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1. Introduction.

  The 52-dimensional exceptional Lie group F4 is in some respects the most exceptional of all Lie groups.

J. Frank Adams devoted his 1980 Nuffield lecture1 to the relationship between F4, the Lie group Spin(8) that is the simply-connected 2-fold covering group of SO(8), and the triality automorphism of Spin(8). F4 is the group of automorphisms of the exceptional Jordan algebra H3(O) of 3x3 Hermitian matrices of octonions, or Cayley numbers, denoted by O. An introductory description of octonions can be found in Kantor and Solodnikov2.

As Adams explains, the 28-dimensional adjoint representation of the Lie group Spin(8) can be embedded in F4 so that the remaining 24 dimensions correspond to the 8-dimensional vector representation of Spin(8), denoted here by V8, and the two mirror image 8-dimensional half-spinor representations of Spin(8), denoted here by S8+ and S8-.

The triality automorphism is just the sequence of isomorphisms

V8 -> S8+ -> S8- -> V8. Each of the 8-dimensional Spin(8) representations are locally isomorphic to the octonions.

The pseudonymous author Arthur L. Besse devotes section 3.G of his first book3 to the Cayley projective plane OP2, which is shown to be the compact rank one symmetric space F4/Spin(9). The local equivalence of OP2 and S8+ 7 S8- is used by Green, Schwartz, and Witten4 in their remark (written by Witten) that the Lie algebra F4 can be built by adding the 16-dimensional spinor representation of Spin(9), which is just

S8± = S8+ 7 S8-, to the adjoint representation of Spin(9).

Since Spin(9)/Spin(8) = S8 = OP1, the structure of F4 is given by the two global Lie group fibrations Spin(8) -> Spin(9) -> S8 = OP1 and

Spin(9) -> F4 -> OP2 = Cayley projective plane.

 

 

F4 is the automorphism group of H3(O), the 27-dimensional space of 3 x 3 octonionic matrices.71,72 If a,b,c ΠR and O+,Ov,O- ΠO, then H3(O) is given by the 3 x 3 matrix

 

a O+ Ov

`O+ b O-

`Ov `O- c

H3(O) with the product (1/2)(AB+BA) is an exceptional Jordan algebra that is commutative but not associative.

The triality subgroup of F4 acts on H3(O) by permuting the three octonionic spaces O+, Ov, and O- spanned by O+, Ov, and O-.

There are three Spin(8) subgroups of F4. They are not independent because they are related by triality.71 The action of each of the three Spin(8) groups on H3(O) is on each of the three octonionic spaces O+, Ov, and O-.

Any action of Spin(8) on any octonion o can be represented by octonionic multiplication:71

Spin(8): o -> xo , o -> ox, or o -> xox; x ΠO.

If the Spin(8) acting on O+ is represented by left-multiplication

o -> xo, then by triality the Spin(8) acting on O- is represented by right-multiplication o -> ox , and the Spin(8) acting on Ov is represented by two-sided multiplication o -> xox.

In the F4 model, O+ represents the first generation fermion particles,

O- represents the first generation antiparticles, and Ov represents the 8-dimensional spacetime V8.

To define the state space for the F4 model quantum theory, take H3(O) to represent the Jordan algebra of quantum operators with Jordan product AoB = (1/2)(AB+BA) where AB is the matrix product.

For a general description of quantum theory, including the Jordan algebra approach and others, see Sudbery70.

To make the F4 model, first note that an element A of H3(O) (obviously) acts by the Jordan product on the other elements of H3(O). Therefore, the first candidate for the quantum state space is H3(O) itself.

This is NOT the conventional approach, which would be to take the state space to be the 3 x 1 (octonionic) vector space with the action of H3(O) 3 x 3 matrices being the ordinary matrix product.

If H3(O) is both the observable operator Jordan algebra and

the state space, then the observable operators can be interpreted as "states" of the "experimental observer", which act on the "observed" state

to produce the "result of observation" state, and there is a symmetry between observer and observed that is missing in the interpretation of quantum mechanics in which "observers" are "classical" and different from "observed" "quantum" states.

By the Jordan product, A: H3(O) -> H3(O) and defines an

automorphism of H3(O) (except for singular A, such as A = 0).

The full automorphism group of H3(O) is 52-dimensional F4, and

is therefore much bigger than 27-dimensional H3(O).

Now, try to represent A: H3(O) -> H3(O) as an element of

a subspace of F4.

Do not try to represent A as an element of a subGROUP

of F4. A simpler analogy is the octonions O. An octonion x:O -> O by octonionic product defines an automorphism of O (except for x=0).

The full automorphism group of O is 14-dimensional G2.72

G2 has a 7-dimensional representation on Im(O), the subspace of O that is spanned by the imaginary octonions, but Im(O) is not a subgroup of O.

To see that Im(O) is not a subgroup, note that the spheres S0, S1, and S3 are the only spheres that are Lie groups, and that S7 (generated by Im(O)) is not a Lie group. The analog of Im(O) is what is needed.

To find the analog of Im(O), note that H3(O) has a scalar product defined by (X,Y) = Tr(XoY), where Tr is trace,

and a scalar triproduct (X,Y,Z) = Tr((XoY)oZ) = (XoY,Z),

and that the automorphisms of F4 leave invariant the corresponding quadratic and cubic forms.

In particular, an automorphism A: H3(O) -> H3(O) leaves invariant

the trace of an element of H3(O).

Therefore, the 26-dimensional space of traceless 3 x 3 hermitian

real (not yet complexified) octonionic matrices, denoted by M3(O),

is a representation space for the automorphism Lie group F4.

M3(O), the second candidate for a state space, is

 

a O+ Ov

 

O+* b O- a,c,b real

a+b+c=0

Ov* O-* c

 

Identify O+ with first generation fermion particles, Ov with 8-dimensional spacetime, and O- with first generation antiparticles; so that an element of the state space describes the fermion particles and antiparticles at a point of spacetime.

Since the identification of O+, Ov, and O- should remain consistent after action by an observable operator, do not allow mixing up of O+, Ov, and O- by the operator A in the automorphism group F4.

The subgroup of F4 that leaves invariant the O+, Ov, and O-

subspaces of M3(O) is Spin(8), so the gauge group of the F4 model is Spin(8).

Therefore the third candidate for state space is the 24-dimensional subspace

 

0 O+ Ov

 

O+* 0 O-

Ov* O-* 0

 

which I denote by OP1OP2, to suggest that (uncomplexified) Ov is octonionic projective 1-space, OP1 = S8 and (uncomplexified) O+ O- is octonionic projective 2-space, OP2 = Moufang plane, or Cayley projective plane.

By triality, the action of Spin(8) on OP1OP2 can be defined by left (La) or right (Ra) octonionic multiplication by a unit octonion a, as

La on O-, La*Ra* on O+, and Ra on Ov.71

 

The automorphism group F4 is a Lie group, not an algebra.

Of course, there is a Lie algebra (also called F4) and an adjoint

map ad: F4(Lie algebra) -> F4(Lie group).

The exponential map exp: F4(Lie algebra) -> F4(Lie group) is

the "inverse" of the adjoint map ad: F4(Lie algebra) -> F4(Lie group),

so that exp(ad(A)) = A and ad(exp(a)) = a .

The F4 Lie algebra acts as the algebra of derivations of H3(O).

That is, for ad(A) in the F4 Lie algebra and states X,Y in H3(O),

ad(A)(XoY) = ad(A)(X) oY + Xo ad(A)(Y).

Therefore, ad(A)(IoI) = ad(A)(I) oI + Io ad(A)(I) = 2 ad(A)(I),

implying that ad(A)(I) = 2 ad(A)(I), so that ad(A): I -> 0 .

The F4 Lie algebra of derivations of H3(O) annihilates the trace,

so the 26-dimensional space of traceless 3 x 3 hermitian real (uncomplexified) octonionic matrices, denoted by M3(O), is a representation space for the automorphism Lie group F4.

 

 

 

 

 

 

 

 

 

 

 

OP1OP2 is

 

0 O+ Ov

 

O+* 0 O- O+,O-,Ov octonion

Ov* O-* 0

 

To have a quantum theory with more than one particle, requires construction of a Fock space, also called the process of second quantization. A Fock space is just the direct sum of tensor products of

all degrees of a single-particle state space (plus a vacuum - also, by distinguishing between fermions and bosons it is sometimes necessary to use only antisymmetric or symmetric subspaces of the tensor space).

A tensor product of a number of copies of OP1OP2 is defined with

respect to a scalar division algebra: either the real numbers R,

the complex numbers C, the quaternions H, or the octonions O.

To have a well-defined tensor product, objects such as

aA ƒ bB ƒ cC = abc AƒBƒC , where A,B,C are in OP1OP2 and

a,b,c are scalars, must be well-defined.

Since OP1OP2 is 3-dimensional over the octonions O,

it is natural to guess that the scalars should be octonions.

However, the octonion scalar product abc is not well-defined,

because the octonions are not associative, so generally (ab)c a(bc).

I do not claim to be able to prove that it is absolutely impossible

to add ordering distinctions to the definition of tensor product spaces, such as (AƒB)ƒC Aƒ(BƒC), and therefore construct a logically consistent "octonionic quantum theory". I just claim that it is, to my taste, "unnatural and unphysical". Therefore, I rule out octonion scalars.

The quaternion H scalar product abc in aA ƒ bB ƒ cC = abc AƒBƒC is associative, so (ab)c = a(bc). However, quaternions are not commutative, either with themselves or with octonions, so generally

aA ƒ bB Aa ƒ bB and ab AƒB is not well-defined.

Again, I do not claim to be able to prove that it is not possible to construct a quaternionic quantum theory. There have been efforts in that direction, such as in Finkelstein, Jauch, Schiminovich, and Speiser73, but

neutron interferometer experiments by Kaiser,George and Werner74 show that there is no quaternionic noncummutative contribution to the neutron-nuclear scattering amplitude to 1 part in 30,000. Therefore, I also rule out quaternion scalars.

The real numbers R are ruled out by requiring quantum theory to have superpositions of states (unless R-quantum theory has added to it a superposition principle that in effect makes the R-quantum theory into a C-quantum theory)73.

Therefore, quantum theory requires complex scalars.

In effect, OP1OP2 must be complexified, and the space M3(O) from which OP1OP2 is constructed should be complexified, to be M3(CƒO).

The complexification is an irreducible Kähler 54-dimensional bounded complex homogeneous domain M3(CƒO) that is isomorphic to the symmetric space E7/E6 x U(1), and is the set of (CƒO)P2's in (HƒO)P2 (Besse22).

In the F4 model, let f be the Fock space defined by M3(CƒO),

a be the space of connections on the generalized principal fibre bundle P, and g0 be the gauge transformation group with fixed base point.

P is generalized because it not only has a gauge group Spin(8) and a base manifold V8, but it also has spinor manifolds S8+ and S8-.

If the 3 octonionic parts of M3(CƒO) are fixed as

Ov=V8, O+=S8+, and O-=S8-,then Aut(P) is not F4 = Aut(H3(O)), which has triality operations interchanging V8, S8+, and S8-, but is smaller.

Aut(P) includes both the space of sections

g0 = G(V8,(P x AdSpin(8) Spin(8))) = G(V8, AdP) and

the space of diffeomorphisms Diff(V8).

g0 is analytic.

Diff(V8) is smooth but not analytic,so the exponential map does not give canonical coordinates near the identity.

Therefore quantum field theory should be formulated on the tangent space of V8, which is Minkowski-like, as suggested by Mayer23.

f is a bundle over a and f/g0 is a natural candidate for the Fock space bundle over the space a/g0 of physical gauge orbit states. It is necessary to reduce the Fock space f to its projective version fP to form the bundle fP over a/g0. (Nash54) Therefore the compact projective Fock space fP, based on the compact symmetric space E7/E6 x U(1) that is the set of (CƒO)P2's in (HƒO)P2, is the physical state space of the F4 model.

When M3(CƒO) is used to define the state space, the fermion particle-antiparticle part S8± of the state space should not be

OP2 = F4/Spin(9), but the Silov boundary of a complex subspace of M3(CƒO) representing the O+ fermion particles and the O- fermion antiparticles. Taking the Silov boundary gives a real 16-dimensional space that contains all the information of the 32-dimensional complex subspace of M3(CƒO) that represents O+ and O-. The 32-dimensional subspace is (CƒO)P2, the complexification of OP2. It is a Kähler manifold, (CƒO)P2 = E6/Spin(10) x U(1), called Rosenfeld's elliptic projective plane. (Besse22) The Silov boundary of (CƒO)P2 is two copies of RP1 x S7, or S7 x S1 x S7.

Similarly, the M3(CƒO) state space implies that the 8-dimensional V8 spacetime should not be S8= Spin(9)/Spin(8), but the Silov boundary of a complex subspace of M3(CƒO) representing the Ov spacetime. The 16-dimensional subspace is (CƒO)P1, the complexification of OP1. (CƒO)P1 = Spin(10)/Spin(8) x U(1). It is the set of RP1s in RP9, and is a Kähler manifold. (Besse22) The Silov boundary of (CƒO)P1 is RP1 x S7 (Hua11). It is the correct compact manifold for fermions and 8-dimensional spacetime in the F4 model.

 

 

The mathematics of quantum field theories with operators acting on a phase space that is a Hermitian symmetric space is studied by Andre and Julianne Unterbeger68 (particularly with respect to the Cartan domains of type BD I (q=2) such as Spin(10)/Spin(8) x Spin(2)) and by Harald Upmeier69.

The plan of this paper is to start in Chapter 2 with an

8-dimensional pure gauge theory with gauge group Spin(8), then construct a model of the fermions, and then add the fermionic interactions to the pure gauge Lagrangian. In the F4 splitting

F4 = Spin(8) V8 S8+ S8- , the gauge group is Spin(8),

fermion particles and antiparticles are S8+ and S8-, and spacetime is V8. The resulting 8-dimensional Lagrangian action is of the form

LF4 = V8 [ -F8Ÿ*F8 + ¯S8± (g) S8± + gg + hÝ (dg/dA) h ]

where V8 is an 8-dimensional spacetime, F8 is an Ad28 Spin(8) curvature 2-form, *F is the dual 6-form, S8± is a spinor fermion field, g is the Spin(8) Dirac operator, gg is a gauge-fixing term, and hÝ (dg/dA) h is a ghost term.

The gauge-fixing and ghost terms reduce the symmetry of the Lagrangian from Spin(8) gauge group symmetry to the BRS symmetry of the cohomology of the group of gauge transformations.

Following Nash54, let a be the space of connections in the F4 model, g the group of gauge transformations, and g‚ the subgroup of gauge transformations acting as the identity at a base point. If V8 is considered to be S8, g0 = W8Spin(8), the maps S8 -> Spin(8).

The space of physical fields, or gauge orbits, in the F4 model is

a/g0 = Bg0, the classifying space of g0. Since B and W are inverse,

a/g0 = Bg0 = BW8Spin(8) = W7Spin(8). Since Bg0 is the universal base manifold for bundles over g0, Bg0 should have the dimension of physical spacetime. As H*(Spin(8);R) = L(x3, x7, x11, x7'), the cohomology of Bg0 is generated by two forms of degree 0, and one form of degree 4.

If the degree 4 form defines the physical spacetime,

such as by being a Cayley calibration 4-form over V8 as described by

Lawson and Michelsohn20 and Harvey21, then the 8-dimensional spacetime of the F4 model is reduced to a physically realistic

4-dimensional spacetime.

Prior to the dimensional reduction, there is only one generation of fermions. If S8+ is given an octonionic basis {1,i,j,k,e,ie,je,ke}, then S8+ is identified with fermion particles as follows:

1 - electron neutrino n;

i, j, k - red, blue, and green up quarks ru, bu, and gu;

e - electron e; and

ie, je, ke - red, blue, and green down quarks rd, bd, and gd.

S8- is similarly identified with fermion antiparticles.

There are associated with S8+ and S8- compact manifolds Q8+ and Q8-, such that Q8+ = Q8- = RP1 x S7. Q8+ is called the spinor manifold of the F4 model. Since Q8+ is the Silov boundary of the bounded complex domain D8+, which is isomorphic to the 16-dimensional symmetric space D8+ = Spin(10)/Spin(8) x U(1), Q8+ has a local symmetry of Spin(8) for the gauge group and a local symmetry U(1) for the complex phase of a propagator amplitude.

A similar structure exists for Q8-.

After the dimensional reduction, the 8 massless fermion particles

(1 Weyl and 7 Dirac) and 8 massless fermion antiparticles become three generations of 8 particles and 8 antiparticles each. The Dirac fermions can get mass at tree level from the Higgs mechanism, but the Weyl fermions, the neutrinos, are massless at tree level and can only get mass by a radiative process.

The dimensional reduction mechanism acts on the 28 infinitesimal generators of the gauge group Spin(8) to reduce it to the 28 infinitesimal generators of Spin(5) x SU(3) x Spin(4) x U(1)4.

The gauge group reduction is not a straightforward isomorphism between Spin(8) and a Cartesian product of subgroups, because Spin(8) is not equal to Spin(5) x SU(3) x Spin(4) x U(1)4.

The group reduction is accomplished by first reducing the vector space of the Lie algebra Spin(8) to R10 R8 R6 R4, then using the Casimir operators of Spin(8) and their Weyl group symmetries to introduce onto R10 R8 R6 R4 the Lie algebra structure

Spin(5) + SU(3) + Spin(4) + U(1)4 , finally leading to the Lie group structure Spin(5) x SU(3) x Spin(4) x U(1)4.

Spin(5) = Sp(2). Below the Planck level, the renormalizable Spin(5) Yang-Mills type Lagrangian is transformed into a non-renormalizable Spin(5) Lagrangian that gives Einstein gravity and a cosmological term by the MacDowell-Mansouri mechanism5,6,7 as described in Chapter 4.

The SU(3) gives the color force.

Spin(4) = SU(2) x SU(2). One of the SU(2)'s survives to give the weak force, and the other SU(2) is transformed into an SU(2) complex doublet Higgs field (F+ , F0) that gives mass to the weak bosons and fermions, leaving one real surviving physical Higgs scalar H. The Higgs mechanism is described in Chapter 5.

The 4 infinitesimal generators of U(1)4 become the x, y, z, and t polarizations of the covariant photon of electromagnetism.

 

The Laplace-Beltami operator for the propagator of each of the four forces is defined on a 4-dimensional compact symmetric space:

Spin(5) on S4

SU(3) on CP2

Spin(4) = SU(2) x SU(2) on S2 x S2

U(1)4 on T4 .

The F4 model force strengths appear in the curvature term of the F4 model Lagrangian by way of the covariant derivative of the potential. The potential is a connection, providing parallel transport globally over the entire spacetime base manifold. Therefore the relative volumes of the compact forms of the spacetime base manifold over which the gauge groups act globally are one factor in the relative strengths of the forces.

In the F4 model each of the four forces must not only act on the spacetime base manifold, but it must also act on a characteristic part Q of the spinor manifold Q8+ = RP1 x S7, the Silov boundary of a bounded domain D8+ which is isomorphic to the symmetric space

D8+ = Spin(10)/Spin(8) x U(1).

To see how a given force acts on the basis fermions

{n, e, ru, gu, bu, rd, gd, bd} of Q8+ = RP1 x S7, consider Sn à S7, where S7 has basis fermions {e, ru, gu, bu, rd, gd, bd}, as a sphere Sn in the tangent space R7 of S7. Then there is a subspace Rn+1 of R7 such that

Sn à Rn+1, so that the tangent bundle of Sn includes n+1 independent linear combinations of the basis fermions {e, ru, gu, bu, rd, gd, bd} of S7.

 

SU(3) acts on S5 = [SU(4)/S(U(3) x U(1))]

S5 Ã R6 ´ {ru, gu, rd, gd, bu, bd}

SU(2) acts on RP1 x S2 = [Spin(5)/SU(2) x U(1)]

RP1 x S2 Ã RP1 x R5 ´ {n, e, (r+g+b)u, (r+g+b)d}

U(1) acts on S1

S1 Ã R2 ´ {e, (r+g+b)(u+d)}

Spin(5) acts on RP1 x S4 = [Spin(7)/Spin(5) x U(1)]

RP1 x S4 Ã RP1 x R5 ´ {n, e, (r+g+b)u, (r+g+b)d, ru, gu} and

{n, e, (r+g+b)u, (r+g+b)d, ru, gu} generates

{n, e, ru, gu, bu, rd, gd, bd} = RP1 x S7 by bu = (r+g+b)u - ru - gu and the individual d quarks by transforming (r+g+b)d to (r+g+b)u, then producing the individual u quark, and then transforming back to the d quark space. Therefore Spin(5) gravity effectively acts on all the fermions.

Force strengths are calculated in Chapter 3. from the relative volumes of those geometric structures related to gauge groups and fermions in the F4 model.

To calculate ratios of volumes of the structures, compact structures are used. For example, a 4-dimensional spacetime base manifold might be taken to be S4 rather than Minkowski spacetime, so that a finite volume V(S4) = 82/3 can be calculated.

Let M be the irreducible m-dimensional manifold on which the gauge group acts globally as a part of 4-dimensional spacetime = M4/m. Let Q be the part of the spinor manifold Q8+ with natural local action by the gauge group, D be the bounded complex domain of which Q is the Silov boundary, and µ be the mass scale for effective theories below symmetry breaking at µ.

Then the force strength is given by [1/µ2] [V(M)] [V(Q)/V(D)1/m] where V denotes volume.

Fermion masses are calculated in Chapter 6. In the F4 model, quark constituent masses are calculated and quark current masses are fundamental.

Fermion masses are calculated as a product of four factors:

V(Q) x N(Graviton) x N(octonion) x Sym

V(Q) is the volume of the part of the half-spinor fermion particle manifold Q8+ = RP1 x S7 that is related to the fermion particle by photon, weak boson, and gluon interactions.

N(Graviton) is the number of types of graviton related to the

fermion. The 10 gravitons correspond to the 10 infinitesimal generators of Spin(5) = Sp(2). 2 of them are in the Cartan subalgebra. 6 of them carry color charge, and may therefore be considered as corresponding to quarks. The remaining 2 carry no color charge, but may carry electric charge and so may be considered as corresponding to electrons.

N(octonion) is an octonion number factor relating up-type quark masses to down-type quark masses in each generation, since 2nd and 3rd generation fermions can be considered to correspond to pairs or triples of first generation fermions, which can be considered to be octonions.

Sym is an internal symmetry factor, relating 2nd and 3rd generation massive leptons to first generation fermions.

Kobayashi-Maskawa parameters are related to the sums of masses of fermions in each generation, Smf1, Smf2, and Smf3, and are calculated in Chapter 7. In the Chau-Keung parameterization, the K-M parameters are: phase angle e = /2; sin(a) = Smf1/(Smf12 + Smf22) ;

sin(b) = Smf1/(Smf12 + Smf32) ; and

sin(g) = [Smf2/(Smf22 + Smf32)] [(Smf2 / Smf1) ] .

In the F4 model, the same K-M mixing angles apply to both leptons and quarks. Also, neutrinos can get mass by radiative processes related to the Planck mass. Therefore, the F4 model has a natural MSW mechanism that is applied to the solar neutrino problem in Chapter 8.

The tree-level results of all these calculations for tree-level force strengths, particle masses (constituent masses for quarks), and

K-M parameters are :

U(1) electromagnetism aE = 1/137.03608

SU(2) weak force GF = GWmproton2  = 1.02 x 10-5

SU(3) aC=0.629 at0.24; 0.168 at 5.3; 0.122 at 34; 0.106 at 91GeV Spin(5) gravity GGmproton2 = (mproton/mPlanck)2 Higgs scalar mass = 260.8 GeV

weak boson masses: mw+ = mw- = 80.9 GeV and mz = 92.4 GeV

Weinberg angle sin2qw = 1 - (mw±/mz)2 = 0.233

me is assumed to be 0.5110 MeV

me-neutrino = 0; md = 312.8 MeV; mu = 312.8 MeV

mµ = 104.8 MeV; mµ-neutrino = 0; ms =  625 MeV; mc = 2.09 GeV

m t = 1.88 Gev; m t-neutrino = 0; mb = 5.63 GeV; mt = 130 GeV

mPlanck (rough estimate) 1-1.6 x 1019 GeV

the Kobayashi-Maskawa Parameters are: phase angle e = /2:

d s b

u 0.975 0.222 -0.00461 i

c -0.222 0.974 0.0423

-0.000190 i -0.0000434 i

t 0.00941 -0.0413 0.999

-0.00449 i -0.00102 i

 

 

The F4 model calculated tree-level values are in rough, but not exact, agreement with experiment; its MSW mechanism works; and the force and particle content of the F4 model is realistic.

 

The price paid for the realistic results is that the structure of the F4 model is not entirely standard, and requires some unusual applications of mathematics to physics.

 

 


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