Orbifold cohomology and quantum cohomology

of orbifolds

ICTP, Trieste, September 2002 lectures by Dan Abramovich

Part 3:

Stringy Orbifold Cohomology: Theory and Examples

• Work over ℂ

• This work with Tom Graber and Angelo Vistoli • Quantum products based on W. Chen + Y. Ruan

• global quotient case by Fantechi + Göttsche.

Equivariant Riemann-Roch Let C be a smooth twisted curve, E a vector bundle on C. Let x be a twisted marking, locally C is [V/μ_{r}], V has coordinate z, Say μ_{r} acts on T_{V,x} via v ↦ ζ_{r} · v. (Then μ_{r} ants on z via z ↦ ζ_{r}^{−1} · z.) say μ_{r} acts on E_{x} with eigenvalues e^{(2πi)·r_{i}}, with 0 ≤ r_{i} < 1. the age is

a(E, x) = ∑ r_{i}

Theorem.

χ(C, E) = d(E) − rk(E)(g − 1) − ∑_{x} a(E, x)

Exercise: quantum product is graded.

Stringy Orbifold cohomology: We defined

γ_{1} ∗ γ_{2} = ∑_{β∈H_{2}(X)} ⟨γ_{1}, γ_{2}, ∗⟩_{0,β} q^{β}.

Now we can plug in q^{β} ↦ 0 and obtain an associative ring structure on H_{orb}^{∗}(X , ℚ) Exercise: This ring has unit 1 ∈ H^{∗}(X , ℚ). Defined ⟨γ_{1}, γ_{2}, ∗⟩_{0,β} using \overline{M}_{0,3}(X , β)

Claim: when β = 0 the three gerbes are trivial, and there are evaluation maps

K_{0,3}(X , β) → X_{1}.

These maps are representable So the above lifts to a product

γ_{1} ⌣ γ_{2}

on H_{orb}^{∗}(X , ℤ)

Theorem [ℵGV]: this is well defined for arbitrary smooth separated Deligne–Mumford stack, independent of choices and associative.

Proof of associativity is much more delicate.

The Fantechi-Göttsche ring In case X = [Y/G] write

Ỹ = ⨆_{g∈G} Y^{g}

The group G acts by sending y ∈ Y^{g} to h(y) ∈ Y^{hgh^{−1}}. So we have I(X) ≃ [Ỹ/G] Notation:

H^{∗}(Y, G) = H^{∗}(Ỹ, ℚ).

Theorem [Fantechi-Göttsche]:

1. There is a (natural) associative (noncommutative) ring structure on H^{∗}(Y, G), with an action of G.

2. There is a ring isomorphism H^{∗}(Y, G)^{G} ≃ H_{orb}^{∗}(Y) The construction and proof is very much analogous to the one for H_{orb}^{∗}(Y)

Example 0

X = BG

I(X) = [G /^{adj} G] =

⨆_{(g)} B(C(g)) =: ⨆_{(g)} X_{(g)}

H_{orb}^{∗}(BG, ℤ) = ⨁_{(g)} H^{∗}(B(C(g)), ℤ)

T_{BG} = 0 ⟹ a(Z) = 0

Component K_{g,h} of K_{0,3}(BG, 0) with g, h, (gh)^{−1}: K_{g,h} ≃ B(C(g) ∩ C(h))

product:

x_{(g)} ⌣ x_{(h)} = ∑_{(g′,h′)} |\frac{C(g′h′)}{C(g′) ∩ C(h′)}| x_{(gh)}

H_{orb}^{∗}(BG, ℚ) = C(ℚ[G]) = (ℚ[G])^{G} (H^{∗}({pt}, G) = ℚ[G])

H_{orb}^{∗}(B(μ_{r}), ℤ) = ℤ[s, t]/(s^{r} − 1, rt)

Example 1

X = [ 𝔸^{1} / μ_{r}]

I(X) = X ⊔ ⨆_{k=1}^{r−1} B(μ_{r})

H^{∗}(X_{i}, ℤ) = ℤ[t_{i}]/(rt_{i})

Age of X_{i} is i/r Components:

K_{i,j} = \{ \begin{matrix}X & i = j = 0 \\ B(μ_{r}) & otherwise \end{matrix}

[K_{i,j}]^{v} = [K_{i,j}] whenever i+j ≤ r by grading. Product:

x_{i} ⌣ x_{j} = \{ \begin{matrix}x_{i+j} & i + j < r \\ t_{0} & i + j = r \end{matrix}
t_{0} ⌣ x_{i} = t_{i}

H_{orb}^{∗}(X , ℤ) = ℤ[x_{1}]/(rx_{1}^{r})

Example 2 The wighted projective stack

ℙ^{1}[a, b] = [ (𝔸^{2} ∖ {0}) /𝔾_{m}] with the action (x, y) ↦ (t^{a}x, t^{b}y), a, b > 0. Let g = gcd(a, b), l = ab/g.

I(X) = ⨆_{i=0}^{g−1} X ⊔ ⨆_{i=0}^{a−g} B(μ_{a}) ⊔ ⨆_{i=0}^{b−g} B(μ_{b})

= ⨆_{i=0}^{g−1} X_{i} ⊔ a−1 ; ⨆ ; i = 1_{gcd(i,a/g)=1} A_{i} ⊔ b−1 ; ⨆ ; i = 1_{gcd(i,b/g)=1} B_{i}

H^{∗}(X, ℤ) = ℤ[t]/(abt^{2})

Age of A_{i} is ig/a, of B_{i} is ig/b Ring: H_{orb}^{∗}(X , ℤ) =

\frac{ℤ[X, A, B, T]}{(X^{g} − 1, aAT, bBT, A^{\frac{a}{g}} − bXT, B^{\frac{b}{g}} − aXT, AB)}

Exercise: Find H_{orb}^{∗}(X , ℚ) and H_{orb}^{∗}(X , ℤ) when X = ℙ^{2}[1, 1, a] Example:

X = [ V^{n} / S_{n}]

Notation: σ ∈ S_{n} has l(σ) = length, o(σ) = number of cycles = n − l(σ). Also dim V = d.

I(X) = ⨆_{(σ)} [ (V^{n})^{σ} / C(σ)]

(V^{n})^{σ} ≃ V^{o(σ)} ⟹ dim X_{(σ)} = d · o(σ)

Age: (σ) = (σ^{−1})

⟹ a(X_{(σ)}) = \frac{d · l(σ)}{2}

Components:

K_{(σ,τ)} = [(V^{n})^{⟨σ,τ⟩} / C(σ) ∩ C(τ)]

Notation: O(σ, τ) = set of orbits of ⟨σ, τ⟩. W = ℂ^{n} standard representation of S_{n}. W_{I} = representation of ⟨σ, τ⟩ on the subspace with coordinates in I ⊂ {1, …, n}, I ∈ O(σ, τ).

Have C →^{π} K_{(σ,τ)} → X

f^{∗}T_{X} = “ ⨁_{I∈O(σ,τ)}(W_{I} ⊗ π^{∗}T_{V_{I}})”

H^{0}(C, W_{I} ⊗^{G} O_{C}) = 1

χ(C, W_{I} ⊗^{G} O_{C}) =

|I| − a(g, W_{I}) − a(h, W_{I}) − a((gh)^{−1}, W_{I})

so r_{I} = H_{I}^{1} = \frac{|I|+1−o_{I}(g)−o_{I}(h)−o_{I}(gh)}{2}. So W_{I} contributes a factor

1  if r_{i} = 0
c_{top}(T_{V}) if r_{i} = 1
0  otherwise

Finally there is a constant factor of |\frac{C(gh)}{C(g)∩C(h)}|.