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Manin's 2009 IHES talk: (Quantum) Motives

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Slides of Yuri I. Manin's talk at the Grothendieck conference. Video:
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IHES, Grothendieck 80 Conference, Jan. 2009
Yu. I. Manin

• The category of classical motives M otK , with coefficients
in a Q–algebra K, is the target of a functor h : V ark → M otK
which is a universal cohomology theory, with values in a tensor
K–linear category.
• Morphisms X → Y in V ark are represented by correspon-
dences, algebraic cycles on X × Y .
Besides objects h(V ), M otK contains their direct pieces and
their twists by Tate’s motive.
• What is special about “total motives” h(V ) ? For example,
they are naturally commutative algebras in M otk.
• But total motives (after a completion) possess a much richer
structure: they are algebras over the cyclic modular operad
HM(n) :=
h(M g,n)
This is the motivic core of Quantum Cohomology.

Grothendieck’s handwritten notes:
Standard Conjectures
(as explained to Yu. Manin in May 1967)

• The category of HM–algebras QCAlg in M otK admits a
natural symmetric tensor product compatible with × in V ark.
However, the map h : (V ark, ×) → (QCAlg, ⊗) is functorial in
a naive sense only wrt automorphisms in V ark.
Can one define more general “correspondences” in QCAlg?
We expect that they might be some kind of Morita mor-
• Moreover, an object in QCAlg is generally a complicated
infinitary algebraic structure; we would like to reduce it to bet-
ter studied differential geometric structures, such as Frobenius
manifolds, without losing motivic functoriality.
What geometric objects in the Frobenius manifolds world are
translations of motivic correspondences?
Plan of the talk:
1. Classical motives: a reminder
2. Motivic Quantum Cohomology
3. Cohomological Field Theories
4. Numerical invariants of Quantum Cohomology
5. F –manifolds and Frobenius manifolds
6. Semisimplicity and reconstruction

— Adequate intersection theory:
(i) Smooth projective (or proper) X → A∗(X), A∗(X) = al-
gebraic (or Hodge) cycles on X modulo an adequate equivalence
relation, over a coefficient ring Z, Q, C, Ql . . . .
(ii) Inverse/direct image functors: f : X → Y induces
f ∗ : A∗(Y ) → A∗(X),
f∗ : A∗(X) → A∗(Y )
(iii) Compatibility with product:
A∗(X) ⊗ A∗(Y ) → A∗(X × Y )
(iv) The diagonal map ∆X : X → X × X induces on A∗(X)
a structure of graded commutative ring with multiplication in-
duced by ∆∗ . It satisfies the projection formula f
∗(x · f ∗(y)) =
f∗(x) · y.
— Correspondences as graded morphisms:
If X is of pure dimension d, put
Corrr(X, Y ) := Ad+r(X × Y ).

Corrr(X, Y ) ⊗ Corrs(Y, Z) → Corrr+s(X, Z) :
f ⊗ g → g ◦ f := pXZ∗(p∗
(f ) · p∗ (g))
— Category of graded correspondences:
Objects: varieties (smooth projective manifolds)/k;
Morphisms: Corr∗(X × Y ).
— Monoidal category of classical motives (M otk, ⊗):
Objects: (X, p, m), X a variety,
p = p2 ∈ Corr0(X, X), m ∈ Z
HomMot ((Y, q, n), (X, p, m)) := q◦Corrn−m(X, Y )◦p ⊂ Corr∗(X, Y ).
Monoidal (tensor) product (on objects):
(X, p, m) ⊗ (Y, q, n) = (X × Y, p ⊗ q, m + n)
Another monoidal structure: ⊕ extending

—Motives as target category of a cohomology theory:
h : V aropp → M ot
k :
h(X) := (X, id, 0),
h(ϕ : X → Y ) := [Γϕ] ∈ Corr0(X, Y ) = HomMot (h(Y ), h(X)).
— Unit and Lefschetz motives:
1 := (Spec k, id, 0),
L := (Spec k, id, −1).
h(Pn) ∼
= 1 ⊕ L ⊕ ... ⊕ L⊗n.
— The Tate twist:
X(n) := X ⊗ L−n

Classical motives are obtained from varieties by :
| • “Linearizing morphisms”: {f } => {
ifi∗f ∗
| • Adding kernels/cokernels of projectors .
| • Twisting by L⊗n, n ∈ Z.
Consider “total (classical) motives”: h(V ) where V ∈ V ark.
Question. What additional motivic structures are naturally
supported by total motives rather than arbitrary ones?
Example. Each total motive is in a natural way a unital
commutative algebra in the monoidal category of motives: Γ∆X
induces the multiplication
∪ : h(V ) ⊗ h(V ) → h(V )

This structure is immensely generalized by the following
Each total motive in a natural way is an algebra over the
cyclic modular operad
HM(n − 1) :=
h(M g,n)
in the monoidal category of ind–motives.
This means that we have for any V canonical correspondences
Ig,n(V ) ∈ Corr∗(M g,n × V n)
which, when considered as morphisms in in M otk, satisfy a
host of identities: axioms of a modular operad and its rep-

• Explanation 1. Why ind–motives rather than simply mo-
tives? Two reasons:
(i) For each “arity” n, we have infinitely many genera g.
(ii) Each Ig,n(V ) is in fact an infinite sum of cycles indexed
by effective numerical equivalence classes β of curves in V :
Ig,n =
Ig,n(V, β)
• Warning 2. Moduli spaces of stable curves of genus g with
n + 1 marked points M g,n+1 are not smooth varieties, they are
smooth proper Deligne–Mumford stacks/orbifolds.
For orbifolds, there are two different Chow rings functors (co-
inciding upon V ark): A∗ (A. Vistoli et al.), A∗ (B. To¨en):
A. Vistoli. Intersection theory on algebraic stacks and their
moduli spaces. Inv. Math. 97 (1989), 613–669.
B. To¨
en. On motives for Deligne–Mumford stacks. IMRN,
17 (2000), 909–928.
Used as correspondences, these constructions give rise to two
a priori different categories of classical motives of orbifolds.
In fact, the categories are the same, but the respective motivic
cohomologies differ.

Manin's 2009 IHES talk: (Quantum) Motives



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