Minimal 3-regular Penny Graph
We prove that a 3-regular penny graph has at least 16 vertices and show that such a graph with 16 vertices exists.
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arXiv subjects
Alexander Karabegov
Publications and source records attributed to Alexander Karabegov.
Lagrangian fields, Calabi functions, and local symplectic groupoids
A Lagrangian field on a symplectic manifold $M$ is a family $Λ=\{Λ_x|x \in M\}$ of pointed Lagrangian submanifolds of $M$. This notion is a generalization of a real Lagrangian polarization for which each $Λ_x$ is the leaf containing $x$. Two Lagrangian fields $Λ$ and $\tilde Λ$ are called transversal if $Λ_x$ intersects $\tildeΛ_x$ transversally at $x$ for every $x$. Two transversal Lagrangian fields determine an almost para-Kähler structure on $M$. We construct a local symplectic groupoid on a neighborhood of the zero section of $T^\ast M$ from two transversal Lagrangian fields on $M$. The Lagrangian manifold of $n$-cycles of this groupoid in $(T^\ast M)^n$ has a generating function whose germ around the diagonal of $M^n$ is given by the $n$-point cyclic Calabi function of a closed (1,1)-form on a neighborhood of the diagonal of $M^2$ obtained from the symplectic form on $M$.
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An algebra of distributions related to a star product with separation of variables
Given a star product with separation of variables $\star$ on a pseudo-Kähler manifold $M$ and a point $x_0 \in M$, we construct an associative algebra of formal distributions supported at $x_0$. We use this algebra to express the formal oscillatory exponents of a family of formal oscillatory integrals related to the star product $\star$.
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Formal oscillatory distributions
We introduce the notion of an oscillatory formal distribution supported at a point. We prove that a formal distribution is given by a formal oscillatory integral if and only if it is an oscillatory distribution that has a certain nondegeneracy property. We give an algorithm that recovers the jet of infinite order of the integral kernel of a formal oscillatory integral at the critical point from the corresponding formal distribution. We also prove that a star product $\star$ on a Poisson manifold $M$ is natural in the sense of Gutt and Rawnsley if and only if the formal distribution $f \otimes g \mapsto (f \star g)(x)$ is oscillatory for every $x \in M$.
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Formal oscillatory integrals and deformation quantization
Following [14] and [12], we formalize the notion of an oscillatory integral interpreted as a functional on the amplitudes supported near a fixed critical point $x_0$ of the phase function with zero critical value. We relate to an oscillatory integral two objects, a formal oscillatory integral kernel and the full formal asymptotic expansion at $x_0$. The formal asymptotic expansion is a formal distribution supported at $x_0$ which is applied to the amplitude. In [12] this distribution itself is called a formal oscillatory integral (FOI). We establish a correspondence between the formal oscillatory integral kernels and the FOIs based upon a number of axiomatic properties of a FOI expressed in terms of its formal integral kernel. Then we consider a family of polydifferential operators related to a star product with separation of variables on a pseudo-Kähler manifold. These operators evaluated at a point are FOIs. We completely identify their formal oscillatory kernels.
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A heat kernel proof of the index theorem for deformation quantization
We give a heat kernel proof of the algebraic index theorem for deformation quantization with separation of variables on a pseudo-Kahler manifold. We use normalizations of the canonical trace density of a star product and of the characteristic classes involved in the index formula for which this formula contains no extra constant factors.
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Deformation quantization with separation of variables on a super-Kähler manifold
We construct deformation quantizations with separation of variables on a split super-Kähler manifold and describe their canonical supertrace densities.
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On the phase form of a deformation quantization with separation of variables
Given a star product with separation of variables on a pseudo-Kaehler manifold, we obtain a new formal (1,1)-form from its classifying form and call it the phase form of the star product. The cohomology class of a star product with separation of variables equals the class of its phase form. We show that the phase forms can be arbitrary and they bijectively parametrize the star products with separation of variables. We also describe the action of a change of the formal parameter on a star product with separation of variables, its formal Berezin transform, classifying form, phase form, and canonical trace density.
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Deformation quantization with separation of variables of an endomorphism bundle
Given a holomorphic Hermitian vector bundle and a star-product with separation of variables on a pseudo-Kaehler manifold, we construct a star product on the sections of the endomorphism bundle of the dual bundle which also has the appropriately generalized property of separation of variables. For this star product we prove a generalization of Gammelgaard's graph-theoretic formula.
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On Gammelgaard's formula for a star product with separation of variables
We show that Gammelgaard's formula expressing a star product with separation of variables on a pseudo-Kaehler manifold in terms of directed graphs without cycles is equivalent to an inversion formula for an operator on a formal Fock space. We prove this inversion formula directly and thus offer an alternative approach to Gammelgaard's formula which gives more insight into the question why the directed graphs in his formula have no cycles.
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Felix Alexandrovich Berezin and his work
This is a survey of Berezin's work focused on three topics: representation theory, general concept of quantization, and supermathematics.
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An invariant formula for a star product with separation of variables
We give an invariant formula for a star product with separation of variables on a pseudo-Kahler manifold.
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An explicit formula for a star product with separation of variables
For a star product with separation of variables * on a pseudo-Kaehler manifold we give a simple closed formula of the total symbol of the left star multiplication operator L_f by a given function f. The formula for the star product f * g can be immediately recovered from the total symbol of L_f.
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Infinitesimal deformations of a formal symplectic groupoid
Given a formal symplectic groupoid $G$ over a Poisson manifold $(M, π_0)$, we define a new object, an infinitesimal deformation of $G$, which can be thought of as a formal symplectic groupoid over the manifold $M$ equipped with an infinitesimal deformation $π_0 + επ_1$ of the Poisson bivector field $π_0$. The source and target mappings of a deformation of $G$ are deformations of the source and target mappings of $G$. To any pair of natural star products $(\ast, \tilde\ast)$ having the same formal symplectic groupoid $G$ we relate an infinitesimal deformation of $G$. We call it the deformation groupoid of the pair $(\ast, \tilde\ast)$. We give explicit formulas for the source and target mappings of the deformation groupoid of a pair of star products with separation of variables on a Kaehler- Poisson manifold. Finally, we give an algorithm for calculating the principal symbols of the components of the logarithm of a formal Berezin transform of a star product with separation of variables. This algorithm is based upon some deformation groupoid.
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Star products with separation of variables admitting a smooth extension
Given a complex manifold $M$ with an open dense subset $Ω$ endowed with a pseudo-Kaehler form $ω$ which cannot be smoothly extended to a larger open subset, we consider various examples where the corresponding Kaehler-Poisson structure and a star product with separation of variables on $(Ω, ω)$ admit smooth extensions to $M$. We suggest a simple criterion of the existence of a smooth extension of a star product and apply it to these examples.
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A covariant Poisson deformation quantization with separation of variables up to the third order
We give a simple formula for the operator C_3 of the standard deformation quantization with separation of variables on a Kähler manifold M. Unlike C_1 and C_2, this operator can not be expressed in terms of the Kähler-Poisson tensor on M. We modify C_3 to obtain a covariant deformation quantization with separation of variables up to the third order which is expressed in terms of the Poisson tensor on M and thus can be defined on an arbitrary complex manifold endowed with a Poisson bivector field of type (1,1).
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