications. Karabegov =-=[21]-=- constructed an algebra of Toeplitz elements that is isomorphic to the algebra of Berezin-Toeplitz quantization. Ma and Marinescu [29] developed the theory of Toeplitz operators on symplectic manifold... |

3 |
Deformation quantization of Kähler manifolds, L.D
- Reshetikhin, Takhtajan
- 2000
(Show Context)
Citation Context ...rticular, the Berezin-Toeplitz star product is of Wick type and the Berezin star product is of anti-Wick type. They are dual opposite to each other. Motivated by the work of Reshetikhin and Takhtajan =-=[31]-=-, Gammelgaard [16] obtained a remarkable universal formula for any star product with separation of variables corresponding to a given classifying Karabegov form. Using Karabegov-Schlichenmaier’s ident... |

3 | Berezin-Toeplitz quantization and star products for compact Kähler manifolds
- Schlichenmaier
(Show Context)
Citation Context ...ned, building on the works of Englǐs [13], Loi [26], and related results of Charles [8]. At a first look, acyclic graphs and strongly connected graphs are quite different. As noted by Schlichenmaier =-=[35]-=-, it shall be interesting to clarify their relations. We will give a purely graph theoretic proof of Theorem 1.1 in §4 by developing a technique of computing the inverse to the Berezin transform. Our ... |

2 |
Berezin-Toeplitz quantization of the moduli space of flat SU(N) connections
- Schlichenmaier
(Show Context)
Citation Context ...tween Berezin-Toeplitz quantization and quantum chaos. Andersen [1] proved asymptotic faithfulness of the mapping class groups action on Verlinde bundles by using Berezin-Toeplitz technique (cf. also =-=[33]-=-). 3. Differential operators encoded by graphs Throughout this paper, a digraph, or simply a graph, G = (V,E) is defined to be a directed multi-graph, i.e. it has finite number of vertices and edges w... |