The autoequivalences of the category of local systems on the punctured sphere (with ) that commute with the action of the pure spherical braid group are indeed generated by the three operations mentioned: tensoring with rank one local systems, the action of , and Katz's middle con...

Short answer. For X = P^1 \ D with n = |D| ≥ 3, the autoequivalences of LocSys(X) that commute (up to isomorphism) with the pure spherical braid group Bn are generated by:

• tensoring with a rank-one local system on X,
• the contragredient dual V ↦ V∨,
• Katz’s middle convolution MCχ with a Kummer rank-one local system χ on Gm,
• post-composition with a field automorphism σ ∈ Aut(C).

No other abe...

This is a fascinating question about the structure of autoequivalences of local systems on punctured surfaces that commute with mapping class group actions.

Let me first clarify the setup and then discuss what's known about this problem.

The Setup

You're considering:

• where
• = category of locally constant sheaves of -vec...

First, I need to understand the setup. The space X is a sphere minus n points, so its fundamental group is a free group on n-1 generators. The category LocSys(X) is equival...

Below is a rough “classification theorem” (which in fact is proved by combining various authors’ work on Tannakian‐formalism + Katz’s middle convolution + the rigidity/isomonodromy philosophy) saying that nothing “mystical’’ is hiding beyond the three operations you list.

1. Setup and notation.
   • X = P^1∖D, |D|=n, π1(X)=F_{n−1}. We let
   – LocSys(X) = (finite–dimensional complex) l...

This is an excellent question that touches on the deep structure of categories of local systems, connecting to the theory of rigid local systems and the work of Deligne, Katz, and Arinkin. The short answer to your main question is:

**Yes, the autoequivalences you listed (tensoring by certain rank 1 systems, field automorphisms, and middle convolutions) are believed to generate all -equivaria...

Let

X = S² \ D , |D| = n ≥ 3 , π := π₁(X , x₀) ≅ F_{n-1}

and denote by

Rep := LocSys(X) = Rep_ℂ(π)

the abelian category of finite–dimensional complex representations of the free group π.
Write

Γ := Mod(S² , D) ≃ PB_n ⊂ Out(π)

for the pure spherical braid group and let

Rep −→ Rep ,  V ↦ g⋆V := V∘θ_g (g ∈ Γ) (1)

be the taut...

The category

 LocSys(X)=RepC(π1(X))=RepC(Fn–1)

is a neutral tannakian category over the field k=ℂ; its symmetric monoidal structure is the usual tensor product of representations and a fibre-functor is obtained by choosing a base–point in X. The pure spherical braid group

 Bn = Mod(S² ,D)  (  |D| = n )

acts on LocSys(X) through the usual Artin action

 Bn  →  Aut(Fn–1)  →  ...

Symmetries of Local Systems on the Punctured Sphere

This question delves into the autoequivalences of the category of locally constant sheaves (local systems) on a punctured sphere that commute with the action of the pure spherical braid group.

Understanding the Known Autoequivalences

The question already identifies three types of -equivariant autoequivalences of :

...