Braids, trees, and operads

§1, The bubbletree operad and quantum cohomology

§2, The mosaic operad and Fukaya’s Lagrangian cohomology

§3, Operads in groups and groupoids §1 The bubbletree operad and quantum

cohomology

Work on conformal field theories leads physicists to an interest in configuration spaces=20

Config ^{n+1} ℂP_{1} ∼ Config ^{n}ℂ .

of points on the complex projective line. They are most interested=20 in the quotients of these spaces by the action of PGl_{2}(ℂ).=20 The points are noncoincident, so both the spaces and the group are noncompact, and taking the quotient is tricky: it leads naturally to a compactification

ℳ_{0,n}(ℂ) ∼ Config ^{n}(ℂP_{1})/PGl_{2}(ℂ) .

The physicists discovered a repulsive potential among these points: pushing two together creates a bubble onto which they escape [cf. Parker et al].=20 Thus ℳ_{0,n}(ℂ) is the moduli space of marked genus zero=20 stable algebraic curves (which have (at worst) double points,=20 and at least three marked points on each irreducible component).=20

Example: ℳ_{0,4} (ℂ) ≅ ℂP_{1}

via the classical cross-ratio. Note, ℳ_{0,3} (ℂ) is a point: a configuration of three points on ℂP_{1} is rigid.=20

These spaces are very nice in some ways: they are compact manifolds, with cohomology concentrated in even dimension, and no torsion.=20

Operads, by example:

An operad 𝒪_{∗} = 3D{𝒪_{k}, k ≥ 1} is a collection of spaces together with some composition maps

𝒪_{n} × 𝒪_{i_{1}} × … × 𝒪_{i_{n}} → 𝒪_{i}

(where i = 3D ∑ i_{k}) satisfying some axioms …

ex. i) ℳ_{0,∗+1} (ℂ)

ex. ii) End_{n}(X) = 3DMaps (X^{n}, X) is the endomorphism operad of an object X in a monoidal category. Composition is defined by=20

X^{i} = 3DX^{i_{1}} × … X^{i_{n}} → X × … × X → X .

Def’n a morphism 𝒪_{∗} → End_{∗}(X) of operads makes X into an 𝒪_{∗}-algebra.

ex. iii) Br_{n} =3D Artin’s braid group on n strings defines the braid operad, with cabling=20

Br_{n} × Br_{i_{1}} × … Br_{i_{n}} → Br_{i}

as composition.=20

Observation Monoidal functors preserve operads; hence the homology of an operad (in spaces) is an operad in graded modules.

Theorem (WDVV, Kontsevich, …): The (rational) homology of a smooth projective algebraic variety V is an H_{∗}(ℳ_{0,∗+1} (ℂ))-operad algebra.=20

[This led to the solution of the 19th-century enumerative geometry problem of classification of lines with specified incidence in ℂP_{2}.]

The construction uses Gromov-Witten invariants:

∃ a (compact) moduli spaces GW_{k}(V) of holomorphic maps from genus zero stable curves with k marked points, to V.

These spaces have many components, indexed by degree [h : C → V] ∈ H_{2}(V, ℤ). There is also an evaluation map

GW_{k}(V) → ℳ_{0,k}(ℂ) × V^{k}

=20 which defines a cycle=20

GW_{k} ∈ H_{∗}(ℳ_{0,k}(ℂ)) ⊗ H_{∗}(V)^{⊗k} .

=20 [Actually the coefficients lie in the Novikov ring Λ = 3D = 20ℚ[H_{2}(V, ℤ)], but this will be supressed.] Using Poincaré=20 duality, we can rewrite GW_{k+1} as an element of=20

Hom (H_{∗}(ℳ_{0,k+1} (ℂ)), Hom (H_{∗}(V)^{⊗k}, H_{∗}(V)))

which then defines a morphism

H_{∗}(ℳ_{0,k+1} (ℂ)) → End_{k}(H_{∗}(V))

of operads, QED.=20

In particular, the point ℳ_{0,3} (ℂ) defines a quantum multiplication

H_{∗}(V, Λ) ⊗_{Λ} H_{∗}(V, Λ) → H_{∗}(V, Λ)

which is usually not standard … §2, the mosaic operad and Fukaya’s Lagrangian cohomology

The moduli space [cf. Devadoss]

Config ^{n}(ℝP_{1})/PGl_{2}(R) ∼ ℳ_{0,n}(ℝ)

of configurations of points on the circle can be pictured as a space of trees or mosaics of hyperbolic polygons.=20

Unlike the complex case, the points have an intrinsic cyclic order, and {ℳ_{0,∗}(ℝ)} is naturally a cyclic operad [Getzler-Kapranov].=20

Fukaya considers a compact symplectic manifold (M, ω) together with an oriented Lagrangian submanifold L (i.e. of half the dimension of M, such that ω|_{L} = 3D0; some subtle issues involving the Stiefel-Whitney class w_{2}(L) will be ignored.)

I conjecture that the following is a theorem. Something slightly weaker (cf. below) has been proved by Fukaya and his school:

For a generic almost-complex structure compatible with ω, ∃ compact oriented moduli spaces F O_{k} of pseudo-holomorphic hyperbolic polygons=20

(P, ∂P) → (M, L)

together with evaluation maps

F O_{k} → ℳ_{0,k}(ℝ) × L^{k}

which define an action of H_{∗}(ℳ_{0,∗+1} (ℝ)) on H_{∗}(L, Λ) (where now Λ = 3Dℚ[H_{2}(M, ℤ)]).

Note, the (co)homology of these spaces is not known, cf. [Yoshida]. However, we can draw some pictures.=20 Grothendieck [Esquisse] says ℳ_{0,5} (ℂ) is ‘un petit joyaux’. Its real points ℳ_{0,5} (ℝ) map to ℳ_{0,4} (ℝ) × ℳ_{0,4} (ℝ) = 3DT^{2} by selecting two distinct subsets of four points. To get the full space, we need to blow up (ie, add crosscaps) at the three configurations defined by triple coincidences, resulting in T^{2}#3ℝP^{2}.

Here is a more symmetric picture, defined by blowing up four points on ℝP^{2}. Both pictures are tesselated by pentagons, though this is easier to see in the second picture.=20 This is a regular polytope with twelve pentagonal faces: it is the dodecahedron’s ‘evil twin’.=20

In general, there is a surjective map

Σ_{k} ×_{D_{k}} K_{k−3} → ℳ_{0,k}(ℝ)

(where D_{k} is the dihedral group of order 2k) which is 2^{n} to=20 1 on codimension n faces: in general, these moduli spaces are=20 tesselated by Stasheff associahedra.

There is a commutative diagram

Config ^{∗}(ℝ) ⟶ Config ^{∗}(ℂ)
↓  ↓
= ℳ_{0,∗+1} (ℝ) ⟶ ℳ_{0,∗+1} (ℂ)

The space in the upper right corner is homotopyequivalent to the little disks operad, and the space in the upper left corner is the classical A_{∞} operad {Σ_{∗} × K_{∗−1}} (made permutative, ie endowed with an action of the symmetric group. The left vertical map defines the tesselation; thus the mosaic operad is a kind of (quasicommutative) quotient of the A_{∞} operad. Fukaya shows that the A_{∞}=20 operad acts on Floer cohomology, but I believe that action passes through this quotient. The diagram above is a fiber product of spaces,=20 but it is not quite a fiber product of operads.=20 Theorem of Davis, Januszkiewicz, and Scott: the tesselation defines a piecewise negatively curved metric on ℳ_{0,∗}(ℝ); these spaces are therefore K(π, 1)’s!

Remark: The quotient of the Fulton-MacPherson compactification of Config ^{∗}(ℝP_{1}) by the circle group 𝕋 is also aspherical!

§3 Operads in groups (and groupoids)

Observation: π_{1} of an operad is an operad in groups … provided you’re careful about basepoints.=20

Example {1, …, n ∈ ℂ} (with that order) defines a basepoint ∗ ∈ Config ^{n}(ℂ); but the natural action of Σ_{n} moves it around (by changing the order).=20

Recall that a space has a fundamental groupoid, with respect to a system of basepoints: it is a category, with the points as objects, and homotopy classes of paths between them as morphisms.=20

Note also that a surjective homomorphism ϕ : G → H of groups defines a groupoid [H/G] with H as set of objects, and

mor (h_{0}, h_{1}) = 3D{g ∈ G | ϕ(g)h_{0} = 3Dh_{1}} as morphisms. With this notation,

π(Config ^{n}(ℂ) rel Σ_{n}(∗)) ≅ [Σ_{n}/Br_{n}]

where Br_{n} → Σ_{n} is the standard homomorphism. Thus the=20 fundamental groupoid of the little disks operad is the braid operad.=20

Def’n the braid category B has integers n as objects, with Br_{n} as its endomorphisms. [There are no morphisms between distinct integers.] This is a (universal) braided monoidal category, with tensor product B × B → B=20 defined by juxtaposition (n, m) ↦ n + m.=20

A standard construction [cf. Kassels] defines a functor

[Σ_{n}/Br_{n}] → Func (B^{n}, B)

which makes the category B an algebra over the braid operad. = More generally, any braided monoidal category is an algebra over the braid operad.=20 The operad ℳ_{0,∗+1} (ℝ) defines a similar category: there is an exact sequence

π_{1}(ℳ_{0,∗+1} (ℝ)) ↣= π_{1}(ℳ_{0,∗+1} (ℝ)_{hΣ_{∗}} ) = 20 ↠ Σ_{∗}

in which the fundamental group of the homotopy quotient plays the role of the braid group. There is a similar tensor category D, which is a kind of universal example of an algebra over the associated operad in groupoids.=20 Here are some questions and speculations:

i) do these fundamental groups act in some natural way on Fukaya’s cohomology?=20

ii) does D have an interpretation in terms of cyclic=20 operads with a trace?

iii) are these fundamental groups in some sense Galois groups for solutions of Calogero-Moser systems [of points moving on the line] analogous to the role played by the braid groups in the Knizhnik-Zamolodchikov equations?

iv) Does the rank of H_{1}(ℳ_{0,k+1} (ℝ)) equal \binom{k}{3}?