Linear connections on Lie groups The affine space of linear - - PDF document

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 Levi-Civita connection of a bi-invariant Riemannian metric. Problem (Dan Freed).– Give a differential-geometric interpretation of the connection 2 3 ∇L + 1 3 ∇R . [This connection arises in the “cubic Dirac equation” introduced by Slabarski and rediscovered by B. Kostant and also in the non-commutative Weil algebra

• f A. Alexeev and Meinnenken.]

1

Hitchin representations of fundamental groups

• f surfaces

Let S be an orientable compact 2-dimensional surface. A Fuchsian rep- resentation of π1(S) in PSL(n, R) is a representation which factors through a cocompact representation of π1(S) in PSL(2, R) and the irreducible repre- sentation 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 RepH(π1(S), SL(n, R)) the moduli space of Hitchin repre- sentations, 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

In 1990, N. Hitchin gave explicit parametrisations of Hitchin components: If J is a complex structure J over S, he produced a homeomorphism HJ : H0(K2

J) ⊕ . . . ⊕ H0(Kn J ) → RepH(π1(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

Here is a more equivariant construction (with respect to the action of the Mapping Class Group): Let E(n) be the vector bundle over Teichm¨ uller space whose fibre above the complex structure J is E(n)

J

= H0(K3

J) ⊕ . . . ⊕ H0(Kn J ) .

We observe that the dimension of the total space of E(n) is the same as that of RepH(π1(S), SL(n, R)) since the dimension of the “missing” quadratic differentials in E(n)

J

accounts for the dimension of Teichm¨ uller space. We now define the Hitchin map H

• E(n)

→ RepH(π1(S), SL(n, R)) (J, ω) → HJ(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

Conjecture (Fran¸ cois Labourie).– If ρ is a Hitchin representation, then there exists a unique ρ-equivariant minimal surface in SL(n, R)/SO(n, R). Hence the space RepH(π1(S), PSL(n, R))/M(S) is homeomorphic to the vector bundle over the Riemann moduli space whose fibre at a point J is H0(K3

J) ⊕ . . . ⊕ H0(Kn 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

Problem (William Goldman).– Which are the surface group representa- tions ρ : π − → PSL(2, R) that correspond to branched hyperbolic structures? For each Riemann surface Σ of genus g with fundamental group π, con- sider rank 2 stable Higgs pairs (V, Φ) where the Higgs field Φ has no com- ponent in Ω1(Σ, K2 ⊗ 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 (non-surjective) homo- topy 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

Let G be an R-split semi-simple Lie group. Let H be the Hitchin com- ponent of Hom(π1(S), G))/G. Problem (William Goldman).– Interpret H as locally homogeneous geo- metric structures (in the sense of Ehresmann and Thurston) on fiber bundles

• ver S.

[For example, when G = PGL(2, R), H is in one-to-one correspondence with hyperbolic structures on S. When G = PGL(3, R), H is in one-to-one 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 T1(S) corresponding to H.] 7

Harmonic maps of higher genus

There is a well-developed theory of integrable systems for harmonic maps from a 2-torus to a Lie group G. Formally the equations for harmonic maps

• f 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 con- text of the previous problem, for maps of surfaces of higher genus? 8

Metrics with special holonomy

Problem (Nigel Hitchin).– Find explicit descriptions of a Calabi-Yau met- ric 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 hyperk¨ ahler quotient of the group U(R)Γ (Γ-invariant unitary transformations) acting on the quaternionic vector space (R ⊗ H)Γ. Replace Γ by a discrete subgroup

• f hyperk¨

ahler isometries of C2 and do the same thing. 9

In particular consider Γ to be the extension of a finite group by transla- tions Zn → Γ → Z2 (with n ≤ 4) instead of just Z2, which gives the Eguchi-Hanson metric. With n = 4, the quotient can be interpreted as the moduli space of Z2- 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 hyperk¨ ahler moment map for the gauge group which requires 3 parameters for each singular point. Together with the 10 parameters for the lattice Z4 this gives 48 + 10 = 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

The Clemens-Friedmann construction of non-K¨ ahler 3-folds yields com- plex structures with non-vanishing holomorphic 3-forms on connected sums

• f S3 × S3. This is an analogue of the study of complex structures on con-

nected sums of S1 × S1 – Teichm¨ uller theory. The local geometry of the complex structure moduli space is known and is like that of an ordinary Calabi-Yau. Problem (Nigel Hitchin).– What about the global structure and its bound- ary or the analogue of the mapping class group? Is there a natural metric with skew torsion on such a 3-fold? 11

Problem (Nigel Hitchin).– Find bounds on the topology of Calabi-Yau 3-folds. [It is conjectured that the Euler characteristic of such manifolds is bounded by 960. In the case of hyperk¨ ahler 4-folds there are bounds due to Guan.] 12

Problem (Robert Bryant).– In a Calabi-Yau 3-manifold is the singular locus of a special Lagrangian 3-cycle a semi-analytic 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

Problem (Simon Salamon).– Classify metrics with holonomy equal to G2 admitting a 2-torus of isometries [extending work of V. Apostolov, S. Salamon et al.]. Problem (Simon Salamon).– Are compact manifolds with exceptional holonomy G2 or Spin(7) necessarily formal? [Partial results by G. Cavalcanti, M. Fernandez, M. Verbitsky.] 14

Problem (Simon Salamon).– Are there metrics with holonomy G2 as- sociated in some way to the (twistor spaces of) self-dual 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 hyperk¨ ahler 8-manifold

• ther than the two spaces found by A. Beauville?

[cf. O’Grady examples in dimensions 12 and 20]. 15

Special geometries

Problem (Joel Fine),– Do there exist two homeomorphic 4-manifolds, only

• ne of which admits an anti-self-dual metric?

[Claude LeBrun has shown this is true if one replaces “anti-self-dual” by “scalar-flat and anti-self-dual”.] Problem (Simon Salamon).– Is every compact nearly-K¨ ahler 6-manifold homogeneous? 16

We now know that not every co-closed G2-structure on a 7-manifold 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 condi- tions? Is every (local) co-closed G2-structure on a 7-manifold the boundary

• f a smooth Spin(7)-manifold? (This is the “one-sided” version of the em-

bedding problem.) What are the conditions on a co-closed G2-structure on the 7-sphere that determine that it is the boundary of a smooth Spin(7)-holonomy Riemannian 8-ball? 17

Gerbes versus connections

Problem (Nigel Hitchin).– Is the connection important or the four-form Trace F 2? [This 4-form is the curvature of a canonical 2-gerbe defined by a connection

• n a principal G-bundle.

In Strominger’s equations the difference of two such forms is dH where H is the 3-form flux. This is like a change of 2-gerbe

• 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 so-called information metric on the moduli space of instantons.] 18

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 3-folds, the integrability forces constraints – Λ2T is a rank 3 bundle on a complex 3-dimensional manifold but (essentially because

• f Bott vanishing) when a section is integrable it is forced to vanish at least
• n a curve and not just at isolated points.]

19

Quantum conjectures

Consider Witten’s interprettion of the Geometric Langlands Programme. The subvariety of flat PSL(2,C)-connections which extend over a bounding 3-manifold 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 mod- uli space should induce constraints on the quantum SU(2)-Chern-Simons state associated to the bounding 3-manifold. Problem (Jørgen Ellegaard Andersen).– Construct these constraints. Problem (Jørgen Ellegaard Andersen).– A closed oriented 3-manifold 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