415 research outputs found

    Hyperbolic Structures and Root Systems

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    We discuss the construction of a one parameter family of complex hyperbolic structures on the complement of a toric mirror arrangement associated with a simply laced root system. Subsequently we find conditions for which parameter values this leads to ball quotients

    Abelian covers of surfaces and the homology of the level L mapping class group

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    We calculate the first homology group of the mapping class group with coefficients in the first rational homology group of the universal abelian Z/LZ\Z / L \Z-cover of the surface. If the surface has one marked point, then the answer is \Q^{\tau(L)}, where τ(L)\tau(L) is the number of positive divisors of LL. If the surface instead has one boundary component, then the answer is \Q. We also perform the same calculation for the level LL subgroup of the mapping class group. Set HL=H1(Σg;Z/LZ)H_L = H_1(\Sigma_g;\Z/L\Z). If the surface has one marked point, then the answer is \Q[H_L], the rational group ring of HLH_L. If the surface instead has one boundary component, then the answer is \Q.Comment: 32 pages, 10 figures; numerous corrections and simplifications; to appear in J. Topol. Ana

    Virasoro constraints and the Chern classes of the Hodge bundle

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    We analyse the consequences of the Virasoro conjecture of Eguchi, Hori and Xiong for Gromov-Witten invariants, in the case of zero degree maps to the manifolds CP^1 and CP^2 (or more generally, smooth projective curves and smooth simply-connected projective surfaces). We obtain predictions involving intersections of psi and lambda classes on the compactification of M_{g,n}. In particular, we show that the Virasoro conjecture for CP^2 implies the numerical part of Faber's conjecture on the tautological Chow ring of M_g.Comment: 12 pages, latex2

    Monodromy of Cyclic Coverings of the Projective Line

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    We show that the image of the pure braid group under the monodromy action on the homology of a cyclic covering of degree d of the projective line is an arithmetic group provided the number of branch points is sufficiently large compared to the degree.Comment: 47 pages (to appear in Inventiones Mathematicae

    Forgetful maps between Deligne-Mostow ball quotients

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    We study forgetful maps between Deligne-Mostow moduli spaces of weighted points on P^1, and classify the forgetful maps that extend to a map of orbifolds between the stable completions. The cases where this happens include the Livn\'e fibrations and the Mostow/Toledo maps between complex hyperbolic surfaces. They also include a retraction of a 3-dimensional ball quotient onto one of its 1-dimensional totally geodesic complex submanifolds

    Ground states of supersymmetric Yang-Mills-Chern-Simons theory

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    We consider minimally supersymmetric Yang-Mills theory with a Chern-Simons term on a flat spatial two-torus. The Witten index may be computed in the weak coupling limit, where the ground state wave-functions localize on the moduli space of flat gauge connections. We perform such computations by considering this moduli space as an orbifold of a certain flat complex torus. Our results agree with those obtained previously by instead considering the moduli space as a complex projective space. An advantage of the present method is that it allows for a more straightforward determination of the discrete electric 't Hooft fluxes of the ground states in theories with non-simply connected gauge groups. A consistency check is provided by the invariance of the results under the mapping class group of a (Euclidean) three-torus.Comment: 18 page

    Algebraic entropy and the space of initial values for discrete dynamical systems

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    A method to calculate the algebraic entropy of a mapping which can be lifted to an isomorphism of a suitable rational surfaces (the space of initial values) are presented. It is shown that the degree of the nnth iterate of such a mapping is given by its action on the Picard group of the space of initial values. It is also shown that the degree of the nnth iterate of every Painlev\'e equation in sakai's list is at most O(n2)O(n^2) and therefore its algebraic entropy is zero.Comment: 10 pages, pLatex fil

    The modular geometry of Random Regge Triangulations

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    We show that the introduction of triangulations with variable connectivity and fluctuating egde-lengths (Random Regge Triangulations) allows for a relatively simple and direct analyisis of the modular properties of 2 dimensional simplicial quantum gravity. In particular, we discuss in detail an explicit bijection between the space of possible random Regge triangulations (of given genus g and with N vertices) and a suitable decorated version of the (compactified) moduli space of genus g Riemann surfaces with N punctures. Such an analysis allows us to associate a Weil-Petersson metric with the set of random Regge triangulations and prove that the corresponding volume provides the dynamical triangulation partition function for pure gravity.Comment: 36 pages corrected typos, enhanced introductio
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