carmen caprau- twin tqfts and frobenius algebras

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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
http://find/http://goback/


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Outline

Category of singular 2-cobordisms; generators and relations

Twin Frobenius AlgebrasTwin Topological Quantum Field Theories (twin TQFTs)


Example



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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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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.


Th Si 2C b


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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:


Th t Si 2C b


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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


The category Sing 2Cob


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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


The category Sing 2Cob


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Examples of morphisms

: (0, 0) (0), : (1) (1, 1)

cozipper : (1) (0), zipper : (0) (1)

= : (0) (1, 1)


The category Sing-2Cob


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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.




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The category Sing 2CobTwin Frobenius Algebras and TQFTs


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.




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The category Sing 2CobTwin Frobenius Algebras and TQFTs


Generators

Proposition

The monoidal category Sing-2Cob is generated under compositionand disjoint union by the following cobordisms:




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g y gTwin Frobenius Algebras and TQFTs


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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Relations

3. The zipper forms an algebra homomorphism.

= =

4. The cozipper is dual to the zipper.

=


The category Sing-2CobT i F b i Al b d TQFT


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Relations

5. The image of the circle under the zipper is a central element.

=

6. The genus-one relation.

=


The category Sing-2CobT i F b i Al b d TQFT


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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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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.




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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.




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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).




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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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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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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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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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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 = .



T i TQFT T i F b i Al b


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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).



T i TQFT T i F b i Al b


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THANK YOU!


carmen caprau- twin tqfts and frobenius algebras

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