Analysis of the algebraic structure

5. Two-dimensional lattice topological field theory

b= c−1 = N = N^ε = ε

µ= η= ∆ = ε =

Figure 5.2.: Abbreviated graphical notation for frequently used morphisms. The pairingband copairingc−1 give a duality onA, see (5.15). The Nakayama automorphismN is defined in (5.16). The product µ is given in (5.21), the existence of the unitη as assumed in Assumption 2, the coproduct ∆ and counitεare defined in (5.27).

agree, by Theorem 4.18 we must have (d_[1], s₁) ∼_fix (d_[2b], s_2b), where d_[1] := (d¹_[1]0, d²_[1]0) and dito for d_[2b]. By Relations 1–3, T_triang is constant on equivalence classes for ∼_fix. Thus finally

T_triang(Λ_2b) = T_triang(Λ₁) . (5.14)

5.3. Analysis of the algebraic structure One of the two computations to verify this is a follows:

N⁻¹◦N = ^deform= ^(5.15)= ^(5.15)= idA . (5.19)

The map N has an additional important property.

Lemma 5.2. Forr <∞, the map N^r acts trivially on the map t:

t◦(N^r⊗idA⊗A) = t◦(id_A⊗N^r⊗id_A) = t◦(idA⊗A⊗N^r) =t . (5.20)

Proof. We verify the first equality. The other cases then follow from the cyclic property of t stated in (5.9). We have:

t

N^r

(1)=

b t b b

c−1 c−1 c−1

N^r−1

(2)=

b t b b

c₀ c₀ c₀ N^r−1

(3)=

t b b

c0 c0

N^r−1

(4)=

t b b

c−1+r c−1+r

(5)= t

In step 1 we used the non-degeneracy of c−1 (Assumption 1) to insert two pairs b, c−1. We also replaced one of the N’s by its definition in (5.16). Step 2 is the leaf exchange (5.6). In step 3 we use the edge exchange, Equation (5.5), on the leftmost b₁,c₀ pair and replace it by id_A using Assumption 1. In step 4 the steps 2 and 3 are replicated r−1 more times. Finally, in step 5 the remaining pairs b₁, c₁ are cancelled via Assumption 1.

5. Two-dimensional lattice topological field theory

We would like to cast the data t, c_k into a more standard algebraic form, namely that of a Frobenius algebra. We recall that aFrobenius algebra (in a monoidal category) is a unital associative algebra and a counital coassociative coalgebra such that the coproduct is a map of bimodules.

We start by introducing a product: Let

µ= (t⊗id_A)◦(idA⊗A⊗c−1) : A⊗A→A . (5.21) In graphical notation, this reads

µ= t . (5.22)

Forµ we will use the graphical shorthand listed in Figure 5.2.

Lemma 5.3. The map µ:A⊗A→A is associative.

Proof. Using non-degeneracy ofc−1 and (5.5), we can rewrite the cyclicity property (5.9) as an identity of morphisms A^⊗3 →1S:

t

= t

(5.23)

To see this write outN and add pairs ofc−1,b on the right hand side. The same can be done for the Pachner moves (5.10) and (5.11); we omit the details. We then compute:

=

t t

c₋₁ c₋₁

(5.9)⁻¹

=

t t

c₀ c₋₁

(5.24)

(5.23)

=

t t

c₀ c−1

(5.10)

=

t t

c₀ c−2

(5.6)

=

t t

c₀ c₋₁

(5.9)⁻¹

=

t t

c₋₁ c₋₁

=

64

5.3. Analysis of the algebraic structure In the second and third equality we used the cyclicity property (in two different versions). The fourth equality is the relation from the Pachner 2-2 move: Start with the right hand side, expand the Ns and then use (5.10). The remaining cs are removed by (5.5) and nondegeneracy. The fifth equality is (5.6) used on both ts with N = 1.

Lemma 5.4. The pairingb is invariant with respect to the product µ, i.e.

b◦(µ⊗id_A) = b◦(id_A⊗µ) . (5.25)

Proof. By direct computation:

=

t b

c−1

(5.9)⁻¹

= b t

c0

(5.26)

nondeg.

=

t

nondeg.

=

t b

c−1

=

To proceed, we need to make our second assumption:

Assumption 2: The algebra (A, µ) has a unitη :1S →A.

The graphical notation we use for the unit is listed in Figure 5.2. By Lemma 5.4 and Assumption 2, A together with the (non-degenerate) pairing b is a Frobenius algebra.

One may now define a coalgebra structure on the Frobenius algebraAin the standard way, so that the coproduct is a map of A-A-bimodules. Explicitly, the coproduct ∆ : A →A⊗A and counitε :A →1 are given by

∆ = , ε= . (5.27)

The asymmetry in this definition is only apparent, since

∆^nondeg.= ^ass.= ^nondeg.= ,

ε^unit= ^inv.= ^unit= .

(5.28)

5. Two-dimensional lattice topological field theory

It is now trivial to see that ε is indeed a counit. Coassociativity is easily checked by combining the two expressions for ∆:

(∆⊗id_A)◦∆ = = = (id_A⊗∆)◦∆ . (5.29)

The Frobenius property, which states that ∆ is a bimodule map, namely

= = , (5.30)

is equally straightforward to check. We omit the details. Finally, note that

b =ε◦µ , c−1 = ∆◦η . (5.31)

From hereon we consider A as a Frobenius algebra with structure morphisms µ, η,∆, ε as described above. By definition, the morphism N defined in (5.16) is the Nakayama automorphism of A, see e.g. [FSt]. For completeness we state

Proposition 5.5. The Nakayama automorphism is a unital algebra automorphism and a counital coalgebra automorphism of a Frobenius algebra.

Proof. ThatN◦η =η andε◦N =ε is straightforward. Compatibility with the product follows from

N ◦µ= ^ass.= ^Frob.= ^coass.=

def. of ∆

= ^deform= =µ◦(N ⊗N).

(5.32)

To see compatibility with the coproduct, first note that N⁻¹

= = = =

N

. (5.33)

When combining this with the definition of the coproduct in terms of the product and copairing in (5.27), the compatibility of N with the coproduct follows from the already established result that N is an algebra homomorphism.

66

5.3. Analysis of the algebraic structure

The following identity will be used frequently in the calculations below:

Lemma 5.6. LetA be a Frobenius algebra and N its Nakayama automorphism. Then id⊗(µ◦σ_A,A)

◦

∆⊗id

=

µ⊗id

◦

id⊗σ_A,A

◦

∆⊗N

. (5.34)

Graphically, this reads

= . (5.35)

Proof. By direct calculation:

Frob.

= ^coass.= ^Frob.= ^deform= =

Proposition 5.5 and Lemma 5.6 hold in general. For the Frobenius algebra A con-structed above from t, c_k we have in addition:

Lemma 5.7. Forr <∞, the Nakayama automorphism of A satisfiesN^r = id_A. Proof. We use the unit property, nondegeneracy and Lemma 5.2:

N^r = (t⊗id_A)◦(N^r⊗η⊗c−1)^5.2= (t⊗id_A)◦(id_A⊗η⊗c−1) = id_A . (5.36)

We call a Frobenius algebra ∆-separable if µ◦∆◦η =η, i.e. ∆◦η is a separability idempotent. We have:

Lemma 5.8. A is ∆-separable.

Proof. First note that

(5.26)

= t

. (5.37)

The statement follows from the identities

Frob.

= ^nondeg.= (5.38)

5. Two-dimensional lattice topological field theory

unit=

=

c₀ c−1 c₀

t t t

(5.11)

=

t

= = .

We have now arrived at the desired algebra structure encoding t, c_k and their proper-ties. As described above, under Assumptions 1 and 2, the datat, c_k, subject to relations 1–5 in Section 5.2, give rise to a ∆-separable Frobenius algebra whose Nakayama auto-morphism fulfils N^r= id for r <∞. The following result shows that the converse holds as well; the proof can be found in Appendix C.

Proposition 5.9. Let A be a ∆-separable Frobenius algebra whose Nakayama auto-morphism is an involution. Set

t =ε◦µ◦(µ⊗id_A) , c_k= (id_A⊗N^k+1)◦∆◦η . (5.39) Then t, c_k fulfil relations 1–5 in Section 5.2.

With the tools assembled so far, we can prove the main result of this thesis. To state the result, we need a little bit more notation. Let B be the number of boundary components of the givenr-spin surface. We would like to think of the morphism assigned to this r-spin surface as a “correlator”, that is, we prefer to write it as a morphism A^⊗3B → 1S rather than the other way around as is the case for T_triang in (5.3). We use the map b to achieve this and define

T_A(Σ) := (b^⊗3B)◦τ◦ id_A^⊗3B ⊗T_triang(C,ϕ,˜ Σ)

, (5.40)

whereτ is a permutation (A^⊗3B⊗A^⊗3B)→(A^⊗2)^⊗3B, which connects thei’th factor of Ain the first (resp. second) copy ofA^⊗3B in the source object to the first (resp. second) copy of A in the i’th factor ofA^⊗2 in the target object.

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5.4. Behaviour of the morphisms under gluing ofr-spin surfaces

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