6,017 research outputs found
The zero section of the universal semiabelian variety, and the double ramification cycle
We study the Chow ring of the boundary of the partial compactification of the
universal family of principally polarized abelian varieties (ppav). We describe
the subring generated by divisor classes, and compute the class of the partial
compactification of the universal zero section, which turns out to lie in this
subring. Our formula extends the results for the zero section of the universal
uncompactified family.
The partial compactification of the universal family of ppav can be thought
of as the first two boundary strata in any toroidal compactification of the
moduli space of ppav. Our formula provides a first step in a program to
understand the Chow groups of toroidal compactifications of the moduli of ppav,
especially of the perfect cone compactification, by induction on genus. By
restricting to the locus of Jacobians of curves, our results extend the results
of Hain on the double ramification (two-branch-point) cycle.Comment: Section 6, dealing with the Eliashberg problem for moduli of curves,
rewritten. A discussion of the extension of the Abel-Jacobi map added, the
resulting formula corrected. Final version, to appear in Duke Math.
An Exceptional Sector for F-theory GUTs
D3-branes are often a necessary ingredient in global compactifications of
F-theory. In minimal realizations of flavor hierarchies in F-theory GUT models,
suitable fluxes are turned on, which in turn attract D3-branes to the Yukawa
points. Of particular importance are ``E-type'' Yukawa points, as they are
required to realize a large top quark mass. In this paper we study the
worldvolume theory of a D3-brane probing such an E-point. D3-brane probes of
isolated exceptional singularities lead to strongly coupled N = 2 CFTs of the
type found by Minahan and Nemeschansky. We show that the local data of an
E-point probe theory determines an N = 1 deformation of the original N = 2
theory which couples this strongly interacting CFT to a free hypermultiplet.
Monodromy in the seven-brane configuration translates to a novel class of
deformations of the CFT. We study how the probe theory couples to the Standard
Model, determining the most relevant F-term couplings, the effect of the probe
on the running of the Standard Model gauge couplings, as well as possible
sources of kinetic mixing with the Standard Model.Comment: v2: 32 pages, 1 figure, references added, appendix remove
On heterotic model constraints
The constraints imposed on heterotic compactifications by global consistency
and phenomenology seem to be very finely balanced. We show that weakening these
constraints, as was proposed in some recent works, is likely to lead to
frivolous results. In particular, we construct an infinite set of such
frivolous models having precisely the massless spectrum of the MSSM and other
quasi-realistic features. Only one model in this infinite collection (the one
constructed in arXiv:hep-th/0512149) is globally consistent and supersymmetric.
The others might be interpreted as being anomalous, or as non-supersymmetric
models, or as local models that cannot be embedded in a global one. We also
show that the strongly coupled model of arXiv:hep-th/0512149 can be modified to
a perturbative solution with stable SU(4) or SU(5) bundles in the hidden
sector. We finally propose a detailed exploration of heterotic vacua involving
bundles on Calabi-Yau threefolds with Z_6 Wilson lines; we obtain many more
frivolous solutions, but none that are globally consistent and supersymmetric
at the string scale.Comment: 38 page
Prepotentials for local mirror symmetry via Calabi-Yau fourfolds
In this paper, we first derive an intrinsic definition of classical triple
intersection numbers of K_S, where S is a complex toric surface, and use this
to compute the extended Picard-Fuchs system of K_S of our previous paper,
without making use of the instanton expansion. We then extend this formalism to
local fourfolds K_X, where X is a complex 3-fold. As a result, we are able to
fix the prepotential of local Calabi-Yau threefolds K_S up to polynomial terms
of degree 2. We then outline methods of extending the procedure to non
canonical bundle cases.Comment: 42 pages, 7 figures. Expanded, reorganized, and added a theoretical
background for the calculation
Tropical secant graphs of monomial curves
The first secant variety of a projective monomial curve is a threefold with
an action by a one-dimensional torus. Its tropicalization is a
three-dimensional fan with a one-dimensional lineality space, so the tropical
threefold is represented by a balanced graph. Our main result is an explicit
construction of that graph. As a consequence, we obtain algorithms to
effectively compute the multidegree and Chow polytope of an arbitrary
projective monomial curve. This generalizes an earlier degree formula due to
Ranestad. The combinatorics underlying our construction is rather delicate, and
it is based on a refinement of the theory of geometric tropicalization due to
Hacking, Keel and Tevelev.Comment: 30 pages, 8 figures. Major revision of the exposition. In particular,
old Sections 4 and 5 are merged into a single section. Also, added Figure 3
and discussed Chow polytopes of rational normal curves in Section
Quantum computing and the brain: quantum nets, dessins d'enfants and neural networks
In this paper, we will discuss a formal link between neural networks and
quantum computing. For that purpose we will present a simple model for the
description of the neural network by forming sub-graphs of the whole network
with the same or a similar state. We will describe the interaction between
these areas by closed loops, the feedback loops. The change of the graph is
given by the deformations of the loops. This fact can be mathematically
formalized by the fundamental group of the graph. Furthermore the neuron has
two basic states (ground state) and (excited state).
The whole state of an area of neurons is the linear combination of the two
basic state with complex coefficients representing the signals (with 3
Parameters: amplitude, frequency and phase) along the neurons. Then it can be
shown that the set of all signals forms a manifold (character variety) and all
properties of the network must be encoded in this manifold. In the paper, we
will discuss how to interpret learning and intuition in this model. Using the
Morgan-Shalen compactification, the limit for signals with large amplitude can
be analyzed by using quasi-Fuchsian groups as represented by dessins d'enfants
(graphs to analyze Riemannian surfaces). As shown by Planat and collaborators,
these dessins d'enfants are a direct bridge to (topological) quantum computing
with permutation groups. The normalization of the signal reduces to the group
and the whole model to a quantum network. Then we have a direct
connection to quantum circuits. This network can be transformed into operations
on tensor networks. Formally we will obtain a link between machine learning and
Quantum computing.Comment: 17 pages, 3 Figures, accepted for the proceedings of the QTech 2018
conference (September 2018, Paris
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