145,953 research outputs found
Geometry of the arithmetic site
We introduce the Arithmetic Site: an algebraic geometric space deeply related
to the non-commutative geometric approach to the Riemann Hypothesis. We prove
that the non-commutative space quotient of the adele class space of the field
of rational numbers by the maximal compact subgroup of the idele class group,
which we had previously shown to yield the correct counting function to obtain
the complete Riemann zeta function as Hasse-Weil zeta function, is the set of
geometric points of the arithmetic site over the semifield of tropical real
numbers. The action of the multiplicative group of positive real numbers on the
adele class space corresponds to the action of the Frobenius automorphisms on
the above geometric points. The underlying topological space of the arithmetic
site is the topos of functors from the multiplicative semigroup of non-zero
natural numbers to the category of sets. The structure sheaf is made by
semirings of characteristic one and is given globally by the semifield of
tropical integers. In spite of the countable combinatorial nature of the
arithmetic site, this space admits a one parameter semigroup of Frobenius
correspondences obtained as sub-varieties of the square of the site. This
square is a semi-ringed topos whose structure sheaf involves Newton polygons.
Finally, we show that the arithmetic site is intimately related to the
structure of the (absolute) point in non-commutative geometry.Comment: 43 page
Higgs bundles, pseudo-hyperbolic geometry and maximal representations
These notes are an extended version of a talk given by the author in the
seminar "Theorie Spectrale et Geometrie" at the Institut Fourier in No- vember
2016. We present here some aspects of a work in collaboration with B. Collier
and N. Tholozan (arXiv:1702.08799). We describe how Higgs bundle theory and
pseudo-hyperbolic geometry interfere in the study of maximal representations
into Hermitian Lie groups of rank 2
Holographic Entanglement in a Noncommutative Gauge Theory
In this article we investigate aspects of entanglement entropy and mutual
information in a large-N strongly coupled noncommutative gauge theory, both at
zero and at finite temperature. Using the gauge-gravity duality and the
Ryu-Takayanagi (RT) prescription, we adopt a scheme for defining spatial
regions on such noncommutative geometries and subsequently compute the
corresponding entanglement entropy. We observe that for regions which do not
lie entirely in the noncommutative plane, the RT-prescription yields sensible
results. In order to make sense of the divergence structure of the
corresponding entanglement entropy, it is essential to introduce an additional
cut-off in the theory. For regions which lie entirely in the noncommutative
plane, the corresponding minimal area surfaces can only be defined at this
cut-off and they have distinctly peculiar properties.Comment: 28 pages, multiple figures; minor changes, conclusions unchange
Gravity duals to deformed SYM theories and Generalized Complex Geometry
We analyze the supersymmetry conditions for a class of SU(2) structure
backgrounds of Type IIB supergravity, corresponding to a specific ansatz for
the supersymmetry parameters. These backgrounds are relevant for the AdS/CFT
correspondence since they are suitable to describe mass deformations or
beta-deformations of four-dimensional superconformal gauge theories. Using
Generalized Complex Geometry we show that these geometries are characterized by
a closed nowhere-vanishing vector field and a modified fundamental form which
is also closed. The vector field encodes the information about the
superpotential and the type of deformation - mass or beta respectively. We also
show that the Pilch-Warner solution dual to a mass-deformation of N =4 Super
Yang-Mills and the Lunin-Maldacena beta-deformation of the same background fall
in our class of solutions.Comment: LaTex, 29 page
Supersymmetric AdS Backgrounds in String and M-theory
We first present a short review of general supersymmetric compactifications
in string and M-theory using the language of G-structures and intrinsic
torsion. We then summarize recent work on the generic conditions for
supersymmetric AdS_5 backgrounds in M-theory and the construction of classes of
new solutions. Turning to AdS_5 compactifications in type IIB, we summarize the
construction of an infinite class of new Sasaki-Einstein manifolds in dimension
2k+3 given a positive curvature Kahler-Einstein base manifold in dimension 2k.
For k=1 these describe new supergravity duals for N=1 superconformal field
theories with both rational and irrational R-charges and central charge. We
also present a generalization of this construction, that has not appeared
elsewhere in the literature, to the case where the base is a product of
Kahler-Einstein manifolds.Comment: LaTeX, 35 pages, to appear in the proceedings of the 73rd Meeting
between Physicists and Mathematicians "(A)dS/CFT correspondence", Strasbourg,
September 11-13, 200
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