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26
Noncommutative geometry, dynamics and ∞adic Arakelov geometry, preprint arXiv:math.AG/0205306
"... We dedicate this work to Yuri Manin, with admiration and gratitude In Arakelov theory a completion of an arithmetic surface is achieved by enlarging the group of divisors by formal linear combinations of the “closed fibers at infinity”. Manin described the dual graph of any such closed fiber in term ..."
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Cited by 27 (11 self)
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We dedicate this work to Yuri Manin, with admiration and gratitude In Arakelov theory a completion of an arithmetic surface is achieved by enlarging the group of divisors by formal linear combinations of the “closed fibers at infinity”. Manin described the dual graph of any such closed fiber in terms of an infinite tangle of bounded geodesics in a hyperbolic handlebody endowed with a Schottky uniformization. In this paper we consider arithmetic surfaces over the ring of integers in a number field, with fibers of genus g ≥ 2. We use Connes ’ theory of spectral triples to relate the hyperbolic geometry of the handlebody to Deninger’s Archimedean cohomology and the cohomology of the cone of the local monodromy N at arithmetic infinity as introduced by the first author of this paper. First, we consider derived (cohomological) spectral data (A, H · (X ∗),Φ), where the algebra is obtained from the SL(2, R) action on the cohomology of the cone, induced by the presence of a polarized Lefschetz module structure, and its restriction to the group ring of a Fuchsian Schottky group. In this setting we recover the alternating product of the Archimedean factors from a zeta function of a spectral triple. Then, we introduce a different construction, which is related to Manin’s description of the dual graph of the fiber at infinity. We
Rational conformal field theories and complex multiplication,” arXiv:hepth/0203213
"... We study the geometric interpretation of two dimensional rational conformal field theories, corresponding to sigma models on CalabiYau manifolds. We perform a detailed study of RCFT’s corresponding to T 2 target and identify the Cardy branes with geometric branes. The T 2 ’s leading to RCFT’s admit ..."
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Cited by 15 (2 self)
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We study the geometric interpretation of two dimensional rational conformal field theories, corresponding to sigma models on CalabiYau manifolds. We perform a detailed study of RCFT’s corresponding to T 2 target and identify the Cardy branes with geometric branes. The T 2 ’s leading to RCFT’s admit “complex multiplication ” which characterizes Cardy branes as specific D0branes. We propose a condition for the conformal sigma model to be RCFT for arbitrary CalabiYau nfolds, which agrees with the known cases. Together with recent conjectures by mathematicians it appears that rational conformal theories are not dense in the space of all conformal theories, and sometimes appear to be finite in number for CalabiYau nfolds for n> 2. RCFT’s on K3 may be dense. We speculate about the meaning of these special points in the moduli spaces of CalabiYau nfolds in connection with freezing geometric moduli. March
Liouville action and WeilPetersson metric on deformation spaces, global Kleinian reciprocity and holography
 COMMUNICATIONS IN MATHEMATICAL PHYSICS
, 2003
"... We rigorously define the Liouville action functional for the finitely generated, purely loxodromic quasiFuchsian group using homology and cohomology double complexes naturally associated with the group action. We prove that classical action – the critical value of the Liouville action functional, ..."
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Cited by 10 (1 self)
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We rigorously define the Liouville action functional for the finitely generated, purely loxodromic quasiFuchsian group using homology and cohomology double complexes naturally associated with the group action. We prove that classical action – the critical value of the Liouville action functional, considered as a function on the quasiFuchsian deformation space, is an antiderivative of a 1form given by the difference of Fuchsian and quasiFuchsian projective connections. This result can be considered as global quasiFuchsian reciprocity which implies McMullen’s quasiFuchsian reciprocity. We prove that the classical action is a Kähler potential of the WeilPetersson metric. We also prove that the Liouville action functional satisfies holography principle, i.e., it is a regularized limit of the hyperbolic volume of a 3manifold associated with a quasiFuchsian group. We generalize these results to a large class of Kleinian groups including finitely generated, purely loxodromic Schottky and quasiFuchsian groups, and their free combinations.
Les Houches lectures on strings and arithmetic
"... These are lecturenotes for two lectures delivered at the Les Houches workshop on Number Theory, Physics, and Geometry, March 2003. They review two examples of interesting interactions between number theory and string compactification, and raise some new questions and issues in the context of those e ..."
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Cited by 8 (1 self)
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These are lecturenotes for two lectures delivered at the Les Houches workshop on Number Theory, Physics, and Geometry, March 2003. They review two examples of interesting interactions between number theory and string compactification, and raise some new questions and issues in the context of those examples. The first example concerns the role of the Rademacher expansion of coefficients of modular forms in the AdS/CFT correspondence. The second example concerns the role of the “attractor mechanism ” of supergravity in selecting certain arithmetic CalabiYau’s as distinguished compactifications.
”New” Veneziano amplitudes from ”old” Fermat (hyper)surfaces
, 2003
"... The history of the discovery of bosonic string theory is well documented. This theory evolved as an attempt to find a multidimensional analogue of Euler’s beta function to describe the multiparticle Veneziano amplitudes. Such an analogue had in fact been known in mathematics at least in 1922. Its ma ..."
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Cited by 3 (2 self)
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The history of the discovery of bosonic string theory is well documented. This theory evolved as an attempt to find a multidimensional analogue of Euler’s beta function to describe the multiparticle Veneziano amplitudes. Such an analogue had in fact been known in mathematics at least in 1922. Its mathematical meaning was studied subsequently from different angles by mathematicians such as Selberg, Weil and Deligne among others. The mathematical interpretation of this multidimensional beta function that was developed subsequently is markedly different from that described in physics literature. This work aims to bridge the gap between the mathematical and physical treatments. Using some results of recent publications (e.g. J.Geom.Phys.38 (2001) 81; ibid 43 (2002) 45) new topological, algebrogeometric, numbertheoretic and combinatorial treatment of the multiparticle Veneziano amplitudes is developed. As a result, an entirely new physical meaning of these amplitudes is emerging: they are periods of differential forms associated with homology cycles on Fermat (hyper)surfaces. Such (hyper)surfaces are considered as complex projective varieties of Hodge type. Although the computational formalism developed in this work resembles that used in mirror symmetry calculations, many additional results from mathematics are used along with their suitable physical interpretation. For instance, the Hodge spectrum of the Fermat (hyper)surfaces is in onetoone correspondence with the possible spectrum of particle masses. The formalism also allows us to obtain correlation functions of both conformal field theory and particle physics using the same type of the PicardFuchs equations whose solutions are being interpreted in terms of periods.
Limiting modular symbols and their fractal geometry
"... ABSTRACT. In this paper we use fractal geometry to investigate boundary aspects of the first homology group for finite coverings of the modular surface. We obtain a complete description of algebraically invisible parts of this homology group. More precisely, we first show that for any modular subgro ..."
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Cited by 2 (1 self)
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ABSTRACT. In this paper we use fractal geometry to investigate boundary aspects of the first homology group for finite coverings of the modular surface. We obtain a complete description of algebraically invisible parts of this homology group. More precisely, we first show that for any modular subgroup the geodesic forward dynamic on the associated surface admits a canonical symbolic representation by a finitely irreducible shift space. We then use this representation to derive an ‘almost complete ’ multifractal description of the higher– dimensional level sets arising from Manin–Marcolli’s limiting modular symbols. 1.
Hyperbolic Space Forms and Orbifold Compactification
 In MTheory, in the Proceedings of the ”Fourth International Winter Conference on Mathematical Methods in Physics”, PoS(WC2004)017
"... Abstract: We analyze solutions of string theory and supergravity which involve real hyperbolic spaces. Examples of string compactifications are given in terms of hyperbolic coset spaces of finite volume Γ\H N, where Γ is a discrete group of isometries of H N. We describe finite flux and the tensor k ..."
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Abstract: We analyze solutions of string theory and supergravity which involve real hyperbolic spaces. Examples of string compactifications are given in terms of hyperbolic coset spaces of finite volume Γ\H N, where Γ is a discrete group of isometries of H N. We describe finite flux and the tensor kernel associated with hyperbolic spaces. The case of arithmetic geometry of Γ = SL(2, Z+iZ)/{±Id}, where Id is the identity matrix, is analyzed. We discuss supersymmetry surviving for supergravity solutions involving real hyperbolic space factors, stringsupergravity correspondence and holography principle for a class of conformal field theories. – 1 –
MODULAR SHADOWS AND THE LÉVY–MELLIN ∞–ADIC TRANSFORM
, 2007
"... Abstract. This paper continues the study of the structures induced on the “invisible boundary ” of the modular tower and extends some results of [MaMar1]. We start with a systematic formalism of pseudo–measures generalizing the well– known theory of modular symbols for SL(2). These pseudo–measures, ..."
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Abstract. This paper continues the study of the structures induced on the “invisible boundary ” of the modular tower and extends some results of [MaMar1]. We start with a systematic formalism of pseudo–measures generalizing the well– known theory of modular symbols for SL(2). These pseudo–measures, and the related integral formula which we call the Lévy–Mellin transform, can be considered as an “∞–adic ” version of Mazur’s p–adic measures that have been introduced in the seventies in the theory of p–adic interpolation of the Mellin transforms of cusp forms, cf. [Ma2]. A formalism of iterated Lévy–Mellin transform in the style of [Ma3] is sketched. Finally, we discuss the invisible boundary from the perspective of non–commutative geometry. When the theory of modular symbols for the SL(2)–case had been conceived in the 70’s (cf. [Ma1], [Ma2], [Sh1], [Sh2]), it was clear from the outset that it dealt with the Betti homology of some basic moduli spaces (modular curves, Kuga varieties, M1,n, and alike), whereas the theory of modular forms involved the de