Linear connections on Lie groups


 Randolph Chase
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1 Linear connections on Lie groups The affine space of linear connections on a compact Lie group G contains a distinguished line segment with endpoints the connections L and R which make left (resp. right) invariant vector fields parallel. The midpoint is the LeviCivita connection of a biinvariant Riemannian metric. Problem (Dan Freed). Give a differentialgeometric interpretation of the connection 2 3 L R. [This connection arises in the cubic Dirac equation introduced by Slabarski and rediscovered by B. Kostant and also in the noncommutative Weil algebra of A. Alexeev and Meinnenken.] 1
2 Hitchin representations of fundamental groups of surfaces Let S be an orientable compact 2dimensional surface. A Fuchsian representation of π 1 (S) in PSL(n, R) is a representation which factors through a cocompact representation of π 1 (S) in PSL(2, R) and the irreducible representation of PSL(2, R) in PSL(n, R). A Hitchin representation is a representation which may be deformed into a Fuchsian representation. We denote by Rep H (π 1 (S), SL(n, R)) the moduli space of Hitchin representations, which is by definition a connected component of the space of all representations. It can be shown that a Hitchin representation is discrete and faithful and that the Mapping Class Group M(S) acts properly on the Hitchin component. 2
3 In 1990, N. Hitchin gave explicit parametrisations of Hitchin components: If J is a complex structure J over S, he produced a homeomorphism H J : H 0 (K 2 J)... H 0 (K n J ) Rep H (π 1 (S), SL(n, R)). He uses the identification of representations with harmonic mappings as in K. Corlette s seminal paper and the fact that a harmonic mapping taking values in a symmetric space gives rise to holomorphic differentials in a manner similar to that in which a connection gives rise to differential forms in Chern Weil theory. In particular, this construction breaks the invariance under the Mapping Class Group. 3
4 Here is a more equivariant construction (with respect to the action of the Mapping Class Group): Let E (n) be the vector bundle over Teichmüller space whose fibre above the complex structure J is E (n) J = H 0 (K 3 J)... H 0 (K n J ). We observe that the dimension of the total space of E (n) is the same as that of Rep H (π 1 (S), SL(n, R)) since the dimension of the missing quadratic differentials in E (n) J accounts for the dimension of Teichmüller space. We now define the Hitchin map { E (n) Rep H H (π 1 (S), SL(n, R)) (J, ω) H J (0, ω). (This terminology is awkward since this Hitchin map is some kind of an inverse of what is usually called the Hitchin fibration). 4
5 Conjecture (François Labourie). If ρ is a Hitchin representation, then there exists a unique ρequivariant minimal surface in SL(n, R)/SO(n, R). Hence the space Rep H (π 1 (S), PSL(n, R))/M(S) is homeomorphic to the vector bundle over the Riemann moduli space whose fibre at a point J is H 0 (K 3 J )... H0 (K n J ). [This conjecture is known to be true for n = 2 where it reduces to the Riemann uniformisation theorem. F. Labourie and J. Loftin proved it for n = 3. Moreover, F. Labourie also proved that the Hitchin map is surjective: it amounts to proving the existence of the above mentioned minimal surface.] 5
6 Problem (William Goldman). Which are the surface group representations ρ : π PSL(2, R) that correspond to branched hyperbolic structures? For each Riemann surface Σ of genus g with fundamental group π, consider rank 2 stable Higgs pairs (V, Φ) where the Higgs field Φ has no component in Ω 1 (Σ, K 2 D) with D is an effective divisor satisfying the degree condition deg(d) < 2g 2. (It says that the harmonic metric is holomorphic.) Taking the union over all surfaces, this gives a universal symmetric power which maps into Hom(π, PSL(2, R))/PSL(2, R) by a (nonsurjective) homotopy equivalence. Problem (William Goldman). What is the image of this symmetric power in Hom(π, PSL(2, R))/PSL(2, R)? Does it contain all [ρ] with dense image? 6
7 Let G be an Rsplit semisimple Lie group. Let H be the Hitchin component of Hom(π 1 (S), G))/G. Problem (William Goldman). Interpret H as locally homogeneous geometric structures (in the sense of Ehresmann and Thurston) on fiber bundles over S. [For example, when G = PGL(2, R), H is in onetoone correspondence with hyperbolic structures on S. When G = PGL(3, R), H is in onetoone correspondence with convex RP 2 structures on S. When G = PGL(4, R), O. Guichard and A. Weinhard have identified a class of RP 3 structures on the unit tangent bundle T 1 (S) corresponding to H.] 7
8 Harmonic maps of higher genus There is a welldeveloped theory of integrable systems for harmonic maps from a 2torus to a Lie group G. Formally the equations for harmonic maps of a surface look like the Higgs bundle equations but with a change of sign. Problem (Nigel Hitchin). Is there a Nahm transform, in the same context of the previous problem, for maps of surfaces of higher genus? 8
9 Metrics with special holonomy Problem (Nigel Hitchin). Find explicit descriptions of a CalabiYau metric on a K3 surface. [Twistor theory tells us that, if we do that, then we can describe explicitly complex structures which are far from algebraic ones which does not sound too hopeful. However, here is a possible scenario. Kronheimer s ALE construction takes a finite subgroup Γ SU(2) and considers the vector space R of functions on Γ, then does a hyperkähler quotient of the group U(R) Γ (Γinvariant unitary transformations) acting on the quaternionic vector space (R H) Γ. Replace Γ by a discrete subgroup of hyperkähler isometries of C 2 and do the same thing. 9
10 In particular consider Γ to be the extension of a finite group by translations Z n Γ Z 2 (with n 4) instead of just Z 2, which gives the EguchiHanson metric. With n = 4, the quotient can be interpreted as the moduli space of Z 2  invariant SU(2) instantons on a flat torus with a certain type of singularity at the 16 fixed points. The thorny issue is the nature of that singularity, but formally there should be a hyperkähler moment map for the gauge group which requires 3 parameters for each singular point. Together with the 10 parameters for the lattice Z 4 this gives = 58 parameters. The construction would be explicit in the sense that in principle we know how to solve the ASD equations on a flat torus by twistor theory.] 10
11 The ClemensFriedmann construction of nonkähler 3folds yields complex structures with nonvanishing holomorphic 3forms on connected sums of S 3 S 3. This is an analogue of the study of complex structures on connected sums of S 1 S 1 Teichmüller theory. The local geometry of the complex structure moduli space is known and is like that of an ordinary CalabiYau. Problem (Nigel Hitchin). What about the global structure and its boundary or the analogue of the mapping class group? Is there a natural metric with skew torsion on such a 3fold? 11
12 Problem (Nigel Hitchin). Find bounds on the topology of CalabiYau 3folds. [It is conjectured that the Euler characteristic of such manifolds is bounded by 960. In the case of hyperkähler 4folds there are bounds due to Guan.] 12
13 Problem (Robert Bryant). In a CalabiYau 3manifold is the singular locus of a special Lagrangian 3cycle a semianalytic set? Is there a way to resolve singularities of such objects? [By Almgren s regularity theorem, it is known to have Hausdorff dimension at most 1, but nothing else appears to be known about it.] 13
14 Problem (Simon Salamon). Classify metrics with holonomy equal to G 2 admitting a 2torus of isometries [extending work of V. Apostolov, S. Salamon et al.]. Problem (Simon Salamon). Are compact manifolds with exceptional holonomy G 2 or Spin(7) necessarily formal? [Partial results by G. Cavalcanti, M. Fernandez, M. Verbitsky.] 14
15 Problem (Simon Salamon). Are there metrics with holonomy G 2 associated in some way to the (twistor spaces of) selfdual structures on the connected sum of n 2 copies of CP 2? [Question of M.F. Atiyah and E. Witten.] Problem (Simon Salamon). Is there a compact hyperkähler 8manifold other than the two spaces found by A. Beauville? [cf. O Grady examples in dimensions 12 and 20]. 15
16 Special geometries Problem (Joel Fine), Do there exist two homeomorphic 4manifolds, only one of which admits an antiselfdual metric? [Claude LeBrun has shown this is true if one replaces antiselfdual by scalarflat and antiselfdual.] Problem (Simon Salamon). Is every compact nearlykähler 6manifold homogeneous? 16
17 We now know that not every coclosed G 2 structure on a 7manifold can be induced by an immersion into a Spin(7)manifold. Analyticity is sufficient but not necessary. Problem (Robert Bryant). Can one give necessary and sufficient conditions? Is every (local) coclosed G 2 structure on a 7manifold the boundary of a smooth Spin(7)manifold? (This is the onesided version of the embedding problem.) What are the conditions on a coclosed G 2 structure on the 7sphere that determine that it is the boundary of a smooth Spin(7)holonomy Riemannian 8ball? 17
18 Gerbes versus connections Problem (Nigel Hitchin). Is the connection important or the fourform Trace F 2? [This 4form is the curvature of a canonical 2gerbe defined by a connection on a principal Gbundle. In Strominger s equations the difference of two such forms is dh where H is the 3form flux. This is like a change of 2gerbe connection. In 4 dimensions it is possible that Trace F 2 for an instanton determines the connection up to gauge equivalence this is the basis of the socalled information metric on the moduli space of instantons.] 18
19 Holomorphic Poisson manifolds Compact holomorphic symplectic manifolds are much studied even though there seem to be few of them. Problem (Nigel Hitchin). What about Poisson manifolds? [There are analogues of the symplectic cases, such as Hilbert schemes of Poisson surfaces and there is a quite simple classification of Poisson surfaces. In the case of Poisson 3folds, the integrability forces constraints Λ 2 T is a rank 3 bundle on a complex 3dimensional manifold but (essentially because of Bott vanishing) when a section is integrable it is forced to vanish at least on a curve and not just at isolated points.] 19
20 Quantum conjectures Consider Witten s interprettion of the Geometric Langlands Programme. The subvariety of flat PSL(2,C)connections which extend over a bounding 3manifold is an (A, B, A)brane on Hitchin s PSL(2,C)Higgs bundle moduli space. The dual brane on the Langlands dual group SL(2,C)Higgs bundle moduli space should induce constraints on the quantum SU(2)ChernSimons state associated to the bounding 3manifold. Problem (Jørgen Ellegaard Andersen). Construct these constraints. Problem (Jørgen Ellegaard Andersen). A closed oriented 3manifold M is simply connected if and only if M is a quantum sphere. [Simply connectedness does imply that the sequence of partition functions of the manifold is that of the sphere.] 20
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