.
MOTIVES
AND QUANTUM COHOMOLOGY
IHES, Grothendieck 80 Conference, Jan. 2009
Yu. I. Manin
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2
A SUMMARY
• The category of classical motives M otK , with coeﬃcients
k
in a Q–algebra K, is the target of a functor h : V ark → M otK
k
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
k
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)
g
This is the motivic core of Quantum Cohomology.
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Grothendieck’s handwritten notes:
Standard Conjectures
(as explained to Yu. Manin in May 1967)
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• The category of HM–algebras QCAlg in M otK admits a
k
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
phisms.
• 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
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INTRODUCTION AND OVERVIEW
• CLASSICAL MOTIVES
— 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
∗(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 ).
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Composition:
Corrr(X, Y ) ⊗ Corrs(Y, Z) → Corrr+s(X, Z) :
f ⊗ g → g ◦ f := pXZ∗(p∗
(f ) · p∗ (g))
XY
Y Z
— 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
Morphisms:
HomMot ((Y, q, n), (X, p, m)) := q◦Corrn−m(X, Y )◦p ⊂ Corr∗(X, Y ).
k
Monoidal (tensor) product (on objects):
(X, p, m) ⊗ (Y, q, n) = (X × Y, p ⊗ q, m + n)
Another monoidal structure: ⊕ extending
.
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—Motives as target category of a cohomology theory:
h : V aropp → M ot
k
k :
h(X) := (X, id, 0),
h(ϕ : X → Y ) := [Γϕ] ∈ Corr0(X, Y ) = HomMot (h(Y ), h(X)).
k
— Unit and Lefschetz motives:
1 := (Spec k, id, 0),
L := (Spec k, id, −1).
Fact:
h(Pn) ∼
= 1 ⊕ L ⊕ ... ⊕ L⊗n.
— The Tate twist:
X(n) := X ⊗ L−n
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A SUMMARY:
———————————————————————

Classical motives are obtained from varieties by :

 • “Linearizing morphisms”: {f } => {
a
}.

i
ifi∗f ∗
i
 • Adding kernels/cokernels of projectors .

 • Twisting by L⊗n, n ∈ Z.

———————————————————————
A QUESTION:
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 )
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This structure is immensely generalized by the following
BASIC DISCOVERY
OF QUANTUM COHOMOLOGY:
————————————————————————
Each total motive in a natural way is an algebra over the
cyclic modular operad
HM(n − 1) :=
h(M g,n)
g
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
resentations.
————————————————————————
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EXPLANATIONS AND WARNINGS
• 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)
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):
χ
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.
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