Picture at beginning of Matt's talk, line 2898
Figure
1. Open-Closed
field TFT,
theories.
Pictures describing
assignations
of OC
middle one corresponds to map on
1. Open-Closed Field Theories, Matt Young, SUNY Stony
Brook
We’ll go through open-closed field theories in two dimensions in increasingly complex situations, starting by generalizing Atiyah’s notion of TFT and
working up to Costello’s theorem about open-closed TCFTs. Everything here
will be a 2-1 theory.
An open-closed TFT studies cobordisms between manifolds with boundary.
What this means practically is that we have three types of boundaries to
our bordisms: closed boundaries (which are homeomorphic to disjoint unions
of circles), open boundaries (which are homeomorphic to disjoint unions of
closed intervals) and free boundaries (which is what remains after removing
open and closed parts from the boundary). The need to include open and
closed boundaries comes from string theory. The “usual” (i.e. closed) notion
of field theory and cobordism is for closed strings and open-closed stuff should
describe open strings. Now, if you take the physics seriously, open strings
should end on some D-brane, which in the language above is a free boundary.
Label the set of D-branes by Λ.
So let’s define a category MΛ whose objects are closed circles or intervals
with endpoints labeled by D-branes and morphisms are as usual, with free
boundaries labeled by D-branes in a manner which agrees with the labeling
of open boundaries. This is easily seen to be a symmetric monoidal category.
An open closed TFT is a symmetric monoidal functor Z from this category
to V ectC :
Z ∈ F un⊗ (MΛ , V ectC ).
Then Z(S 1 ) = H is a commutiative Frobenius algebra for the same reasons
as in closed TFT. Let Z(Iba ) = Oab , where a, b are labels for D-branes. In
other words, Iba is an open boundary, with incoming boundary labeled by b
and outgoing boundary labeled by a. Then
Oab ⊗ Oba → Oaa → C
1
Pictures describing assignations of OC TFT, middle one corresponds to map on line 29
2
Figure 2. Data of an open-closed field theory.
is a perfect pairing. In particular, Oaa is also a Frobenius algebra, but it need
not be commutative!
A whistle is an example of a morphism from a closed string to an open
one. Notice that we need some free boundary to make this possible. Then
i
a
Z(whistle) : H →
Oaa ,
and we also have the dual, ia . The claim is that ia is an algebra homomorphism, and moreover, that it is central. We also have the Cardy condition,
which says
πba = ib ◦ ia
To explain the map πba : Oaa → Obb , fix a basis ψµ of Oba and ψ µ for the dual
Oab (use the perfect pairing above). Then
X
πba (ψ) =
ψµ ψψ µ ,
µ
see the picture.
Theorem 1.1 (Sewing). This data (together with some additional compatibility relations) is the same as an open-closed two dimensional TFT.
Picture for Cardy condition, line 2927
3
Figure 3. Cardy condition.
There is a D-brane category B, whose objects are elements of Λ and whose
morphisms are
HomB (b, a) = Oab .
Double complex in Kevin's talk, line
Corollary 1.2. B is a CY category, which just means there is a non-degenerate
pairing coming from the Frobenius pairing:
HomB (a, b) ⊗ HomB (b, a) → C.
(Maybe this should also be called a Frobenius category.)
We can think of this theorem as the open-closed analogue of the sewing
theorem involving commutative Frobenius algebras and closed TFT. So, can
we classify/construct these things? For example, is there a way of getting
an open-closed field theory from a closed one? A good reference for this is
Moore-Segal or Lazaroiu (the latter being the physics perspective).
Theorem 1.3. Given H a semisimple commutative Frobenius algebra, we can
reconstruct B ' V ect(X) where X = Spec(H), which is unique up to tensoring
with a line bundle on X. We can also recover the Frobenius structure on B.
Conversely, given a semisimple CY category B (it’s hom-spaces are semisimple), we can reconstruct H as the ring of endomorphisms of the identity functor,
H = EndB (id).
In the above, if B is not semisimple, we may run into issues with defining
a trace on H, giving existence and uniqueness problems. Notice when H is
semisimple, Spec(H) is a finite space. The notion of twisting already shows
up in the simple example consider here. It is explained in the above theorem
through the ambiguity in the isomorphism B ' V ect(X); we have a bundle
of categories on X where the identification between categories over different
fibres is tensoring by a line bundle. This gives a gerbe on X classified by a
class in H 3 (X; Z), which in our case is trivial since X is finite.
To see why it is natural to consider endomorphisms of the identity functor,
say we are given an open-closed theory. Let ξ ∈ H and put ξa = ia (ξ) ∈ Oaa .
4
Then we have
ξa ◦ η = η ◦ ξb ,
for η ∈ Oab , which gives an element {ξa } of EndB (id). Notice that H is
commutative, as follows by naturality.
Now let’s look at an example. Let G be a finite group, and
B = RepC (G),
the category of finite dimensional complex representations of G. Now if Σ is
a closed orientable 2-manifold, we put
X
1
Z(Σ) =
|Aut(P )|
{P →Σ}/∼
where P → Σ is a principle G-bundle. We take
H = C[G]G = C[G/G].
Say now Σ is a 2-manifold with ∂Σ ' S 1 , for simplicity. Then
X
1
.
Z(Σ)([g]) =
|Aut(P
)|
bundles with
prescribed holonomy [g]
This suffices to define a closed TFT. To define an open-closed theory, let
OAA = EndG A.
Consider the pair of pants where the boundaries are free boundaries labeled
by Ai and we’ve shrunk the open strings to points. Then
X Q χAi (holP (Ci ))
i
Z(Σ, decorations) =
.
|Aut(P )|
P →Σ/∼
It is clear how to generalize the functor Z to 2-manifolds with open and closed
boundaries.
Now let’s look at open-closed TCFTs. To do this, we replace V ectC by
ChainC . Again, this generalization can be motivated by physics. The closed
string algebra should be the space of states of a single closed string. The space
of states is graded by ghost number and is acted on by a BRST differential,
which increases ghost number by one. For example, in some twisted N = 2
theories the closed string algebra is (a suitable class of) differential forms on
the loop space of some Kähler manifold, with BRST differential the usual
differential.
Now,
)
OC Λ := C• (Mconf
Λ
where Mconf
is the conformal analogue of MΛ considered above. This is again
Λ
a dg symmetric monoidal category in a natural way.
Definition 1.4. An open-closed TCFT is an h-split symmetric monoidal functor
Z : OC Λ → ChainC .
5
That Z is h-split means
Z(a ⊗ b) → Z(a) ⊗ Z(b)
is a quasi-isomorphism for all objects a and b.
There are restrictions of this to functor to OΛ and C, the corresponding
open or closed categories. It turns out by pullback we also get field theories (the inclusions of the categories make things h-split still) but there is no
obvious way to pushforward open field theries, unlike the topological case.
Theorem 1.5 (Costello).
(1) The category of open-closed TCFTs with Dbranes Λ is homotopy equivalent to the category of unital extended CY
A∞ -categories with set of objects Λ.
(2) Also, there is a way to make the pushforward above exact so that we
get a functor
Open T CF T → Open − Closed T CF T.
where the map is given by
Z 7→ j ∗ Li! Z.
This map is in a certain sense universal.
(3) Finally
HH∗ (Z) = H∗ (j ∗ Li! Z(S 1 ))
where i is the inclusion of open stuff into open-closed.
The unital part comes from a disc with a single open boundary labeled by
a, and gives C → Oaa .
Recall that A is an A∞ algebra if it is a Z-graded vector space
A = ⊕p∈Z Ap
and there are degree 2−d linear maps md : A⊗d → A, where m1 is a differential,
m2 is a multiplication map for which m1 is a graded derivation (up to sign),m3
measures how much m2 fails to be associate, etc.
Definition 1.6. A is an A∞ -category if there are maps
mn : HomA (A0 , A1 )⊗HomA (A1 , A2 )⊗· · ·⊗HomA (An−1 , An ) → HomA (A0 , An )
satisfying the analogous properties to an A∞ -algebra.
For example an A∞ -category with a single object is just an A∞ -algebra.
Definition 1.7. A CY A∞ -category is an A∞ -category together with a nondegenerate pairing
< , >A,B : HomA (A, B) ⊗ HomA (B, A) → C
for each pair of objects A, B that is cyclic symmetric, < md (ϕ0 , . . . ϕd−1 ), ϕd >=
± < md (ϕ1 , . . . , ϕd ), ϕ0 >.
6
Note that the CY category in the Costello theorem is the category of Dbranes.
Now we need to talk about HH∗ of an A∞ -category.
Recall the cobar construction for an associative or dg C-algebra. We can do
this exactly the same way for a dg-category. To compare with Moore-Segal,
we just need to check that HH0 is just the endomorphisms of the identity
functor. Note that if the category is semi-simple then HHi = 0 for i > 0. We
can choose a quasi-isomorphic dg-category to our A∞ -category and use this
to define the Hochschild homology. Alternatively, there is some A∞ -category
construction to calculate HH∗ without choosing some auxiliary dg category.
One motivation for this increase in abstraction is to understand string
topology. IT WAS UNCLEAR FROM THE DISCUSSION HOW CLOSE
WE ARE. THERE WERE SOME INTERESTING AND INTRICATE COMMENTS HERE. JACOB MAY OR MAY NOT HAVE CONSTRUCTED THIS,
CLAIMS SOMETHING LIKE IT IN HIS SURVEY? PERHAPS THE ISSUE
IS WHETHER OR NOT HIS OPERATIONS CORRESPOND WITH THE
OPERATIONS DEFINED BY CHAS-SULLIVAN.
As another example, let’s talk about the B-model. Take X to be a compact Calabi-Yau. Define P erf (X) or P (X) as having objects complexes of
holomorphic vector bundles on X and morphisms
HomP (X) (E, F ) = Ω0,• (E ∗ ⊗ F )
This is naturally a dg-category and you can get a non-degenerate pairing on
the homology of this category using the holomorphic volume form.
Theorem 1.8 (Kadeishvili, Kontsevich-Soibelman). Let A be an A∞ -algebra.
Then H• (A) can be made into an A∞ algebra such that m1 = 0, m2 = mA
2,
with a quasi-isomorphism of A∞ -algebras
H∗ (A) → A.
The corresponding theorem for A∞ -categories is also true.
Using this theorem we get a CY A∞ structure on
b
(X)
H∗ (P (X)) =: D∞
This gives an easy way to get an interesting A∞ structures on the derived
category of coherent sheaves on X. So then we take this, apply Costello’s
b
theorem to D∞
(X) to get an open-closed TCFT, and then push this forward
to get a closed TCFT. This closed theory is the B-model, though it is still
unclear exactly how this relates to the physicists B-model.