There is no "Theory of Everything" inside E8
aa r X i v : . [ m a t h . R T ] O c t THERE IS NO “THEORY OF EVERYTHING” INSIDE E JACQUES DISTLER AND SKIP GARIBALDIA
BSTRACT . We analyze certain subgroups of real and complex forms of the Lie group E ,and deduce that any “Theory of Everything” obtained by embedding the gauge groups ofgravity and the Standard Model into a real or complex form of E lacks certain representation-theoretic properties required by physical reality. The arguments themselves amount to rep-resentation theory of Lie algebras in the spirit of Dynkin’s classic papers and are writtenfor mathematicians.
1. I
NTRODUCTION
Recently, the preprint [1] by Garrett Lisi has generated a lot of popular interest. It boldlyclaims to be a sketch of a “Theory of Everything”, based on the idea of combining the localLorentz group and the gauge group of the Standard Model in a real form of E (necessarilynot the compact form, because it contains a group isogenous to SL(2 , C ) ). The purpose ofthis paper is to explain some reasons why an entire class of such models—which includethe model in [1]—cannot work, using mostly mathematics with relatively little input fromphysics.The mathematical set up is as follows. Fix a real Lie group E . We are interested insubgroups SL(2 , C ) and G of E so that:(ToE1) G is connected, compact, and centralizes SL(2 , C ) We complexify and then decompose
Lie(E) ⊗ C as a direct sum of representations of SL(2 , C ) and G . We identify SL(2 , C ) ⊗ R C with SL , C × SL , C and write(1.1) Lie(E) = M m,n ≥ m ⊗ n ⊗ V m,n where m and n denote the irreducible representation of SL , C of that dimension and V m,n is a complex representation of G ⊗ R C . (Physicists would usually write and ¯2 instead of ⊗ and ⊗ .) Of course, m ⊗ n ⊗ V m,n ≃ n ⊗ m ⊗ V m,n and since the action of SL(2 , C ) · G on Lie(E) is defined over R , we deduce that V m,n ≃ V n,m . We further demand that V m,n = 0 if m + n > , and(ToE2) V , is a complex representation of G .(ToE3)We recall the definition of complex representation and explain the physical motivation forthese hypotheses in the next section. Roughly speaking, (ToE1) is a trivial requirementbased on trying to construct a Theory of Everything along the lines suggested by Lisi,(ToE2) is the requirement that the model not contain any “exotic” higher-spin particles, and Mathematics Subject Classification.
Primary 22E47; secondary 17B25, 81R05, 83E15.Report (ToE3) is the statement that the gauge theory (with gauge group G ) is chiral , as requiredby the Standard Model. In fact, physics requires slightly stronger hypotheses on V m,n , for m + n = 4 . We will not impose the stronger version of (ToE2). Definition 1.2. A candidate ToE subgroup of a real Lie group E is a subgroup generated bya copy of SL(2 , C ) and a subgroup G such that (ToE1) and (ToE2) hold. A ToE subgroup is a candidate ToE subgroup for which (ToE3) also holds.Our main result is:
Theorem 1.3.
There are no ToE subgroups in (the transfer of) the complex E nor in anyreal form of E . Notation.
Unadorned Lie algebras and Lie groups mean ones over the real numbers. Weuse a subscript C to denote complex Lie groups—e.g., SL , C is the (complex) group of2-by-2 complex matrices with determinant 1. We can view a d -dimensional complex Liegroup G C as a d -dimensional real Lie group, which we denote by R ( G C ) . (Algebraistscall this operation the “transfer” or “Weil restriction of scalars”; geometers, and manyphysicists, call this operation “realification.”) We use the popular notation of SL(2 , C ) forthe transfer R (SL , C ) of SL , C ; it is a double covering of the “restricted Lorentz group”,i.e., of the identity component SO(3 , of SO(3 , . Our strategy for proving Theorem 1.3 will be as follows.We will first catalogue, up to conjugation, all possible embeddings of
SL(2 , C ) satisfyingthe hypotheses of (ToE2). The list is remarkably short. Specifically, for every candidateToE subgroup of E , the group G is contained in the maximal compact, connected subgroup G max of the centralizer of SL(2 , C ) in E . The proof of Theorem 1.3 shows that the onlypossibilities are:(1.5) E G max V , E − Spin(11) 32E
Spin(5) × Spin(7) (4 , − Spin(9) × Spin(3) (16 , R (E , C ) E R (E , C ) Spin(12) 32 ⊕ ′ R (E , C ) Spin(13) 64 We then note that the representation V , of G max (and hence, of any G ⊆ G max ) has aself-conjugate structure. In other words, (ToE3) fails.2. P HYSICS BACKGROUND
One of the central features of modern particle physics is that the world is described bya chiral gauge theory . Let M be a four-dimensional pseudo-Riemannian manifold, of signature (3 , , whichwe will take to be oriented, time-oriented and spin. Let G be a compact Lie group. Thedata 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 , bundle. A choice of spin structure defines a lift to a principal Spin(3 , = SL(2 , C ) bundle. Let S ± → M be the irreducible spinor bundles, asso-ciated, via the defining two-dimensional representation and its complex conjugate, to this SL(2 , C ) principal bundle. HERE IS NO “THEORY OF EVERYTHING” INSIDE E The fermions of our gauge theory are denoted ψ ∈ Γ( S + ⊗ V ) , ψ ∈ Γ( S − ⊗ V ) where V → M is a vector bundle associated to a (typically reducible) representation R of G . Definition 2.2.
Consider, V , a unitary representation of G over C —i.e., a homomorphism G → U( V ) —and an antilinear map J : V → V that commutes with the action of G . Themap J is called a real structure on V if J = 1 ; physicists call a representation possessinga real structure real . The map J is called a quaternionic structure on V if J = − ;physicists call a representation possessing a quaternionic structure pseudoreal .Subsuming these two subcases, we will say that V has a self-conjugate structure ifthere exists an antilinear map J : V → V commuting with the action of G and satisfying J = 1 . Physicists call a representation V that does not possess a self-conjugate structure complex . Remark . We sketch how to translate the above definition into the language of algebraicgroups and Galois descent as in [2] and [3, § X.2]. Let G be an algebraic group over R andfix a representation ρ : G ⊗ C → GL( V ) for some complex vector space V . Let J be anantilinear map V → V that satisfies(2.4) ρ ( g ) = J − ρ ( g ) J for g ∈ G ( C ) .We define real, quaternionic, etc., by copying the second and third sentences verbatim fromDefinition 2.2.(In the special case where G is compact, there is necessarily a positive-definite invarianthermitian form on V and ρ arises by complexifying some map G → U( V ) ; this puts usback in the situation of Def. 2.2. In the special case where G is connected, the hypothesisfrom Def. 2.2 that J commutes with G ( R ) —which is obviously implied by (2.4)—is ac-tually equivalent to (2.4). Indeed, both sides of (2.4) are morphisms of varieties over C ,so if they agree on G ( R ) —which is Zariski-dense by [2, 18.2(ii)]—then they are equal on G ( C ) .)If V has a real structure J , then the R -subspace V ′ of elements of V fixed by J is a realvector space and V is canonically identified with V ′ ⊗ C so that J ( v ′ ⊗ z ) = v ′ ⊗ z for v ′ ∈ V ′ and z ∈ C ; this is Galois descent. Because ρ commutes with complex conjugation(which acts in the obvious manner on G ( C ) and via J on V ), it is the complexification ofa homomorphism ρ ′ : G → GL ( V ′ ) defined over R by [2, AG.14.3]. Conversely, if thereis a representation ( V ′ , ρ ′ ) whose complexification is ( V, ρ ) , then taking J to be complexconjugation on V = V ′ ⊗ C defines a real structure on ( V, ρ ) .If V has a quaternionic structure J , then we define a real structure ˆ J on ˆ V := V ⊕ V via ˆ J ( v , v ) := ( Jv , − Jv ) .Finally, suppose that G is reductive and V is irreducible (as a representation over C ,of course). Then by [4, § W whosecomplexification W ⊗ C contains V as a summand. By Schur, End G ( W ) is a divisionalgebra, and we have three possibilities: • End G ( W ) = R , W ⊗ C ≃ V , and V has a real structure. • End G ( W ) = H , W ⊗ C ≃ V ⊕ V , and V has a quaternionic structure. • End G ( W ) = C , W ⊗ C ≃ V ⊕ V where V V , and V is complex.We have stated this remark for G a group over R , but all of it generalizes easily to the casewhere G is reductive over a field F and is split by a quadratic extensions K of F . JACQUES DISTLER AND SKIP GARIBALDI
Definition 2.5.
A gauge theory, with gauge group G , is said to be chiral if the representa-tion 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-conjugatestructure.Note that whether a gauge theory is chiral depends crucially on the choice of G . Agauge theory might be chiral for gauge group G , but nonchiral for a subgroup H ⊂ G .That is, there can be a self-conjugate structure on R compatible with H , even though nosuch structure exists that is compatible with the full group G .Conversely, suppose that a gauge theory is nonchiral for the gauge group G . It is alsonecessarily nonchiral for any gauge group H ⊂ G . GUTs.
The Standard Model is a chiral gauge theory with gauge group G SM := (SU(3) × SU(2) × U(1)) / ( Z / Z ) Various grand unified theories (GUTs) proceed by embedding G SM is some (usually sim-ple) group, G GUT . Popular choices for G GUT are
SU(5) [5],
Spin(10) , E , and the Pati-Salam group, (Spin(6) × Spin(4)) / ( Z / Z ) [6].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) . Therepresentation R from 2.1 is the direct sum of three copies of R = ∧ W ⊕ W Each such copy is called a “generation” and is 15-dimensional. One identifies each of the15 weights of R with left-handed fermions: 6 quarks (two in a doublet, each in threecolors), two leptons (e.g., the electron and its neutrino), 6 antiquarks, and a positron. Withthree generations, R is 45-dimensional. Definition 2.6.
As a generalization, physicists sometimes consider the n -generation Stan-dard Model , which is defined in similar fashion, but with R = R ⊕ n . The n -generationStandard Model is a chiral gauge theory, for any positive n . Particle physics, in the realworld, is described by “the” Standard Model, which is the case n = 3 .For the other choices of GUT group, the analogue of a generation ( R ) is higher-dimensional, containing additional fermions that are not seen at low energies. When de-composed under G SM ⊂ G GUT , the representation decomposes as R + R ′ , where R ′ isa real representation of G SM . In Spin(10) , a generation is the 16-dimensional half-spinorrepresentation. In E , it is a 27-dimensional representation, and for the Pati-Salam groupit is the (4 , , ⊕ (4 , , representation. In each case, these representations are complexrepresentations (in the above sense) of G GUT , and the complex-conjugate representationis called an “anti-generation.”3. L
ISI ’ S PROPOSAL FROM [1]In the previous section, 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 connectionon M and the gauge connection on P into a single dynamical framework. This motivatesDefinition 1.2 of a ToE subgroup.More precisely, following [1], we fix subgroups SL(2 , C ) and G — say, with G = G SM — satisfying (ToE1) in some real Lie group E . The action of the central element − ∈ SL(2 , C ) provides a Z / Z -grading on the Lie algebra of E . This Z / Z -grading allows oneto define a sort of superconnection associated to E (precisely what sort of superconnection HERE IS NO “THEORY OF EVERYTHING” INSIDE E is explained in a blog post by the first author [7]). In the proposal of [1], we are supposedto identify each of the generators of Lie(E) as either a boson or a fermion. (See Table 9 in[1] for an identification of the 240 roots.)The Spin-Statistics Theorem [8] says that fermions transform as spinorial representa-tions of
Spin(3 , ; bosons transform as “tensorial” representations (representation whichlift to the double cover, SO(3 , ). To be consistent with the Spin-Statistics Theorem, wemust, therefore, require that the fermions belong to the − -eigenspace of the aforemen-tioned Z / Z action, and the bosons to the +1 -eigenspace.In fact, to agree with 2.1, we should require that the − -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. Interpretations of V m,n and (ToE2). In the notation of (1.1), the V m,n , with m + n odd,correspond to fermions; those with m + n even correspond bosons. In Lisi’s setup, thebosons are 1-forms on M , with values in a vector bundle associated to the aforementioned Spin(3 , principal bundle via the m ⊗ n representation (with m + n even). The V m,n with m + n = 4 are special; they correspond to the gravitational degrees of freedom inLisi’s theory. (3 ⊗ ⊕ (1 ⊗ is the adjoint representation of SL(2 , C ) ; these correspondto the spin connection. The 1-form with values in the ⊗ representation is the vierbein .It is a substantial result from physics (see sections 13.1, 25.4 of [9]) that a unitaryinteracting theory is incompatible with massless particles in higher representation ( m + n ≥ ). Our hypothesis (ToE2) reflects this and also forbids gravitinos ( m + n = 5 ). In § Explanation of (ToE3).
Our hypothesis (ToE3) says that the candidate “Theory of Ev-erything” one obtains from subgroups
SL(2 , C ) and G as in (ToE1) must be chiral in thesense of Definition 2.5. In private communication, Lisi has indicated that he objects to our condition (ToE3),because he no longer wishes to identify all 248 generators of
Lie(E) as particles (eitherbosons or fermions). In his new—and unpublished—formulation, only a subset are to beidentified as particles. In particular, V , is typically a reducible representation of G and,in his new formulation, only a subrepresentation corresponds to particles (fermions). Thisis not the approach followed in [1], where all 248 generators are identified as particles andwhere, moreover, 20-odd of these are claimed to be new as-yet undiscovered particles—aprediction of his theory. As recently as April 2009, Lisi reiterated this prediction in anessay published in the Financial Times , [11].Our paper assumes that the approach of [1] is to be followed, and that all 248 generatorsare to be identified as particles, hence (ToE3). In any case, even if one identifies only asubset of the generators as particles, all the fermions must come from the ( − -eigenspace,which is too small to accommodate 3 generations, as we now show. In making this identification, we have tacitly assumed that V , is one-dimensional. This is, in fact, requiredfor a unitary interacting theory. We will not, however, impose this additional constraint. Suffice to say that it isnot satisfied by any of the candidate ToE subgroups (per Definition 1.2) of E . Of course, there are many other features of the Standard Model that a candidate Theory of Everything mustreproduce. We have chosen to focus on the requirement that the theory be chiral for two reasons. First, it is “phys-ically robust”: Whatever intricacies a quantum field theory may possess at high energies, if it is non -chiral, there isno known mechanism by which it could reduce to a chiral theory at low energies (and there are strong arguments[10] that no such mechanism exists). Second, chirality is easily translated into a mathematical criterion—our(ToE3). This allows us to study a purely representation-theoretic question and side-step the difficulties of makingsense of Lisi’s proposal as a dynamical quantum field theory.
JACQUES DISTLER AND SKIP GARIBALDI
No-go based on dimensions.
The fermions of Lisi’s theory correspond to weight vectorsin V m,n , with m + n odd. In particular, the weight vectors in V , and V , correspond(as in § ×
15 =45 known fermions of each chirality, V , must be at least 45-dimensional, and similarlyfor V , . Thus, the − -eigenspace of the central element of SL(2 , C ) , which contains (2 ⊗ ⊗ V , ) ⊕ (1 ⊗ ⊗ V , ) , must have dimension at least × ×
45 = 180 .When E is a real form of E , the − -eigenspace has dimension 112 or 128 (this isimplicit in Elie Cartan’s classification of real forms of E as in [12, p. 518, Table V]), sono 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.These dimensional considerations do not, however, rule out the possibility of accom-modating a 1- or 2-generation Standard Model (per Definition 2.6) in a real form of E .That requires more powerful considerations, which are the subject of our main theorem.We now turn to the proof of that theorem.4. sl SUBALGEBRAS AND THE D YNKIN INDEX
In [15, § index of an inclusion f : g ֒ → g of simple complexLie algebras as follows. Fix a Chevalley basis of the two algebras, so that the Cartansubalgebra h of g is contained in the Cartan subalgebra h of g . The Chevalley basisidentifies h i with the complexification Q ∨ i ⊗ C of the coroot lattice Q ∨ i of g i , and theinclusion f gives an inclusion Q ∨ ⊗ C ֒ → Q ∨ ⊗ 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 theinclusion is the ratio ( f ( α ∨ ) , f ( α ∨ )) / ( α ∨ , α ∨ ) where α ∨ is a short coroot of g . Forexample, the irreducible representation sl → sl n has index (cid:0) n +13 (cid:1) by [15, Eq. (2.32)]. We now consider the case g = sl and write simply g and Q ∨ for g and Q ∨ .The coroot lattice of sl is Z and the image of 1 under the map Z ֒ → Q ∨ is an element h ∈ h called the defining vector of the inclusion. In § § VIII.11]),Dynkin proved that, after conjugating by an element of the automorphism group of g , onecan assume that the defining vector h satisfies the strong restrictions: h = X δ ∈ ∆ p δ δ ∨ for p δ real and non-negative [15, Lemma 8.3],where ∆ denotes the set of simple roots of g and further that(4.3) δ ( h ) ∈ { , , } for all δ ∈ ∆ .But note that for each simple root δ , the fundamental irreducible representation of g withhighest weight dual to δ ∨ restricts to a representation of sl with p δ as a weight, hence p δ is an integer. As a consequence of these generalities and specifically [15, Lemma 8.2], one can iden-tify an sl subalgebra of g up to conjugacy by writing the Dynkin diagram of g and puttingthe number δ ( h ) from (4.3) at each vertex; this is the marked Dynkin diagram of the sl subalgebra.Here is an alternative formula for computing the index of an sl subalgebra from itsmarked Dynkin diagram. Write κ g and m ∨ for the Killing form and dual Coxeter number Alternatively, Serre’s marvelous bound on the trace from [13, Th. 3] or [14, Th. 1] implies that for ev-ery element x of order 2 in a reductive complex Lie group G , the − -eigenspace of Ad( x ) has dimension ≤ (dim G + rank G ) / . In particular, when G is a real form of E , the − -eigenspace has dimension ≤ . HERE IS NO “THEORY OF EVERYTHING” INSIDE E of g . We have:(4.4) ( Dynkin index ) = 12 ( h, h ) = 14 m ∨ κ g ( h, h ) = 12 m ∨ X positive roots α of g α ( h ) , where the second equality is by, e.g., [17, § κ g . Onecan calculate the number α ( h ) by writing α as a sum of positive roots and applying themarked Dynkin diagram for h . Lemma 4.5.
For every simple complex Lie algebra g , there is a unique copy of sl in g ofindex , up to conjugacy. This is (equivalent to) Theorem 2.4 in [15]. We give a different proof for the conve-nience of the reader.
Proof.
The index of an sl -subalgebra is ( h, h ) / , where the defining vector h belongs tothe coroot lattice Q ∨ . If g is not of type B, then the coroot lattice is not of type C, and theclaim amounts to the statement that the vectors of minimal length in the coroot lattice areactually coroots. This follows from the constructions of the root lattices in [18, § g has type B and is so n for some odd n ≥ . The conjugacy class of an sl -subalgebra is determined by the restriction of the natural n -dimensional representation;they are parameterized by partitions of n (i.e., P n i = n ) so that the even n i occur witheven multiplicity and some n i > , see [19, 5.1.2] or [20, § sl → so n → sl n is then P (cid:0) n i +13 (cid:1) ; we must classify those partitions suchthat this sum equals the Dynkin index of so n → sl n , which is 2. The unique such partitionis · · · + 1 > . (cid:3) In the bijection between conjugacy classes of sl subalgebras and orbits of nilpotentelements in g from [19, 3.2.10], the unique orbit of index 1 sl ’s corresponds to the minimalnilpotent orbit described in [19, 4.3.3].If g has type C, F , or G , then the argument in the proof of the lemma shows thatthere is up to conjugacy a unique copy of sl in g with index 2, 2, or 3 respectively. For g of type B n with n ≥ , there are two conjugacy classes of sl -subalgebras of index 2.This amounts to the fact that there are vectors in the C n root lattice that are not roots buthave the same length as a root—specifically, sums of two strongly orthogonal short roots,cf. Exercise 5 in §
12 of [18].5. C
OPIES OF sl , C IN THE COMPLEX E We now prove some facts about copies of sl , C in the complex Lie algebra e of type E . Of course, the 69 conjugacy classes of such are known—see [15, pp. 182–185] or [21,pp. 430–433]—but we do not need this information.Fix a pinning for e ; this includes a Cartan subalgebra h , a set of simple roots ∆ := { α i | ≤ i ≤ } (numbered(5.1) 1 3 4 5 6 7 82as in [22]), and fundamental weights ω i dual to α i . As all roots of the E root system havethe same length, we can and do identify the root system with its coroot system (also calledthe “inverse” or “dual” root system). JACQUES DISTLER AND SKIP GARIBALDI
Example 5.2.
Taking any root of E , one can generate a copy of sl , C in e with index 1.Doing this with the highest root gives an sl , C with marked Dynkin diagramindex 1: 0 0 0 0 0 0 10Every index 1 copy of sl in e is conjugate to this one by Lemma 4.5. Example 5.3.
One can find a copy of sl , C × sl , C in e by taking the first copy to begenerated by the highest root of E and the second copy to be generated by the highestroot of the obvious E subsystem. If you embed sl , C diagonally in this algebra, you finda copy of sl , C with index 2 and marked Dynkin diagramindex 2: 1 0 0 0 0 0 00 Proposition 5.4.
The following collections of copies of sl , C in e are the same: (1) copies such that ± are weights of e (as a representation of sl , C ) and no otherodd weights occur. (2) copies such that every weight of e is in { , ± , ± } . (3) copies such that the inclusion sl , C ⊂ e has Dynkin index or . (4) copies of sl , C conjugate to one of those defined in Examples 5.2 or 5.3.Proof. 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 P α i ( h ) ω i . By (4.4), the index of h satisfies: X α α ( h ) = 160 X α X i α i ( h ) h ω i , α i ! ≥ X i α i ( h ) X α h ω i , α i ! where the sums vary over the positive roots. We calculate for each fundamental weight ω i the number P α h ω i , α i / :(5.5) 2 7 15 10 6 3 14As the numbers α i ( h ) are all 0, 1, or 2, the numbers (5.5) show that h for an sl , C withDynkin index 1 or 2 must be ω (index 2) or ω (index 1).For (2), the highest root ˜ α of E is ˜ α = P i c i α i , where c = c = 2 and the other c i ’sare all at least 3. As ˜ α ( h ) is a weight of e relative to a given copy of sl , C , we deduce thatan sl , C as in (2) must have h = ω or ω , as claimed.Suppose now that we are given an h for a copy of sl , C as in (1). As ± occur asweights, 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 , P i ∈ S α i isa root [22, § VI.1.6, Cor. 3b]. If the number of 1’s in the marked diagram of h is at leastthree, then one can pick S so that it meets exactly three of the α i ’s with α i ( h ) = 1 , inwhich case P 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 ,say, corresponding to simple roots α i and α j with i < j . For each i, j , one can find a root β in the list of roots of E of large height in [22, Plate VII] such that the coefficients of α i and α j in β have opposite parity and sum at least 3. (Merely taking β to be the highest root HERE IS NO “THEORY OF EVERYTHING” INSIDE E suffices for many ( i, j ) .) This contradicts (1), so there is a unique 1 in the marked diagramfor h , i.e., α i ( h ) = 1 for a unique i .If α i ( h ) = 1 for some i = 1 , , then we find a contradiction because there is a root α of E with α i -coordinate 3. Therefore α i ( h ) = 1 only for i = 1 or 8 and not for both. By thefact used two paragraphs above, β := P i α i is a root of E , so β ( h ) = P α i ( h ) is oddand must be 1. It follows that h = ω or ω . (cid:3) The sl , C of index 1 in e has centralizer the obviousregular subalgebra e of type E . (A subalgebra is regular if it is generated by the rootsubalgebras corresponding to a closed sub-root-system [15, no. 16].) Indeed, it is clear that e centralizes this sl , C . Conversely, the centralizer of sl , C is contained in the centralizerof h = ω —i.e., e ⊕ C h —but does not contain h . e . Suppose we are given a copy of sl , C in e specified by a definingvector h . By applying the 240 roots of e to h (and throwing in also 0 with multiplicity 8),we obtain the weights of e as a representation of sl , C and therefore also the decompositionof e into irreducible representations of sl , C as in, e.g., [18, § sl , C × sl , C in e , where the two sum-mands are specified by defining vectors in h . (Here we want the defining vectors to spanthe Cartan subalgebras in the images of the two sl , C ’s. In particular, they need not benormalized in the sense of (4.3).) Computing as in the previous paragraph, we can decom-pose e as a direct sum of irreducible representations m ⊗ n of sl , C × sl , C . It is easy towrite code from scratch to make a computer algebra system perform this computation. Weremark that applying this recipe in the situation from the introduction gives the dimensionof V m,n as the multiplicity of m ⊗ n .6. I NDEX COPIES OF sl , C IN THE COMPLEX E Lemma 6.1.
The centralizer of the index 2 sl , C in e from Example 5.3 is a copy of so contained in the regular subalgebra so of e .Proof. The centralizer of the sl , C of index 2 in e is contained in the centralizer of thedefining vector h ; this centralizer is reductive with semisimple part the regular subalgebra so of type D . The centralizer of sl , C contains the centralizer of the sl , C × sl , C fromExample 5.3, which is the regular subalgebra so of type D , as can be seen by the recipefrom [15, pp. 147, 148]. Computing as in 5.7, we see that the centralizer of sl , C hasdimension 78 (as is implicitly claimed in the statement of the lemma), so it lies properlybetween the regular so and the regular so .For concreteness, let us suppose that the structure constants for e are as in [23]. Definea copy of sl , C by sending ( ) to the sum of the elements in the Chevalley basis of e spanning the root subalgebras corresponding to − α and the highest root in the obvious D subdiagram. This copy of sl , C has defining vector α + α + 2 α + 2 α + 2 α + 2 α . Onechecks using the structure constants that this sl , C centralizes the index 2 sl , C we startedwith, and that together with so it generates a copy of so . In particular, the coroot latticeof this so has basis β ∨ , . . . , β ∨ , embedded in the (co)root lattice of e as in the table:(6.2) so β ∨ β ∨ β ∨ β ∨ β ∨ β ∨ e α α α α α − α − α − α − α − α − α We remark that the numbering of the coroots β ∨ , . . . , β ∨ corresponds to a numbering ofthe simple roots of so as in the diagram β β β β β β > r r r r r r Dimension count shows that this so is the centralizer. (cid:3) The claim of the lemma is already in [24, p. 125]. We gave the details of a proof becauseit specifies an inclusion of so in e and a comparison of the pinnings of the two algebrasas in (6.2).The index 2 sl and the copy of so give an sl × so subalgebra of e . We nowdecompose e into irreducible representations of sl × so . We can do this from firstprinciples by restricting the roots of e to the Cartan sublagebras of sl (using the markedDynkin diagram from Example 5.3) and so (using (6.2)). Alternatively, we can readthe decomposition off the tables in [25] as follows. As in the proof of Lemma 6.1, sl iscontained in the regular subalgebra sl × sl × so of e , and the tables on pages 301 and305 of ibid. show that e decomposes as a sum of(6.3) the adjoint representation, ⊗ ⊗ S + , ⊗ ⊗ S − , and ⊗ ⊗ V, where S ± denotes the half-spin representations of so and V is the vector representation.We can restrict the representations of sl × sl to the diagonal sl subalgebra to obtain adecomposition of e into representations of sl × so . Consulting the tables in ibid. forrestricting representations from type B to D allows us to deduce the decomposition(6.4) ⊗ so , C ⊕ ⊗ ( spin ) ⊕ ⊗ ⊕ ⊗ ( vector ) of e as a representation of sl × so . From this it is obvious that so , C is the Lie algebraof a copy of Spin in E .The main result of this section is the following: Lemma 6.5.
Up to conjugacy, there is a unique copy of SL , C × SL , C in E , C so thateach inclusion of SL , C in E , C has index 2. The centralizer of this SL , C × SL , C hasidentity component Sp , C × Sp , C .Proof. As in the proof of Lemma 4.5 (or by the method used to prove Prop. 5.4), there aretwo index 2 copies of sl in so , coresponding to the partitions(a) · · · + 1 and (b) · · · + 1 of 13. The recipe in [19, § sl ’s, which we can rewritein terms of the E simple roots using (6.2):(6.6) (a) β ∨ + 2 β ∨ + 2 β ∨ + 2 β ∨ + 2 β ∨ + β ∨ = − α + α (b) β ∨ + 2 β ∨ + 3 β ∨ + 4 β ∨ + 4 β ∨ + 2 β ∨ = − α − α − α − α We can pair each of (a) and (b) with the copy of sl from Example 5.3 to get an sl × sl subalgebra of e where both sl ’s have index 2. Clearly, these represent the only two E -conjugacy classes of such subalgebras. With (6.6) in hand, we can calculate the multi-plicities of the irreducible representations of sl × sl in e as in 5.7.In case (a), every irreducible summand m ⊗ n has m + n even. Therefore, this copyof sl × sl is the Lie algebra of a subgroup of E isomorphic to (SL × SL ) / ( − , − .(An alternative way to see this is to note that the simple roots with odd coefficients are thesame in (6.6a) and the defining vector in Example 5.3.) HERE IS NO “THEORY OF EVERYTHING” INSIDE E In case (b), we have the following table of multiplicities for m ⊗ n :(6.7) m n In particular, it is the Lie algebra of a copy of SL × SL in E . The centralizer of (b) in Spin has been calculated in [26, IV.2.25], and the identity component is Sp × Sp , asclaimed. (cid:3) We can decompose e into a direct sum of irreducible representations of the sl × sl × sp × sp subalgebra from Lemma 6.5 by combining the decomposition of e into irre-ducible representations of sl × so from (6.4) with the tables in [25]. Specifically, werestrict representations from so to an sp × so subalgebra and then from so to sp × sl ,where this sl also has index 2. Recall that sp has two fundamental irreducible represen-tations: 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:(6.8) V , ≃ ⊗ , V , ≃ ⊗ , V , ≃ ⊗ , V , ≃ ⊗ , and V , ≃ ⊗ .
7. C
OPIES OF
SL(2 , C ) IN A REAL FORM OF E Suppose now that we have a copy of
SL(2 , C ) inside a real Lie group E of type E .Over the complex numbers, we decompose Lie(E) ⊗ C into a direct sum of irreduciblerepresentations of SL(2 , C ) ⊗ C ≃ SL , C × SL , C ; each irreducible representation can bewritten as m ⊗ n where m and n denote the dimension of an irreducible representation ofthe first or second SL , C respectively. The goal of this section is to prove: Proposition 7.1.
Maintain the notation of the previous paragraph. If
Lie(E) ⊗ C containsno irreducible summands m ⊗ n with m + n > , then the identity component Z of thecentralizer of SL(2 , C ) in E is a subgroup isomorphic to (1) Spin(7 , if E is split; or (2) Spin(9 , or Spin(11 , if the Killing form of Lie(E) has signature − .In either case, Lie( Z ) ⊗ C is the regular so subalgebra of Lie(E) ⊗ C .Proof. Complexifying the inclusion of
SL(2 , C ) in E and going to Lie algebras gives aninclusion of sl , C × sl , C in the complex Lie algebra e . The hypothesis on the irreduciblesummands m ⊗ n implies that each of the two sl , C ’s has index 1 or 2 by Proposition 5.4.As complex conjugation interchanges the two components, they must have the same index.Suppose first that both sl ’s have index 2. When we decompose e as in 1.1, we find therepresentation ⊗ with positive multiplicity 4 by (6.7), which violates our hypothesis onthe SL(2 , C ) subgroup of E .Therefore both sl ’s have index 1. Lemma 4.5 (twice) gives that this sl × sl is con-jugate to the one generated by the highest root of E from Example 5.2 (so the second sl belongs to the centralizer of type E ) and by the highest root of the E subsystem andmakes up the first two summands of an sl × sl × so subalgebra, the same one used tofind (6.3). That is, so centralizes sl × sl . Conversely, the centralizer of the definingvectors of the two copies of sl has semisimple part so ; it follows that Lie( Z ) ⊗ C isisomorphic to so .From this and the decomposition (6.3), we see that Z is a real form of Spin . As Lie(E) is a real representation of Z , 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 observationshows that Z is Spin(12 − a, a ) for some ≤ a ≤ . The second shows that a must be 1,3, or 5, as claimed in the statement of the proposition.It remains to prove the correspondence between a and the real forms of E . For a = 5 ,this is clear: the subgroup generated by SL(2 , C ) and Spin(7 , has real rank 6, so it canonly be contained in the split real form.Now suppose that a = 3 or 1 and that SL(2 , C ) is in the split E ; we will show thatthe Killing form of E has signature − . Over C , SL(2 , C ) is conjugate to the copy of SL , C × SL , C in E , C generated by the highest root of E and the highest root of thenatural subsystem of type E . Writing out these two roots in terms of the E simple roots,we see that α and α are the only simple roots whose coefficients have different parities.It follows that the element − ∈ SL(2 , C ) —equivalently, ( − , − ∈ SL × SL —is h α ( − h α ( − in the notation of [27], where h α i : C × → E ⊗ C is the cocharactercorresponding to the coroot α ∨ i . Now, α and α are the only simple roots with oddcoefficients in the fundamental weight ω , so the subgroup of E ⊗ C fixed by conjugationby this − is generated by root subgroups corresponding to roots α such that h ω , α i iseven. These roots form the natural D subsystem of E , and in this way we see SL(2 , C ) · Spin(12 − a, a ) as a semisimple subgroup of maximal rank in a copy of a half-spin group H in 16 dimensions—the identity component of the centralizer of − .We claim that H is isogenous to SO(12 , . As H is a half-spin group with a half-spinrepresentation defined over R , it is isogenous to SO(16 − b, b ) for b = 0 , , or or itis quaternionic; these possibilities have Killing forms of signature − , − , , or − respectively, as can be looked up in [28], for example. The adjoint representation of H ,when restricted to SL(2 , C ) · Spin(12 − a, a ) , decomposes as the adjoint representation of SL(2 , C ) · Spin(12 − a, a ) and ⊗ ⊗ V by (6.3). The Killing form on H restricts to apositive multiple of the Killing form on SL(2 , C ) · Spin(12 − a, a ) (as can be seen over C by the explicit formula on p. E-14 of [26])—i.e., has signature − or − for a = 1 or 3—and a form of signature ± − a ) on ⊗ ⊗ V ; the sum of these has signature , − , or − since a = 1 or . Comparing the two lists verifies that H is isogenous to SO(12 , .The Killing form on H has signature − . The invariant bilinear form on the half-spinrepresentation is hyperbolic (because H is isogenous to spin of an isotropic quadratic formof dimension divisible by 8, see [29, 1.1]). As a representation of H , Lie( E ) is a sum ofthese two representations, and we conclude that the Killing form on Lie( E ) has signature − , as claimed. (cid:3) Remark . We can determine the centralizer and the real form of E also in the excludedcase in the proof where both sl ’s have index 2. As in Lemma 6.5, the centralizer is areal form of Sp , C × Sp , C . The decomposition (6.8) shows that complex conjugationinterchanges the two Sp , C terms, so the centralizer is R (Sp , C ) . Complex conjugationinterchanges the irreducible representations appearing in (1.1) in pairs (contributing 0 tothe signature of the Killing form κ E of E ), except for ⊗ ⊗ V , , which has dimension . This last piece breaks up into a 36-dimensional even subspace, and a 28-dimensionalodd subspace, contributing 8 to the signature of κ E and proving that the resulting real formof E is the split one. HERE IS NO “THEORY OF EVERYTHING” INSIDE E
8. N O T HEORY OF E VERYTHING IN A REAL FORM OF E In the decomposition (1.1) of
Lie(E) ⊗ C , the integers m, n are positive, so (ToE2)implies(ToE2’) V m,n = 0 if m ≥ or n ≥ .We prove the following strengthening of the real case of Theorem 1.3: Lemma 8.1.
If subgroups
SL(2 , C ) and G of a real form E of E satisfy (ToE1) and(ToE2’), then V , is a self-conjugate representation of G , i.e., (ToE3) fails.Proof. As in the proof of Proposition 7.1, over the complex numbers we get two copies of sl that embed in E with the same index, which is 1 or 2.If the index is 1, we are in the case of that proposition. The − -eigenspace in Lie(E) (of the element − in the center of SL(2 , C ) ) is a real representation of SL(2 , C ) · G , and G is contained in a copy of Spin(12 − a, a ) for a = 1 , 3, or 5. As in the proof of theproposition, there is a representation W of SL(2 , C ) × Spin(12 − a, a ) defined over R thatis isomorphic to (2 ⊗ ⊗ S + ) ⊕ (1 ⊗ ⊗ 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 twohalf-spin representations of Spin(12 − a, a ) to the compact subalgebra are equivalent [25,p. 264], and we see that in each case the restriction is quaternionic . (To see this, one usesthe standard fact that the spin representation of so (2 ℓ + 1) is real for ℓ ≡ , andquaternionic for ℓ ≡ , .) That is, the restrictions of S + , S − , and their complexconjugates to the maximal compact subgroup are all equivalent (over C ), hence the sameis true for their further restrictions to G , and (ToE3) fails.If the index is 2, then G is contained in a real form of Sp , C × Sp , C by Lemma 6.5.When we decompose e as in (1.1), we find V , and V , as in (6.8). As complex con-jugation interchanges these two representations, it follows that complex conjugation in-terchanges the two Sp , C factors, i.e., the centralizer of SL(2 , C ) has identity componentthe transfer R (Sp , C ) of Sp , C . Its maximal compact subgroup is the compact form of Sp , C (also known as Spin(5) ), all of whose irreducible representations are self-conjugate.Therefore, (ToE3) fails. (cid:3)
Remark . It is worthwhile noting that, in each of the three cases in Proposition 7.1(the three cases where (ToE2) holds), it is possible to embed G SM in the centralizer, thusshowing that (ToE1) is satisfied. Given such an embedding, a simple computation verifiesexplicitly that S + has a self-conjugate structure as a representation of G SM .First consider Spin(11 , . There is an obvious embedding of G GUT := Spin(10) .Under this embedding, S + decomposes as the direct sum of the two half-spinor represen-tations, i.e., as a generation and an anti-generation.For Spin(7 , , there is an obvious embedding of the Pati-Salam group, G GUT :=(Spin(6) × Spin(4)) / ( Z / Z ) . Again, S + decomposes as the direct sum of a generationand an anti-generation.Finally, Spin(3 , contains (SU(3) × SU(2) × SU(2) × U(1)) / ( Z / Z ) as a subgroup.Under this subgroup, S + = (3 , , / ⊕ (3 , , − / + (1 , , − / + (1 , , / where the subscript indicates the U(1) weights, and the overall normalization is chosento 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 anembedding of G SM ⊂ Spin(3 , where, again S + has a self-conjugate structure as arepresentation of G SM .9. N O T HEORY OF E VERYTHING IN COMPLEX E We now complete the proof of Theorem 1.3 by proving the following strengthening ofthe complex case.
Lemma 9.1.
If subgroups
SL(2 , C ) and G of R ( E , C ) satisfy (ToE1) and (ToE2’), then V , is a self-conjugate representation of G , i.e., (ToE3) fails. First, recall the definition of the transfer R ( H C ) of a complex group H C as described,e.g., in [30, § H C × H C , where complexconjugation acts via ( h , h ) = ( h , h ) . One can view R ( H C ) as the subgroup of the complexification consisting of elements fixedby complex conjugation.Now consider an inclusion φ : SL(2 , C ) = R (SL , C ) ֒ → R (E , C ) . Complexifying, weidentify R (SL , C ) ⊗ C with SL , C × SL , C and similarly for R ( E , C ) and write out φ as(9.2) φ ( h , h ) = ( φ ( h ) φ ( h ) , ψ ( h ) ψ ( h )) for some homomorphisms φ , φ , ψ , ψ : SL , C → E , C . As φ is defined over R , wehave: φ ( h , h ) = φ ( h , h ) = ( ψ ( h ) ψ ( h ) , φ ( h ) φ ( h )) , and it follows that ψ ( h ) = φ ( h ) and ψ ( h ) = φ ( h ) . Conversely, given any twohomomorphisms φ , φ : SL , C → E , C (over C ) with commuting images, the same equa-tions define a homomorphism φ : SL(2 , C ) → R (E , C ) defined over R . Proof of Lemma 9.1.
Write Z for the identity component of the centralizer of the imageof the map φ × φ : SL , C × SL , C → E , C from (9.2). Clearly, G is contained in thetransfer R ( Z ) of Z . In each of the cases below, we verify that(9.3) Z is semisimple and − is in the Weyl group of Z .It follows from this that the maximal compact subgroup of R ( Z ) is the compact real form Z R of Z and that Z R is an inner form. Hence every irreducible representation of Z R is realor quaternionic, hence every representation of Z R is self-conjugate. That is, (ToE3) fails,which is the desired contradiction.Case1: φ or φ istrivial. Consider the easiest-to-understand case where φ or φ isthe zero map, say φ . In the notation of (9.2), φ ( h , h ) = ( φ ( h ) , φ ( h )) , i.e., φ is thetransfer of the homomorphism φ : SL , C → E , C . By Proposition 5.4, φ has index 1 or2. If φ has index 1, then Z is simple of type E by 5.6, hence (9.3) holds. If φ has index2, then Lie( Z ) is isomorphic to so , C by Lemma 6.1, and again (9.3) holds.Case2: Neither φ nor φ istrivial.Now suppose that neither φ nor φ is trivial. Again,Proposition 5.4 implies that φ and φ have Dynkin index 1 or 2.If φ and φ both have index 1, then (over C ) the homomorphism φ × φ is the onefrom the proof of Proposition 7.1 and Z is the standard D subgroup of E , C and (9.3)holds.Now suppose that φ and φ both have index 2. As φ is an injection, it is not possiblethat φ and φ both vanish on − ∈ SL , C , and it follows from the proof of Lemma 6.5 HERE IS NO “THEORY OF EVERYTHING” INSIDE E that φ × φ is an injection as in the statement of Lemma 6.5. In particular, Z has Liealgebra sp , C × sp , C of type B × B and (9.3) holds. Note that (ToE2) fails in this caseby (6.7).Suppose finally that φ has index 1 and φ has index 2. We conjugate so that φ ( sl ) isthe copy of sl from Example 5.3, and (by Lemma 4.5 for the centralizer so of φ ( sl ) )we can take φ ( sl ) to be a copy of sl generated by the highest root of so . Calculating asdescribed in 5.7 gives the following table of multiplicities for the irreducible representation m ⊗ n of sl × sl in e :(9.4) m n In particular, the A × B subgroup of Spin that centralizes the image of φ × φ is allof the identity component Z of the centralizer in E . Again (9.3) holds. (Of course, (9.4)shows that (ToE2) fails.) (cid:3)
10. R
ELAXING (T O E2) TO (T O E2’)Combining Lemmas 8.1 and 9.1 gives a proof not only of Theorem 1.3, but of thefollowing stronger statement.
Theorem 10.1.
There are no subgroups
SL(2 , C ) · G satisfying (ToE1), (ToE2’), and (ToE3)in the (transfer of the) complex E or any real form of E . (cid:3) We retained hypothesis (ToE2) in the introduction because that is what is demandedby physics. Technically, it is possible for V , and V , to be nonzero in an interactingtheory—so (ToE2) is false but (ToE2’) still holds—but only in the presence of local su-persymmetry (i.e., in supergravity theories) [31]. Lisi’s framework is not compatible withlocal supersymmetry, so we excluded this possibility above.For real forms of E , weakening (ToE2) to (ToE2’) only adds the case of E , with G max = Spin(5) , where we find(10.2) V , ≃ V , = 4 , V , ≃ V , = 4 ⊕ and we have indicated the irreducible representations of Spin(5) by their dimensions. Be-cause the gravitinos transform nontrivially under G max and because of their multiplicity,the only consistent possibility would be a gauged N = 4 supergravity theory (for a re-cent review of such theories, see [32]). Unfortunately, the rest of the matter content (itsuffices to look at V , ) is not compatible with N = 4 supersymmetry. Even if it were, N = 4 supersymmetry would, of course, necessitate that the theory be non-chiral, makingit unsuitable as a candidate Theory of Everything.To summarize the results of this section, the previous subsection, and Remark 7.2, weak-ening (ToE2) to (ToE2’) adds only three additional entries to Table 1.5.(10.3) E G max V , V , E Spin(5) 4 4 ⊕ R (E , C ) Spin(5) × Spin(5) (4 , ⊕ (1 ,
4) (4 , ⊕ (5 , R (E , C ) SU(2) × Spin(9) (2 ,
1) (2 , ⊕ (2 , In each case the fermion representations, V , ≃ V , and V , ≃ V , , are pseudorealrepresentations of G max .
11. C
ONCLUSION
In paragraph 3 above, we observed by an easy dimension count that no proposed Theoryof Everything constructed using subgroups of a real form E of E has a sufficient numberof weight vectors in the − -eigenspace to identify with all known fermions. The proof ofour Theorem 1.3 was quite a bit more complicated, but it also gives much more. It showsthat you cannot obtain a chiral gauge theory for any candidate ToE subgroup of E , whether E is a real form or the complex form of E . In particular, it is impossible to obtain eventhe 1-generation Standard Model (in the sense of Definition 2.6) in this fashion. Acknowledgments.
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