187,714 research outputs found
Enumerative Real Algebraic Geometry
Enumerative Geometry is concerned with the number of solutions to a
structured system of polynomial equations, when the structure comes from
geometry. Enumerative real algebraic geometry studies real solutions to such
systems, particularly a priori information on their number. Recent results in
this area have, often as not, uncovered new and unexpected phenomena, and it is
far from clear what to expect in general. Nevertheless, some themes are
emerging.
This comprehensive article describe the current state of knowledge,
indicating these themes, and suggests lines of future research. In particular,
it compares the state of knowledge in Enumerative Real Algebraic Geometry with
what is known about real solutions to systems of sparse polynomials.Comment: Revised, corrected version. 40 pages, 18 color .eps figures. Expanded
web-based version at http://www.math.umass.edu/~sottile/pages/ERAG/index.htm
Non-extremal black hole solutions from the c-map
We construct new static, spherically symmetric non-extremal black hole
solutions of four-dimensional supergravity, using a systematic
technique based on dimensional reduction over time (the c-map) and the real
formulation of special geometry. For a certain class of models we actually
obtain the general solution to the full second order equations of motion,
whilst for other classes of models, such as those obtainable by dimensional
reduction from five dimensions, heterotic tree-level models, and type-II
Calabi-Yau compactifications in the large volume limit a partial set of
solutions are found. When considering specifically non-extremal black hole
solutions we find that regularity conditions reduce the number of integration
constants by one half. Such solutions satisfy a unique set of first order
equations, which we identify.
Several models are investigated in detail, including examples of
non-homogeneous spaces such as the quantum deformed model. Though we
focus on static, spherically symmetric solutions of ungauged supergravity, the
method is adaptable to other types of solutions and to gauged supergravity.Comment: 57 pages. Minor changes to the introduction, typos corrected and
references added. Accepted for publication in JHE
The Hesse potential, the c-map and black hole solutions
We present a new formulation of the local c-map, which makes use of the real
formulation of special Kahler geometry and the associated Hesse potential. As
an application we use the temporal version of the c-map to derive the black
hole attractor equations from geometric properties of the scalar manifold, and
we construct various stationary solutions for four-dimensional vector
multiplets by lifting instanton solutions of the time-reduced theory.Comment: 76 pages. Second revised version: substantial extension. Further
references added and discussion extended. Construction of axion-free non-BPS
extremal solutions for a class of non-homogeneous target spaces added.
Accepted for publication in JHE
- …