Sketch of proof

Let Z denote the 2+1-dimensional TQFT associated to the super pivotal category C. Let Y be a closed spin surface with Hilbert space Z(Y), and let p|q = dim(Z(Y)). Then we have

p+q = tr(id :Z(Y)→Z(Y)) =Z(Y ×S1

B) (5.47)

and

p−q = tr((−1)F :Z(Y)→Z(Y)) =Z(Y ×S_N1). (5.48) More generally, ifY has nonempty boundary and the boundary components are labeled by

a1, . . . , ak (minimal idempotents of the tube category), then

p+q = tr(id :Z(Y;a1, . . . , ak)→Z(Y;a1, . . . , ak)) (5.49) =Z(Y ×S_B1)(clB(a1)t · · · tclB(ak)) (5.50)

and

p−q= tr((−1)F :Z(Y;a1, . . . , ak)→Z(Y;a1, . . . , ak)) (5.51) =Z(Y ×S1_N)(clN(a1)t · · · tclN(ak)). (5.52)

Here clX(ai) denotes the element ofZ(T_{U X}2 ) obtained by closing up the idempotentai, and

U is the spin structure (B orN) on thei-th boundary component. Note that ifaiis q-type, then clN(ai)∈Z(T_{N N}2 ) is zero. But in this case we also know that p−q= 0, since the odd endomorphism ofai maps the even part ofZ(Y;a1, . . . , ak) isomorphically to the odd part.

Recall that we can define reduced 1+1-dimensional TQFTs Z_S1

B andZSN1 via Z_S1 B(M) =Z(M ×S 1 B) ZS1 N(M) =Z(M×S 1 N), (5.53) where M is a manifold of dimension 0, 1 or 2. Combining the above we have, for closed spin surfacesY, p+q =Z_S1 B(Y), (5.54) and p−q=Z_S1 N(Y), (5.55)

and there are similar formulas whenY has boundary.

We can now outline the remainder of the proof of the dimension formula. We have just seen that the super dimension p|q can be calculated entirely in terms of the reduced 1+1-dimensional TQFTs Z_S1

B and ZSN1. Because 1 is a small number, we can completely classify 1+1-dimensional spin TQFTs and give an explicit expression for the path integral in terms of basic structure constants of the 1+1-dimensional TQFT. Then all that remains to be done is express the structure constants of the 1+1-dimensional TQFTs in terms of the structure constants of the original 2+1-dimensional TQFT Z. It turns out that the only structure constants we will need are the list of minimal idempotents of the tube category, their types (m or q), and the S-matrix.

Spin 1+1-dimensional TQFTs are determined by two pieces of data: the cylinder cate- gory of a point, which is a linear super categoryC, and a non-degenerate trace on C. The trace is the path integral of the diskD2_{; iff is an endomorphism ofC, then}

tr(f) =ZC,tr(D2)(cl(f)), (5.56)

where, as usual, cl(f) denotes the closure off, an element of the (pre-dual) Hilbert space associated to S1

B = ∂D2. The non-degenerate trace implies that C is semisimple. Up to Morita equivalence, there are only two indecomposable semisimple super categories, the trivial algebraCand the complex Clifford algebraC`1.

We first consider the case C =C. Let ebe the unique minimal idempotent of C (i.e.

e= 1∈_C). Let λ= tr(e). Let Y be a spin surface with kboundary components, and let cl(e)t · · · tcl(e) denote the boundary condition given by placing cl(e) on each boundary component ofY. The path integral for the TQFT determined by (C, λ) is

Z_C,λ(Y)(cl(e)t · · · tcl(e)) =λχ(Y), (5.57)

whereχ(Y) denotes the Euler characteristic ofY.

Next we consider the case C =C`1. Again let e be the unique minimal idempotent of

C`1. Letλ= tr(e). LetY be a spin surface withk boundary components. We will assume

case we will need for the dimension formula. Let

ˆ

e= √1

2e (5.58)

be the normalized idempotent. (The norm of cl(ˆe) in A(S1

B) is 1.) The path integral for the TQFT determined by (C`1, λ) is Z<sub>C</sub>`1,λ(Y)(cl(ˆe)t · · · tcl(ˆe)) = (−1) Arf(Y) λ √ 2 χ(Y) , (5.59)

whereχ(Y) denotes the Euler characteristic ofY, and Arf(Y) is the Arf invariant ofY with its boundary components capped off by disks.

A general 1+1-dimensional spin TQFT is a direct sum of instances of the two theories described above.

All that remains to be done is to write the reduced theories Z_S1

B and ZSN1 as a direct sum of the (C, λ) and (C`1, λ) theories described above.

The first task is to obtain a list of the idempotents (and their type, m or q) of the minimal idempotents ofZ_S1

B and ZS1N. This is easily done: the category whichZSB1 assigns to a point is TubeB, and the category which Z_S1

N assigns to a point is Tube

N_{. The m-} type idempotents correspond to (C, λ) theories, and the q-type idempotents correspond to (C`1, λ) theories.

The second task is to determine, for each idempotent a in TubeB and TubeN, the number λ above (path integral of the disk evaluated on a closed-up idempotent). This is exactly theS-matrix entryS1a ifais m-type. Ifais q-type, then (since we have normalized theS-matrix)S1a is equal toλ/

√

2, but this is the value we need for (5.59).

The third and final task is to convert the cl(ai) boundary conditions from the beginning of 5.4.1 to the cl(e) and cl(ˆe) boundary conditions appearing in (5.57) and (5.59). Both of these boundary conditions are (after undoing the dimensional reduction along S1

B or

S1

N) vectors in A(T2), and both boundary conditions are closed-up idempotents (possibly normalized with a factor of 1/√2). But the cl(ai) boundary condition cuts the torus along a longitude, while the cl(e) and cl(ˆe) boundary conditions cut the torus along a meridian. So we need to apply the S-matrix (actuallyS0_{, because ˆeis normalized while a}

C2 SO(3)6/ψ 1₂E6/y g= 1,Arf = 0 3|0 4|0 3|0 g= 1,Arf = 1 0|3 2|2 1|2 g= 2,Arf = 0 10|0 40|24 19|8 g= 2,Arf = 1 0|10 32|32 11|16 g= 3,Arf = 0 36|0 1184|1120 281|232 g= 3,Arf = 1 0|36 1152|1152 241|272 g= 4,Arf = 0 136|0 51328|51072 5755|5504 g= 4,Arf = 1 0|136 51200|51200 5531|5728 g= 5,Arf = 0 528|0 2368000|2366976 126449|125056 g= 5,Arf = 1 0|528 2367488|2367488 125137|126368 Figure 5.3: Hilbert space dimensions for closed surfaces in various theories.

change basis:

cl(ai) =

X

x

S_a0_ixcl(ˆx), (5.60) where for convenience we have defined ˆx=x ifx is m-type.

Combining (5.54), (5.55), (5.57), (5.59), and (5.60) yields the dimension formula.