PROBABLE EVALUATION OF

TOPOLOGICAL FIELD THEORIES

Frank Quinn September 1996

Abstract. Machine computations are used to show that some topological quantum field theories defined on 2-complexes are probably determined by the homology of the 2-complex. The conclusion is “probably true” in a stronger sense than just that many cases have been checked. It certainly gives a good test to determine if a theory is worth further exploration. The method gives a model for descriptions of other field theories.

0. Introduction

Topological quantum field theories were introduced in the mid 80s with great fanfare. The analytic cousins (eg. the Seiberg-Witten equation) have been very successful, but algebraic versions [RT, TV, W, KM, Y, Q2, T] have been less productive. This is at least in part because calculation has been difficult. There have been a few abstract comparisons with classical invariants, some numerical work, and many conjectures, but not much general information.

Here we consider the “baby model” field theories defined on 2-dimensional CW complexes [Q1, Q2, Q4]. These use the tensor categories defined by Gelfand and Kazhdan [GK] from mod p representations of a simple Lie algebra. Calculations have been carried out for some of the smaller categories, and in these cases invariants of complexes with two 1-cells are probably determined by the homology of the complex. It would certainly be interesting to extend this to three 1-cells, but complexes with two 1-cells are sufficiently varied and rich that this probably gives a reasonably good picture of the field theory. Indeed it is more complete than could have been expected, and it raises hopes that similar methods might apply to more complicated field theories. On the other hand the outcome is disappointing since the objective of the project is to detect properties much more subtle than homology [Q2]. The limitation to small categories is likely to be more significant than the limitation to two 1-cells so larger categories may give more interesting theories. Calculations are being undertaken to test this. Programs, data files, and results of further calculation will be posted on the project home page www.math.vt.edu/quantumtopology.

The first section gives a precise statement of the result, and the meaning of “probably determined” is explained in the proof. The second section describes a more general and flexible tool for investigating such theories, and gives the results of applying it to the computed examples.

Typeset by AMS-TEX

Ivelina Bobtcheva wrote the programs that determine the structure of the categories, cf. [B]. Luoqi Zhang recoded into C the program that evaluates field theories on specific examples, and then used this to check the hypotheses of the main result. Nathan Smith implemented much of [Q3], and the output of that program suggested the determination presented here.

1. The results

The Gelfand-Kazhdan categories used as input depend on three parameters. First, a simple Lie algebra. Next, a prime p greater than the Coxeter number of the algebra. Finally the “order of the center.” Technically this is a sublattice of the weight lattice that contains the root lattice of the algebra. The root lattice has finite index, and the parameter is the index. For example A_{2} has root lattice of index 3 in the weight lattice so there are two possibilities: the root lattice, with index 1, and the whole weight lattice with index 3. Another point of view is that representations with weights in such a sublattice correspond to representations of a Lie group realizing the given algebra, and the number in question is the order of the center of this group.

The output of the abstract construction is a topological field theory that associates vector spaces over Z/p to 1-complexes, and homomorphisms of these to 2-complexes; see [Q1, Q2, Q4]. The invariants of interest are numbers in Z/p associated to a 2-complex by considering it as a “bordism” from the empty 1-complex to the empty 1-complex.

Result. For the categories A_{2} center 1 mod 5 or 7, A_{2} center 3 mod 5, C_{2} center 1 mod 7, and G_{2} center 1 mod 11, and a 2-complex X with two 1-cells, then it is highly probable that

(1) the invariant Z_{X} is trivial if there are three or more 2-cells, or two 2-cells and H_{1}(X; Z/p) ≠ 0;

(2) in other cases with two 2-cells H_{1}(X; Z) is finite, and Z_{X} depends on the this as follows: let d denote the order of this group modulo pc (p is the prime, c is the order of the center), then

algebra  c  p  0 1 2 3 4 5 6 7 8 9 10
A_{2}  1  5  0 1 4 4 1 0
A_{2}  1  7  0 1 1 1 1 1 1 0
C_{2}  1  7  0 1 2 4 4 2 1 0
G_{2}  1  11  0 1 3 5 9 4 4 9 5 3 1
A_{2}  3  5  0 1 4 {2 \atop 1} 1 0 {3 \atop 4} 4 4 {3 \atop 4} 0

Several clarifications are needed. First, the invariants are symmetric with respect to reflection in pc/2, and this determines the invariants in the cases where the table does not go all the way up to pc. Second, there are two entries for A_{2} center 3 mod 5, and d congruent to c. In these cases the homology H_{1}(X; Z/c) is nontrivial, and has rank either 1 or 2. The two entries in the table correspond to these two possibilities. Finally the number d can be most easily computed as the determinant of the boundary homomorphism in the chain complex: see the remark at the end of the section.

For example, consider X = ⟨a, b | a^{2}b^{3}aba^{−2}b, aba^{2}b^{2}a^{3}b^{3}ab^{3}⟩. This has the same invariants as ⟨a, b | ab^{5}, a^{7}b^{9}⟩. The determinant is det ( \begin{matrix}1 & 5 \\ 7 & 9 \end{matrix} ) = −26. This is congruent to 2 mod 7, so the invariants defined by A_{2} and C_{2}, center 1 mod 7, are 1 and 2 respectively. Similarly the A_{2} center 3 mod 5 invariant is 1, and the G_{2} center 1 mod 11 invariant is 9. These statements are highly probable, but should be checked before use in life-threatening situations.

Reason. We focus on the two 2-cell case. Let R, S be attaching maps for the 2-cells (i.e. words in letters corresponding to the two 1-cells). We can write the 2-complex as a union with one 2-cell in each piece:

X = (2S^{1} ∪_{R} D^{2}) ∪_{2S^{1}} (2S^{1} ∪_{S} D^{2})

Thinking of one piece as a bordism from the empty set to 2S^{1} gives a homomorphism Z_{2S^{1}∪_{R}D^{2}} : Z(ϕ) → Z(2S^{1}). Denote the image of the unit in Z(ϕ) = Z/p under this by

Z_{R}(v) := Z_{2S^{1}∪_{R}D^{2}}(1).

General properties of a field theory provide a symmetric bilinear form λ: Z(2S^{1}) ⊗ Z(2S^{1}) → Z/p, and express the total invariant of the union in terms of the form applied to the pieces:

Z_{X} = λ(Z_{R}(v), Z_{S}(v)).

The next ingredient is “abelianization.” The abelianization of a word in two letters is the word with length two and the same total exponents. For instance the abelianization of a^{2}b^{3}a^{−1}b^{2}a^{3} is a^{r}b^{s}, with r = 2 − 1 + 3 = 4, and s = 3 + 2 = 5. Denote the abelianization nof R by “ab (R).” We want to show that the invariant of a 2-complex is the same as the invariant of the complex obtained by abelianizing the attaching maps of the cells.

Denote by J the subspace of Z(2S^{1}) generated by images Z_{R}(v) − Z_{ab (R)}(v) for all R. The probabilistic aspect of the statement involves mechanical determination of J, and will be discussed below. Suppose that

(1) λ(J, J) = 0; and

(2) λ(Z_{A}(v), J) = 0 for all abelian words A. We calculate

0 = λ(Z_{R}(v) − Z_{ab (R)}(v), Z_{S}(v) − Z_{ab (S)}(v))

= λ(Z_{R}(v), Z_{S}(v)) − λ(Z_{R}(v), Z_{ab (S)}(v)) + λ(Z_{ab (R)}(v), Z_{S}(v) − Z_{ab (S)}(v))

The first line vanishes by (1) above, while the last term vanishes by (2). This implies λ(Z_{R}(v), Z_{S}(v)) = λ(Z_{R}(v), Z_{ab (S)}(v)). Interchanging R and S and applying this again shows that this is the same as λ(Z_{ab (R)}(v), Z_{ab (S)}(v)), as required.

The computation leading to the main result is organized as follows:

(1) Determine the subspace J;

(2) verify the conditions on the form λ; and

(3) compute the invariants of abelianized complexes.

The second and third are finite computations. J is finite dimensional, and as noted in [Q1] for the categories listed the “circulator” turns out to have order pc. This implies that invariants only depend on exponents modulo pc. In particular all abelian words can be tested by testing a^{r}b^{s} for 0 ≤ r < p.

The first step, the determination of J, is not a finite problem. We proceed as follows: begin with J = 0. Then for small words R find Z_{R}(v) − Z_{ab (R)}(v) and see if it is in the currently known part of J. If not, enlarge J to include it. We find that this process stabilizes rapidly, and we guess that the result is all of J. This cannot be proved, but it can be tested. Imagine that this known part is actually a proper subspace of J. A random element will probably not be in the subspace. Therefore we compute Z_{R}(v) − Z_{ab (R)}(v) for some random words R. If they all land in the known part then it is almost certain that we know all of J.

On a huristic level the probability can be quantified. These are vector spaces over Z/p, so a proper subspace contains at most 1/p of the elements of a space. The probability that a random element will be in the subspace is thus 1/p. If we check 1,000 random elements then the probability that they are all in the subspace is 1/p^{1000}. Thus we can be quite confident that we do have the whole space. This type of test is much better than extensive searches. Here if the hypothesis is wrong we have a high probability of discovering that immediately, while searches may take a long time to find low-probability events. For example for A_{2} center 3 the second d ≅ 0 mod c case (when H_{1}(X; Z/c) has rank 2) is quite rare, but the calculations reveal it with complete certainty if all of J is known.

An algebraic topology remark: the coefficients in the abelianized words, taken mod pc, give the boundary homomorphism for the mod pc chain complex

C_{2}(X; Z/pc) ⟶^{∂} C_{1}(X; Z/pc).

There is a great temptation to think that since the invariant depends only on this, it depends only on the mod pc homology. The table shows this is wrong: it depends on integral information that can be recovered from the entries in this matrix.

2. Induced homomorphisms as traces

The demonstration of the main result can be described as: “the invariants have an unexpectedly simple structure. Once we were lucky enough to guess it, the guess could be verified with high probability.” In this section we describe the more general tool that was used to make the guess. A detailed description of the algorithms is given in [Q3].

Consider a 2-complex as a bordism from its 1-skeleton to itself. As such it induces an endomorphism of the state space of the 1-skeleton. In particular if R is a word in two letters then the complex 2S^{1} ∪_{R} D^{2} induces

Z_{R}: Z(2S^{1}) ⟶ Z(2S^{1}),

or in other words a function from the free group to the endomorphism

Z: F^{2} ⟶ End (Z2S^{1})).

Topological properties of the field theory, particularly invariance under homotopy of attaching maps of 2-cells, immediately imply:

(1) Z_{R} and Z_{S} commute;

(2) Z_{R} = Z_{R^{−1}};

(3) Z_{RS} = Z_{SR}; and

(4) Z_{R}Z_{S} = Z_{RS}Z_{S}. These together with the fact that the trivial word induces multiplication by 0 imply that Z_{R}^{2} = 0 for any R.

Now let R denote the subring of the endomorphism ring End (Z(2S^{1})) generated by the identity matrix and the image of Z. The properties above imply that R is a commutative nilpotent ring. Recall that nilpotent means there is an ideal I of codimension 1 (here generated by the image of Z) with I^{n} = 0 for some n. Z, as a function F^{2} → R, can be thought of as a sort of trace function (cf. property (3) above). In fact it can be described as a sort of character for a representation of F^{2} on Z(3S^{1}) as a module over a commutative nilpotent ring, possibly larger than R. In any case this provides an efficient way to record calculations, since R is very much smaller than the full endomorphism ring of Z(2S^{1}). Indeed in the cases studied it is substantially smaller than Z(2S^{1}).

As with the subspace J of section 1, we do not have a finite algorithm for identifying R. However as with J we can quickly identify a good candidate and use random tests to establish that it is extremely likely to be the full space.

The argument of the last section can be reformulated using images of Z_{R} and Z_{R}−Z_{ab (R)} in R instead of Z_{R}(v) etc. in Z(2S^{1}). In principle this can provide more information, but it is a corollary of the calculations that in the cases discussed in section 1 the homomorphism R → Z(2S^{1}) defined by evaluation on v is an injection. Further, in these cases I^{2} is 1-dimensional and I^{3} = 0. Invariants of 2-complexes with three cells are obtained from 3-fold products in I, so all such invariants are 0. The 1-dimensionality implies that for two 2-cells the entire homomorphism on the 1-skeleton Z(2S^{1}) is determined by the single number Z_{X}. This is a strong clue that nothing much is going on.

To give an idea of the size of the spaces involved we give a table of dimensions of Z(kS^{1}) for k = 1, 2, 3, 4 and of R. Recall, however, that these are the very small examples.

\begin{matrix}algebra & c & p & S^{1} & 2S^{1} & 3S^{1} & 4S^{1} & R \\ A_{2} & 1 & 5 & 2 & 5 & 15 & 50 & 4 \\ A_{2} & 1 & 7 & 5 & 56 & 1372 & 44,933 & 16 \\ C_{2} & 1 & 7 & 3 & 14 & 98 & 883 & 9 \\ G_{2} & 1 & 11 & 5 & 55 & 1331 & 1,449,459 & 25 \\ A_{2} & 3 & 5 & 6 & 45 & 405 & 4,050 & 20 \end{matrix}

References

[B] I. Bobtcheva, Numerical generation of semisimple tortile categories coming from quantum groups, Virginia Tech thesis,1996.

[BQ] I. Bobtcheva and F. Quinn, Numerical presentations of tortile categories, preprint 1996.

[GK] S. Gelfand and D. Kazhdan, Examples of tensor categories, Invent. Math. 109 (1992), 595–617.

[KM] R. Kirby and P. Melvin, The 3-manifold invariants of Witten and Reshetikhin-Turaev for sl(2, C), Inventiones Math. 105 (1991), 473–545.

[Q1] Frank Quinn, Lectures on axiomatic topological quantum field theory, IAS/Park City Mathematical series 1 (1995), American Math. Society, 325–453.

[Q2] , Topological quantum invariants and the Andrews-Curtis conjecture, progress report 1993.

[Q3] , Parameterization of induced homomorphisms in nilpotent field theories, Quantum Topology Project internal report 1996.

[Q4] , Categories in quantum topology I, algebra, preprint 1996.

[RT] N. Yu. Reshetikhin and V. G. Turaev, Invariants of 3-manifolds via link polynomials and quantum groups, Invent. Math. 103 (1991), 547–597.

[T] V. G. Turaev, Quantum invariants of knots and 3-manifolds, W. de Gruyter, Berlin, 1994.

[RT] V. G. Turaev and O. Viro, State sum invariants of 3-manifolds and quantum 6j-symbols, Topology.

[Y] D. Yetter, State-sum invariants of 3-manifolds associated to artinian semisimple tortile categories, Topology and Appl. 58 (1994), 47–80.