6,548 research outputs found
Algebraic varieties with automorphism groups of maximal rank
We confirm, to some extent, the belief that a projective variety X has the
largest number (relative to the dimension of X) of independent commuting
automorphisms of positive entropy only when X is birational to a complex torus
or a quotient of a torus. We also include an addendum to an early paper though
it is not used in the present paper.Comment: Mathematische Annalen (to appear
Isometric actions of simple Lie groups on pseudoRiemannian manifolds
Let M be a connected compact pseudoRiemannian manifold acted upon
topologically transitively and isometrically by a connected noncompact simple
Lie group G. If m_0, n_0 are the dimensions of the maximal lightlike subspaces
tangent to M and G, respectively, where G carries any bi-invariant metric, then
we have n_0 \leq m_0. We study G-actions that satisfy the condition n_0 = m_0.
With no rank restrictions on G, we prove that M has a finite covering \hat{M}
to which the G-action lifts so that \hat{M} is G-equivariantly diffeomorphic to
an action on a double coset K\backslash L/\Gamma, as considered in Zimmer's
program, with G normal in L (Theorem A). If G has finite center and
\rank_\R(G)\geq 2, then we prove that we can choose \hat{M} for which L is
semisimple and \Gamma is an irreducible lattice (Theorem B). We also prove that
our condition n_0 = m_0 completely characterizes, up to a finite covering, such
double coset G-actions (Theorem C). This describes a large family of double
coset G-actions and provides a partial positive answer to the conjecture
proposed in Zimmer's program.Comment: 29 pages, published versio
Symmetries and invariances in classical physics
Symmetry, intended as invariance with respect to a transformation (more precisely, with respect to a transformation group), has acquired more and more importance in modern physics. This Chapter explores in 8 Sections the meaning, application and interpretation of symmetry in classical physics. This is done both in general, and with attention to specific topics. The general topics include illustration of the distinctions between symmetries of objects and of laws, and between symmetry principles and symmetry arguments (such as Curie's principle), and reviewing the meaning and various types of symmetry that may be found in classical physics, along with different interpretative strategies that may be adopted. Specific topics discussed include the historical path by which group theory entered classical physics, transformation theory in classical mechanics, the relativity principle in Einstein's Special Theory of Relativity, general covariance in his General Theory of Relativity, and Noether's theorems. In bringing these diverse materials together in a single Chapter, we display the pervasive and powerful influence of symmetry in classical physics, and offer a possible framework for the further philosophical investigation of this topic
Geometric Langlands Twists of N = 4 Gauge Theory from Derived Algebraic Geometry
We develop techniques for describing the derived moduli spaces of solutions
to the equations of motion in twists of supersymmetric gauge theories as
derived algebraic stacks. We introduce a holomorphic twist of N=4
supersymmetric gauge theory and compute the derived moduli space. We then
compute the moduli spaces for the Kapustin-Witten topological twists as its
further twists. The resulting spaces for the A- and B-twist are closely related
to the de Rham stack of the moduli space of algebraic bundles and the de Rham
moduli space of flat bundles, respectively. In particular, we find the
unexpected result that the moduli spaces following a topological twist need not
be entirely topological, but can continue to capture subtle algebraic
structures of interest for the geometric Langlands program.Comment: 55 pages; minor correction
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