THERE IS NO “THEORY OF EVERYTHING” INSIDE E_{8}

JACQUES DISTLER AND SKIP GARIBALDI

Abstract. We analyze certain subgroups of real and complex forms of the Lie group E_{8}, and deduce that any “Theory of Everything” obtained by embedding the gauge groups of gravity and the Standard Model into a real or complex form of E_{8} lacks certain representationtheoretic properties required by physical reality. The arguments themselves amount to representation theory of Lie algebras along the lines of Dynkin’s classic papers and are written for mathematicians.

1. Introduction

Recently, the preprint [1] by Garrett Lisi has generated a lot of popular interest. It boldly claims to be a sketch of a “Theory of Everything”, based on the idea of combining the local Lorentz group and the gauge group of the Standard Model in a real form of E_{8} (necessarily not the compact form, because it contains a group isogenous to SL(2, ℂ)). The purpose of this paper is to explain some reasons why an entire class of such models—which include the model in [1]—cannot work, using mostly mathematics with relatively little input from physics.

The mathematical set up is as follows. Fix a real Lie group E. We are interested in subgroups SL(2, ℂ) and G of E so that:

(ToE1)  G is connected, reductive, compact, and centralizes SL(2, ℂ)

We complexify and decompose Lie(E) ⊗ ℂ as a direct sum of representations of SL(2, ℂ) and G. We identify SL(2, ℂ) × ℂ with SL_{2,ℂ} × SL_{2,ℂ} and write

(1.1)  Lie(E) = ⨁_{m,n≥1} m ⊗ n ⊗ V_{m,n}

where m and n denote the irreducible representation of SL_{2,ℂ} of that dimension and V_{m,n} is a complex representation of G × ℂ. (Physicists would usually write 2 and ¯2 instead of 2 ⊗ 1 and 1 ⊗ 2.) Of course,

\overline{m ⊗ n ⊗ V_{m,n}} ≃ n ⊗ m ⊗ \overline{V_{m,n}}

and since the action of SL(2, ℂ).G on Lie(E) is defined over ℝ, we deduce that \overline{V_{m,n}} ≃ V_{n,m}. We further demand that

(ToE2) V_{2,1} is a complex representation of G, and

V_{m,n} = 0 if m + n > 4. (ToE3)

We recall the definition of complex representation and explain the physical motivation for these hypotheses in the next section. Roughly speaking, (ToE1) is a trivial requirement based on trying to construct a Theory of Everything along the lines suggested by Lisi, (ToE2) is the statement that the gauge theory (with gauge group G) is chiral, as required by the Standard Model, and (ToE3) is the requirement that the model not contain any “exotic” higher-spin particles. In fact, physics requires slightly stronger hypotheses on V_{m,n}, for m + n = 4. We will not impose the stronger version of (ToE3).

2000 Mathematics Subject Classification. Primary 22E47; secondary 17B25, 81R05, 83E15.

Report #: UTTG-05-09, TCC-020-09.

Definition 1.2. A ToE subgroup of a real Lie group E is a subgroup generated by a copy of SL(2, ℂ) and a subgroup G such that (ToE1), (ToE2), and (ToE3) hold.

Our main result is:

Theorem 1.3. There are no ToE subgroups in (the transfer of) the complex E_{8} nor in any real form of E_{8}.

Notation. Unadorned Lie algebras and Lie groups mean ones over the real numbers. We use a subscript ℂ to denote complex Lie groups—e.g., SL_{2,ℂ} is the (complex) group of 2-by-2 complex matrices with determinant 1. We can view a d-dimensional complex Lie group G_{ℂ} as a 2d-dimensional real Lie group, which we denote by R(G_{ℂ}). (Algebraists call this operation the “transfer” or “Weil restriction of scalars”.) We use the popular notation of SL(2, ℂ) for the transfer R(SL_{2,ℂ}) of SL_{2,ℂ}; it is a double covering of the “restricted Lorentz group”, i.e., of the identity component SO(3, 1)_{0} of SO(3, 1). 1.4. Strategy and main results. Our strategy for proving Theorem 1.3 will be as follows. We will first catalogue, up to conjugation, all possible embedding of SL(2, ℂ) satifying the hypotheses of (ToE3). The list is remarkably short. Specifically, every ToE subgroup of E is contained in SL(2, ℂ) · G_{max}, where G_{max} is the maximal compact, connected, reductive subgroup of the centralizer of SL(2, ℂ) in E. The proof of Theorem 1.3 shows that the only possibilities are:

E  G_{max}
E_{8(8)}  Spin(5) × Spin(7)
E_{8(−24)} Spin(11) or Spin(9) × Spin(3)
R(E_{8,ℂ})  E_{7}, Spin(13) or Spin(12).

We then note that the representation, V_{2,1}, of G_{max} (and hence, of any G ⊆ G_{max}) has a self-conjugate structure. In other words, (ToE2) fails.

2. Physics background

One of the central features of modern particle physics is that the world is described by a chiral gauge theory. 2.1. Let M a four dimensional pseudo-Riemannian manifold, of signature (3, 1), which we will take to be oriented, time-oriented and spin. Let G be a compact Lie group. The data of a gauge theory on M, with gauge group G consists of a connection, A, on a principal G-bundle, P → M, and some “matter fields” transforming as sections of vector bundle(s) associated to unitary representations of G.

Of particular interest are the fermions of the theory. The orthonormal frame bundle of M is a principal SO(3, 1)_{0} bundle. A choice of spin structure defines a lift to a principal Spin(3, 1)_{0} = SL(2, ℂ) bundle. Let S_{±} → M be the irreducible spinor bundles, associated, via the defining two-dimensional representation and its complex conjugate, to this SL(2, ℂ) principal bundle.

The fermions of our gauge theory are denoted

ψ ∈ Γ(S_{+} ⊗ V), \overline{ψ} ∈ Γ(S_{−} ⊗ \overline{V})

where V → M is a vector bundle associated to a (typically reducible) representation, R, of G.

Definition 2.2. A real structure on a representation V (over ℂ) is an antilinear map, J : V → V, satisfying J^{2} = 1. Physicists call a representation possessing a real structure real.

A quaternionic structure on a representation V (over ℂ) is an antilinear map, J : V → V, satisfying J^{2} = −1. Physicists call a representation possessing a quaternionic structure pseudoreal.

Subsuming these two subcases, we will say that a representation V (over ℂ) has a selfconjugate structure if there is an antilinear map J : V → V, satisfying J^{4} = 1. Physicists call a representation V, which does not possess a self-conjugate structure, complex.

For an alternative view, we suppose that V is an irreducible representation (over ℂ) of a real reductive Lie group G and exploit [2, §7]. If V has a real structure J, then the subset V′ of elements of V fixed by J is a real vector space that is a representation of G such that End_{G}(V′) = ℝ and V′ ⊗ ℂ is canonically identified with V. A quaternionic structure on V defines a real structure on V ⊕ V via (v_{1}, v_{2}) ↦ (Jv_{2}, −Jv_{1}) such that (V ⊕ V)′ is irreducible with End_{G}((V ⊕ V)′) ≅ ℍ. If V is complex, then V ⊕ \overline{V} has an essentially unique real structure and End_{G}((V ⊕ \overline{V})′) ≅ ℂ.

Definition 2.3. A gauge theory, with gauge group G, is said to be chiral if the representation, R by which the fermions (2.1) are defined, is complex in the above sense. By contrast, a gauge theory is said to be nonchiral if the representation R in 2.1 has a self-conjugate structure.

Note that whether a gauge theory is chiral depends crucially on the choice of G. A gauge theory might be chiral for gauge group G, but nonchiral for a subgroup H ⊂ G, because there exists a self-conjugate structure on R, compatible with H, even though no such structure exists, compatible with the full group G.

Conversely, suppose that a gauge theory is nonchiral for the gauge group G. It is also necessarily nonchiral for any gauge group H ⊂ G.

2.4. GUTs. The Standard Model is a chiral gauge theory with gauge group

G_{SM} := (SU(3) × SU(2) × U(1))/(ℤ/6ℤ)

Various grand unified theories (GUTs) proceed by embedding G_{SM} is some (usually simple) group, G_{GUT}. Popular choices for G_{GUT} are SU(5) [3], Spin(10), E_{6}, and the Pati-Salam group, (Spin(6) × Spin(4))/(ℤ/2ℤ) [4].

It is easiest to explain what the fermion representation of G_{SM} is after embedding G_{SM} in G_{GUT} := SU(5). Let W be the five-dimensional defining representation of SU(5). The representation R from 2.1 is the direct sum of three copies of

R_{0} = ∧^{2}W ⊕ \overline{W}

Each such copy is called a “generation” and is 15-dimensional. One identifies each of the 15 weights of R_{0} with left-handed fermions: 6 quarks (two in a doublet, each in three colors), two leptons (e.g., electron and its neutrino), 6 antiquarks, and a positron. With three generations, R is 45-dimensional.

For the other choices of GUT group, the analogue of a generation (R_{0}) is higherdimensional, containing additional fermion which are not seen at low energies. When decomposed under G_{SM} ⊂ G_{GUT}, the representation decomposes as R_{0} + R′, where R′ is a real representation of G_{SM}. In Spin(10), a generation is the 16-dimensional half-spinor representation. In E_{6}, it is the 27, and for the Pati-Salam group, it is the (4, 1, 2) ⊕ (\overline{4}, 2, 1) representation. In each case, these representations are complex representations (in the above sense) of G_{GUT}, and the complex-conjugate representation is called an “anti-generation.”

So far, we have described a chiral gauge theory in a fixed (pseudo) Riemannian structure on M. Lisi’s proposal [1] is to try to combine the spin connection on M, and the gauge connection on P into a single dynamical framework. This motivates Definition 1.2 of a ToE subgroup.

Fix a ToE subgroup—say, with G = G_{SM}—in some real Lie group E. The action of central element −1 ∈ SL(2, ℂ) provides a ℤ/2ℤ grading on the Lie algebra of E. This ℤ/2ℤ grading allows one to define a sort of superconnection associated to E (precisely what sort of superconnection is explained in a blog post by the first author [5].) In the proposal of [1], we are supposed to identify each of the generators of Lie(E) as either a boson or a fermion. The Spin-Statistics Theorem [6] says that fermions transform as spinorial representations of Spin(3, 1); bosons transform as “tensorial” representations (representation which lift to the double cover, SO(3, 1)). To be consistent with the Spin-Statistics Theorem, we must, therefore, require that the fermions belong to the −1-eigenspace of the aforementioned ℤ/2ℤ action, and the bosons to the +1-eigenspace.

In fact, to agree with 2.1, we should require that the −1 eigenspace (when tensored with ℂ) decompose as a direct sum of two-dimensional representations (over ℂ) of SL(2, ℂ), corresponding to “left-handed” and “right-handed” fermions, in the sense of 2.1.

Remark 2.5. In the language of (ToE3), m + n = odd are fermions and m + n = even are bosons. In Lisi’s setup, the bosons are 1-forms on M, with values in a vector bundle associated to the aforementioned Spin(3, 1)_{0} principal bundle via the m⊗n representation (with m + n even). The case m + n = 4 is special; these correspond to the gravitational degrees of freedom in Lisi’s theory. (3 ⊗ 1) ⊕ (1 ⊗ 3) is the adjoint representation; these correspond to the spin connection. The 1-form with values in the 2 ⊗ 2 representation is the vierbein. It is a serious result from physics (see sections 13.1, 25.4 of [7]) that a unitary interacting theory is incompatible with massless particles in higher representation (m + n ≥ 6). But, in light of the difficulties in making physical sense of the bosonic sector of Lisi’s theory, it would be cleaner — meaning demanding less input from physics— to focus on the fermionic sector and forbid the presence of gravitinos (m + n = 5) or yethigher spin fermionic fields. (ToE3), as stated, forbids both. In §9, we will revisit the possibility of admitting gravitinos.

2.6. Dimension considerations. Elaborating on the discussion above, in a Theory of Everything one wishes to identify weight vectors in V_{2,1} and V_{1,2} with left- and righthanded fermions. As there are 3 × 15 = 45 known fermions of each chirality, we find that the −1-eigenspace must have dimension at least 2 × 2 × 45 = 180.

In case E is a real form of E_{8}, the −1-eigenspace has dimension 112 or 128 (this is implicit in Elie Cartan’s classification of real forms of E_{8} as in [8, p. 518, Table V]),^{1} so no identification of the fermions as distinct weight vectors in Lie(E) (as in Table 9 in [1]) can be compatible with the Spin-Statistics Theorem and the existence of three generations.

1Alternatively, the marvelous bound on the trace from [9, Th. 3] implies that for every element x of order 2 in a reductive complex Lie group G, the −1-eigenspace of Ad(x) has dimension ≤ (dim G + rank G)/2. In particular, when G is a real form of E_{8}, the −1-eigenspace has dimension ≤ 128.

3. 𝔰𝔩_{2} subalgebras and the Dynkin index

3.1. In [10, §2], Dynkin defined the index of an inclusion f : 𝔤_{1} ↪ 𝔤_{2} of simple complex Lie algebras as follows. Fix a Chevalley basis of the two algebras, so that the Cartan 𝔥_{1} of 𝔤_{1} is contained in the Cartan 𝔥_{2} of 𝔤_{2}. The Chevalley basis identifies 𝔥_{i} with the complexification Q_{i}^{∨} ⊗ ℂ of the coroot lattice Q_{i}^{∨} of 𝔤_{i}, and the inclusion f gives an inclusion Q_{1}^{∨} ⊗ ℂ ↪ Q_{2}^{∨} ⊗ ℂ. Fix the Weyl-invariant inner product ( , )_{i} on Q_{i}^{∨} so that (α^{∨}, α^{∨})_{i} = 2 for short coroots α^{∨}. Then the Dynkin index of the inclusion is the ratio (f(α^{∨}), f(α^{∨}))_{2}/(α^{∨}, α^{∨})_{1} where α^{∨} is a short coroot of 𝔤_{1}. For example, the irreducible representation 𝔰𝔩_{2} → 𝔰𝔩_{n} has index \binom{n+1}{3} by [10, Eq. (2.32)]. 3.2. We now consider the case where 𝔤_{1} = 𝔰𝔩_{2} and write simply 𝔤 and Q^{∨} for 𝔤_{2} and Q_{2}^{∨}. In §8 of that same paper (or see [11, §VIII.11]), Dynkin proved that after conjugating by an element of the automorphism group of 𝔤, one can assume that the Cartan subalgebra of 𝔰𝔩_{2} is contained in the given Cartan subalgebra of 𝔤 and that the image h of a simple root of 𝔰𝔩_{2} in Q^{∨} ⊗ ℂ satisfies the strong restrictions:

h = ∑_{δ∈Δ} p_{δ}δ^{∨} for p_{δ} real and non-negative [10, Lemma 8.3],

where Δ denotes the set of simple roots of 𝔤 and further that

δ(h) ∈ {0, 1, 2} for all δ ∈ Δ.

But note that for each simple root δ, the fundamental irreducible representation of 𝔤 with highest weight dual to δ has weight p_{δ} (as a representation of 𝔰𝔩_{2}), hence p_{δ} is an integer.

As a consequence of this and specifically [10, Lemma 8.2], one can identify an 𝔰𝔩_{2}- subalgebra of 𝔤 up to conjugacy by writing the Dynkin diagram of 𝔤 and putting the number δ(h) at each vertex; this is the marked Dynkin diagram of the 𝔰𝔩_{2} subalgebra.

Here is an alternative formula for computing the index of an 𝔰𝔩_{2} subalgebra from its marked Dynkin diagram. Write κ_{𝔤} and m^{∨} for the Killing form and dual Coxeter number of 𝔤. We have:

(3.3) (Dynkin index) = \frac{1}{2}(h, h) = \frac{1}{4m^{∨}} κ_{𝔤}(h, h) = \frac{1}{2m^{∨}} ∑_{positive roots α of 𝔤}α(h)^{2},

where the second equality is by, e.g., [12, §5], and the third is by the definition of κ_{𝔤}. One can calculate the number α(h) by writing α as a sum of positive roots and applying the marked Dynkin diagram for h.

Lemma 3.4. For every simple complex Lie algebra 𝔤, there is a unique copy of 𝔰𝔩_{2} in 𝔤 of index 1, up to conjugacy.

Proof. The index of an 𝔰𝔩_{2}-subalgebra is (h, h)/2, where the defining vector h belongs to the coroot lattice Q^{∨}. If 𝔤 is not of type B, then the coroot lattice is not of type C, and the claim amounts to the statement that the vectors of minimal length in the coroot lattice are actually roots. This follows from the constructions of the root lattices in [13, §12.1].

Otherwise 𝔤 has type B and is 𝔰𝔬_{n} for some odd n ≥ 5. The conjugacy class of an 𝔰𝔩_{2}- subalgebra is determined by the restriction of the natural n-dimensional representation; they are parameterized by partitions of n (i.e., ∑ n_{i} = n) so that n_{1} ≥ ⋯ ≥ n_{s} > 0, n_{1} > 1, and there are an even number of even n_{i}’s [14, §6.2.2]. The index of the composition 𝔰𝔩_{2} → 𝔰𝔬_{n} → 𝔰𝔩_{n} is then ∑ \binom{n_{i}+1}{3}; we must classify those partitions such that this sum equals the Dynkin index of 𝔰𝔬_{n} → 𝔰𝔩_{n}, which is 2. The unique such partition is 2 ≥ 2 ≥ 1 ≥ ⋯ ≥ 1 > 0. □

If 𝔤 has type C, F_{4}, or G_{2}, then the same argument shows that there is up to conjugacy a unique copy of 𝔰𝔩_{2} in 𝔤 with index 2, 2, or 3 respectively. We will see below that for 𝔤 of type B_{6}, there are two conjugacy classes of 𝔰𝔩_{2}-subalgebras of index 2. This amounts to the fact that there are vectors in the C_{6} root lattice that are not roots but have the same length as a root, cf. Exercise 5 in §12 of [13].

4. Copies of 𝔰𝔩_{2,ℂ} in the complex E_{8}

We first prove some facts about copies of 𝔰𝔩_{2,ℂ} in the complex Lie algebra 𝔢_{8} of type E_{8}. The 69 conjugacy classes of such are listed in Table 20 on p. 182 of [10]; we are interested in just a few of these, without using Dynkin’s table.

Fix a pinning for 𝔢_{8}; this includes a maximal toral subalgebra 𝔥, a set of simple roots D := {α_{i} | 1 ≤ i ≤ 8} (numbered

(4.1)  1 3 4 5 6 7 8 2

as in [15]), and fundamental weights ω_{i} dual to α_{i}. As all roots of the E_{8} root system have the same length, we can and do identify the root system with its inverse root system.

Example 4.2. Taking any root of E_{8}, one can generate a copy of 𝔰𝔩_{2,ℂ} in 𝔢_{8} with index 1. Doing this with the highest root gives an 𝔰𝔩_{2,ℂ} with marked Dynkin diagram

index 1: 0 0 0 0 0 0 1 0

Every index 1 copy of 𝔰𝔩_{2} in 𝔢_{8} is conjugate to this one by Lemma 3.4.

Example 4.3. One can find a copy of 𝔰𝔩_{2,ℂ} × 𝔰𝔩_{2,ℂ} in 𝔢_{8} by taking the first copy to be generated by the highest root of E_{8} and the second copy to be generated by the highest root of the obvious E_{7} subsystem. If you embed 𝔰𝔩_{2,ℂ} diagonally in this algebra, you find a copy of 𝔰𝔩_{2,ℂ} with index 2 and marked Dynkin diagram

index 2: 1 0 0 0 0 0 0 0

Proposition 4.4. The following collections of copies of 𝔰𝔩_{2,ℂ} in 𝔢_{8} are the same:

(1) copies such that ±1 are weights of 𝔢_{8} (as a representation of 𝔰𝔩_{2,ℂ}) and no other odd weights occur.

(2) copies such that every weight of 𝔢_{8} is in {0, ±1, ±2}.

(3) copies such that the inclusion 𝔰𝔩_{2,ℂ} ⊂ 𝔢_{8} has Dynkin index 1 or 2.

(4) copies of 𝔰𝔩_{2,ℂ} conjugate to one of those defined in Examples 4.2 or 4.3.

The equality of (3) and (4) says that 𝔰𝔩_{2,ℂ}’s of index 1 or 2 are determined (up to conjugacy) by their index. This is a small part of the data contained in Dynkin’s Table 20, which is not used in our proof.

Proof of Proposition 4.4. One easily checks that (4) is contained in (1)–(3); we prove the opposite inclusion.

For (3), we identify 𝔥 with the complexification Q ⊗ ℂ of the (co)root lattice Q, hence h with ∑ α_{i}(h)ω_{i}. By (3.3), the index of h satisfies:

\frac{1}{60} ∑_{α} α(h)^{2} = \frac{1}{60} ∑_{α} (∑_{i} α_{i}(h)⟨ω_{i}, α⟩)^{2} ≥ ∑_{i} (α_{i}(h)^{2} ∑_{α} \frac{⟨ω_{i}, α⟩^{2}}{60} )

where the sums vary over the positive roots. We calculate for each fundamental weight ω_{i} the number ∑_{α} ⟨ω_{i}, α⟩^{2}/60:

(4.5)  2 7 15 10 6 3 1 4

As the numbers α_{i}(h) are all 0, 1, or 2, the numbers (4.5) show that h for an 𝔰𝔩_{2,ℂ} with Dynkin index 1 or 2 must be ω_{1} (index 2) or ω_{8} (index 1).

For (2), the highest root α̃ of E_{8} is α̃ = ∑_{i} c_{i}α_{i}, where c_{1} = c_{8} = 2 and the other c_{i}’s are all at least 3. As α̃(h) is a weight of 𝔢_{8} relative to a given copy of 𝔰𝔩_{2,ℂ}, we deduce that an 𝔰𝔩_{2,ℂ} as in (2) must have h = ω_{1} or ω_{8}, as claimed.

Suppose now that we are given an h for a copy of 𝔰𝔩_{2,ℂ} as in (1). As ±1 occur as weights, there is at least one 1 in the marked Dynkin diagram.

But note that there cannot be three or more 1’s in the marked Dynkin diagram for h. Indeed, for every connected subset S of vertices of the Dynkin diagram of E_{8}, ∑_{i∈S} α_{i} is a root [15, §VI.1.6, Cor. 3b]. If the number of 1’s in the marked diagram of h is at least three, then one can pick S so that it meets exactly three of the α_{i}’s with α_{i}(h) = 1, in which case ∑_{i∈S} α_{i}(h) is odd and at least 3, violating the hypothesis of (1).

For sake of contradiction, suppose that there are two 1’s in the marked diagram for h. If α_{8}(h) and α_{i}(h) are the two 1’s, then one can find a root α of E_{8} where ⟨ω_{i} + ω_{8}, α⟩ is odd and at least 3, i.e., the coefficients of α_{i} and α_{8} in α have opposite parities and sum at least 3. This is a contradiction, so we may assume that both 1’s in the marked Dynkin diagram lie in the E_{7} subdiagram. Repeating this argument with smaller and smaller subsystems shows the contradiction.

We are left with the case the there is exactly one 1 in the marked diagram for h. If α_{i}(h) = 1 for some i ≠ 1, 8, then we find a contradiction because there is a root α of E_{8} with α_{i}-coordinate 3. Therefore α_{i}(h) = 1 only for i = 1 or 8 and not for both. By the fact used two paragraphs above, β := ∑_{i} α_{i} is a root of E_{8}, so β(h) = ∑ α_{i}(h) is odd and must be 1. It follows that h = ω_{1} or ω_{8}. □

4.6. Centralizer for index 1. The 𝔰𝔩_{2,ℂ} of index 1 in 𝔢_{8} has centralizer the regular subalgebra 𝔢_{7} of type E_{7}. Indeed, it is clear that 𝔢_{7} centralizes this 𝔰𝔩_{2,ℂ}, and conversely 𝔢_{7} × 𝔰𝔩_{2,ℂ} is a maximal proper subalgebra of 𝔢_{8}, so the centralizer cannot be any larger.

5. Index 2 copies of 𝔰𝔩_{2,ℂ} in the complex E_{8}

Lemma 5.1. The centralizer of the index 2 𝔰𝔩_{2,ℂ} in 𝔢_{8} from Example 4.3 is a copy of 𝔰𝔬_{13} contained in the regular subalgebra 𝔰𝔬_{14} of 𝔢_{8}.

Proof. The centralizer of the 𝔰𝔩_{2,ℂ} of index 2 in 𝔢_{8} is contained in the centralizer of the defining vector h; this centralizer is reductive with semisimple part the regular subalgebra 𝔰𝔬_{14} of type D_{7}. The centralizer of 𝔰𝔩_{2,ℂ} contains the centralizer of the 𝔰𝔩_{2,ℂ} × 𝔰𝔩_{2,ℂ} from Example 4.3, which is the regular subalgebra 𝔰𝔬_{12} of type D_{6}. Furthermore, using the standard representation theory of 𝔰𝔩_{2,ℂ}, it is easy to see that that the the centralizer of 𝔰𝔩_{2,ℂ} has dimension 78 (as is implicitly claimed in the statement of the lemma).

The regular 𝔰𝔬_{14} does not centralize this 𝔰𝔩_{2,ℂ}. On the other hand, the centralizer is strictly bigger than 𝔰𝔬_{12} because it includes a copy of 𝔰𝔩_{2,ℂ} obtained by embedding diagonally in the 𝔰𝔩_{2,ℂ} × 𝔰𝔩_{2,ℂ} generated by nilpotents corresponding to the highest root of the D_{7} subsystem and −α_{8}. (To check that this 𝔰𝔩_{2,ℂ} centralizes the index 2 copy we started with, it is helpful to have a table of structure constants as in [16].) Together with 𝔰𝔬_{12}, this generates an 𝔰𝔬_{13} subalgebra of 𝔰𝔬_{14}. Dimension count gives the claim. □

5.2. Implicit in the proof above is an inclusion of 𝔰𝔬_{13} in 𝔢_{8} and a comparison of the pinnings of the two algebras, and in particular an inclusion of coroot lattices in terms of those pinnings. Number the simple roots of 𝔰𝔬_{13} according to the diagram

[figure: 1; 2; 3; 4; 5; 6; >]

We write β_{i}^{∨} for the simple coroot corresponding to the simple root i. Here is a translation table between the coroots of 𝔰𝔬_{13} and the (co)roots of 𝔢_{8}:

\begin{matrix}𝔰𝔬_{13} & β_{1}^{∨} & β_{2}^{∨} & β_{3}^{∨} & β_{4}^{∨} & β_{5}^{∨} & β_{6}^{∨} \\ 𝔢_{8} & α_{3} & α_{4} & α_{5} & α_{6} & α_{7} & α_{2} + α_{3} + 2α_{4} + 2α_{5} + 2α_{6} + 2α_{7} \end{matrix}

The index 2 𝔰𝔩_{2,ℂ} and the copy of 𝔰𝔬_{13,ℂ} give an 𝔰𝔩_{2,ℂ} × 𝔰𝔬_{13,ℂ} subalgebra of 𝔢_{8,ℂ}, and as a representation, 𝔢_{8,ℂ} decomposes as a direct sum of irreducibles:

1 ⊗ 𝔰𝔬_{13,ℂ} ⊕ 2 ⊗ (spin) ⊕ 3 ⊗ 1 ⊕ 3 ⊗ (vector)

Example 5.3. By Dynkin’s game of adding the highest root to the Dynkin diagram and deleting a vertex as in [10, §5], 𝔰𝔬_{13,ℂ} contains a maximal subalgebra 𝔰𝔬_{8,ℂ}×𝔰𝔭_{4,ℂ}. In turn, 𝔰𝔬_{8} contains a maximal subalgebra 𝔰𝔩_{2} × 𝔰𝔭_{4} [17, Th. 1.4]. This gives an 𝔰𝔩_{2} × 𝔰𝔭_{4} × 𝔰𝔭_{4} subalgebra of 𝔰𝔬_{13}.

We remark that this 𝔰𝔩_{2} has index 2 in 𝔢_{8}. As the inclusions 𝔰𝔬_{8} ⊂ 𝔰𝔬_{13} ⊂ 𝔢_{8} have index 1, it suffices to check that 𝔰𝔩_{2} has index 2 in 𝔰𝔬_{8}. This follows from the fact that the adjoint representation of 𝔰𝔬_{8} has index 12, whereas its restriction to 𝔰𝔩_{2} decomposes as six copies of the 3-dimensional irreducible representation and a 10-dimensional trivial representation [18, p. 260], so has index 6 · 4 + 10 · 0 = 24.

The main result of this section is the following:

Proposition 5.4. Up to conjugacy, there is a unique copy of SL_{2,ℂ} × SL_{2,ℂ} in E_{8,ℂ} so that each inclusion of SL_{2,ℂ} in E_{8,ℂ} has index 2. The Lie algebra of the centralizer of this SL_{2,ℂ} × SL_{2,ℂ} is an 𝔰𝔭_{4,ℂ} × 𝔰𝔭_{4,ℂ} subalgebra of 𝔰𝔬_{13,ℂ} in 𝔢_{8}.

Proof. An analysis similar to the one in the proof of Proposition 4.4—but more complicated because there is more than one root length and Spin_{13} has a nontrivial center—shows that there are two conjugacy classes of copies of 𝔰𝔩_{2} in 𝔰𝔬_{13} of index 2, corresponding to marked Dynkin diagrams

(a) 2 0 0 0 0 0 and (b) 0 0 0 1 0 0

We can pair each of (a) and (b) with the copy of 𝔰𝔩_{2} from Example 4.3 to get an 𝔰𝔩_{2} × 𝔰𝔩_{2} subalgebra of 𝔢_{8} where both 𝔰𝔩_{2}’s have index 2. Clearly, these represent the only two E_{8}- conjugacy classes of such subalgebras. With the marked Dynkin diagram in hand, it is not difficult to calculate the decomposition of 𝔢_{8} into a direct sum of irreducible representations of 𝔰𝔩_{2} × 𝔰𝔩_{2}.

In case (a), every irreducible summand m ⊗ n has m + n even. Therefore, this copy of 𝔰𝔩_{2} × 𝔰𝔩_{2} is the Lie algebra of a subgroup of E_{8} isomorphic to (SL_{2} × SL_{2})/(−1, −1).

In case (b), we have the following table of multiplicities for m ⊗ n:

(5.5)  \begin{aligned}& & 1 & 2 & 3 & m \\ & 1 & 20 & 20 & 6 \\ n & 2 & 20 & 16 & 4 \\ & 3 & 6 & 4 & 0 \end{aligned}

In particular, it is the Lie algebra of a copy of SL_{2} × SL_{2} in E_{8}. Combining the 𝔰𝔩_{2}’s from Examples 4.3 and 5.3 gives an 𝔰𝔩_{2} × 𝔰𝔩_{2} subalgebra with both 𝔰𝔩_{2}’s of index 2 and the same decomposition of 𝔢_{8} into irreducible representations, so it is in the same conjugacy class. Clearly, its centralizer contains the copy of 𝔰𝔭_{4} × 𝔰𝔭_{4} from Example 5.3, and the number 20 in the upper-left corner of (5.5) shows that 𝔰𝔭_{4} × 𝔰𝔭_{4} is the whole centralizer. □

5.6. We can decompose 𝔢_{8} into a direct sum of irreducible representations of the 𝔰𝔩_{2} × 𝔰𝔩_{2} × 𝔰𝔭_{4} × 𝔰𝔭_{4} subalgebra from the proposition by combining the decomposition of 𝔢_{8} into irreducible representations of 𝔰𝔩_{2} × 𝔰𝔬_{13} from 5.2 with the tables in [18]. Recall that 𝔰𝔭_{4} has two fundamental irreducible representations: one that is 4-dimensional symplectic and another that is 5-dimensional orthogonal; we denote them by their dimensions. With this notation and 1.1, we find:

V_{2,1} = 5 ⊗ 4, V_{1,2} = 4 ⊗ 5, V_{2,3} = 1 ⊗ 4, and V_{3,2} = 4 ⊗ 1.

6. Copies of SL(2, ℂ) in a real form of E_{8}

Suppose now that we have a copy of SL(2, ℂ) inside a real Lie group E of type E_{8}. Over the complex numbers, we decompose Lie(E) ⊗ ℂ into a direct sum of irreducible representations of SL(2, ℂ) × ℂ ≅ SL_{2,ℂ} × SL_{2,ℂ}; each irreducible representation can be written as m ⊗ n where m and n denote the dimension of an irreducible representation of the first or second SL_{2,ℂ} respectively. The goal of this section is to prove:

Proposition 6.1. Maintain the notation of the previous paragraph. If Lie(E) contains no irreducible summands m⊗n with m+n > 4, then the identity component of the centralizer of SL(2, ℂ) in E is a

(1) a regular subgroup Spin(7, 5) if E is split; or

(2) a regular subgroup Spin(9, 3) or Spin(11, 1) if the Killing form of Lie(E) has signature −24.

Proof. Complexifying the inclusion of SL(2, ℂ) in E and going to Lie algebras gives an inclusion of 𝔰𝔩_{2,ℂ} × 𝔰𝔩_{2,ℂ} in the complex Lie algebra 𝔢_{8} from §4. The hypothesis on the irreducible summands m⊗n amounts to the statement that each of the two 𝔰𝔩_{2,ℂ}’s has index 1 or 2 by Proposition 4.4. As complex conjugation interchanges the two components, they must have the same index.

Suppose first that both 𝔰𝔩_{2}’s have index 1. Lemma 3.4 (twice) gives that this 𝔰𝔩_{2} × 𝔰𝔩_{2} is conjugate to the one generated by the highest root of E_{8} from Example 4.2 (so the second 𝔰𝔩_{2} belongs to the centralizer of type E_{7}) and by the highest root of the E_{7} subsystem and makes up the first two summands of an 𝔰𝔩_{2} × 𝔰𝔩_{2} × 𝔰𝔬_{12} subalgebra as in [10, pp. 147, 148]. As this subalgebra has rank 8, it follows that the Lie algebra of the centralizer 𝔷 of SL(2, ℂ) in E is a real form of 𝔰𝔭𝔦𝔫_{12}.

We can decompose Lie(E)⊗ℂ into irreducible representations of (𝔰𝔩_{2}×𝔰𝔩_{2}×𝔰𝔭𝔦𝔫_{12})⊗ℂ using the tables in [18] to find the adjoint representation plus

2 ⊗ 1 ⊗ S_{+}, 1 ⊗ 2 ⊗ S_{−}, and 2 ⊗ 2 ⊗ V,

where S_{±} denotes the half-spin representations of 𝔰𝔭𝔦𝔫_{12} and V is the vector representation. But Lie(E) is a real representation of 𝔷, so we deduce that V is also a real representation of 𝔷 but S_{+} and S_{−} are not; they are interchanged by the Galois action. The first observation shows that 𝔷 is 𝔰𝔭𝔦𝔫(12 − a, a) for some 0 ≤ a ≤ 6. The second shows that a must be 1, 3, or 5, as claimed.

It remains to prove the correspondence between a and the real forms of E_{8}. For a = 5, this is clear: the subgroup generated by SL(2, ℂ) and Spin(7, 5) has real rank 6, so it can only be contained in the split real form. Now suppose that a = 3 or 1 and that SL(2, ℂ) is in the split E_{8}; we will obtain a contradiction. Over ℂ, SL(2, ℂ) is conjugate to the copy of SL_{2,ℂ} × SL_{2,ℂ} in E_{8,ℂ} generated by the highest root of E_{8} and the highest root of the natural subsystem of type E_{7}. One calculates using the tables in [15], e.g., that the element −1 ∈ SL(2, ℂ), equivalently, (−1, −1) ∈ SL_{2} × SL_{2} is h_{α_{2}}(−1) h_{α_{3}}(−1) where h_{α_{i}} is the cocharacter corresponding to the coroot α_{i}^{∨} as in [19]. That is, the subgroup fixed by conjugation by this −1 has root system consisting of roots α such that ⟨ω_{1}, α⟩ is even. These roots form the natural D_{8} subsystem of E_{8} and our SL(2, ℂ) · Spin(12 − a, a) is a standard subgroup. The vector representation of this D_{8} restricts to a sum of the vector representation of Spin(12 − a, a) and the 4-dimensional vector representation of SL(2, ℂ) (which factors through SO(3, 1)). In particular, this representation is real, so this D_{8} subgroup is, according to [20, p. 161], isogenous to SO(8, 8). But then the invariant symmetric bilinear form on the vector representation has signature 0 but restricts to have signature ±(12 − 2a) ∈ {±6, ±10} and ±2 on each summand, which cannot add up to get 0. This is a contradiction, so for a = 3 or 1, the real form of E_{8} is neither split nor compact.

Now suppose that both 𝔰𝔩_{2}’s have index 2. When we decompose 𝔢_{8} as in 1.1, we find the representation 2⊗3 with positive multiplicity 4 by (5.5), which violates our hypothesis on the SL(2, ℂ) subgroup of E. □

7. No Theory of Everything in a real form of E_{8}

We now prove the second claim in Theorem 1.3, namely that each real form E of E_{8} contains no ToE subgroups. Suppose E contains a copy of SL(2, ℂ) (so E is non-compact) and a subgroup G satisfying (ToE1) and (ToE3). We will show that (ToE2) fails.

The −1-eigenspace in Lie(E) is a real representation of SL(2, ℂ) · G. By Proposition 6.1, G is contained in a copy of Spin(12 − a, a) for a = 1, 3, or 5. As in the proof of Proposition 6.1, there is a representation W of SL(2, ℂ) × Spin(12 − a, a) defined over ℝ that is isomorphic to

(2 ⊗ 1 ⊗ S_{+}) ⊕ (1 ⊗ 2 ⊗ S_{−})

over ℂ. Now G is contained in the maximal compact subgroup of Spin(12 − a, a), i.e., Lie(G) is a subalgebra of 𝔰𝔬(11), 𝔰𝔬(9)×𝔰𝔬(3), or 𝔰𝔬(7)×𝔰𝔬(5). The restriction of the two half-spin representations of Spin(12 − a, a) to the compact subalgebra are equivalent [18, p. 264], and we see that in each case the restriction is quaternionic. (To see this, one uses the standard fact that the spin representation of 𝔰𝔬(2ℓ+1) is real for ℓ ≡ 0, 3 (mod 4) and quaternionic for ℓ ≡ 1, 2 (mod 4).) That is, the restrictions of S_{+}, S_{−}, and their complex conjugates to the maximal compact subgroup are all equivalent (over ℂ), hence the same is true for their further restrictions to G, and (ToE2) fails. □

Remark 7.1. It is worthwhile noting that, in each of the three cases, it is possible to embed G_{SM} in the centralizer, thus showing that (ToE1) is satisfied. Given such an embedding, a simple computation verifies explicitly that S_{+} has a self-conjugate structure as a representation of G_{SM}.

First consider Spin(11, 1). There is an obvious embedding of G_{GUT} := Spin(10). Under this embedding, S_{+} decomposes as the direct sum of the two half-spinor representations, i.e. as a generation and an anti-generation.

For Spin(7, 5), there is an obvious embedding of the Pati-Salam group, G_{GUT} := (Spin(6) × Spin(4))/(ℤ/2ℤ). Again, S_{+} decomposes as the direct sum of a generation and an anti-generation.

Finally, Spin(3, 9) contains (SU(3) × SU(2) × SU(2) × U(1))/(ℤ/6ℤ) as a subgroup. Under this subgroup,

S_{+} = (3, 2, 2)_{1/6} ⊕ (\overline{3}, 2, 2)_{−1/6} + (1, 2, 2)_{−1/2} + (1, 2, 2)_{1/2}

where the subscript indicates the U(1) weights, and the overall normalization is chosen to agree with the physicists’ convention for the weights of the Standard Model’s U(1)_{Y}. Embedding the SU(2) of the Standard model in one of the two SU(2)s, we obtain an embedding of G_{SM} ⊂ Spin(3, 9) where, again S_{+} has a self-conjugate structure as a representation of G_{SM}.

8. No Theory of Everything in complex E_{8}

We now complete the proof of Theorem 1.3by showing that there are no ToE subgroups in the transfer E := R(E_{8,ℂ}) of the complex Lie group of type E_{8}. 8.1. First, recall the transfer R(G_{ℂ}) of a complex group G_{ℂ} as described, e.g., in [21, §2.1.2]. Its complexification can be viewed as G_{ℂ} × G_{ℂ}, where complex conjugation acts via

(8.2)  \overline{(g, g′)} = (\overline{g′}, \overline{g}).

One can view R(G_{ℂ}) as the subgroup of the complexification consisting of elements fixed by (8.2).

Now consider an inclusion ϕ: SL(2, ℂ) = R(SL_{2,ℂ}) ↪ R(E_{8,ℂ}). Complexifying, we identify R(SL_{2,ℂ}) × ℂ with SL_{2,ℂ} × SL_{2,ℂ} and similarly for R(E_{8,ℂ}) and write out ϕ as:

(8.3)  ϕ(g_{1}, g_{2}) = (ϕ_{1}(g_{1})ϕ_{2}(g_{2}), ψ_{1}(g_{1})ψ_{2}(g_{2}))

for some homomorphisms ϕ_{1}, ϕ_{2}, ψ_{1}, ψ_{2} : SL_{2,ℂ} → E_{8,ℂ}. As ϕ is defined over ℝ, we have:

ϕ(g_{1}, g_{2}) = \overline{ϕ(\overline{g_{2}}, \overline{g_{1}})} = (\overline{ψ_{1}(\overline{g_{2}})ψ_{2}(\overline{g_{1}})}, \overline{ϕ_{1}(\overline{g_{2}})ϕ_{2}(\overline{g_{1}})}),

and it follows that ψ_{1}(g_{1}) = \overline{ϕ_{2}(\overline{g_{1}})} and ψ_{2}(g_{2}) = \overline{ϕ_{1}(\overline{g_{2}})}. Conversely, given any two homomorphisms ϕ_{1}, ϕ_{2} : SL_{2,ℂ} → E_{8,ℂ} (over ℂ) with commuting images, the same equations define a homomorphism ϕ: SL(2, ℂ) → R(E_{8,ℂ}) defined over ℝ. 8.4. Plan of the proof. Now suppose that we have a subgroup SL(2, ℂ) · G of R(E_{8,ℂ}) satisfying (ToE1) and (ToE3). Write C for the identity component of the centralizer of the image of the map (ϕ_{1}, ϕ_{2}) : SL_{2,ℂ} × SL_{2,ℂ} → E_{8,ℂ} from (8.3). Clearly, G is contained in the transfer R(C) of C. In each of the cases below, we verify that

(8.5)  C is semisimple,

so the maximal compact subgroup of R(C) is the compact real form C_{ℝ} of C. Furthermore, in each of the cases below, we will observe that

(8.6)  −1 is in the Weyl group of C.

It follows that C_{ℝ} is an inner form, hence every irreducible representation of C_{ℝ} is real or quaternionic, hence every representation of C_{ℝ} is self-conjugate. That is, (ToE2) fails, which is the desired contradiction.

8.7. Case 1: ϕ_{1} or ϕ_{2} is trivial. Consider the easiest-to-understand case where ϕ_{1} or ϕ_{2} is the zero map, say ϕ_{2}. That is, ϕ is the transfer of a homomorphism ϕ_{1} : SL_{2,ℂ} → E_{8,ℂ}, which by Proposition 4.4 has index 1 or 2.

If ϕ_{1} has index 1, then C is simply connected of type E_{7}, hence (8.5) and (8.6) hold. If ϕ_{1} has index 2, then C is isogenous to Spin_{13,ℂ} by Lemma 5.1, and again (8.5) and (8.6) hold.

8.8. Case 2: ϕ_{1} and ϕ_{2} are injections. Finally, we consider the case where ϕ_{1} and ϕ_{2} are both injections. Again, (ToE3) and Proposition 4.4 implies that ϕ_{1} and ϕ_{2} have Dynkin index 1 or 2.

If ϕ_{1} and ϕ_{2} both have index 1, then over ℂ this is the same embedding of SL_{2,ℂ} × SL_{2,ℂ} in E_{8,ℂ} as the one in the proof of Proposition 6.1. The centralizer C is the standard D_{6} subgroup of E_{8,ℂ}.

If ϕ_{1} and ϕ_{2} both have index 2, then ϕ_{1} × ϕ_{2} gives an embedding as in Proposition 5.4, and C has Lie algebra 𝔰𝔭_{4,ℂ} × 𝔰𝔭_{4,ℂ} of type B_{2} × B_{2}. Note that (ToE3) fails in this case by (5.5).

Suppose finally that ϕ_{1} has index 1 and ϕ_{2} has index 2. We conjugate so that ϕ_{2}(𝔰𝔩_{2}) is the copy of 𝔰𝔩_{2} from Example 4.3, and (by Lemma 3.4 for the centralizer 𝔰𝔬_{13} of ϕ_{2}(𝔰𝔩_{2})) we can take ϕ_{1}(𝔰𝔩_{2}) to be a copy of 𝔰𝔩_{2} generated by the highest root of Spin_{13}. Calculating the weights of the representation of this 𝔰𝔩_{2} × 𝔰𝔩_{2} on 𝔢_{8} gives the following decomposition into irreducibles:

(8.9)  \begin{aligned}& & 1 & 2 & 3 & m \\ & 1 & 39 & 18 & 1 \\ n & 2 & 32 & 16 & 0 \\ & 3 & 10 & 2 & 0 \end{aligned}

with the same notation as (5.5). In particular, the A_{1} × B_{4} subgroup of Spin_{13} that centralizes the image of ϕ_{1} × ϕ_{2} is all of the identity component C of the centralizer in E_{8}. Again (8.5) and (8.6) hold. (Of course, (8.9) shows that (ToE3) fails anyway.)

This completes the proof of Theorem 1.3. □

9. Relaxing (ToE3)

Technically, (m, n) = (2, 3) and(3, 2) are possible in an interacting theory, but only in the presence of local supersymmetry (i.e., in supergravity theories) [22]. Lisi’s framework is not compatible with local supersymmetry, so we excluded this possibility above. If we relax (ToE3) by replacing it with

(ToE3’)  V_{1,4} = V_{4,1} = 0 and V_{m,n} = 0 if m + n > 6

then we still don’t find anything further. More precisely, we have the following strengthening of Theorem 1.3.

Theorem 9.1. There are no subgroups SL(2, ℂ)·G satisfying (ToE1), (ToE2), and (ToE3’) in the (transfer of the) complex E_{8} or any real form of E_{8}.

Proof. Note that (ToE3’) still forces that in the decomposition of Lie(E), the representation V_{m,n} = 0 if m > 3 or n > 3, so each of the two 𝔰𝔩_{2,ℂ} summands in the complexification of SL(2, ℂ) have index 1 or 2 by Proposition 4.4. Looking back, we see that we already proved the theorem for the transfer of the complex E_{8} in §8.

Suppose we have an SL(2, ℂ) · G subgroup of a real form E of E_{8}. Imitating the proof in §7, we appeal to Proposition 6.1. We now have the additional possibility that the complexification SL_{2,ℂ} × SL_{2,ℂ} of the SL(2, ℂ) subgroup is such that both SL_{2,ℂ}’s have index 2 as in §5. The centralizer of such an SL(2, ℂ) is a real form of 𝔰𝔭_{4,ℂ} × 𝔰𝔭_{4,ℂ} by Proposition 5.4. When we decompose 𝔢_{8} as in 1.1, we find V_{2,1} and V_{1,2} as in 5.6. As complex conjugation interchanges these two representations, it follows that complex conjugation interchanges the two 𝔰𝔭_{4,ℂ} factors, i.e., the centralizer of SL(2, ℂ) has identity component the transfer R(Sp_{4,ℂ}) of Sp_{4,ℂ}. Its maximal compact subgroup is the compact form of Sp_{4,ℂ} (also known as Spin(5)), all of whose irreducible representations are selfconjugate. Therefore, (ToE2) fails. □

Remark 9.2. As we have already mentioned, weakening (ToE3) to (ToE3’) is only consistent in supergravity theories. In the case at hand, with G_{max} = Spin(5), we find

(9.3)  V_{3,2} ≃ V_{2,3} = 4,  V_{2,1} ≃ V_{1,2} = 4 ⊕ 16

where we have indicated the irreducible representations of Spin(5) by their dimensions. Since the gravitinos transform nontrivially under G_{max} and, given their multiplicity, the only consistent possibility would be a gauged N = 4 supergravity theory (for a recent review of such theories, see [23]). Unfortunately, the rest of the matter content (it suffices to look at V_{2,1}) is not compatible with N = 4 supersymmetry. Even if it were, N = 4 supersymmetry would, of course, necessitate that the theory be non-chiral, making it unsuitable as a candidate Theory of Everything.

10. Conclusion

In paragraph 2.6 above, we observed by an easy dimension count that no proposed Theory of Everything constructed using subgroups of a real form E of E_{8} has a sufficient number of weight vectors in the −1-eigenspace to identify with all known fermions. The proof of our Theorem 1.3 was quite a bit more complicated, but it also gives much more. It shows that you cannot obtain a chiral gauge theory for any ToE subgroup of E, whether E is a real form or the complex form of E_{8}. In particular, it is impossible to obtain even the one-generation Standard Model in this fashion.

Acknowledgments. The second author thanks Fred Helenius for pointing out a reference and Patrick Brosnan for suggesting this project. The authors’ research was supported by the National Science Foundation under grant nos. PHY-0455649 (Distler) and DMS-0653502 (Garibaldi).

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(Distler) Theory Group, Department of Physics, and Texas Cosmology Center, University of Texas, Austin, TX 78712

   E-mail address: distler@golem.ph.utexas.edu

(Garibaldi) Department of Mathematics & Computer Science, 400 Dowman Dr., Emory University, Atlanta, GA 30322

   E-mail address: skip@member.ams.org