415 research outputs found

### Hyperbolic Structures and Root Systems

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

We calculate the first homology group of the mapping class group with
coefficients in the first rational homology group of the universal abelian $\Z
/ L \Z$-cover of the surface. If the surface has one marked point, then the
answer is \Q^{\tau(L)}, where $\tau(L)$ is the number of positive divisors of
$L$. If the surface instead has one boundary component, then the answer is
\Q. We also perform the same calculation for the level $L$ subgroup of the
mapping class group. Set $H_L = H_1(\Sigma_g;\Z/L\Z)$. If the surface has one
marked point, then the answer is \Q[H_L], the rational group ring of $H_L$.
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

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

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

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

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

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 $n$th 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 $n$th iterate of every
Painlev\'e equation in sakai's list is at most $O(n^2)$ and therefore its
algebraic entropy is zero.Comment: 10 pages, pLatex fil

### The modular geometry of Random Regge Triangulations

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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