Download Topological Field Theories

Topological Field Theories
RTG Graduate Summer School “Geometry of Quantum Fields and Strings”
University of Pennsylvania, Philadelphia
June 15-19, 2009
Lectures by John Francis
Notes by Alberto Garcia-Raboso
1
Cobordism
Historically, it all began with cobordism, before topological field theories were even defined. Pontrjagin invented
cobordism theory in the 1930’s in order to compute homotopy groups of spheres. Back then they knew that
πi S n = 0 for i < n (proof by cellular approximation) and that πn S n ∼
= Z, classified by degree (for n = 1 it is the
∼
=
1
n
winding number; the (n − 1)th suspension takes π1 S −
→ πn S ).
For spheres of different dimensions, there are no obvious invariants: a map S n → S m for n 6= m is zero at
the level of homology. Let f : S n+k → S n be a smooth map (can always homotope to one) and x ∈ S n a regular
value of f (surjective on tangent spaces; the set of regular values is a dense subset of S n by Sard’s theorem) so
that f −1 (x) is a smooth submanifold of S n+k . Take a homotopy H between two regular values x and y of f and
pull back to S n+k :
f −1 H(0) 
o
? _ f −1 H(1)
Oo
/ f −1 H([0, 1]) o
_
W
Mx
{0} 
x
My
~
H
y
/ S n+k
f
S n+k
? _ {1}
rR
/ [0, 1] o
_
l
Then W is a manifold with boundary ∂W = Mx q My : this is a cobordism between Mx and My . There is extra
structure: take Dε a disc around x ∈ S n and pull it back to f −1 Dε , a tubular neighborhood of Mx diffeomorphic to
the normal bundle Ni of i : Mx ,→ S n+k . Moreover, we can produce a global trivialization Ni ∼
= Mx ×Rn from the
fr
map f , called a framing. We have constructed a map πn+k S n → Ωfr
k , where Ωk consists of k-manifolds embedded
n+k
in R
with a framing of the normal bundle, modulo cobordism. Pontrjagin proved this is an isomorphism. He
also erroneously calculated some homotopy groups to be zero; later he corrected his mistake.
Problem: most manifolds are not framed. How do we modify the right hand side to get rid of the framing,
or incorporate more information (e.g., orientation, spin structure,. . . )? Accordingly, we have to modify the left
hand side. This was solved by René Thom in his thesis (1950’s).
fn
Definition 1.1. “Extra structure” is a sequence of maps Bn −→ BO(n) (= Grn R∞ ) such that the following
diagram commutes:
/ Bn+1
Bn
fn
BO(n) 
/ BO(n + 1)
1
Definition 1.2. Let V → M be a vector bundle of rank k, with classifying map ν : M → BO(k). A Bk -structure
on V is the homotopy class of a lift of ν to Bk :
<
M
Bk
fk
/ BO(k)
ν
A (stable) B-structure on V is the equivalence class of a Bk -structure on V , where g, g 0 : M → Bk are equivalent
if they coincide as Bk+m -structures on V ⊕ Rm for some m ≥ 0.
f n R∞ , oriented n-planes in R∞ ): a Bk -structure on V is an orientation
Examples 1.3. a) Bn = BSO(n) (= Gr
of V .
b) Bn = BSpin(n): given an orientation on V , a Bk -structure is a spin structure.
c) B2n = B2n+1 = BU (n) (= Grn C∞ ) codifies complex structures on real vector bundles (remember the
inclusion U (n) ,→ O(2n), yielding BU (n) → BO(2n)).
ν
d) Bn ' pt (make Bn → BO(n) into a fibration in the homotopy category): if you can lift M −
→ BO(n), then
V has to be trivial, so a Bk -structure is a global trivialization of V .
Definition 1.4. Let B be “extra structure”. We define ΩB
n to be the cobordism classes of n-manifolds with Bν
n+k
structure, i.e., closed compact n-manifolds M ,→ R
with a Bk -structure on Nν , modulo cobordism and stable
is
equivalent
to
zero
if
there
exists W n+1 such that
equivalence: we say M ∈ ΩB
n

W n+1 η
/ Rn+k+1
O
O
?

∂W ∼
=M
ν
?
/ Rn+k
and Nη has a B-structure that restricts to the B-structure on Nν .
Theorem 1.5 (Thom). πn M B ∼
= ΩB
n , where M B is the Thom spectrum of {Bn }
Let us now define the elements appearing in the statement of this theorem.
Definition 1.6. A spectrum is a sequence of pointed spaces {X(n)}n∈Z with maps ΣX(n) → X(n + 1)
The importance of spectra stems from the fact that they are in one-to-one correspondence with generalized
homology and cohomology theories.
Examples 1.7. a) X(n) = K(Z, n), the Eilenberg-Maclane spectrum, denoted HZ. We know the homotopy
equivalences ΩK(Z, n + 1) ' K(Z, n); taking the adjoint gives ΣK(Z, n) → K(Z, n + 1) (warning: these
are not homotopy equivalences; look at examples K(Z, 0) ' Z, K(Z, 1) ' S 1 , K(Z, 2) ' CP∞ , . . . ). The
associated generalized cohomology theory is good ole singular cohomology H n (Y ) = [Y, K(Z, n)].
∼
=
→ S n+1 . The associated homology theory
b) X(n) = S n , the sphere spectrum S, with the obvious maps ΣS n −
is stable homotopy theory: πns Y := limk→∞ πn+k Σk Y .
2
c) X(n) = M B(n), the Thom spectrum of manifolds with B-structure, which we will define shortly. The
ν
n+k with B-structure and a map
associated homology theory is ΩB
n (Y ), consisting of n-manifolds M ,→ R
f
M −
→ Y , modulo cobordism and stable equivalence: define M ∼ 0 if there is W n+1 , a cobordism with
f˜
B-structure compatible with that of M ∼
→ Y such that f˜|∂W ' f . What we defined
= ∂W and a map W −
B
before, ΩB
n , is evidently Ωn (pt).
Definition 1.8. Let V → X be a vector bundle. The Thom space of V , denoted Th(V ), is the disc bundle of V
with its boundary identified to a point. If X is compact, Th(V ) is equivalent to the one-point compactification of
V.
We have Th(X × Rn ) ' Σn X+ and Th(V ⊕ Rn ) ' Σn Th(V ) (exercise: check these). If γ n → Grn R∞ is the
universal n-plane bundle, the spaces Th(γ n ) assemble into a spectrum M O (which will be the Thom spectrum
for Bn = BO(n)): the calssifying map Grn R∞ → Grn+1 R∞ of the bundle γn ⊕ R → Grn R∞ induces a lengthpreserving bundle map
γn ⊕ R
/ γ n+1
/ Grn+1 R∞
Grn R∞
which gives a map ΣTh(γ n ) ' Th(γ n ⊕ R) → Th(γ n+1 ). More generally, there is a natural map
fn∗ γ n ⊕ R
/ f ∗ γ n+1
n+1
/ Bn+1
Bn
∗ γ n+1 ) (remember f : B → BO(n)). This defines the Thom spectrum M B of
inducing ΣTh(fn∗ γ n ) → Th(fn+1
n
n
{Bn }.
Definition 1.9. Let E be a spectrum. We define πn E := limk→∞ πn+k E(k).
Example 1.10. πn S = πns S 0 ∼
= πn+k S k for sufficiently large k (Freudhental suspension theorem).
Definition 1.11. Let E be a spectrum. The generalized homology theory associated to E assigns to a topological
space X the abelian groups
E∗ X := lim π∗+k E(k) ∧ X+
k→∞
Remark 1.12. π∗ E ∼
= E∗ (pt)
∼
Theorem 1.13 (Thom). ΩB
n (X) = (M B)n X = limk→∞ πn+k M B(k) ∧ X+ .
We sketch the proof of the previous version of the theorem, which corresponds to the case X ' pt in the latter
formulation.
ν
n+k with a
Proof. First we construct a map ΩB
n → limk→∞ πn+k M B(k). We begin with an n-manifold M ,→ R
stable B-structure on the normal bundle Nν , i.e., a lift to Bk of the classifying map M → BO(k). Identifying
3
Nν with a tubular neighborhood of M , there is a natural map j defined by the following diagram:
j
Nν
ν
M

∼
=
/ T Rn+k
/ Rn+k × Rn+k
+
/ Rn+k
/ Rn+k
For ε small enough, j is an embedding; we thus get a map S n+k → Th(Nν ) by taking p ∈ S n+k to j −1 (p) if
p ∈ im j or to the distinguished point of Th(Nν ) otherwise. On the other hand, there is a pullback diagram
Nν
/ f ∗γk
k
/ Bk
M
which induces a map Th(Nν ) → Th(fk∗ γ k ) = M B(k). Composing these maps yields an element of πn+k M B(k)
(exercise: check that this construction is stable, and that it only depends on the B-cobordism class of M ).
α
To construct an inverse to this map, choose a representative S n+k −
→ M B(k) of limk→∞ πn+k M B(k) and
compose it with the map M B(k) = Th(fk∗ γ k ) → Th(γ k ) = M O(k) coming from the bundle map
/ γk
fk∗ γ k
Bk
fk
/ BO(k)
Deform this composition to make it transverse to the zero section of the map M O(k) → Grk R∞ . Then the
inverse image of this zero section is a manifold of dimension n embedded in S n+k (exercise: check this); a lifting
argument gives the desires B-structure.
Now we give a general recipe for defining cobordism invariants. A spectrum E defines both a generalized
α
homology theory E∗ and a generalized cohomology theory E ∗ . Given a map of spectra M B −
→ E (i.e., maps
M B(n) → E(n) commuting with suspensions up to homotopy; this gives natural transformations of the associated
homology and cohomology theories) we can produce cobordism invariants of B-manifolds in E∗ (pt) (e.g., for the
Eilenberg-Maclane spectrum, E∗ (pt) = H0 (pt) ∼
= Z). We need two ingredients:
• The fundamental class: for M n a B-manifold there is a distinguished point in ΩB
n (M ), namely M itself
with the identity map, which we denote by M . Using the map of spectra α we can push its class forward:
α∗
ΩB
∗ (M ) −→ E∗ (M ).
• Let B = limk→∞ Bk . For any class p ∈ E ∗ (B), consider the pullback ν ∗ p ∈ E ∗ (M ) under the map
ν : M → Bk → B.
h−,−i
Theorem 1.14. p(M ) := hν ∗ p, α∗ [M ]i, where E ∗ (M ) × E∗ (M ) −−−→ E∗ (pt) is the natural pairing, is a
B-cobordism invariant.
4
Proof. Because of the additivity of the definition it suffices to show p(M ) = 0 if M ' ∂W (respecting the
B-structure). Assume such a W exists and consider the relative fundamental class [W, ∂W ] ∈ ΩB
n+1 (W, ∂W ).
Remember the long exact sequence in homology and its functoriality:
ΩB
n+1 (W, ∂W )
δ
α∗
En+1 (W, ∂W )
δ
∂∗
/ ΩB (B)
n
α∗
∂∗
/ En (B)
/ ΩB (W )
n
α∗
/ En (W )
∂
where M ' ∂W ,→ W , and notice that the compatibility of B-structures implies the commutativity of the
following diagram:
ν
/ Bk
M p
∂
ν̃
>
W
Moreover, δ[W, ∂W ] = [M ]. Then,
p(M ) = hν ∗ p, α∗ [M ]i = h∂ ∗ ν̃ ∗ p, α∗ δ[W, ∂W ]i
= hν̃ ∗ p, ∂∗ α∗ δ[W, ∂W ]i
(by naturality of the pairing)
∗
= hν̃ p, α∗ ∂∗ δ[W, ∂W ]i
(by functoriality)
=0
(because ∂∗ δ = 0)
Example 1.15. Take E = HZ to be the Eilenberg-Maclane spectrum. The pairing is then the usual pairing of
α
cocycles and cycles. For B = BSO, a version of the Hurewicz theorem yields
a map M B = M SO −
→ HZ. Recall
Q
n
that H ∗ (BSO)/torsion ∼
= Z[p1 , p2 , . . .] (Pontrjagin classes). Choose p = pi i ; α∗ [M ] is the usual fundamental
∗
class, and ν p is a Pontrjagin class of the stable normal bundle. Then p(M ) are the so-called Pontrjagin numbers
of M . We can do the same with Z/2 coefficients and B = BO, yielding the Stiefel-Whitney numbers of M .
Remark 1.16. In the last example, it is usually the tangent bundle that is used to define the Pontrjagin numbers
of M instead of the stable normal bundle. In the stable category, these bundles are inverses to each other, so
they determine each other’s invariants.
2
TFTs: first attempt
⊗
Definition 2.1. A tensor (a.k.a, symmetric monoidal) structure on a category C is a functor C × C −
→ C
satisfying appropiate symmetry and associativity conditions, and with a unit element. A tensor functor between
tensor categories is a functor respecting ⊗.
Examples 2.2. a) Set can be given two different tensor structures: disjoint union and cartesian product.
b) Vectk with the usual tensor product ⊗k .
c) Let CobB
n be the category whose objects are closed compact (n − 1)-manifolds with B-structure, and whose
morphisms HomCobB
(M n−1 , N n−1 ) are diffeomorphism classes of cobordisms W n (i.e., ∂W = M q N ) with
n
B-structure compatible with the B-structures on M and N . It is a tensor category with respect to disjoint
union of (n − 1)-manifolds.
5
F
→ C, where C is
Definition 2.3 (Atiyah). An n-dimensional topological field theory is a tensor functor CobB
n −
a tensor category.
Remark 2.4. Atiyah’s original definition had C = Vectk .
Our first attempt is to associate to M n−1 the cochain complex of singular cochains on M . This is well-defined
on objects. What about morphisms? We would need W n to give a map C ∗ (M ) → C ∗ (N ).
C ∗ (W )
W
? _
i∗
j
i
/
y
/O
M
j∗
$
C ∗ (M )
N
C ∗ (N )
The map i∗ goes in the wrong direction; we would need something like i∗ . The idea is to do some kind of “pushpull” over cobordisms but, in some sense, complexes are too rigid. By increasing the categorical level we can use
the existence of adjoint functors.
3
TFTs: second attempt
Notice that the singular cochains on a topological space form a commutative dg-algebra.
Definition 3.1. Let A ∈ cdgAlgk . An A-module is a cochain complex M ∈ C ∗ (k) with an action A ⊗k M → M
satisfying appropiate associativity conditions. The category of A-modules is denoted A − Mod.
f
Given a map of dg-algebras A −
→ B, we can define the usual restriction and extension of scalars functors
A − Mod o
Indf
/
B − Mod
Resf
Here we do have a push-pull scheme. Hence, take C = Catk to be the (2-)category of all k-linear categories
(i.e., categories enriched over C ∗ (k)), where there is a notion of ⊗, and define F : Cobn → Catk taking
M 7→ C ∗ (M ) − Mod. Then,
C ∗ (W ) − Mod
W
? _
i
/
M
Resi∗
j
/O
C ∗ (M ) − Mod
N
5
Indj ∗ ◦Resi∗
Indj ∗
)
/ C ∗ (N ) − Mod
The problem is that this construction does distinguish between cobordisms that are diffeomorphic., but they are
identified in Cobn .
g n is the category whose objects are the same as those of Cobn , and whose morphisms
Definition 3.2. Cob
HomCob
g n (M, N ) are classifying spaces for bordisms between M and N .
] n is a topological category, i.e., its Hom-sets are actually topological spaces. Notationally,
Remark 3.3. Cob
this is conveyed by the underlining of Hom’s
To describe the Hom spaces more concretely, notice that π0 HomCob
] n (M, N ) = HomCobn (M, N ). Each connected component is a classifying space for the diffeomorphism group of any given representative of the diffeomorphism class, i.e.,
a
HomCob
(M,
N
)
=
BDiff(W )
]
n
[W ]∈HomCobn (M,N )
6
Let X be a connected topological space. A map f : X → HomCob
] n (M, N ) has image contained in one connected
component, say [W ], so that it can be considered as a map f : X → BDiff(W ). This classifies a bundle
f ∗ EDiff(W ) → X with fiber Diff(W ). Considering the associated bundle E := EDiff(W ) ×Diff(W ) W and its
pullback f ∗ E, we can say that maps f : X → BDiff(W ) classify bundles of bordisms diffeomorphic to W .
Remark 3.4. From this construction, we see diffeomorphism groups acting on C ∗ (−) − Mod. However, these
actions are through homotopy automorphisms. Hence we cannot expect this construction to shed any light on
things like the Poincaré conjecture.
] n → Catk . For consistency, we shall want
We now want to define our TFT as a tensor functor F : Cob
0
to turn Catk into a topological category as well: given D, D ∈ Catk , discard all noninvertible morphisms in
the category HomCatk (D, D0 ) = Fun(D, D0 ), obtaining a groupoid, and consider its classifying space (i.e., the
geometric realization of its nerve).
We are now left with two problems:
• Our definition of HomCob
] n (M, N ) does not make composition associative on the nose: it is necessary to
modify the smooth structures of the cobordisms in a collar neighborhood of their boundary to glue them
to a smooth cobordism. There are two solutions: one is strictification (highly unnatural and contrary to
our philosophy here); the other is to take the ∞-categorical route.
• More importantly, M 7→ C ∗ (M ) − Mod does not respect the tensor structure!
We will now concentrate on solving this last issue by using the topological category structures just defined on
] n and Catk .
Cob
4
Tensor structures
Let X is a topological category, and X ∈ X. Then HomX (X, −) is a functor from X to Top, and we define
− ⊗ X : Top → X as its left adjoint, should it exist. Concretely, given a topological space M we have an object
M ⊗ X of X, together with natural isomorphisms
∼
=
HomX (M ⊗ X, Y ) −−−−→ Map(M, HomX (X, Y ))
for every Y ∈ X. If such a left adjoint exists for every X ∈ X we say that X is a tensored topological category.
Examples 4.1. a) For X = Top, we can take M ⊗ X = M × X.
b) X = Set can be thought of as a topological category, by considering the Hom sets as discrete topological
spaces. Then, M ⊗ X = π0 M × X satisfies the requirements.
c) X = C∗ (Z) is also a topological category: the Hom sets are chain complexes, and we have the following
sequence of functors:
C∗ (Z) o
Γ
N
/
sAb o
forgetful
free
/
sSet o
|−|
/
Top
Sing(−)
The first pair of adjoint functors is the Dold-Kan correspondence; the second one consists of the forgetful
functor and its left adjoint, the free simplicial abelian group functor; in the third pair we have the geometric
realization and the singular set functors. The composition of these functors yields the geometric realization
functor, on one hand, and the functor associating to a topological space M the complex of singular chains
on it, C∗ (M ), on the other. The tensor structure is then given by M ⊗ X = C∗ (M ) ⊗k X.
7
d) Let X = cdgAlgk . There is a forgetful functor to the category of chain complexes over k, with left adjoint
the free commutative dg-algebra functor, Sym(−) : C∗ (k) → cdgAlgk . Since the Hom sets in cdgAlgk are
commutative dg-algebras themselves, a version of the Dold-Kan correspondence (see the last example) gives
a topological category structure. As for the tensor structure, we give one example: for a free commutative
dg-algebra Sym V we have
HomcdgAlgk (M ⊗ Sym V, A) ∼
= Map(M, HomcdgAlgk (Sym V, A))
∼
= Map(M, HomC∗ (k) (V, A))
∼
(C∗ (M ) ⊗k V, A)
= Hom
C∗ (k)
∼
= HomcdgAlgk (Sym(C∗ (M ) ⊗k V ), A)
i.e., M ⊗ Sym V ∼
= Sym(C∗ (M ) ⊗k V ).
Remark 4.2. From this point on, either char k = 0 or we need to substitute commutativity by an E∞ -structure.
Guided by out previous attempt at a TFT, we now study the case of the commutative dg-algebra C ∗ (X).
Proposition 4.3. M ⊗ C ∗ (X) ∼
= C ∗ (Map(M, X))
We need a few elements before outlining the proof.
Remarks 4.4. a) If M is a discrete space and A ∈ cdgAlgk , we have M ⊗A ∼
= ⊗M A, where the last expression
is the coproduct of copies of A indexed by M .
b) − ⊗ A : Top → cdgAlgk preserves (homotopy) colimits.
c) C ∗ (−) takes (homotopy) colimits to (homotopy) limits.
Example 4.5. The sphere S 1 can be realized as a homotopy pushout:


S 0 / pt


S1 ∼
= hocolim  
pt
What this means is that we take the usual colimit after replacing the maps S 0 → pt by cofibrations S 0 → D1 :




0
/ D1
S 0 / pt
S

∼

S1 ∼
= hocolim   = colim  pt
D1
Then,


S1 ⊗ A ∼
= hocolim 
S0 ⊗ A
/ pt ⊗ A
pt ⊗ A


∼
 = hocolim 
A ⊗k A
/ pt ⊗ A

L
∼
=A ⊗ A
A⊗k A
pt ⊗ A
which is the complex calculating the Hochschild homology of A. Abusing notation, we also denote this dg-algebra
by HH∗ (A).
Remark 4.6. More generally, we have

X
ΣX = hocolim  pt
/ pt



and
pt
ΩX = holim 
pt
8


/X
Proof (of the proposition). Notice that the statement is true for M a discrete space. In this case, by 4.4a, M ⊗
C ∗ (X) ∼
= ⊗M C ∗ (X).
= ⊗M C ∗ (X); on the other hand, a Künneth-type formula gives C ∗ (Map(M, X)) ∼
= C ∗ (X M ) ∼
Furthermore, both side preserve homotopy colimits (remarks 4.4b and 4.4c). Hence it is true in general, for any
topological space can be generated (up to homotopy) by homotopy colimits of finite sets (the case of spheres
follows from the remarks above, and we can then use CW approximation).
Example 4.7. H ∗ (Map(S 1 , X)) ∼
= HH∗ (C ∗ (X)).
Proposition 4.8. For A ∈ cdgAlgk , the functor (−) − Mod : cdgAlgk → Catk preserves tensor products.
Proof. First of all, notice that for C ∈ Catk we have Fun⊗ (A − Mod, C) ∼
= HomcdgAlgk (A, EndC (1C )). Indeed,
∼
a tensor functor preserves the unit, so that A 7→ 1C and A = HomcdgAlgk (A, A) 7→ HomC (1C , 1C ). Moreover,
A − Mod is generated under colimits by A, and tensor functors also preserve colimits.
Then,
HomCatk M ⊗ (A − Mod), C ∼
= Map M, HomCatk (A − Mod, C)
∼
= Map M, HomcdgAlgk (A, EndC (1C ))
∼
= HomcdgAlgk M ⊗ A, EndC (1C )
∼
(M ⊗ A) − Mod, C
= Hom
Catk
Corollary 4.9. M ⊗ (C ∗ (X) − Mod) ' C ∗ Map(M, X) − Mod.
5
TFTs: third attempt
] n → Catk by taking M 7→ M ⊗ C. We have
Given a k-linear category we can now define F : Cob
W
> `
.
M
i
j
i⊗C
/O
N
W9 ⊗ Ce
j⊗C
M ⊗C
N ⊗C
The functor j ⊗ C preserves colimits, so by general nonsense it has a right adjoint, giving the desired pull-push
construction. The upshot is that now F preserves the tensor structure:
F (M q M 0 ) = (M q M 0 ) ⊗ C = (M ⊗ C) ⊗ (M 0 ⊗ C)
Example 5.1. Let C = C ∗ (X) − Mod. Writing X M for Map(M, X) to lighten the notation, we have
0
(M q M 0 ) ⊗ C ∗ (X) − Mod ' C ∗ (X M qM ) − Mod
0
' C ∗ (X M × X M ) − Mod
0 ' C ∗ (X M ) ⊗ C ∗ (X M ) − Mod
0
' C ∗ (X M ) − Mod ⊗ C ∗ (X M ) − Mod
' (M ⊗ C ∗ (X) − Mod) ⊗ (M 0 ⊗ C ∗ (X) − Mod)
In retrospect, we see what was wrong with M 7→ C ∗ (M ) − Mod: in a sense, we were trying to define a σ-model
without a target!
9
We now study in detail the following geometric example: let X be algebraic (a scheme or stack), obtained
by gluing affine schemes Uα = Spec Aα , and consider the (∞-version of the derived) category of quasicoherent
sheaves on X:
QCX ' lim QCUα ' lim Aα − Mod
α
α
We have M ⊗ QCUα ' M ⊗ Aα − Mod ' (M ⊗ Aα ) − Mod, where M ⊗ Aα is a commutative dg-algebra;
defining X M by QCX M = M ⊗ QCX , we see that QCX M ' limα (M ⊗ Aα ) − Mod, i.e., X M is a derived
algebraic object (in general, a stack).
ev
A point pt → M gives an evaluation map X M −−→ X pt ' X, which in turn yields a pair of adjoint functors:
ev∗
QCX M o
/
ev ∗
QCX
In the affine case X = Spec A, these functors are nothing but restriction and extension of scalars:
/
Res
M ⊗ A − Mod o
A − Mod
L
(M ⊗A)⊗−
A
This identifies QCX M with the category Modev∗ OX M (QCX ) of elements of QCX with an action of ev∗ OX M .
Example 5.2. If M = S 1 and X = Spec A is affine, example 4.5 gives Γ OX S1 = S 1 ⊗ A ∼
= HH∗ (A), so that
QCX S1 ' HH∗ (A) − Mod. More generally, from the homotopy pushout square
S0
/ pt
/ S1
pt
we obtain the homotopy pullback
XS
and ev∗ OX S1 ∼
= ev∗ ev ∗ OX ∼
= ∆∗ ∆∗ OX
/ X pt ' X
ev
1
ev
∆
/X 'X ×X
X pt ' X
∼
= HH∗ (OX ), the Hochschild homology sheaf, and
∆
S0
QCX S1 ' ModHH∗ (OX ) (QCX )
n
Consider the n-disc Dn as a cobordism from S n−1 to the empty (n − 1)-manifold. Notice that X D ∼
= X and
n s
n−1
X∅ ∼
− X S . Then,
= pt, and denote X D →
s∗
Γ
X
QCX Sn−1 −−−−−→ QCX −−−−
−→ k − Mod
We have the following result.
Proposition 5.3. Let S, T be algebraic and sufficiently nice. Then
∼
=
QCS×T −−−−−→ Fun(QCS , QCT )
In other words, any functor QCS to QCT is of the form F 7→ q∗ (p∗ F ⊗ K) for some K ∈ QCS×T , where
p : S × T → S and q : S × T → T are the canonical projections. These functors are called integral transforms
with kernel K.
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Using the projection formula and the adjunction between s∗ and s∗ , we can calculate the kernel corresponding
to the functor ΓX ◦ s∗ :
ΓX (s∗ F) ∼
= HomO
= HomOX (OX , s∗ F) ∼
= HomOX (s∗ OX Sn−1 , s∗ F) ∼
XS
n−1
(OX Sn−1 , s∗ s∗ F)
∼
= ΓX Sn−1 (F ⊗ s∗ OX )
On the other hand, Dn can also be considered as a cobordism in the opposite direction. It can easily be shown
that it corresponds to the functor − ⊗ s∗ OX : k − Mod → QCX Sn−1 . Moreover, these calculations should
completely determine the whole TFT, since any manifold can be assembled from discs. For example, cutting S n
into two discs and composing the previous calculations yields a functor k − Mod → k − Mod given by tensoring
with ΓX (s∗ s∗ OX ) ∼
= ΓX Sn−1 (s∗ OX ). Alternatively, we can directly calculate the functor associated to a closed
n-manifold to be − ⊗ ΓX M (OX M )
6
Extending TFTs
[Missing]
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