Index estimate by first Betti number of minimal hypersurfaces in compact symmetric spaces

Maximal antipodal sets of compact classical symmetric spaces and their cardinalities. II

The homology of the free and the based loop space of a compact globally symmetric space can be studied through explicit cycles. We use cycles constructed by Bott and Samelson and by Ziller to study the string topology coproduct and the Chas-Sullivan product on compact symmetric spaces. We show that the Chas-Sullivan product for compact symmetric spaces is highly non-trivial for any rank and we prove that there are many non-nilpotent classes whose powers correspond to the iteration of closed geodesics. Moreover, we show that the based string topology coproduct is trivial for compact symmetric spaces of higher rank and we study the implications of this result for the string topology coproduct on the free loop space.

We propose a generalization of the Bogomol’ny–Prasad–Sommerfield Skyrme model [L. A. Ferreira, Exact self-duality in a modified Skyrme model, ] for simple compact Lie groups <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mi>G</a:mi> </a:math> that leads to Hermitian symmetric spaces. In such a theory, the Skyrme field takes its values in <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"> <c:mi>G</c:mi> </c:math> , while the remaining fields correspond to the entries of a symmetric, positive, and invertible <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"> <e:mi>dim</e:mi> <e:mtext> </e:mtext> <e:mi>G</e:mi> <e:mo>×</e:mo> <e:mi>dim</e:mi> <e:mtext> </e:mtext> <e:mi>G</e:mi> </e:math> -dimensional matrix <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"> <g:mi>h</g:mi> </g:math> . We also use the holomorphic map between <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"> <i:msup> <i:mi>S</i:mi> <i:mn>2</i:mn> </i:msup> <i:mo stretchy="false">→</i:mo> <i:mi>G</i:mi> <i:mo>/</i:mo> <i:mi>H</i:mi> <i:mo stretchy="false">⊗</i:mo> <i:mi>U</i:mi> <i:mo stretchy="false">(</i:mo> <i:mn>1</i:mn> <i:mo stretchy="false">)</i:mo> </i:math> proposed in Ferreira and Livramento [Harmonic, holomorphic and rational maps from self-duality, ] to study the self-dual sector of the theory, which generalizes the holomorphic between <o:math xmlns:o="http://www.w3.org/1998/Math/MathML" display="inline"> <o:msup> <o:mi>S</o:mi> <o:mn>2</o:mn> </o:msup> <o:mo stretchy="false">→</o:mo> <o:mi>C</o:mi> <o:msup> <o:mi>P</o:mi> <o:mi>N</o:mi> </o:msup> </o:math> proposed in Ioannidou [Low-energy states in the SU(N) Skyrme models, in International Meeting on Mathematical Methods in Modern Theoretical Physics (ISPM 98) (1998), pp. 91–123, ]. This is constructed using the fact that stable harmonic maps of the two <r:math xmlns:r="http://www.w3.org/1998/Math/MathML" display="inline"> <r:msup> <r:mi>S</r:mi> <r:mn>2</r:mn> </r:msup> </r:math> spheres for compact Hermitian symmetric spaces are holomorphic or antiholomorphic [J. Eells and L. Lemaire, (World Scientific Publishing Company, Singapore, 1995)]. Apart from some special cases, the self-duality equations do not fix the matrix <t:math xmlns:t="http://www.w3.org/1998/Math/MathML" display="inline"> <t:mi>h</t:mi> </t:math> entirely in terms of the Skyrme field, which is completely free, as it happens in the original self-dual Skyrme model for <v:math xmlns:v="http://www.w3.org/1998/Math/MathML" display="inline"> <v:mi>G</v:mi> <v:mo>=</v:mo> <v:mi>S</v:mi> <v:mi>U</v:mi> <v:mo stretchy="false">(</v:mo> <v:mn>2</v:mn> <v:mo stretchy="false">)</v:mo> </v:math> . In general, the freedom of the <z:math xmlns:z="http://www.w3.org/1998/Math/MathML" display="inline"> <z:mi>h</z:mi> </z:math> fields tend to grow with the dimension of <bb:math xmlns:bb="http://www.w3.org/1998/Math/MathML" display="inline"> <bb:mi>G</bb:mi> </bb:math> . The holomorphic enable us to construct an infinite number of exact self-dual Skyrmions for each integer value of the topological charge and for each value of <db:math xmlns:db="http://www.w3.org/1998/Math/MathML" display="inline"> <db:mi>N</db:mi> <db:mo>≥</db:mo> <db:mn>1</db:mn> </db:math> , in case of the <fb:math xmlns:fb="http://www.w3.org/1998/Math/MathML" display="inline"> <fb:mi>C</fb:mi> <fb:msup> <fb:mi>P</fb:mi> <fb:mi>N</fb:mi> </fb:msup> </fb:math> , and for each values of <hb:math xmlns:hb="http://www.w3.org/1998/Math/MathML" display="inline"> <hb:mrow> <hb:mi>p</hb:mi> </hb:mrow> </hb:math> , <jb:math xmlns:jb="http://www.w3.org/1998/Math/MathML" display="inline"> <jb:mrow> <jb:mi>q</jb:mi> <jb:mo>≥</jb:mo> <jb:mn>1</jb:mn> </jb:mrow> </jb:math> in case of <lb:math xmlns:lb="http://www.w3.org/1998/Math/MathML" display="inline"> <lb:mi>S</lb:mi> <lb:mi>U</lb:mi> <lb:mrow> <lb:mo stretchy="false">(</lb:mo> <lb:mi>p</lb:mi> <lb:mo>+</lb:mo> <lb:mi>q</lb:mi> <lb:mo stretchy="false">)</lb:mo> </lb:mrow> <lb:mo>/</lb:mo> <lb:mi>S</lb:mi> <lb:mi>U</lb:mi> <lb:mrow> <lb:mo stretchy="false">(</lb:mo> <lb:mi>p</lb:mi> <lb:mo stretchy="false">)</lb:mo> </lb:mrow> <lb:mo stretchy="false">⊗</lb:mo> <lb:mi>S</lb:mi> <lb:mi>U</lb:mi> <lb:mrow> <lb:mo stretchy="false">(</lb:mo> <lb:mi>q</lb:mi> <lb:mo stretchy="false">)</lb:mo> </lb:mrow> <lb:mo stretchy="false">⊗</lb:mo> <lb:mi>U</lb:mi> <lb:mrow> <lb:mo stretchy="false">(</lb:mo> <lb:mn>1</lb:mn> <lb:mo stretchy="false">)</lb:mo> </lb:mrow> </lb:math> .

Abstract In the 1990s, C.-L. Terng and G. Thorbergsson investigated a natural Riemannian submersion from an infinite dimensional Hilbert space onto a compact Riemannian symmetric space G/K. This map is called the parallel transport map over G/K. Later, N. Koike extended their theory to the case that G/K is a Riemannian symmetric space of non-compact type. In this paper, more generally, we define the parallel transport map over an affine symmetric space and show that it is an affine submersion with horizontal distribution in the sense of Abe and Hasegawa. Based on this result, we prove the Fredholm property of affine immersions into a Hilbertable space lifted by the parallel transport map. Furthermore, we greatly extend the author’s previous result on weakly reflective submanifolds from the case of compact Riemannian symmetric spaces to the case of affine symmetric spaces.

Abstract Bounded symmetric domains carry several natural invariant metrics, for example the Carathéodory, Kobayashi or Bergman metric. We define another natural metric, from the generalized Hilbert metric defined in [4], by considering the Borel embedding of the domain as an open subset of its dual compact Hermitian symmetric space and then its Harish–Chandra realization in projective spaces. We describe this construction for the four classical families of bounded symmetric domains and compute both this metric and its associated Finsler metric. We compare it to the Carathéodory and Bergman metrics and show that, except for the complex hyperbolic space, those metrics differ.

In this work we find a unifying scheme for the known explicit complex-valued eigenfunctions on the classical compact Riemannian symmetric spaces. For this we employ the well-known Cartan embedding for those spaces. This also leads to the construction of new eigenfunctions on the quaternionic Grassmannians.

Sasaki–Einstein orbits in compact Hermitian symmetric spaces

The action of the subgroup G2 of SO(7) (resp. Spin(7) of SO(8)) on the Grassmannian space M = SO(7)/(SO(5)×SO(2)) (resp. M = SO(8)/(SO(5)×SO(3)) ) is still transitive. We prove that the spectrum (i.e. the collection of eigenvalues of its Laplace-Beltrami operator) of a symmetric metric g0 on M coincides with the spectrum of a G2-invariant (resp. Spin(7)-invariant) metric g on M only if g0 and g are isometric. As a consequence, each non-flat compact irreducible symmetric space of non-group type is spectrally unique among the family of all currently known homogeneous metrics on its underlying differentiable manifold.

The exceptional compact hermitian symmetric space EIII is the quotient E6/Spin(10)×Z4U(1). We introduce the Plücker coordinates which give an embedding of EIII into CP26 as a projective subvariety. The subvariety is cut out by 27 Plücker relations. We show that, using Clifford algebra, one can solve this over-determined system of relations, giving local coordinate charts to the space.Our motivation is to understand EIII as the complex projective octonion plane (C⊗O)P2, whose construction is somewhat scattered across the literature. We will see that the EIII has an atlas whose transition functions have clear octonion interpretations, apart from those covering a sub-variety X∞ of dimension 10. This subvariety is itself a hermitian symmetric space known as DIII, with no apparent octonion interpretation. We give detailed analysis of the geometry in the neighbourhood of X∞.We further decompose X=EIII into F4-orbits: X=Y0∪Y∞, where Y0∼(OP2)C is an open F4-orbit and is the complexification of OP2, whereas Y∞ has co-dimension 1, thus EIII could be more appropriately denoted as (OP2)C‾. This decomposition appears in the classification of equivariant completion of homogeneous algebraic varieties by Ahiezer [2].

Cartan–Helgason theorem for quaternionic symmetric and twistor spaces

We define two types of Witten’s zeta functions according to Cartan’s classification of compact symmetric spaces. The type II is the original Witten zeta function constructed by means of irreducible representations of the simple compact Lie group U. The type I Witten zeta functions, we introduce here, are related to the irreducible spherical representations of U. They arise in the harmonic analysis on compact symmetric spaces of the form U/K, where K is the maximal subgroup of U. To construct the type I zeta function we calculate the partition functions of 2d YM theory with broken gauge symmetry using the Migdal–Witten approach. We prove that for the rank one symmetric spaces the generating series for the values of the type I functions with integer arguments can be defined in terms of the generating series of the Riemann zeta-function.

In this paper we construct certain quantum spin systems on moduli spaces of G-connections on a connected oriented finite graph, with G a simply connected compact Lie group. We construct joint eigenfunctions of the commuting quantum Hamiltonians in terms of local invariant tensors. We determine sufficient conditions ensuring superintegrability of the quantum spin system using irreducibility criteria for Harish-Chandra modules due to Harish-Chandra and Lepowsky & McCollum. The resulting class of quantum superintegrable spin systems includes the quantum periodic and open spin Calogero–Moser spin chains as special cases. In the periodic case the description of the joint eigenfunctions in terms of local invariant tensors are multipoint generalized trace functions, in the open case multipoint spherical functions on compact symmetric spaces.

Abstract We construct a sequence of explicit blow-ups and blow-downs on an irreducible compact Hermitian symmetric spaces $X$ which transforms it into a projective space of the same dimension. Moreover, this resolves a birational map given by Landsberg and Manivel. Centers of the blow-ups for $X$ are constructed by loci of chains of minimal rational curves and centers of the blow-ups for the projective space are constructed from the variety of minimal rational tangents of $X$ and its higher secant varieties. The result was known in the special case where $X$ is of rank 2 and could be found in Zak’s monograph “Tangents and secants of algebraic varieties.”

On the free loop space of compact symmetric spaces Ziller introduced explicit cycles generating the homology of the free loop space. We use these explicit cycles to compute the string topology coproduct on complex and quaternionic projective space. The behavior of the Goresky-Hingston product for these spaces then follows directly.

Compact symmetric spaces are probably one of the most prominent class of formal spaces, i.e. of spaces where the rational homotopy type is a formal consequence of the rational cohomology algebra. As a generalization, it is even known that their isotropy action is equivariantly formal. In this paper, we show that [Formula: see text]-symmetric spaces are equivariantly formal and formal in the sense of Sullivan, in particular. Moreover, we give a short alternative proof of equivariant formality in the case of symmetric spaces with our new approach.

Moment maps and isoparametric hypersurfaces in spheres — Grassmannian cases

We identify the standard weighted Bergman kernels of spaces of nearly holomorphic functions, in the sense of Shimura, on bounded symmetric domains. This also yields a description of the analogous kernels for spaces of “invariantly-polyanalytic” functions — a generalization of the ordinary polyanalytic functions on the ball which seems to be the most appropriate one from the point of view of holomorphic invariance. In both cases, the kernels turn out to be given by certain spherical functions, or equivalently Heckman-Opdam hypergeometric functions, and a conjecture relating some of these to a Faraut-Koranyi hypergeometric function is formulated based on the study of low rank situations. Finally, analogous results are established also for compact Hermitian symmetric spaces, where explicit formulas in terms of multivariable Jacobi polynomials are given.

A new characterization is provided for the class of compact rank-one symmetric spaces. Such spaces are the only symmetric spaces of compact type for which the standard vector field xi ^{S} on their sphere bundles is Killing with respect to some invariant Riemannian metric. The set of all these metrics is determined, as well as the set of all those invariant contact metric structures with characteristic vector field xi ^{S}. Moreover, on tangent sphere bundles of compact symmetric spaces with rank greater than or equal to two, a family of invariant contact metric structures, which contains the standard structure, is obtained.

On gap rigidity problems for compact Hermitian symmetric spaces