Holonomy

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Parallel transport on a sphere depends on the path. Transporting from A → N → B yields a vector different from the initial vector. This failure to return to the initial vector is measured by the holonomy of the connection.

In differential geometry, the holonomy of a connection on a smooth manifold is a general geometrical consequence of the curvature of the connection measuring the extent to which parallel transport around closed loops fails to preserve the geometrical data being transported. For flat connections, the associated holonomy is a type of monodromy, and is an inherently global notion. For curved connections, holonomy has nontrivial local and global features.

Any kind of connection on a manifold gives rise, through its parallel transport maps, to some notion of holonomy. The most common forms of holonomy are for connections possessing some kind of symmetry. Important examples include: holonomy of the Levi-Civita connection in Riemannian geometry (called Riemannian holonomy), holonomy of connections in vector bundles, holonomy of Cartan connections, and holonomy of connections in principal bundles. In each of these cases, the holonomy of the connection can be identified with a Lie group, the holonomy group. The holonomy of a connection is closely related to the curvature of the connection, via the Ambrose-Singer theorem.

The study of Riemannian holonomy has led to a number of important developments. In particular, in 1952 Georges de Rham proved the de Rham decomposition theorem, a principle for splitting a Riemannian manifold into a Cartesian product of Riemannian manifolds by splitting the tangent bundle into irreducible spaces under the action of the local holonomy groups. Later, in 1953, M. Berger classified the possible irreducible holonomies. The decomposition and classification of Riemannian holonomy has applications to physics, and in particular to string theory.

[edit] Holonomy of a connection in a vector bundle

Let E be a rank k vector bundle over a smooth manifold M and let ∇ be a connection on E. Given a piecewise smooth loop γ : [0,1] → M based at x in M, the connection defines a parallel transport map . This map is both linear and invertible and so defines an element of GL(E_x). The holonomy group of ∇ based at x is defined as

The restricted holonomy group based at x is the subgroup coming from contractible loops γ.

If M is connected then the holonomy group depends on the basepoint x only up to conjugation in GL(k, R). Explicitly, if γ is a path from x to y in M then

Choosing different identifications of E_x with R^k also gives conjugate subgroups. Sometimes, particularly in general or informal discussions (such as below), one may drop reference to the basepoint, with the understanding that the definition is good up to conjugation.

Some important properties of the holonomy group include:

• Hol⁰(∇) is a connected, Lie subgroup of GL(k, R).
• Hol⁰(∇) is the identity component of Hol(∇).
• There is a natural, surjective group homomorphism π₁(M) → Hol(∇)/Hol⁰(∇) where π₁(M) is the fundamental group of M which sends the homotopy class [γ] to the coset P_γ·Hol⁰(∇).
• If M is simply connected then Hol(∇) = Hol⁰(∇).
• ∇ is flat (i.e. has vanishing curvature) if and only if Hol⁰(∇) is trivial.

[edit] Holonomy of a connection in a principal bundle

The definition for holonomy of connections on principal bundles proceeds in parallel fashion. Let G be a Lie group and P a principal G-bundle over a smooth manifold M which is paracompact. Let ω be a connection on P. Given a piecewise smooth loop γ : [0,1] → M based at x in M and a point p in the fiber over x, the connection defines a unique horizontal lift such that . The end point of the horizontal lift, , will not generally be p but rather some other point p·g in the fiber over x. Define an equivalence relation ~ on P by saying that p~q if they can be joined by a piecewise smooth horizontal path in P.

The holonomy group of ω based at p is then defined as

The restricted holonomy group based at p is the subgroup coming from horizontal lifts of contractible loops γ.

If M and P are connected then the holonomy group depends on the basepoint p only up to conjugation in G. Explicitly, if q is any other chosen basepoint for the holonomy, then there exists a unique g ∈ G such that q ~ p g. With this value of g,

Hol_q(ω) = g ^{− 1}Hol_p(ω)g.

In particular,

Moreover, if p~q then Hol_p(ω) = Hol_q(ω). As above, sometimes one drops reference to the basepoint of the holonomy group, with the understanding that the definition is good up to conjugation.

Some important properties of the holonomy and restricted holonomy groups include:

• Hol⁰_p(ω) is a connected Lie subgroup of G.
• Hol⁰_p(ω) is the identity component of Hol_p(ω).
• There is a natural, surjective group homomorphism π₁(M) → Hol_p(ω)/Hol⁰_p(ω).
• If M is simply connected then Hol_p(ω) = Hol⁰_p(ω).
• ω is flat (i.e. has vanishing curvature) if and only if Hol⁰_p(ω) is trivial.

[edit] Holonomy bundles

Let M be a connected paracompact smooth manifold and P a principal G-bundle with connection ω, as above. Let p ∈ P be an arbitrary point of the principal bundle. Let H(p) be the set of points in P which can be joined to p by a horizontal curve. Then it can be shown that H(p), with the evident projection map, is a principal bundle over M with structure group Hol⁰_p(ω). This principal bundle is called the holonomy bundle (through p) of the connection. The connection ω restricts to a connection on H(p), since its parallel transport maps preserve H(p). Thus H(p) is a reduced bundle for the connection. Furthermore, since no subbundle of H(p) is preserved by parallel transport, it is the minimal such reduction.^[1]

As with the holonomy groups, the holonomy bundle also transforms equivariantly within the ambient principal bundle P. In detail, if q ∈ P is another chosen basepoint for the holonomy, then there exists a unique g ∈ G such that q ~ p g (since, by assumption, M is path-connected). Hence H(q) = H(p) g. As a consequence, the induced connections on holonomy bundles corresponding to different choices of basepoint are compatible with one another: their parallel transport maps will differ by precisely the same element g.

[edit] Monodromy

The holonomy bundle H(p) is a principal bundle for Hol_p(ω), and so also admits an action of the restricted holonomy group Hol⁰_p(ω) (which is a normal subgroup of the full holonomy group). The discrete group Hol_p(ω)/Hol⁰_p(ω) is called the monodromy group of the connection; it acts on the quotient bundle H(p)/Hol⁰_p(ω). There is a surjective homomorphism φ : π₁(M) → Hol_p(ω)/Hol⁰_p(ω), so that φ(π₁(M)) acts on H(p)/Hol⁰_p(ω). This action of the fundamental group is a monodromy representation of the fundamental group.^[2]

[edit] Ambrose-Singer theorem

The Ambrose-Singer theorem relates the holonomy of a connection with its curvature form. To make this theorem plausible, consider the familiar case of an affine connection (or a connection in the tangent bundle — the Levi-Civita connection, for example). The curvature arises when one travels around an "infinitesimal parallelogram".

In detail, if σ : [0,1] × [0,1] → M is a surface in M parametrized by a pair of variables x and y, then a vector V may be transported around the boundary of σ: first along (x,0), then along (1,y), followed by (x,1) going in the negative direction, and then (0,y) back to the point of origin. This is a special case of a holonomy loop: the vector V is acted upon by the holonomy group element corresponding to the lift of the boundary of σ. The curvature enters explicitly when the parallelogram is shrunk to zero, by traversing the boundary of smaller parallelograms over [0,x] × [0,y]. This corresponds to taking a derivative of the parallel transport maps at x=y=0:

where R is the curvature tensor.^[3] So, roughly speaking, the curvature gives the infinitesimal holonomy over a closed loop (the infinitesimal parallelogram). More formally, the curvature is the differential of the holonomy action at the identity of the holonomy group. In other words, R(X,Y) is an element of the Lie algebra of Hol_p(ω).

In general, let g be the Lie algebra of the Lie group G. Then the curvature form of the connection is a g-valued 2-form Ω on P. The Ambrose-Singer theorem states:^[4]

• The Lie algebra of Hol_p(ω) is spanned by all the elements of g of the form Ω_q(X,Y) as q ranges over all points which can be joined to p by a horizontal curve (q ~ p), and X and Y are horizontal tangent vectors at q.

Alternatively, the theorem can be restated in terms of the holonomy bundle:^[5]

• The Lie algebra of Hol_p(ω) is the subspace of g spanned by elements of the form Ω_q(X,Y) where q ∈ H(p) and X and Y are horizontal vectors at q.

[edit] Riemannian holonomy

The holonomy of a Riemannian manifold (M, g) is just the holonomy group of the Levi-Civita connection on the tangent bundle to M. A 'generic' n-dimensional Riemannian manifold has an O(n) holonomy, or SO(n) if it is orientable. Manifolds whose holonomy groups are proper subgroups of O(n) or SO(n) have special properties.

[edit] Reducible holonomy and the de Rham decomposition

Let x ∈ M be an arbitrary point. Then the holonomy group Hol(M) acts on the tangent space T_xM. This action may either be irreducible as a group representation, or reducible in the sense that there is a splitting of T_xM into orthogonal subspaces T_xM = T′_xM ⊕ T′′_xM, each of which is invariant under the action of Hol(M). In the latter case, M is said to be reducible.

Suppose that M is a reducible manifold. Allowing the point x to vary, the bundles T′M and T′′M formed by the reduction of the tangent space at each point are smooth distributions which are integrable in the sense of Frobenius. The integral manifolds of these distributions are totally geodesic submanifolds. So M is locally a Cartesian product M′ × M′′. The (local) de Rham isomorphism follows by continuing this process until a complete reduction of the tangent space is achieved:^[6]

• Let M be a simply connected Riemannian manifold,^[7] and TM = T⁽⁰⁾M ⊕ T⁽¹⁾M ⊕ ... ⊕ T^(k)M be the complete reduction of the tangent bundle under the action of the holonomy group. Suppose that T⁽⁰⁾M consists of vectors invariant under the holonomy group (i.e., such that the holonomy representation is trivial). Then locally M is isometric to a product

where V₀ is an open set in a Euclidean space, and each V_i is an integral manifold for T⁽ⁱ⁾M. Furthermore, Hol(M) splits as a direct product of the holonomy groups of each M_i.

If, moreover, M is assumed to be geodesically complete, then the theorem holds globally, and each M_i is a geodesically complete manifold.^[8]

[edit] The Berger classification

In 1955, M. Berger gave a complete classification of possible holonomy groups for simply connected, Riemannian manifolds which are irreducible (not locally a product space) and nonsymmetric (not locally a Riemannian symmetric space). Berger's list is as follows:

(Berger's original list also included the possibility of Spin(9) as a subgroup of SO(16). Riemannian manifolds with such holonomy were later shown independently by D. Alekseevski and Brown-Gray to be necessarily locally symmetric, i.e., locally isometric to the Cayley plane F₄/Spin(9) or locally flat. See below.) It is now known that all of these possibilities occur as holonomy groups of Riemannian manifolds. The last two exceptional cases were the most difficult to find. See G₂ manifold and Spin(7) manifold.

Note that Sp(n) ⊂ SU(2n) ⊂ U(2n) ⊂ SO(4n), so every hyperkähler manifold is a Calabi-Yau manifold, every Calabi-Yau manifold is a Kähler manifold, and every Kähler manifold is orientable.

The strange list above was explained by Simons's proof of Berger's theorem. A simple and geometric proof of Berger's theorem was given by C. Olmos in 2005. One first shows that if a Riemannian manifold is not a locally symmetric space and the reduced holonomy acts irreducibly on the tangent space, then it acts transitively on the unit sphere. The Lie groups acting transitively on spheres are known: they consist of the list above, together with 2 extra cases: the group Spin(9) acting on R¹⁶, and the group T·Sp(m) acting on R^4m. Finally one checks that the first of these two extra cases only occurs as a holonomy group for locally symmetric spaces (that are locally isomorphic to the Cayley projective plane), and the second does not occur at all as a holonomy group.

Berger's original classification also included non-positive-definite pseudo-Riemannian metric non-locally symmetric holonomy. That list consisted of SO(p,q) of signature (p,q), U(p,q) and SU(p,q) of signature (2p,2q), Sp(p,q) and Sp(p,q)·Sp(1) of signature (4p,4q), SO(n,C) of signature (n,n), SO(n,H) of signature (2n,2n), split G₂ of signature (4,3), G₂(C) of signature (7,7), Spin(4,3) of signature (4,4), Spin(7,C) of signature (7,7), Spin(5,4) of signature (8,8) and, lastly, Spin(9,C) of signature (16,16). The split and complexified Spin(9) are necessarily locally symmetric as above and should not have been on the list. The complexified holonomies SO(n,C), G₂(C), and Spin(7,C) may be realized from complexifying real analytic Riemannian manifolds. The last case, manifolds with holonomy contained in SO(n,H), were shown to be locally flat by R. McLean.^{[citation needed]}

Riemannian symmetric spaces, which are locally isometric to homogeneous spaces G / H have local holonomy isomorphic to H. These too have been completely classified.

Lastly, Berger also classified non-metric holonomy groups of manifolds with merely an affine connection. That list was shown to be incomplete. Non-metric holonomy groups not on Berger's original list are referred to as exotic holonomies and examples have been found by R. Bryant and Chi, Merkulov, and Schwachhofer.^{[citation needed]}

[edit] Special holonomy and spinors

Manifolds with special holonomy play a fundamental role in the theory of spinors, and particularly pure spinors.^[9] In particular, the following facts hold:

• Hol(ω) ⊂ U(n) if and only if M admits a covariantly constant (or parallel) projective pure spinor field.
• If M is a spin manifold, then Hol(ω) ⊂ SU(n) if and only if M admits a parallel pure spinor field. In fact, a parallel pure spinor field determines a canonical reduction of the structure group to SU(n).
• If M is a seven-dimensional spin manifold, then M carries a non-trivial parallel spinor field if and only if the holonomy is contained in G₂.
• If M is an eight-dimensional spin manifold, then M carries a non-trivial parallel spinor field if and only if the holonomy is contained in Spin(7).

The unitary and special unitary holonomies are often studied in connection with twistor theory,^[10] as well as in the study of almost complex structures.^[11]

[edit] Applications to string theory

Riemannian manifolds with special holonomy play an important role in string theory compactifications.^[12] This is because special holonomy manifolds admit covariantly constant (parallel) spinors and thus preserve some fraction of the original supersymmetry. Most important are compactifications on Calabi-Yau manifolds with SU(2) or SU(3) holonomy. Also important are compactifications on G₂ manifolds.

[edit] Local and infinitesimal holonomy

If π:P→M is a principal bundle, and ω is a connection in P, then the holonomy of ω can be restricted to the fibre over an open subset of M. Indeed, if U is a connected open subset of M, then ω restricts to give a connection in the bundle π^-1U over U. The holonomy (resp. restricted holonomy) of this bundle will be denoted by Hol_p(ω, U) (resp. Hol_p⁰(ω, U)) for each p with π(p)∈U.

If U⊂V are two open sets containing π(p), then there is an evident inclusion

The local holonomy group at a point p is defined by

for any family of nested connected open sets U_k with .

The local holonomy group has the following properties:

1. It is a connected Lie subgroup of the restricted holonomy group Hol_p⁰(ω).
2. Every point p has a neighborhood V such that Hol_p*(ω)=Hol_p⁰(ω,V). In particular, the local holonomy group depends only on the point p, and not the choice of sequence U_k used to define it.
3. The local holonomy is equivariant with respect to translation by elements of the structure group G of P; i.e., Hol_pg*(ω)=Ad(g^-1)Hol_p*(ω) for all g∈G. (Note that, by property 1., the local holonomy group is a connected Lie subgroup of G, so the adjoint is well-defined.)

The local holonomy group is not well-behaved as a global object. In particular, its dimension may fail to be constant. However, the following theorem holds:

• If the dimension of the local holonomy group is constant, then the local and restricted holonomy agree: Hol_p*(ω)=Hol_p⁰(ω).

[edit] Notes

1. ^ Kobayashi and Nomizu II.7.
2. ^ Cf Sharpe (1997) 3.7.
3. ^ Spivak (1999) p. 241.
4. ^ From Sternberg (1964), Theorem VII.1.2.
5. ^ Kobayashi and Nomizu, Volume I, II.8.
6. ^ See Kobayashi and Nomizu, IV.5.
7. ^ This theorem generalizes to non-simply connected manifolds, but the statement is more complicated.
8. ^ See Kobayashi and Nomizu, IV.6.
9. ^ See Chapters IV.9-10 in Lawson and Michelsohn (1989).}}
10. ^ See, for instance, Baum (1991).
11. ^ Lawson and Michelsohn, ibid.
12. ^ Gubser (2002), TASI lectures: Special holonomy in string theory and M-theory, arXiv:hep-th/0201114 .

[edit] References

• Ambrose, W. and Singer, I. M. (1953). "A theorem on holonomy". Trans. Amer. Math. Soc. 75: 428–443. doi:10.2307/1990721. 
• Baum, H. (1991). Twistors and killing spinors on Riemannian manifolds. B.G. Teubner. 
• Berger, M. (1953). "Sur les groupes d'holonomie des variétés a connexion affine et des variétés riemanniennes". Bull. Soc. Math. Frace 83: 279–330. 
• Joyce, D. (2000). Compact Manifolds with Special Holonomy. Oxford University Press. ISBN 0-19-850601-5. 
• Kobayashi, S. and Nomizu, K. (1996 (New edition)). Foundations of Differential Geometry, Vol. 1 & 2. Wiley-Interscience. ISBN 0471157333. 
• Lawson, H. B. and Michelsohn, M-L. (1989). Spin Geometry. Princeton University Press. ISBN 0-691-08542-0. 
• Olmos, C. (2005). "A geometric proof of the Berger Holonomy Theorem". Annals of Mathematics 161 (1): 579–588. 
• Sharpe, R.W. (1997). Differential Geometry: Cartan's Generalization of Klein's Erlangen Program. Springer-Verlag, New York. ISBN 0-387-94732-9. 
• Simons, J. (1962). "On the transitivity of holonomy systems". Annals of Mathematics 76 (2): 213–234. 
• Spivak, M. (1999). A comprehensive introduction to differential geometry, Volume II. Houston, Texas: Publish or Perish. ISBN 0914098713. 
• Sternberg, S. (1983). Lectures on differential geometry. New York: Chelsea. ISBN 0-8284-0316-3. 

[edit] Further reading

• Literature about manifolds of special holonomy, a bibliography by Frederik Witt.