The relevant silent objects are obtained via a remarkable interplay between algebraic struc-
tures in finite tensor categories and the geometry of framings. The gluing categories for tadpole
circles are canonically equivalent to (twisted) centers:
T(
Q
A
+
) =
I
0
2
⊠
∼
=
Z
(
A
)
and
T(
Q
A
−
) =
I
0
2
⊠
∼
=
Z
−
4
(
A
)
(5.37)
(recall Definition 3.6 of the twisted center
Z
κ
).
Thus in particular they contain canonical
objects, namely the monoidal unit in
Z
(
A
) and the distinguished invertible object (see page
32) in
Z
−
4
(
A
), respectively.
We define the respective silent objects to be these two specific
objects:
℧
(
Q
A
+
) :=
1
∈ Z
(
A
)
∼
= T(
Q
A
+
)
and
℧
(
Q
A
−
) :=
D
A
∈
Z
−
4
(
A
)
∼
= T(
Q
A
−
)
.
(5.38)
In the second step we consider an arbitrary fillable gluing circle or interval
L
. There exists
(uniquely up to isomorphism) a defect surface
D
tad
L
which is a fillable disk (in the sense of
Definition 5.10) such that
L
is its outer boundary and each of its inner boundary circles is a
tadpole circle
Q
i
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- Fall '17
