arXiv:1504.01618v1 [math-ph] 22 Jan 2015

is known to be Sp(2)/Sp(1) × Sp(1) ∼ S 4 . The projective metric for this space will be constructed and the curvature tensor will be shown to consist of both self-dual and anti-selfdual components. In the central projection, one hemisphere of S 4 is mapped onto the other, such that one hemisphere, excluding the equator, is equivalent to a four dimensional ball. This enables a mapping between the compact geometry of YM theory and relativity, with the double cover of the equator mapping to infinity in the anti-de Sitter (AdS) space. By interpreting the YM geometry (instanton) as the space of interactions between two particles it is suggested that the Sp(2)/Sp(1) × Sp(1) YM quaternionic geometry can be generalized to n fermions, each having an Sp(1) ∼ SU (2) gauge group.

Introduction The BPST[1] solution of the Yang-Mills (YM) functional[2] gives the field strength or curvature tensor as d¯ q ∧ dq (1) F = (1 + q q¯)2 as was shown by Atiyah.[3] Here q¯ is the conjugate of a quaternion q = x0 1+x1 i+x2 j+x3 k ∈ H, q ∧ dq is the exterior product of one-forms. The anti-commuting with q q¯ = x20 + x21 + x22 + x23 , and d¯ 2 quaternion basis elements satisfy i = j2 = k2 = ijk = −1. There are several features of this field strength = curvature two-form and associated metric that are important for the interpretation of the YM theory to be proposed, and to make the presentation self-contained the appropriate differential geometry will be developed. The YM field strength in eq. (1) will be shown to be the curvature form for the coset space Sp(2)/Sp(1) × Sp(1). Here Sp(n) is the symplectic group,[4, 5] which is a unitary group over the quaternions, H, i.e., Sp(n) ∼ U (n, H). This coset (symmetric) space is a cross-section of a principal bundle;[6] it is also a flag manifold.[7] It will be shown that this geometry can be interpreted as a description of the interaction between a pair of particles, each of which has spin with the gauge group (fiber) Sp(1) ∼ SU (2) ∼ SO(3). This has an immediate generalization to a many-body YM theory based on Sp(n)/Sp(1)n . Some of the consequences of this interpretation are discussed.

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Principal Bundle Construction In the expectation that the differential geometry tools to be developed are useful for a variety of physical problems, but without knowing in advance the rank of an interesting space, or which field – real (R), complex (C), or quaternion (H) – might apply to a particular case, the presentation will be general. The quaternion field was displayed in the Introduction, and obviously this will be our ultimate target. However, there are other problems for which real and complex spaces are pertinent. For example, the configuration space of N particles comprising a classical system can be described with a 3 × N matrix of coordinates,

x x ··· 1 2 X = y1 y2 · · · z1 z2 · · ·

xN yN . zN

which is a rank three matrix over R3N . (For the present discussion, N ≥ 4 particles are in general position.) A measure on this space can be defined as ds2 = Tr(dXdX0 ), where X 0 is the transpose of X. This measure is invariant to T : X → XT , where T is a matrix in the orthogonal group O(N ). Furthermore, on extracting the center of symmetry motion for this space followed by the so-called “UDV” or polar decomposition, one is led to a Stiefel manifold of rank three over the reals (the V in the decomposition).[8] Of the 3N dimensions of the configuration space, 3(N − 3) comprise the compact Stiefel manifold, which makes this an interesting geometry for study. Our discussion begins with the Stiefel manifold X = [Xk , Xn ] ⊂ g, which comprises the first k rows of a matrix g ∈ SO(k + n, R), SU (k + n, C), or Sp(k + n) = U (k + n; H); gg ∗ = 1 (for present purposes n ≥ k). In the real case the g ∗ is the transpose of g, and for the complex and quaternionic cases it is the conjugate transpose. Let U (k + n, K) encompass the three groups and corresponding fields under consideration. The Stiefel manifold can be identified with the coset space X ∼ U (k + n; K)/U (n; K).[6] Here Xk is a k × k matrix and Xn is a k × n matrix such that XX ∗ = 1. These are clearly compact spaces. The Xk part of X can be factored: XX ∗ = Xk Xk∗ + Xn Xn∗ = Xk (1 + Y Y ∗ )Xk∗ = 1

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with Y = Xk−1 Xn , from which it follows that 1 + Y Y ∗ = (Xk∗ Xk )−1 ≥ 1. The space Y is a Grassmann manifold. Using ν to denote the real dimension of K, the real dimension of X is νk(n + k) − k − νk(k − 1)/2 = νkn + k(νk/2 + ν/2 − 1). The real dimension of Y is νkn, and the remaining part, with real dimension k(νk/2 + ν/2 − 1), is an “ineffective” part of U (k + n; H) corresponding to the left action of U (k; K) on X, implying that the Grassmannian is isomorphic to U (k + n; K)/U (k; K) × U (n; K) = G/H. 2

Since Xg1 , g1 ∈ U (k + n; K), is a subset of gg1 ∈ U (k + n; K), it follows that g1 acts on X by " ˆ = [Xk , Xn ] g1 : X → X

A∗ −C ∗ −B ∗ D∗

# = [Xk A∗ − Xn B ∗ , −Xk C ∗ + Xn D∗ ],

where the partitioning of g1 is compatible with X. (The reason for the unconventional labeling of ˆ sends Y → Yˆ , such matrix elements will become apparent shortly.) The transformation X → X that g1 : Y → Yˆ = (A∗ − Y B ∗ )−1 (−C ∗ + Y D∗ ). Given that g1 g1∗ = 1, it follows that Yˆ = (A∗ − Y B ∗ )−1 (−C ∗ + Y D∗ ) = (AY + B)(CY + D)−1 .

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The right hand version of this equation is canonical in the literature,[9] which justifies the choice made for the matrix elements in g.

Metric As Y varies infinitesimally around a fixed point, the corresponding excursion in Yˆ , for fixed g, can be evaluated from eq. (3) by differentiation. An easy calculation (using the unitarity of g) gives dYˆ = (A∗ − Y B ∗ )−1 dY (CY + D)−1 . To construct the invariant metric one needs to eliminate terms depending explicitly on g, and this can be done with 1 + Yˆ Yˆ ∗ =(A∗ − Y B ∗ )−1 (1 + Y Y ∗ )(A − BY ∗ )−1 ; 1 + Yˆ ∗ Yˆ =(Y ∗ C ∗ + D∗ )−1 (1 + Y ∗ Y )(CY + D)−1 . The pieces are assembled to give the invariant metric ds2 = Tr[(1 + Y Y ∗ )−1 dY (1 + Y ∗ Y )−1 dY ∗ ],

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where Tr(·) is the trace of the matrix argument. An understanding of the relation between matrix elements and the Y coordinates on a cross section is essential for the general theory of coset spaces and will be put to use in calculating the curvature tensor. Note that the origin, Y = 0, is mapped to −A∗−1 C ∗ = BD−1 by the action of g in eq. (3). This induces the equivalence Y ∼ −A∗−1 C ∗ = BD−1 , which provides an alternative

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expression for the metric. In particular, 1 + Y Y ∗ =(AA∗ )−1

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1 + Y ∗ Y =(DD∗ )−1 .

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dY = dBD−1 + A∗−1 C ∗ dDD−1 = A∗−1 (A∗ dB + C ∗ dD)D−1 = −A∗−1 ω12 D−1

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The differentials become

where ω12 = −(A∗ dB + C ∗ dD) is an element of the connection form[10] " ω=

ω11 ω12 ω21 ω22

#

" = −dgg ∗ = gdg ∗ =

A∗ −C ∗ −B ∗ D∗

#"

dA −dB −dC dD

# .

Combining eq. (5-7) one obtains the line element in the form ∗ ds2 = Tr(ω12 ω12 ).

Note that ω12 is a k × n matrix of K-valued one-forms.

Curvature The Appendix contains a coordinate-free calculation of the metric and curvature tensors for a general flag manifold. For present purposes the flag manifold consists of a single k × n Grassmannian over H. This corresponds to decomposition of the k + n-dimensional quaternionic space into just two subspaces, the k-space and n-space, respectively. The Maurer-Cartan form is written with block components, and the diagonal components yield two curvature forms, Ωii , from the Cartan equation dωii + ωii ∧ ωii − Ωii = 0 (no summation on repeated indices). (The Cartan equation is usually written F = dA + A ∧ A in the physics literature.) Since ω is skew-symmetric, it follows ∗ that ω21 = −ω12 . The two parts of the curvature tensor are[11] ∗ ∗ ∧ ω12 . Ω11 = ω12 ∧ ω12 and Ω22 = ω12

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The magnitudes of these two curvature two-forms are equal, which is important to the interpretation of the relation between curvature and force to be discussed later. From eq. (7) it follows that ω12 = A∗ dY D, giving Ω11 = A∗ dY (1 + Y ∗ Y )−1 ∧ dY ∗ A and Ω22 = D∗ dY ∗ (1 + Y Y ∗ )−1 ∧ dY D. The curvature forms specialize to the YM case in which the ω12 is a single quaternion, so that Y → q. As shown in eq. (16) in the Appendix, one is free to rotate the curvature two-forms by ˆ 11 Aˆ∗ and DΩ ˆ 22 D ˆ ∗ where Aˆ = A/|A| and D ˆ = D/|D| are unit quaternions. Furthermore, AΩ 4

|A| = |D| = (1 + q¯q)−1/2 = (1 + q q¯)−1/2 are scalars, so that with this particular choice of cross-section, ¯ 11 = (1 + q q¯)−2 d¯ Ω22 = Ω q ∧ dq.

Ω11 = (1 + q q¯)−2 dq ∧ d¯ q and

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This demonstrates that Sp(2)/Sp(1) × Sp(1) is the underlying YM geometry. As Atiyah[3] points out, dq ∧ d¯ q is self-dual and d¯ q ∧ dq is anti-self-dual. One goes into the other by reflection of the 3-space (conjugation).

Projective Space An important sequence of Lie group homomorphisms[12, 13] relates the quaternion coset space Sp(2)/Sp(1) × Sp(1) to the real sphere S 4 : Sp(2)/Sp(1) × Sp(1) ∼ SO(5)/SO(3) × SO(3) ∼ SO(5)/SO(4) ∼ S 4 . A real n-dimensional projective space is the space of all lines through the origin in Rn+1 . A line through the origin intersects the unit sphere centered at the origin at x and −x, where xx0 = 1. Here x0 is the transpose of the row x = [x0 , x1 , · · · , xn ]. The line through the point x ∼ −x on the sphere intersects an n-dimensional plane that does not pass through the origin at the point y. A plane that is tangent to the sphere at the south pole, x0 = 1, is a canonical choice for this plane, as shown in Fig. 1 (for convenience, the positive x0 axis is directed downwards in Fig. 1). The similar triangles in the figure yield |ˆ x|/x0 = |y|/1 ˆ = [x1 , x2 , · · · , xn ]. The inhomogeneous coordinates, y = [x1 /x0 , · · · , xn /x0 ], satisfy where x 1 + yy0 = 1/x20 ≥ 1, and are unbounded [see eq. (3)]. (Here the projection from the Stiefel manifold to the Grassmannian gives the Fubini-Study metric.[6]) The projection maps the equator of the sphere to infinity on the y-plane, while the x ∼ −x equivalence folds one hemisphere on the other and half the equator folds onto the other half. This double cover is very important in understanding the relation between YM and relativity. The direct mapping from Sp(2)/Sp(1) × Sp(1) to the sphere is easy – simply set Y = q = tan(θ)u or cot(θ)u in eq. (4), where u is a unit quaternion, i.e., u ∼ S 3 . The metric in eq. (4) then reduces to the metric on S 4 . The use of either tan(θ) or cot(θ) reflects the inversion symmetry of the space.

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Figure 1. Antipodal projection of the sphere onto the plane.

Projection Between YM and Relativity The four-sphere projects onto the YM space by the double cover antipodal projection, as has just been seen. But the antipodal projection is not the only mapping of the sphere onto a plane. The single cover stereographic projection of the sphere may also be considered. The stererographic projection from the north pole projects the southern hemisphere onto a disc or ball, while the other hemisphere projects onto its complement, the inverted ball. The compact topology of the YM metric can be captured in a stereographic projection just as well as the antipodal projection, since only one hemisphere is projected. The topology would be wrong if the entire sphere were projected. This observation is sufficient to map the YM metric into the anti-de Sitter space. Consider the hyperbola (AdS space) y02 − yy0 = a2 , and the sphere x20 + xx0 = b2 , a and b both real, where y and x are rows consisting of the components of n-dimensional real vectors. A branch of the hyperbola and a hemisphere each project onto the unit ball, B n , and hence onto one another, by the remarkably similar projections y/(y0 + a) = x/(x0 + b).

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On the left [y/(y0 + a)][y/(y0 + a)]0 = (y02 − a2 )/(y0 + a)2 = (y0 − a)/(y0 + a) ≤ 1 with the bound valid for the positive branch of the hyperbola, i.e., y0 ≥ 0. And on the right [x/(x0 + b)][x/(x0 + b)]0 = (b2 − x20 )/(b + x0 )2 = (b − x0 )/(b + x0 ) ≤ 1 for the hemisphere on which x0 ≥ 0. This is a stereographic projection anchored at the south

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pole of the northern hemisphere onto B n . Given just the projections, eq. (10), it is not possible to distinguish the hyperbola from the sphere. That information can only be gleaned from the sign of y0 − a or b − x0 respectively, and this implies knowledge of the global space, not just the local. It will be noted that a and b factor from the projections so that the hyperbola and sphere may be taken with a = b = 1. The equator, x0 = 0, of the sphere maps to the boundary of B n , and this maps y0 → ∞ on the hyperbola. The double cover of the equator is not apparent in the geometry of the hyperbola, as the boundary of the ball is at infinity on the hyperbola – this is also at infinity in the q space. To make use of YM geometry in physical theory requires a different boundary condition at infinity than the usual assumption that space is flat at infinity. The four sphere does not possess a uniform vector field, so that the poles are singularities as shown in Fig. (2). One may interpret the singularities as particles. The figure is suggestive of a vacuum fluctuation producing an electron-positron bubble.

Figure 2. Vector fields on S 2 are illustrative of vector fields on S 4 . Reprinted from Figure 10 (p. 14) in H. Blaine Lawson, Jr. The Theory of Gauge Fields in Four Dimensions, CBMS Regional Conference Series in Mathematics Volume 58 (Providence: American c Mathematical Society, 1985). 1985 by the American Mathematical Society.

Interpretation These considerations lead one to interpret S 4 , the YM geometry, as the field created by the two particles at the poles. This “instanton” is the only space available to the theory – the instanton is not imbedded in a hyperbolic space but is projectively equivalent to one. But equally important is the notion that we are dealing with a configuration space, as implicit in the connection between 7

projective geometry and configurations of points. Each point, or particle, treated independently of the other, is invariant to the action of Sp(1) ∼ SU (2) (see the discussion in the Appendix regarding vanishing Grassmannians). Of course, this is the spin degree of freedom with which Yang and Mills started. The surprise is that the solution of the YM functional demands two particles, not just the one that Yang and Mills were describing. The theory is telling us that a field can only exist between two objects – there is no field geometry to be ascribed to a single spin. An isolated particle has only its internal Sp(1) symmetry, and this generates the H = Sp(1) × Sp(1) fiber in the principal bundle G/H for two particles. In this two-particle configuration space the H-valued Grassmannian, Sp(2)/Sp(1) × Sp(1), defines space. Given the two notions of a configuration space and of the interaction between two particles being carried by an H-valued object, we have the tools required to build higher dimensional configuration spaces at will. We may conjecture that Sp(k)/Sp(1)k describes the fields acting amongst k particles. Between any two particles the interaction is represented by an H-valued element of the flag manifold, so four spatial dimensions suffices for each. The tremendous advantage of this structure is that one has a group with all of the elaborate machinery that goes with it to describe the configurations. The group algebra supplies a plethora of operators, but general aspects of the group also have important implications. To illustrate: An irreducible representation of a system with Sp(k) symmetry is not coupled to anything else. A Sp(k) system can be coupled to its surroundings by the Grassmannian Sp(k + n)/Sp(k) × Sp(n), where Sp(n) is the symmetry group of the surroundings. A reducible Sp(k) representation consists of two or more parts that are independent of one another, because the corresponding Grassmannian vanishes. It may be useful to see a little of this machinery in the context of representation theory. A representation Ag of g ∈ Sp(k) has a left-action on functions Ψ(x) by[4] Ag Ψ(x) = Ψ(g −1 x). The coordinates x at our disposal are those of the coset space (flag manifold), G/H ∼ Sp(k)/Sp(1)k , on which g ∈ G acts by gxH → yH. The action of the fiber H has to be factored to understand how the group acts on a cross-section (briefly, for two subspaces as in eq. (2-7), Y ∼ BD−1 = Bh(Dh)−1 , where h ⊂ H). We want functions that are good on any cross-section of the bundle, i.e. independent of the gauge, which suggests averaging over the subgroup H. This is done by means of an integration:[14] Z σ(h)f (h−1 x)dh.

Ψ(x) = H

Here σ(h); h ∈ H is a representation of H and f (x) is a map from the coset space to a Hilbert space of square integrable functions. The construction here is for a left action of g and h on G/H;

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a right action works as well. The action of ξ ∈ H on Ψ(x) is Z Ψ(ξx) =

Z

−1

σ(h)f (h ξx)dh = H

σ(ξη)f (η −1 x)dη,

H

giving Ψ(ξx) = σ(ξ)Ψ(x).

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The gauge group acts on the functions in the manner prescribed by particle theory, but this representation departs from the standard model in that the gauge group is independent of coordinates. In this interpretation the coordinates x are the potentials – they live in the algebra and the group. Since H = Sp(1)k , Ψ(x) is a column vector of k single particle states with values in the quaternion algebra. The single particle states are identified as leptons in a Sp(1 + n)/Sp(1) × Sp(n) representation, but higher order, Sp(k + n)/Sp(k) × Sp(n), k > 1, representations have to be interpreted as composite particles. Higher dimensional representations than the fundamental can be understood to represent excited states, and it will doubtless be the case that the practical applications of the theory will involve the construction and study of higher dimensional irreducible representations. One may note that those representations will involve products of quaternions, and products are in the quaternion algebra. This has implications for the representation of composite particles such as mesons and baryons, which will be considered elsewhere. The two parts of the curvature tensor in eq. (8) can be understood as Newton’s third law: The force acting on a system is equal and opposite to the force that the system exerts on its surroundings. This is captured precisely in: The curvature at particle 1 is the 3-space reflection of the curvature at particle 2. In the general case of a k-dimensional system space and its n-dimensional surroundings, the Appendix shows that the magnitudes of the curvatures are equal. The Appendix also emphasizes the versatility of the Cartan moving frame method in its generalization to any number of subspaces comprising a flag manifold. Equation (11) provides a way to make contact with relativity that does not make use of the projection, but like the projection it only works for elementary interactions taken one at a time. The SU (2) basis can be used for x rather than the quaternion basis; then note that the largest group that preserves the magnitude of x in eq. (11) is ρ ∈ SL(2, C) acting by ρ : x → ρ∗ xρ. The isomorphism SL(2, C)/ ± 1 ∼ SO(1, 3) offers a direct route to relativity via the Lorentz group. Another route to Minkowski space switches the SU (2) basis to the Pauli basis (equivalent √ to multiplication of the quaternion basis by −1), for which the determinant of the argument is Minkowski space. While one might be tempted to pursue these mappings for the general manybody case, this cannot be done without ruining the Sp(n) group structure. The compact Sp(n) group is proposed as the symmetry group of any compact system comprised of particles with spin. This is inconsistent with the Newtonian idea of a boundless absolute 9

three-dimensional space. However, that space is not observable. Any observable part of the natural world is necessarily compact, because we have no means to sensibly or experimentally observe anything but the interactions between material objects. The range of observable interactions is necessarily finite because the speed of light is finite – objects cannot explore the infinite distances inherent in a non-compact space, or at least not and still be observable to one another. To reiterate: All observations of the natural world take place in compact spaces, regardless of whether the natural scales are large or small.

Acknowledgment The author is grateful for several helpful discussions with Profs. John Sullivan and Gerald Folland.

Appendix: Curvature Tensors on Flag Manifolds A flag manifold over a field K = {R, C, H} can be realized either as GL(n, K)/P , where P is a maximal parabolic subgroup of GL(n, K) and n is the dimension of the K-vector space on which, GL(n, K) (the general linear group) operates, or as UG (n, K)/H, where H is a maximal subgroup of the orthogonal/unitary/symplectic group UG (n, K) over, respectively, the real/complex/quaternion field. We will use unitary to signify collectively the orthogonal, unitary, and symplectic cases, and this will be the version of the flag manifold that is our concern. Fix an H as follows: Let {kµ } be a partition of n appropriate to the field K, such that Σµ kµ = n where 1 ≤ µ ≤ m is an index set, so that H = h(k1 ) × h(k2 ) · · · × h(km ). To avoid multiple levels of subscripts in the sequel, k will be suppressed, so that hµ = h(kµ ). The metric for a matrix-valued set of coordinates, g, over the field K is ds2 = Tr(dgdg ∗ ). Here g ∗ is the transpose of g for K = R and is the transpose conjugate for K = C or H. We are interested in the unitary case where gg ∗ = 1; this condition gives dgg ∗ + gdg ∗ = 0 ⇒ dg ∗ = −g ∗ dgg ∗ , so that ds2 = −Tr(dgg ∗ dgg ∗ ) = −Tr(g ∗ dgg ∗ dg) = Tr[(g ∗ dg)(g ∗ dg)∗ ], with the last equality from g ∗ dg + dg ∗ g = 0. Let x be a cross-section of the flag manifold UG (n, K)/H, i.e., g = xH, where H is a fixed subgroup. Then dgg ∗ → dxH(H ∗ x∗ ) = dxx∗ = ω ˆ , so that dˆ s2 = Tr(ˆ ωω ˆ ∗ ) is the rightinvariant metric on cross-sections. However, it is also left invariant to the action of g1 ∈ UG , since Tr[(g1 dg)(g1 dg)∗ ] = Tr[(g1 dg)(dg∗ g1∗ )] = Tr(dgdg∗ ). Define ω = g ∗ dg, which emphasizes the left-invariance of ω, while ω ˆ = dxx∗ does the same for the right invariance of x. Note that ω = g ∗ dg = −dg ∗ g = −ω ∗ is skew-symmetric. The definition of the metric for the compact space UG (n, K) is not the only role for ω. Define a vector e = (e1 , e2 , · · · , en ) of orthonormal K-basis-vectors (frame) on the tangent space to a 10

manifold M n (K). The frame at any point p ∈ M n can be pulled back to the origin, p0 , by the action of g −1 ∈ UG (n, K). That is, e = e0 g, where e denotes the K-vector-valued set of bais vectors at the point p, while e0 is this vector at p0 . The change in the basis near p as one moves around an infinitesmal neighborhood of p is de = e0 dg = eg −1 dg, and for the unitary case this is de = eg ∗ dg = eω.

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That is, ω is the connection form on the tangent bundle of M n . The exterior derivative of ω is dω = d(g ∗ dg) = dg ∗ ∧ dg = −g ∗ dgg ∗ ∧ dg = −g ∗ dg ∧ g ∗ dg = −ω ∧ ω, giving dω + ω ∧ ω = 0. (13) This is the second Maurer-Cartan equation (MCII).[15] It implies that the affine group acts on the tangent space of M n , i.e., the tangent space is horizontal.[6] This construction fits into the setting of the flag manifold as follows. Corresponding to the {kµ } partition of n, denote the blocks of g ∈ UG by gαβ . Similarly, ω = (ωαβ ), 1 ≤ α, β ≤ m is the matrix written in block form. With this partitioning in mind, MCII becomes dωµν + Σm α=1 ωµα ∧ ωαν = 0 and in particular, a block on the diagonal is dωµµ + ωµµ ∧ ωµµ + Σα6=µ ωµα ∧ ωαµ = 0.

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(Note that the summation convention is not used.) Now suppose that g = H. The corresponding connection form reduces: ω → ω ¯=ω ¯ 11 × ω ¯ 22 × · · ·×ω ¯ mm . All off-diagonal blocks of ω vanish, and one is left with d¯ ωµµ + ω ¯ µµ ∧ ω ¯ µµ = 0 for all µ. The tangent bundles for the m-subspaces are all horizontal, just as is tangent bundle for M n . Thus the tangent bundles are independent of one another – all subspaces with vanishing MCII equations are independent of the other subspaces. However, in our general flag manifold the off-diagonal elements do not vanish, and when dω + ω ∧ ω 6= 0, Cartan[15] identifies the obstruction as the curvature two-form Ω, i.e., dω + ω ∧ ω = Ω. The diagonal part, dωµµ + ωµµ ∧ ωµµ in eq. (14), is horizontal and the off-diagonal part, which couples the basis vectors that are orthogonal to the kµ -subspace, is vertical.[6] It follows from eq. (14) that there is a curvature two-form ∗ Ωµµ = −Σα6=µ ωµα ∧ ωαµ = Σα6=µ ωµα ∧ ωµα ,

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associated to every subspace in an irreducible UG . The skew-symmetry of ω was used to get the second equality. The cuvature two-forms are clearly non-negative definite. ˆ = eh, which corresponds to a difFor the given kµ -partitioning, consider a change of basis e ferent selection of cross-section of G/H (h ⊂ H).[10] The associated connection form is defined ˆω ˆ is by dˆ e=e ˆ . The exterior derivative of e ˆω dˆ e=e ˆ = ehˆ ω = deh + edh = eωh + edh giving hˆ ω = ωh + dh. The exterior derivative of this equation gives ˆ = dˆ Ω ω+ω ˆ ∧ω ˆ = h∗ (dω + ω ∧ ω)h = h∗ Ωh

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∗ forms. which can be an aid in simplifying the Ωµµ = Σν6=µ ωµν ∧ ωµν The off-diagonal blocks in the µ-th row and column of ω couple the kµ -subspace to the remaining systems, so in this sense the connection form “connects” different subspaces to one another. The off-diagonal blocks of UG /H are Grassmannians, so these are geometrical objects that represent the interactions between different subspaces. To see this, consider H = ha × hb , with ka + kb = n. The coset space UG /ha × hb defines a Grassmannian, so one may interpret the subspaces as configuration spaces for a system (ka ) and its surroundings (kb ), with the Grassmannian representing all interactions between the two. For a space consisting of just two points, ha ∼ hb , so these subgroups are simply equivalent to the ±1 for K = R, U (1) for K = C, and Sp(1) for K = H. The real case is uninteresting, the complex case is U (2)/U (1) × U (1), and for H we have Sp(2)/Sp(1) × Sp(1), which is the YM geometry. The curvature forms for the latter case are ∗ Ω11 = ω12 ∧ ω12

and

∗ ∗ Ω22 = ω21 ∧ ω21 = ω12 ∧ ω12 ,

which are evaluated in terms of the Grassmannian coordinates in the text.

References [1] A. A. Belavin, A. M. Polyakov, A. S. Schwartz and Yu. S. Tyupkin, Phys. Lett. 59B, 85 (1975). [2] C. N. Yang and R. L. Mills, Phys. Rev. 96, 191 (1954). [3] M. F. Atiyah, The Geometry of Yang-Mills Fields, Lezioni Fermiane: Scuola Normale Sup., Pisa, 1979

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[4] B. Simon, Representations of Finite and Compact Groups, Providence, RI: Amer. Math. Soc., 1991 [5] W. Fulton and J. Harris, Representation Theory, New York: Springer, 1991 [6] S. Kobayashi and K. Nomizu, Foundations of Differential Geometry, Vol I and II, New York: Interscience, 1963 [7] A.-L. Mare, ”Equivariant cohomology of quaternionic flag manifolds” J. Alg., 319, 2830 (2008) [8] B. E. Eichinger ”Shape Distributions of Gaussian Molecules” Macromolecules, 18, 211 (1985). [9] L. K. Hua, Harmonic Analysis of Functions of Several Complex Variables in the Classical Domains, Trans. Math. Monographs, Vol. 6; Providence RI: Amer. Math. Soc., 1963 [10] S.S. Chern, Complex Manifolds Without Potential Theory, 2nd Ed. New York: Springer, 1995 [11] S. S. Chern, Bull. Amer. Math. Soc. 52, 1 (1946) [12] H. B. Lawson, Jr., The Theory of Gauge Fields in Four Dimensions, Providence RI: Amer. Math. Soc., 1980 [13] S. Helgason, Differential Geometry and Symmetric Spaces, NewYork: Academic Press, 1962 [14] G. B. Folland, A Course in Abstract Harmonic Analysis, Boca Raton: CRC Press, 1995 [15] Cartan, E., Riemannian Geometry in an Orthogonal Frame, Trans. V. V. Goldberg, World Scientific, NJ, (2001)

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