Publications of the Research Institute for Mathematical Sciences

Schubert Varieties and the Fusion Products

Boris Feigin, Evgeny Feigin

2004 Volume 40 Issue 3 Pages 625-668
Published: 2004
Released on J-STAGE: January 22, 2009

DOIhttps://doi.org/10.2977/prims/1145475487

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For each A∈\mathbb{N}ⁿ we define a Schubert variety sh_A as a closure of the SL₂(\mathbb{C}[t])-orbit in the projectivization of the fusion product M^A. We clarify the connection of the geometry of the Schubert varieties with an algebraic structure of M^A as \mathfrak{sl}₂⊗\mathbb{C}[t] modules. In the case, when all the entries of A are different, sh_A is smooth projective complex algebraic variety. We study its geometric properties: the Lie algebra of the vector fields, the coordinate ring, the cohomologies of the line bundles. We also prove that the fusion products can be realized as the dual spaces of the sections of these bundles.

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A Parallel between Brownian Bridges and Gamma Bridges

Michel Émery, Marc Yor

2004 Volume 40 Issue 3 Pages 669-688
Published: 2004
Released on J-STAGE: January 22, 2009

DOIhttps://doi.org/10.2977/prims/1145475488

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Some properties of the Gamma bridges (obtained by conditioning the Gamma subordinator to take a given value at a given time) are investigated; similarities with the Brownian bridges are emphasized.

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Specialization of Zero Cycles

János Kollár

2004 Volume 40 Issue 3 Pages 689-708
Published: 2004
Released on J-STAGE: January 22, 2009

DOIhttps://doi.org/10.2977/prims/1145475489

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Toward the Exact WKB Analysis for Higher-Order Painlevé Equations — The Case of Noumi-Yamada Systems —

Yoshitsugu Takei

2004 Volume 40 Issue 3 Pages 709-730
Published: 2004
Released on J-STAGE: January 22, 2009

DOIhttps://doi.org/10.2977/prims/1145475490

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As the first step toward the exact WKB analysis for higher-order Painlevé equations, we study the Stokes geometry of the Noumi-Yamada system. It is shown that there are intriguing relations, similar to those for traditional Painlevé equations, between the Stokes geometry of the Noumi-Yamada system and that of its underlying Lax pair.

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Temperature as Order Parameter of Broken Scale Invariance

Izumi Ojima

2004 Volume 40 Issue 3 Pages 731-756
Published: 2004
Released on J-STAGE: April 24, 2009

DOIhttps://doi.org/10.2977/prims/1145475491

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In algebraic quantum field theory the (inverse) temperature is shown to be a macroscopic order parameter to parametrize mutually disjoint thermal sectors arising from the broken scale invariance under renormalization-group transformations.
This is accomplished in a mathematical formalism for the consistent treatment of explicitly broken symmetries such as broken scale invariance, on the basis of a clear-cut criterion for the symmetry breakdown in a unified scheme for sectors proposed recently by the author.

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Crystal Bases for Quantum Classical Algebras and Nakajima's Monomials

Seok-Jin Kang, Jeong-Ah Kim, Dong-Uy Shin

2004 Volume 40 Issue 3 Pages 757-791
Published: 2004
Released on J-STAGE: April 24, 2009

DOIhttps://doi.org/10.2977/prims/1145475492

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Using Nakajima's monomials, we construct a new realization of crystal bases for finite dimensional irreducible modules over quantum classical algebras. We also give an explicit bijection between the monomial realization and the Young tableau realization of crystal bases.

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Heat Kernel Estimates and Parabolic Harnack Inequalities for Graphs and Measure Metric Spaces on Graphs and Resistance Forms

Takashi Kumagai

2004 Volume 40 Issue 3 Pages 793-818
Published: 2004
Released on J-STAGE: January 22, 2009

DOIhttps://doi.org/10.2977/prims/1145475493

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We summarize recent work on heat kernel estimates and parabolic Harnack inequalities for graphs, where the time scale is the β-th power of the space scale for some β≥2. We then discuss self-adjoint operators induced by resistance forms. Using a resistance metric, we give a simple condition for detailed heat kernel estimates and parabolic Harnack inequalities. As an application, we show that on trees a detailed two-sided heat kernel estimate is equivalent to some volume growth condition.

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The Geometry of Anabelioids

Shinichi Mochizuki

2004 Volume 40 Issue 3 Pages 819-881
Published: 2004
Released on J-STAGE: April 24, 2009

DOIhttps://doi.org/10.2977/prims/1145475494

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Correspondence and Analyticity

Henry P. Stapp

2004 Volume 40 Issue 3 Pages 883-903
Published: 2004
Released on J-STAGE: April 24, 2009

DOIhttps://doi.org/10.2977/prims/1145475495

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The analyticity properties of the S-matrix in the physical region are determined by the correspondence principle, which asserts that the predictions of classical physics are generated by taking the classical limit of the predictions of quantum theory. The analyticity properties deducible in this way from classical properties include the locations of the singularity surfaces, the rules for analytic continuation around these singularity surfaces, and the analytic character (e.g., pole, logarithmic, etc.) of these singularities. These important properties of the S-matrix are thus derived without using stringent locality assumptions, or the Schroedinger equation for temporal evolution, except for freely moving particles. Sum-over-all-paths methods that emphasize paths of stationary action tend to produce the quantum analogs of the contributions from classical paths. These quantum analogs are derived directly from the associated classical properties by reverse engineering the correspondence-principle connection.

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Functional Equations and Fusion Matrices for the Eight Vertex Model

Klaus Fabricius, Barry M. McCoy

2004 Volume 40 Issue 3 Pages 905-932
Published: 2004
Released on J-STAGE: January 22, 2009

DOIhttps://doi.org/10.2977/prims/1145475496

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We derive sets of functional equations for the eight vertex model by exploiting an analogy with the functional equations of the chiral Potts model. From these equations we show that the fusion matrices have special reductions at certain roots of unity. We explicitly exhibit these reductions for the 3, 4 and 5 order fusion matrices and compare our formulation with the algebra of Sklyanin.

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Filtrations on Chow Groups and Transcendence Degree

Morihiko Saito

2004 Volume 40 Issue 3 Pages 933-948
Published: 2004
Released on J-STAGE: January 22, 2009

DOIhttps://doi.org/10.2977/prims/1145475497

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For a smooth complex projective variety X defined over a number field, we have filtrations on the Chow groups depending on the choice of realizations. If the realization consists of mixed Hodge structure without any additional structure, we can show that the obtained filtration coincides with the filtration of Green and Griffiths, assuming the Hodge conjecture. In the case the realizations contain Hodge structure and etale cohomology, we prove that if the second graded piece of the filtration does not vanish, it contains a nonzero element which is represented by a cycle defined over a field of transcendence degree one. This may be viewed as a refinement of results of Nori, Schoen, and Green-Griffiths-Paranjape. For higher graded pieces we have a similar assertion assuming a conjecture of Beilinson and Grothendieck's generalized Hodge conjecture.

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A Cabling Formula for the 2-Loop Polynomial of Knots

Tomotada Ohtsuki

2004 Volume 40 Issue 3 Pages 949-971
Published: 2004
Released on J-STAGE: April 24, 2009

DOIhttps://doi.org/10.2977/prims/1145475498

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The 2-loop polynomial is a polynomial presenting the 2-loop part of the logarithm of the Kontsevich invariant of knots. We show a cabling formula for the 2-loop polynomial of knots. In particular, we calculate the 2-loop polynomial for torus knots.

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From Exact-WKB towards Singular Quantum Perturbation Theory

André Voros

2004 Volume 40 Issue 3 Pages 973-990
Published: 2004
Released on J-STAGE: January 22, 2009

DOIhttps://doi.org/10.2977/prims/1145475499

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We use exact WKB analysis to derive some concrete formulae in singular quantum perturbation theory, for Schrödinger eigenvalue problems on the real line with polynomial potentials of the form (q^M+gq^N), where N>M>0 even, and g>0. Mainly, we establish the g→0 limiting forms of global spectral functions such as the zeta-regularized determinants and some spectral zeta functions.

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The Uniformity Principle on Traced Monoidal Categories

Masahito Hasegawa

2004 Volume 40 Issue 3 Pages 991-1014
Published: 2004
Released on J-STAGE: January 22, 2009

DOIhttps://doi.org/10.2977/prims/1145475500

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The uniformity principle for traced monoidal categories has been introduced as a natural generalization of the uniformity principle (Plotkin's principle) for fixpoint operators in domain theory. We show that this notion can be used for constructing new traced monoidal categories from known ones. Some classical examples like the Scott induction principle are shown to be instances of these constructions. We also characterize some specific cases of our constructions as suitable enriched limits.

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Applications of Discrete Convex Analysis to Mathematical Economics

Akihisa Tamura

2004 Volume 40 Issue 3 Pages 1015-1037
Published: 2004
Released on J-STAGE: April 24, 2009

DOIhttps://doi.org/10.2977/prims/1145475501

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Discrete convex analysis, which is a unified framework of discrete optimization, is being recognized as a basic tool for mathematical economics. This paper surveys the recent progress in applications of discrete convex analysis to mathematical economics.

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The Second Painlevé Hierarchy and the Stationary KdV Hierarchy

Nalini Joshi

2004 Volume 40 Issue 3 Pages 1039-1061
Published: 2004
Released on J-STAGE: April 24, 2009

DOIhttps://doi.org/10.2977/prims/1145475502

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It is well known that soliton equations such as the Korteweg-de Vries equation are members of infinite sequences of PDEs known as hierarchies. Here we consider infinite sequences of ODEs associated with the Painlevé equations. We review methods of constructing such hierarchies, specifically the second Painlevé hierarchy, as reductions of PDE hierarchies. We also show that in the large-parameter limit, the solutions of the second Painlevé hierarchy are given by the periodic solutions of the stationary KdV hierarchy.

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String and Vortex

Toshiya Kawai

2004 Volume 40 Issue 3 Pages 1063-1091
Published: 2004
Released on J-STAGE: April 24, 2009

DOIhttps://doi.org/10.2977/prims/1145475503

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We discuss how the geometry of D2-D0 branes may be related to Gromov-Witten theory of Calabi-Yau threefolds.

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On Twisted Microdifferential Modules I. Non-existence of Twisted Wave Equations

Andrea D'Agnolo, Pierre Schapira

2004 Volume 40 Issue 3 Pages 1093-1111
Published: 2004
Released on J-STAGE: April 24, 2009

DOIhttps://doi.org/10.2977/prims/1145475504

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Using the notion of subprincipal symbol, we give a necessary condition for the existence of twisted \mathcal{D}-modules simple along a smooth involutive submanifold of the cotangent bundle to a complex manifold. As an application, we prove that there are no generalized massless field equations with non-trivial twist on grassmannians, and in particular that the Penrose transform does not extend to the twisted case.

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