aa r X i v : . [ h e p - t h ] F e b S pin (7) instantons in eight dimensions.
A.V. Smilga
SUBATECH, Universit´e de Nantes, 4 rue Alfred Kastler, BP 20722, Nantes 44307, France.
Abstract
We explicitly construct topologically nontrivial 8-dimensional gauge field configura-tions that belong to the algebra spin (7) and are associated with the homotopy group π [ Spin (7)] = Z . Similar constructions for other algebras in different dimensions arebriefly discussed. Four dimensions
We recall here the well-known construction of SU (2) instantons in four dimensions [1].Consider a 4-dimensional Euclidean gauge field A ( x ) = A µ ( x ) dx µ ∈ su (2). Assume that thefield density form F ( x ) = dA ( x ) − iA ( x ) ∧ A ( x ) ≡ F µν dx µ ∧ dx ν , (1.1)where F µν = ∂ µ A ν − ∂ ν A µ − i [ A µ , A ν ], decays fast enough at x → ∞ , so that the integral q = 18 π Z R Tr { F ∧ F } (1.2)converges. As is very well known, in this case the Pontryagin topological charge (1.2) is quan-tized to be integer. This integer is associated with the nontrivial π [ SU (2)] = Z .Indeed, the form Tr { F ∧ F } is exact:Tr { F ∧ F } = dω , where ω = Tr (cid:26) A ∧ F + i A ∧ A ∧ A (cid:27) . (1.3)Then the integral (1.2) boils down to the surface term, q = 18 π Z S ω , (1.4)where S is a 3-sphere at large distance from the origin. Bearing in mind that F rapidly decaysat large x , the potential A acquires at large distances a pure gauge form A = ig − dg (1.5)with g ∈ SU (2). We derive q = 124 π Z S Tr { g − dg g − dg g − dg } . (1.6)The simplest nontrivial mapping S → SU (2) may be presented as g = x − ix m σ m r , (1.7)where is the unity matrix, σ m =1 , , are the Pauli matrices and x µ = ( x m , x ) marks a pointon the distant sphere of radius r = √ x µ x µ . The gauge field (1.5) acquires the form A µ = η aµν x ν σ a r , (1.8)where η aµν is the ‘t Hooft symbol [2]: η am = − η a m = δ am , η amn = ε amn . (1.9)1o calculate the integral (1.6), we represent it in a little bit more explicit form q = 124 π ε µνρσ Z S x µ r Tr { g − ∂ ν g g − ∂ ρ g g − ∂ σ g } dV S (1.10)with the convention ε = 1. Pick up the north pole of S , x µ = ( ~ , r ). Then the integrandin (1.10) reads − i π r ε jkl Tr { σ j σ k σ l } = 12 π r . (1.11)Due to rotational symmetry, the value of the integrand is the same at all other points of thesphere. Multiplying (1.11) by V S = 2 π r , we reproduce the result q = 1.Any 4-dimensional gauge field with the asymptotics (1.8) has the topological charge q = 1.But a particular physical interest represent the configurations realizing the minimum of theaction functional S = Z R Tr { F ∧ ∗ F } = 12 Z d x Tr { F µν F µν } . (1.12)Such configurations with q = 1 satisfy the self-duality condition, F = ∗ F or F µν = 12 ε µνρσ F ρσ . (1.13)The standard BPST self-dual instanton with the center at the origin has the form A µ = η aµν x ν σ a r + ρ , (1.14)where ρ is a parameter having the meaning of the instanton size.There are also self-dual configurations with an arbitrary positive or negative topologicalcharge q [3]. Note (it will be helpful for us later) that one can also describe the mapping S → SU (2) in analternative way: g = exp { iα m σ m } = cos α + i sin αα α m σ m , (1.15)where 0 ≤ α = k ~α k ≤ π . The relation to the standard parameterization (1.7) is α m = − x m k ~x k arccos (cid:16) x r (cid:17) . (1.16)The 2-sphere k ~α k = π maps onto one point of S —its south pole, g = − .In these variables, q = 14 π Z α ≤ π d α Tr { g − ∂ g g − ∂ g g − ∂ g } , (1.17)2here g − ∂ m g = − iA m = i (cid:20) cos α sin αα σ m + sin αα α k ε kml σ l + α m α k α (cid:18) − cos α sin αα (cid:19) σ k (cid:21) . (1.18)The result of the calculation is, of course, the same: q = 2 π Z π sin α dα = 1 . (1.19) Spin (8) instantons
Let us now go to eight dimensions and consider the gauge fields A M =1 ,..., belonging to thealgebra spin (8). The 8-dimensional counterpart of the topological charge (1.2) is the fourthChern class q = 1384 π Z R Tr { F ∧ F ∧ F ∧ F } . (2.2)The construction is quite parallel to the standard instanton construction of the previoussection. We note that the form Tr { F ∧ F ∧ F ∧ F } is exact:Tr { F } = dω , (2.3)where ω = Tr (cid:26) AF + 2 i F A + i AF A F − F A − i A (cid:27) (2.5)(the products are understood here as wedge products).Let F decay rapidly at large distances so that the integral (2.2) converges. Then A representsat large distances a pure gauge (1.5). The integral is reduced to q = − · · π Z S Tr { ( g − dg ) } . (2.6) A “physical” way to fix the coefficient in (2.2) is to require that the Dirac equation on the gauge fieldbackground A M ( x ) is well defined and has a single zero mode. For the Chern class of order D/ R D , onederives [4] q = ± π ) D/ ( D/ Z R D Tr { D/ z }| { F ∧ · · · ∧ F } (2.1)with the sign depending on the convention. The expression (2.5) follows from the general formula [5] ω n − = n Z dt t n − Tr { A ( dA − itA ) n − } . (2.4) π [ Spin (8)] = Z × Z (2.7)is nontrivial [6], and there are topologically nontrivial embeddings of S into Spin (8). Thesimplest such embedding reads g = exp { iα Γ j } = cos α + i sin αα α j Γ j , (2.8)where Γ j =1 ,..., are purely imaginary antisymmetric 8-dimensional matrices satisfying the Clif-ford algebra, Γ m Γ n + Γ n Γ m = 2 δ mn . (2.9)and the relation Γ Γ Γ Γ Γ Γ Γ = i . (2.10)One of many explicit representations for Γ m isΓ = − σ ⊗ σ ⊗ σ ; Γ = ⊗ σ ⊗ σ ; Γ = ⊗ σ ⊗ σ ; Γ = − σ ⊗ σ ⊗ ;Γ = σ ⊗ σ ⊗ ; Γ = σ ⊗ ⊗ σ ; Γ = σ ⊗ ⊗ σ . (2.11)Seven matrices Γ m are among the generators of the group Spin (8). 21 other generators areΣ mn = i m , Γ n ] . (2.12)One can also define Γ M = ( ~ Γ , i ) , Γ † M = ( ~ Γ , − i ) , (2.13)Then the whole set of the generators of Spin (8) can be presented asΣ MN = i M Γ † N − Γ N Γ † M ) . (2.14)The embedding (2.8) is quite analogous to (1.15), but the difference is that is this case notall the generators of Spin (8) are engaged and it is an injective rather than surjective map.By the variable change α m = x m k ~x k arccos (cid:16) x r (cid:17) , (2.15)(2.8) can be presented in the form analogous to (1.7): g = x + ix m Γ m r (2.16) With this convention, the topological charge (2.6) is positive. r = ( x M ) . The unit vector n M = x M /r gives a natural “round” parameterization of S .The gauge field (1.5) expressed in these terms reads A M = Σ NM x N r (2.17)Now, pick up the north pole of S , x M = ( ~ , r ). At this point, g − ∂ m g = − iA m = i Γ m r . (2.18)The integrand in (2.6) is − ( i ) r Tr { Γ Γ Γ Γ Γ Γ Γ } = − r . (2.19)(the first minus sign in (2.19) comes from ε = − S , V S = π r , (2.20)and substituting in (2.6), we derive q = 1 . (2.21)The property | q | = 1 reflects the fact that the mapping S −→ Spin (8) defined in (2.16) isinjective.There is one particular field A M ( x ∈ R ) in the sector q = 1 where the field density satisfiesthe nonlinear self-duality condition F ∧ F = ⋆ ( F ∧ F ) (2.22)It reads A M ( x ) = Σ NM x N r + ρ , (2.23)going to (2.17) in the large r limit. The gauge configuration (2.23) is what is called Spin (8) or SO (8) instanton [7]. It realizes the minimum of the functional Z R Tr { F ∧ F ∧ ⋆ ( F ∧ F ) } (2.24)in the sector q = 1. The analogy with the BPST instanton (1.14) is straightforward, but thedifference is that the functional (1.12) has a lot of physical meaning — it is the Yang-Millsaction. And therefore the 4-dimensional BPST instantons have a lot of physical applications.On the other hand, the functional (2.24) does not have an interesting physical interpretation andthe Spin(8) instantons have the status of a beautiful, but abstract mathematical construction.Mutltiinstanton solutions characterized by an arbitrary integer value of q , the analogs ofADHM solutions, are also known [9].For completeness, I should also mention another type of a 8-dimensional instanton discussedin the literature. It is the so-called secular instanton where the field satisfies a certain linear constraint, a generalization of the four-dimensional constraint F = ± ⋆ F [8]. This instanton isoutside the scope of our paper. 5 Spin (7) instantons
The generators of
Spin (7) ⊂ Spin (8) are given in (2.12).Note that the embedding (2.8) involves seven generators of
Spin (8) that are not in the set(2.12). In fact, this embedding realizes the fiber bundle
Spin (8)
Spin (7) −→ S . (3.1)It is known, however, that the sphere S represents also a base for another fiber bundle Spin (7) G −→ S . (3.2)With the explicit construction of this fiber bundle in hand, we can construct topologicallynontrivial Spin (7) gauge field configurations along the same lines as it was done in the previoussection for
Spin (8). To this end, we only have to write down an explicit formula for theembedding S ⊂ Spin (7). The topology associated with this embedding is π [ Spin (7)] = Z . Figure 1: Fano graph.Consider the expression g = exp (cid:26) α m f mnk Γ n Γ k (cid:27) , (3.3)where f jkl are the structure constants of the octonion algebra, f = f = f = f = f = f = f = 1 (3.4)and all other nonzero elements of f mnk are restored by antisymmetry (see the Fano graph onFig. 1). It is an element of Spin (7). Its logarithm represents a linear combination of seven
Spin (7) generators, T m = f mnk Σ nk . Explicitly: T = Σ + Σ − Σ , T = Σ + Σ − Σ , T = Σ − Σ − Σ ,T = Σ − Σ − Σ , T = Σ + Σ + Σ ,T = Σ − Σ − Σ , T = Σ + Σ − Σ . (3.5)6 heorem 1. The expression (3.3) considered in the range ≤ k ~α k ≤ π represents a map S −→ Spin (7) associated with the fiber bundle (3.2).Proof.
To prove the first part of this statement: that (3.3) is a map of S in Spin (7), we onlyhave to prove that, for any ~α of norm π , g ( ~α ) = − , as it was for the mappings (1.15) and(2.8). Let first α m = πδ m . Then g = exp { π (Γ Γ − Γ Γ + Γ Γ ) } = exp { π Γ Γ } exp {− π Γ Γ } exp { π Γ Γ } = ( − ) = − . (3.6)We rotate now ~α in the plane { } : α = π cos φ, α = π sin φ, α ,..., = 0 . The exponent in (3.3) readsln g = − iπ [cos φ T + sin φ T ]= π Γ (Γ cos φ + Γ sin φ ) − π (Γ cos φ − Γ sin φ )Γ + π (Γ cos φ − Γ sin φ )Γ = π Γ Γ ′ − π Γ ′ Γ + π Γ ′ Γ . (3.7)We may observe that the matrices Γ , Γ , Γ , Γ ′ , Γ ′ , Γ ′ and Γ ′ = Γ cos φ + Γ sin φ still obey theClifford algebra (2.9), which means that the three terms in (3.7) still commute, the exponentialcan be “disentangled” and we still have g = ( − ) = − . The same is true for rotations inother planes.Any ~α of norm π can be “reached” from α m = πδ m by a set of such elementary rotations,and we conclude that, for such ~α , g ( ~α ) = − , indeed.To prove the second part: that this mapping is associated with the fiber bundle (3.2), weneed to prove that the set of all g ( ~α ) can be considered as the base in this fiber bundle. Andthis is true in view of the property: Lemma 1.
The generators (3.5) are not in the subalgebra g ⊂ spin (7) .Proof. Indeed, G is a group of automorphisms of the octonion algebra. It can be alternativelydefined as the subgroup of SO (7) that involves only the rotations leaving invariant the object I = f jkl A j B k C l = [165] + [134] + [127] + [235] + [246] + [367] + [475] , (3.8)where [165] = ( A B − A B ) C + ( B C − B C ) A + ( C A − C A ) B (3.9)etc., for any triple of 7-vectors ~A, ~B, ~C . The generators of these rotations are˜Σ + ˜Σ , ˜Σ − ˜Σ , ˜Σ − ˜Σ , ˜Σ + ˜Σ , ˜Σ − ˜Σ , ˜Σ + ˜Σ , ˜Σ − ˜Σ , ˜Σ − ˜Σ , ˜Σ + ˜Σ , ˜Σ + ˜Σ , ˜Σ + ˜Σ , ˜Σ − ˜Σ , ˜Σ − ˜Σ , ˜Σ + ˜Σ , (3.10)where ˜Σ mn are the counterparts of (2.12) in the vector representation.Consider e.g. the action of the operator ˜Σ on I . The structures [165], [134] and [127] areleft invariant. But four other structures are not:˜Σ I ∝ [245] − [236] + [467] − [375] . − ˜Σ I and ˜Σ I , so that I doesnot transform under the action of the generators ˜Σ + ˜Σ or ˜Σ − ˜Σ . On the other hand,( ˜Σ + ˜Σ + ˜Σ ) I ∝ − [236] + [467] − [375]) = 0 . The same is true for six other generators in (3.5) and their arbitrary linear combinations.We can now make another look at (3.7) and observe that Γ ′ , , , are obtained from Γ , , , by a rotation in the planes (12) and (45) by the same angle. Such a rotation belongs to G [cf.the last generator in the list in Eq.(3.10)]. Thus, the relation (3.7) is just a manifestation ofthe fact that the exponent in (3.3) is invariant under G .By applying an approriate G rotation, the exponential in (3.3) can be disentangled for anarbitrary ~α , the condition α ≡ k ~α k = π is not necessary. For α m = αδ m , we derive g = (cos α + sin α Γ Γ )(cos α − sin α Γ Γ )(cos α + sin α Γ Γ )= cos α − i cos α sin α T + cos α sin α Γ T − i sin α Γ . (3.11)The third and the fourth term in the R.H.S. of this identity were derived using (2.10).For an arbitrary ~α , we obtain g = cos α − i cos α sin α α m T m α + cos α sin α α m α n Γ m T n α − i sin α α m Γ m α . (3.12)This form is convenient for rewriting g in the Cartesian coordinates x M . Using (1.16) with x → x , we derive g = x − ix x m T m + x x m Γ m x n T n − ix m x m x n Γ n r . (3.13)The expressions for the gauge potentials A M = ig − ∂ M g are rather ugly and we will not quotethem.Our next task is to evaluate the topological charge (2.6) for this mapping. A nonzero valueof the charge will assure the topological nontriviality of the mapping. It would not be correct in this case to use the same simple method that we used in theprevious section: evaluating the integrand at one particular point of S and multiplying theresult by the volume of S . The matter is that the integrand is now not the same at all pointsof S , but depends nontrivially on its seven angles S . An analytical calculation is not possiblehere. We evaluated the integral using the parameterization (3.12) numerically by the MonteCarlo method: adding the contributions of many randomly chosen points. Unfortunately, Note that not any mapping of a sphere into a Lie group is topologically nontrivial. For example, theexpression g = exp { i ( α σ + α σ ) } (3.14)with 0 ≤ p α + α ≤ π realizes a mapping S → SU (2) related to the fiber bundle SU (2) U (1) −→ S . However,there is no topological charge associated with the mapping (3.14) and this mapping is topologically trivial(contractible to a point by a continuous deformation): π [ SU (2)] = 0. q = . ± . . (3.15)The accuracy is not perfect, but the only integer number with which the result (3.15) is com-patible is q = 1. Quite similar constructions of topologically nontrivial gauge fields in 8 dimensions can also berealized for the groups
Spin (6) ≡ SU (4) and Spin (5) ≡ Sp (2). They are associated with thenontrivial homotopies π [ Spin (6)] = π [ Spin (5)] = Z , which in turn are related to the fiberbundles Spin (6) SU (3) −→ S and Spin (5) SU (2) −→ S . The problem is, however, that we are not aware of a nice explicit formula, an analog of (3.3),which would describe the maps S −→ Spin (6) and S −→ Spin (5), from which the gaugefields at the distant sphere S could be deduced and the topological charge ∝ R S Tr { ( g − dg ) } could be calculated.Eight dimensions are not so much distingushed. It is known that π r − [ Sp ( r )] = Z for any r , from which the existence of nontrivial 4 r -dimensional gauge field configurations with thegroup Sp ( r ) follows. Of all these configurations in a given topological class, one can considerdistinguished configurations—the minima of the functional Z R r Tr { r z }| { F ∧ . . . ∧ F ⋆ ( r z }| { F ∧ . . . ∧ F ) } (4.1)that satisfy the generalized self duality conditions r z }| { F ∧ . . . ∧ F = ± ⋆ ( r z }| { F ∧ . . . ∧ F ) . (4.2)One can also construct topologically nontrivial gauge field configurations in R r − . In6 dimensions, they are associated with the nontrivial π [ SU (3)] = Z and the fiber bundle SU (3) SU (2) −→ S . In this case, each fiber represents the subgroup SU (2) ⊂ SU (3) that leavesintact a unit complex vector, V = α α α , | α | + | α | + | α | = 1 , (4.3)and the base S is the set of all such vectors. An explicit expression for the mapping S −→ SU (3) is not so nice as in (3.3), but it is known [10]. If | α | >
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