**Title: \;Black Holes\, arithmetic and modularit
y for families ofCalabi-Yau manifolds**

The seminars are jointly organized by the Department of Mathematics and the Departmen t of Physics and Astronomy\, coffee and tea will be served after the fir st talk at 16:15.

**Abstract:** \;The main goal o
f these two talks is to explore some questions of common interest for ph
ysicists\, number theorists and geometers\, in the context of the arithm
etic of Calabi-Yau 3-folds. There are many such relations\, however we w
ill focus on the rich structure of black hole solutions of type II super
strings on a Calabi-Yau manifolds. We will give a self contained introdu
ction aimed at a mixed audience of physicists and mathematicians. The ma
in quantities of interest in the arithmetic context are the numbers of p
oints of the manifold\, considered as a variety over a finite field. A m
athematician is interested in the computation of these numbers and their
dependence on the moduli of the variety. The surprise for a physicist i
s that the numbers of points over a finite field are also given by expre
ssions that involve the periods of a manifold. These periods determine m
any aspects of the physical theory\, as for example the kinetic terms of
the effective Lagrangian as well as the Yukawa couplings\, but also pro
perties of black hole solutions. For a mathematician\, the number of poi
nts determine the zeta function\, about which much is known in virtue of
the Weil conjectures. We discuss a number of interesting topics related
to the zeta function\, the corresponding L-function\, and the appearanc
e of modularity for one parameter families of Calabi-Yau manifolds. We w
ill focus on an example for which the quartic numerator of the zeta func
tion of a one parameter family factorises into two quadrics at special v
alues of the parameter\, which satisfies an algebraic equation with coef
ficients in Q (so independent of any particular prime)\, and for which t
he underlying manifold is smooth. The significance of these factorisatio
ns in physics is that they are due to the existence of black hole attrac
tor points in the sense of type II supergravity and are related to a spl
itting of the Hodge structure and that at these special values of the pa
rameter. For a mathematician these factorisations of the Hodge structure
are related to the famous Hodge Conjecture. Modular groups and modular
forms arise in relation to these attractor points. To our knowledge\, th
e rank two attractor points that were found by the application of these
number theoretic techniques\, provide the first explicit examples of suc
h attractor points for Calabi-Yau manifolds. Time permitting\, we will d
escribe this scenario also for the mirror manifold in type IIA supergrav
ity.

\;

DESCRIPTION: SUMMARY:21st Geometry and Physics Seminar LOCATION:Ångströmlaboratoriet\, Lägerhyddsvägen 1\, Polhemsalen TZID:Europe/Stockholm DTSTART:20230201T151500 DTEND:20230201T173000 UID:20230201T151500-76842@uu.se END:VEVENT END:VCALENDAR