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The category Sing-2CobTwin Frobenius Algebras and TQFTs
Twin TQFTs Twin Frobenius Algebras
Twin TQFTs and Frobenius Algebras
Carmen CaprauCalifornia State University, Fresno
January 6, 2009
Carmen Caprau California State University, Fresno Twin TQFTs and Frobenius Algebras
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The category Sing-2CobTwin Frobenius Algebras and TQFTs
Twin TQFTs Twin Frobenius Algebras
Outline
Category of singular 2-cobordisms; generators and relations
Twin Frobenius AlgebrasTwin Topological Quantum Field Theories (twin TQFTs)
Twin TQFTs Twin Frobenius Algebras
Example
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The category Sing-2CobTwin Frobenius Algebras and TQFTs
Twin TQFTs Twin Frobenius Algebras
Motivation
The motivation for this project has its roots mainly in:
C. Caprau, The universalsl(2) cohomology via webs andfoams, preprint 2008, arXiv:math.GT/0802.2848.
A. Lauda, H. Pfeiffer, Open-Closed strings: Two-dimensionalextended TQFTs and Frobenius algebras, Topology Appl. 155
(2008) no. 7, 623666. Preprint arXiv:math.AT/0510664.
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The category Sing-2CobTwin Frobenius Algebras and TQFTs
Twin TQFTs Twin Frobenius Algebras
Singular 2-cobordisms
Definition
A singular 2-cobordism is an abstract piecewise oriented smooth2-dimensional manifold with boundary = +, where is with opposite orientation. Both and + are
embedded 1-dimensional closed manifolds and they are called thesource and target boundaries, respectively.
Definition
The two singular 2-cobordisms 1 and 2 are consideredequivalent, and we write 1 = 2, if there exists anorientation-preserving diffeomorphism 1 2 which restricts tothe identity on the boundary.
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Th Si 2C b
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The category Sing-2CobTwin Frobenius Algebras and TQFTs
Twin TQFTs Twin Frobenius Algebras
Singular 2-cobordisms
A singular cobordism has singular arcs and/or singular circleswhere orientations disagree.
Two neighboring facets of a singular cobordism arecompatible oriented, inducing an orientation on the singulararc/circle that they share.
Examples:
Carmen Caprau California State University, Fresno Twin TQFTs and Frobenius Algebras
Th t Si 2C b
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The category Sing-2CobTwin Frobenius Algebras and TQFTs
Twin TQFTs Twin Frobenius Algebras
Objects in the category Sing-2Cob
An object in Sing-2Cob is diffeomorphic to a disjoint union oforiented circles and closed graphs, called bi-webs.
An object consists of a finite sequence n = (n1, n2, . . . , nk)where nj {0, 1}.
0 = 1 = bi-web
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The category Sing 2Cob
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The category Sing-2CobTwin Frobenius Algebras and TQFTs
Twin TQFTs Twin Frobenius Algebras
Morphisms in the category Sing-2Cob
A morphism : n m is an equivalence class of singular2-cobordisms (induced by =) with source boundary n andtarget boundary m.
As a morphism, we read a cobordism from top to bottom,by convention.
Two morphisms 1 : n m and 2 : m k are composedby gluing 1 on top 2, along m:
2 1 : n k
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The category Sing 2Cob
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The category Sing-2CobTwin Frobenius Algebras and TQFTs
Twin TQFTs Twin Frobenius Algebras
Examples of morphisms
: (0, 0) (0), : (1) (1, 1)
cozipper : (1) (0), zipper : (0) (1)
= : (0) (1, 1)
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The category Sing-2Cob
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The category Sing-2CobTwin Frobenius Algebras and TQFTs
Twin TQFTs Twin Frobenius Algebras
The structure of the category Sing-2Cob
The category Sing-2Cob is a symmetric monoidalcategory:
the concatenation n m := (n1, n2, . . . , n|n|, m1, m2, . . . , m|m|)of sequences together with the free union of singular2-cobordisms endows the category Sing-2Cob with thestructure of a symmetric monoidal category.
|n| and |m| above are the lengths of sequences n and m.
Carmen Caprau California State University, Fresno Twin TQFTs and Frobenius Algebras
The category Sing-2Cob
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The category Sing 2CobTwin Frobenius Algebras and TQFTs
Twin TQFTs Twin Frobenius Algebras
Generators and relations
Using Morse theory, we obtain a generators and relationsdescription of the category Sing-2Cob.
Each singular 2-cobordism can be decomposed into singularcobordisms each of which contains exactly one critical point.
The components of such a decomposition are the generators
for the morphisms of this category.
Carmen Caprau California State University, Fresno Twin TQFTs and Frobenius Algebras
The category Sing-2Cob
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The category Sing 2CobTwin Frobenius Algebras and TQFTs
Twin TQFTs Twin Frobenius Algebras
Generators
Proposition
The monoidal category Sing-2Cob is generated under compositionand disjoint union by the following cobordisms:
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The category Sing-2Cob
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g y gTwin Frobenius Algebras and TQFTs
Twin TQFTs Twin Frobenius Algebras
Relations
1. The oriented circle forms a commutative Frobeniusalgebra object.
= = =
= = =
= =
=
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Twin Frobenius Algebras and TQFTsTwin TQFTs Twin Frobenius Algebras
Relations
2. The bi-web forms a symmetric Frobenius algebra object.
= = =
= = =
= =
=
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The category Sing-2Cob
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Twin Frobenius Algebras and TQFTsTwin TQFTs Twin Frobenius Algebras
Relations
3. The zipper forms an algebra homomorphism.
= =
4. The cozipper is dual to the zipper.
=
Carmen Caprau California State University, Fresno Twin TQFTs and Frobenius Algebras
The category Sing-2CobT i F b i Al b d TQFT
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Twin Frobenius Algebras and TQFTsTwin TQFTs Twin Frobenius Algebras
Relations
5. The image of the circle under the zipper is a central element.
=
6. The genus-one relation.
=
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Twin Frobenius Algebras and TQFTsTwin TQFTs Twin Frobenius Algebras
Extended Frobenius Algebras
DefinitionA twin Frobenius algebra C := (C, W, z, z) consists of
a commutative Frobenius algebra C = (C, mC, C, C, C),
a symmetric Frobenius algebra W = (W, mW, W, W, W),
two morphisms z: C W and z : W C
such that z is a homomorphism of algebra objects and
C mC (IdC z) = W mW (z IdW) (duality)
mW(IdW z) = mWW,W(IdW z) (centrality condition)
z mC C z = mW W,W W. (genus-one condition)
In particular, z is a homomorphism of coalgebras in C.
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Twin Frobenius Algebras and TQFTsTwin TQFTs Twin Frobenius Algebras
Extended Frobenius Algebras
Definition
A homomorphism of twin Frobenius algebras
f : (C1, W1, z1, z1 ) (C2, W2, z2, z
2 )
consists of a pair f = (f1, f2) of Frobenius algebra homomorphismsf1 : C1 C2 and f2 : W1 W2 such that z2 f1 = f2 z1 andz2 f2 = f1 z
1 .
Proposition
The category T-Frob of twin Frobenius algebras and theirhomomorphisms is a symmetric monoidal category.
Carmen Caprau California State University, Fresno Twin TQFTs and Frobenius Algebras
The category Sing-2CobTwin Frobenius Algebras and TQFTs
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Twin Frobenius Algebras and TQFTsTwin TQFTs Twin Frobenius Algebras
Extended TQFTs
Consider ModR, the category of R-modules and modulehomomorphisms, where R is some commutative ring.
Definition
A twin Topological Quantum Field Theory (twin TQFT) is asymmetric monoidal functor Sing-2Cob ModR.A homomorphism of twin TQFTs is a monoidal naturaltransformation of such functors.
Definition
We denote by T-TQFT the category of twin TQFTs and theirhomomorphisms.
Carmen Caprau California State University, Fresno Twin TQFTs and Frobenius Algebras
The category Sing-2CobTwin Frobenius Algebras and TQFTs
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Twin Frobenius Algebras and TQFTsTwin TQFTs Twin Frobenius Algebras
Theorem
The category Sing-2Cob of singular 2-cobordisms is equivalent tothe symmetric monoidal category freely generated by a twinFrobenius algebra.
Corollary
The category T-Frob of twin Frobenius algebras is equivalent tothe category T-TQFT of twin TQFTs (as symmetric monoidalcategories).
Carmen Caprau California State University, Fresno Twin TQFTs and Frobenius Algebras
The category Sing-2CobTwin Frobenius Algebras and TQFTs
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Twin Frobenius Algebras and TQFTsTwin TQFTs Twin Frobenius Algebras
Example
Consider the ring R = Z[i][a, h],where i is so that i2 = 1 and a and h are formal variables.
Consider A = R[X]/(X2 hX a) =< 1, X >R.
Construct a twin Frobenius Algebra (AC, AW, z, z)
AC = (A, mC, C, C, C) is commutative Frobenius;AW = (A, mW, W, W, W) is commutative (thus symmetric)Frobenius;
z: AC AW,
z(1) = 1
z(X) = Xis a homomorphism of
Frobenius algebra objects;
z : AW AC,
z(1) = i
z(X) = iXis a homomorphism of
Frobenius coalgebra objects, dual to z.
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The category Sing-2CobTwin Frobenius Algebras and TQFTs
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g QTwin TQFTs Twin Frobenius Algebras
Example
Structure maps on AC and AW:
unit maps:
C : R A, (1) = 1
W : R A, (1) = 1counit maps:
C : A R,
C(1) = 0
C(X) = 1
W : A R,
W(1) = 0W(X) = i
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The category Sing-2CobTwin Frobenius Algebras and TQFTs
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gTwin TQFTs Twin Frobenius Algebras
Example
Structure maps on AC and AW:
multiplication maps: mC,W : A A A
mC = mW : mC(1 X) = mC(X 1) = X
mC(1 1) = 1, mC(X X) = hX + a
comultiplication maps: C,W : A A A
C : C(1) = 1 X + X 1 h1 1
C(X) = X X + a1 1
W :
W(1) = i(1 X + X 1 h1 1)
W(X) = i(X X + a1 1)
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The category Sing-2CobTwin Frobenius Algebras and TQFTs
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Twin TQFTs Twin Frobenius Algebras
Remark on the two structures defined on A
Note that AW is a twisting of AC:
i.e. comultiplication W and counit W are obtained from Cand C by twisting them by the invertible element i A:
W(x) = C(ix) for all x A,
W(x) = C((i)1x) = C(ix) for all x A.
Kadison showed that twisting by invertible elements of A isthe only way to modify the counit and comultiplication inFrobenius extensions.
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Twin TQFTs Twin Frobenius Algebras
Example
Twin Frobenius algebra (AC, AW, z, z) gives rise to a twin TQFT:
T: Sing-2Cob ModR
On objects, T is defined as follows:T(empty manifold) = R.
T(n) = Aki , for n = (n1, n2, . . . , nk) a k-component object,where
Ai = AC for ni = 0 = ,Ai = AW for ni = 1 = .
Carmen Caprau California State University, Fresno Twin TQFTs and Frobenius Algebras
The category Sing-2CobTwin Frobenius Algebras and TQFTs
T i TQFT T i F b i Al b
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Twin TQFTs Twin Frobenius Algebras
Example
On the generating morphisms, T is defined as follows:
T: C T: mC T: C T: C
T: W T: mW T: W T: W
T: z T: IdAC T: z T: IdAW
Remark: (AC, AW, z, z) is almost a twin Frobenius algebra (all
properties of such an algebra are satisfied except fro thegenus-one condition which holds up to a sign).
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The category Sing-2CobTwin Frobenius Algebras and TQFTs
T i TQFT T i F b i Al b
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THANK YOU!
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