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Algebraic spin structures
Volume 150B, number 1,2,3
PHYSICS LETTERS
3 January 1985
ALGEBRAIC SPIN STRUCTURES I_M. BENN a, B.P. DOLAN a and R.W. TUCKER b aDepartment of Natural Philosophy, Glasgow University, Glasgow G12 8QQ, UK bDepartment of Physics, Lancaster University, Lancaster, UK Received 21 June 1984 Revised manuscript received 19 October 1984 We demonstrate that the existence of an algebraic spin structure (a globally defined primitive idempotent in the Clifford bundle of a manifold) is sufficient to describe algebraic spinors globally even when the manifold is not endowed with a spinor structure. The modified Kiihler equation can then be used to provide a global dynamical framework for such spinors.
Attention has focussed recently on the global properties of riemannian (or pseudo-riemannian) manifolds that enable one to formulate the global dynamics of fields, particularly those satisfying spinor field equations. Attempts to incorporate gravity into the basic description of the interactions among the elementary particles have led to the contemplation of theories formulated in spaces with dimensions other than 4 and with non-lorentzian signatures. Despite the recognition that global properties are relevant, the tensor field equations are often discussed in terms of their components with respect to a local tensor basis (constructed from a local basis of vector fields on the manifold). Although such bases always exist locally topological properties of the manifold may prohibit them from existing globally. It is therefore useful to represent one's theory in terms o f differential forms which can be defined globally on a manifold. Such self-evident considerations become less trivial for the description of spinor theories. It is almost universal to fred spinor equations in theoretical physics written in component form with the spinor basis being suppressed. It is tacitly assumed that one restricts to manifolds with a vanishing second Steifel-Whitney class. This ensures the existence of a spinor structure that enables a consistent correlation to be defined between the orthogonal group of the metric and its covering over the whole manifold. Considering the restriction on the topology arising from such a spinor structure it is of interest to investigate carefully a necessity for such constraints 100
when dealing with spinor theories both in space-time and extended spaces (it may be noticed that insisting on a spinor structure in space-time implies that the properties of the far reaches o f the universe must affect the standard spinor description of an electron in the laboratory). In a series of recent articles following K/ihler and Graf the less restricted notion of an algebraic spinor structure has been studied. Roughly speaking an algebraic spinor may be defined on a manifold when a certain inhomogeneous differential form can be found to be idempotent in the Clifford algebra associated with the (pseudo)-riemannian metric of the space. This appears much less restrictive than requiring a spinor structure; there is no necessity to correlate spin transformations on the Clifford ideal projected by this idempotent with the structure group of the orthonormal frame bundle. The dynamics of such an algebraic spinor field may be provided by a modified K~ihler equation [9]. Locally such an equation can be written in spinor components such as to reproduce the usual curved-space Dirac equation. In this letter we make the observation that this modified K~ller equation can have globally defined solutions on a manifold without a standard spinor structure. Algebraic spinors may therefore be expected to be relevant in situations where the global implications of a standard spinor structure are absent. Although the requirement of global hyperbolicity in lorentzian space-time assures one that at least one standard spinor 0370-2693/85/$ 03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
Volume 150B, number 1,2,3
PHYSICS LETTERS
structure exists [2] no similar requirement is mandatory when discussing Kaluza-Klein type theories which naturally lead to considerations of a large class of manifolds with diverse dimensions and properties. Since the spin group conists of certain elements of the Clifford algebra we may represent it on the minimal left ideals of that algebra. (That is, if the Clifford algebra is the algebra o f matrices with entries in R, C or H, then we may take the space o f spinors to be a column, which is a minimal left ideal.) Each minimal left ideal is projected out by an idempotent P. Since there is a correspondence between the Clifford and exterior algebras of the cotangent space we are led naturally to the notion of a spinor being a differential form lying in a minimal left ideal of the Clifford algebra, characterised by an idempotent P [ 3 - 5 ] . The global existence of such a P gives what Graf has called a generalised spin structure [4] which we shall call an algebraic spin structure. (This is to avoid confusion with the different concept of a spin c structure, which has also been referred to as a "generalised spin structure".) An algebraic spin structure makes no overt reference to the spin group; in particular there is no requirement that it should globally cover the orthogonal group, which corresponds to the usual notion o f spinor structure [6]. It is therefore perhaps not surprising that an algebraic spin structure might exist for a manifold which does not admit a spinor structure. In this letter we take the example o f C P 2 to demonstrate that this is in fact the case. We consider CP 2, regarded as a real manifold, endowed with the standard Fubini-Study metric [7]. Since this manifold can be given the structure o f a Kifftler manifold we have the globally defined K~ller two form K, which may be used to choose an orientation given by K ^ K = 2Z,
(1)
where Z is the volume four-form, * 1. The orientability ensures that P -_1 ~ (1 - z )
(2)
is globally defined. The relation o f the Clifford product, v, to the exterior product, ^, is given by associativity and Av¢ =A^¢ + ix¢,
(3)
where A is a one-form and ~b any form. i~ denotes interior multiplication where z~ is the vector metric dual
3 January 1985
to A. The positive definite signature of the metric gives Z v Z = 1 ensuring t h a t P is Clifford idempotent
e v P = e.
(4)
The Kiihler two-form is self-dual i e. K = * K.
(5)
Thus since *K = - K v Z we have KvP =K,
(6)
a n d K lies in the Clifford ideal projected out b y P . The real Clifford algebra associated with the cotangent space is isomorphic to the algebra of 2 X 2 matrices with quaternionic entries [8] C4,0 (R) "" H(2).
(7)
The even sub-algebra is semi-simple, being isomorphic to the sum of two quaternion algebras C], 0 (R) ~ H + H.
(8)
We thus see t h a t P is primitive in C4,0 (R); that is, it projects out a minimal left ideal. SinceP is even it projects out one of the simple components of C~,0(R ). T h u s P and K may be regarded as semi-spinors, both being in a minimal left ideal of a simple component of the even sub-algebra. (We would need two other two-forms to have a globally defined semi-spinor basis.) Since K is closed and co-closed it, like P, will satisfy the massless Kiihler equation, (d +* d *) ~b = 0.
(9)
Since K is a two-form satisfying the massless K~aler equation it has been interpreted as an electromagnetic field on CP 2 [9]. It is thus intriguing to note that it lies in a minimal left ideal of the even subalgebra offering the interpretation o f being a semi-spinor. Since the requirements of a spinor structure and an algebraic spin structure place different global restric. tions on a manifold it is important to decide which is physically necessary. This question is related to how we interpret the covariance of our equations, in particular the role of the spin group. If the dynamics are given by the KLlaler equation [3,4] rather than the Dirac equation then it is perhaps not surprising that we have no need of a spinor structure. However we may modify the K~Lhler equation such that it describes a field lying in one minimal left ideal and, at least locally, corresponds to the usual formulation of the 101
Volume 150B, number 1,2,3
PHYSICS LETTERS
curved~space Dirac equation [ 1]. This formulation of the Dirac equation in terms of differential forms seems to require only an algebraic spin structure. It is the global writing of this equation in matrix components, which corresponds to the usual formulation, that requires a spinor structure. Thus the global properties we require of a manifold depend rather delicately on how we formulate our equations and interpret their covariance.
102
3 January 1985
References [ 1] I.M. Berm and R.W. Tucker The Dirac equation in exterior form, to be published. [21 C. Isham, Proc. Roy, Soc. A364 (1978) 591. [3] E. K~ihler,Rend. Mat. (3-4) 21 (1962) 425. [4] W. Graf, Ann. Inst. Henri Poincar~ XXIX (1978) 85. [5 ] I.M. Benn and R.W Tucker, Commun. Math. Phys. 89 (1983) 341. [6] J.W. Milnor, Enseign. Math. 9 (1963) 198. [7] T. Eguchi, P.B. Gilkey and A,J. Hanson, Phys. Rep. 66 (1980) 213. [8] I R. Porteous, Topological geometry, (Cambridge U.P., London, 1981). [9] A. Trautman, Intern. J. Theor. Phys. 16 (1977) 561.
PHYSICS LETTERS
3 January 1985
ALGEBRAIC SPIN STRUCTURES I_M. BENN a, B.P. DOLAN a and R.W. TUCKER b aDepartment of Natural Philosophy, Glasgow University, Glasgow G12 8QQ, UK bDepartment of Physics, Lancaster University, Lancaster, UK Received 21 June 1984 Revised manuscript received 19 October 1984 We demonstrate that the existence of an algebraic spin structure (a globally defined primitive idempotent in the Clifford bundle of a manifold) is sufficient to describe algebraic spinors globally even when the manifold is not endowed with a spinor structure. The modified Kiihler equation can then be used to provide a global dynamical framework for such spinors.
Attention has focussed recently on the global properties of riemannian (or pseudo-riemannian) manifolds that enable one to formulate the global dynamics of fields, particularly those satisfying spinor field equations. Attempts to incorporate gravity into the basic description of the interactions among the elementary particles have led to the contemplation of theories formulated in spaces with dimensions other than 4 and with non-lorentzian signatures. Despite the recognition that global properties are relevant, the tensor field equations are often discussed in terms of their components with respect to a local tensor basis (constructed from a local basis of vector fields on the manifold). Although such bases always exist locally topological properties of the manifold may prohibit them from existing globally. It is therefore useful to represent one's theory in terms o f differential forms which can be defined globally on a manifold. Such self-evident considerations become less trivial for the description of spinor theories. It is almost universal to fred spinor equations in theoretical physics written in component form with the spinor basis being suppressed. It is tacitly assumed that one restricts to manifolds with a vanishing second Steifel-Whitney class. This ensures the existence of a spinor structure that enables a consistent correlation to be defined between the orthogonal group of the metric and its covering over the whole manifold. Considering the restriction on the topology arising from such a spinor structure it is of interest to investigate carefully a necessity for such constraints 100
when dealing with spinor theories both in space-time and extended spaces (it may be noticed that insisting on a spinor structure in space-time implies that the properties of the far reaches o f the universe must affect the standard spinor description of an electron in the laboratory). In a series of recent articles following K/ihler and Graf the less restricted notion of an algebraic spinor structure has been studied. Roughly speaking an algebraic spinor may be defined on a manifold when a certain inhomogeneous differential form can be found to be idempotent in the Clifford algebra associated with the (pseudo)-riemannian metric of the space. This appears much less restrictive than requiring a spinor structure; there is no necessity to correlate spin transformations on the Clifford ideal projected by this idempotent with the structure group of the orthonormal frame bundle. The dynamics of such an algebraic spinor field may be provided by a modified K~ihler equation [9]. Locally such an equation can be written in spinor components such as to reproduce the usual curved-space Dirac equation. In this letter we make the observation that this modified K~ller equation can have globally defined solutions on a manifold without a standard spinor structure. Algebraic spinors may therefore be expected to be relevant in situations where the global implications of a standard spinor structure are absent. Although the requirement of global hyperbolicity in lorentzian space-time assures one that at least one standard spinor 0370-2693/85/$ 03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
Volume 150B, number 1,2,3
PHYSICS LETTERS
structure exists [2] no similar requirement is mandatory when discussing Kaluza-Klein type theories which naturally lead to considerations of a large class of manifolds with diverse dimensions and properties. Since the spin group conists of certain elements of the Clifford algebra we may represent it on the minimal left ideals of that algebra. (That is, if the Clifford algebra is the algebra o f matrices with entries in R, C or H, then we may take the space o f spinors to be a column, which is a minimal left ideal.) Each minimal left ideal is projected out by an idempotent P. Since there is a correspondence between the Clifford and exterior algebras of the cotangent space we are led naturally to the notion of a spinor being a differential form lying in a minimal left ideal of the Clifford algebra, characterised by an idempotent P [ 3 - 5 ] . The global existence of such a P gives what Graf has called a generalised spin structure [4] which we shall call an algebraic spin structure. (This is to avoid confusion with the different concept of a spin c structure, which has also been referred to as a "generalised spin structure".) An algebraic spin structure makes no overt reference to the spin group; in particular there is no requirement that it should globally cover the orthogonal group, which corresponds to the usual notion o f spinor structure [6]. It is therefore perhaps not surprising that an algebraic spin structure might exist for a manifold which does not admit a spinor structure. In this letter we take the example o f C P 2 to demonstrate that this is in fact the case. We consider CP 2, regarded as a real manifold, endowed with the standard Fubini-Study metric [7]. Since this manifold can be given the structure o f a Kifftler manifold we have the globally defined K~ller two form K, which may be used to choose an orientation given by K ^ K = 2Z,
(1)
where Z is the volume four-form, * 1. The orientability ensures that P -_1 ~ (1 - z )
(2)
is globally defined. The relation o f the Clifford product, v, to the exterior product, ^, is given by associativity and Av¢ =A^¢ + ix¢,
(3)
where A is a one-form and ~b any form. i~ denotes interior multiplication where z~ is the vector metric dual
3 January 1985
to A. The positive definite signature of the metric gives Z v Z = 1 ensuring t h a t P is Clifford idempotent
e v P = e.
(4)
The Kiihler two-form is self-dual i e. K = * K.
(5)
Thus since *K = - K v Z we have KvP =K,
(6)
a n d K lies in the Clifford ideal projected out b y P . The real Clifford algebra associated with the cotangent space is isomorphic to the algebra of 2 X 2 matrices with quaternionic entries [8] C4,0 (R) "" H(2).
(7)
The even sub-algebra is semi-simple, being isomorphic to the sum of two quaternion algebras C], 0 (R) ~ H + H.
(8)
We thus see t h a t P is primitive in C4,0 (R); that is, it projects out a minimal left ideal. SinceP is even it projects out one of the simple components of C~,0(R ). T h u s P and K may be regarded as semi-spinors, both being in a minimal left ideal of a simple component of the even sub-algebra. (We would need two other two-forms to have a globally defined semi-spinor basis.) Since K is closed and co-closed it, like P, will satisfy the massless Kiihler equation, (d +* d *) ~b = 0.
(9)
Since K is a two-form satisfying the massless K~aler equation it has been interpreted as an electromagnetic field on CP 2 [9]. It is thus intriguing to note that it lies in a minimal left ideal of the even subalgebra offering the interpretation o f being a semi-spinor. Since the requirements of a spinor structure and an algebraic spin structure place different global restric. tions on a manifold it is important to decide which is physically necessary. This question is related to how we interpret the covariance of our equations, in particular the role of the spin group. If the dynamics are given by the KLlaler equation [3,4] rather than the Dirac equation then it is perhaps not surprising that we have no need of a spinor structure. However we may modify the K~Lhler equation such that it describes a field lying in one minimal left ideal and, at least locally, corresponds to the usual formulation of the 101
Volume 150B, number 1,2,3
PHYSICS LETTERS
curved~space Dirac equation [ 1]. This formulation of the Dirac equation in terms of differential forms seems to require only an algebraic spin structure. It is the global writing of this equation in matrix components, which corresponds to the usual formulation, that requires a spinor structure. Thus the global properties we require of a manifold depend rather delicately on how we formulate our equations and interpret their covariance.
102
3 January 1985
References [ 1] I.M. Berm and R.W. Tucker The Dirac equation in exterior form, to be published. [21 C. Isham, Proc. Roy, Soc. A364 (1978) 591. [3] E. K~ihler,Rend. Mat. (3-4) 21 (1962) 425. [4] W. Graf, Ann. Inst. Henri Poincar~ XXIX (1978) 85. [5 ] I.M. Benn and R.W Tucker, Commun. Math. Phys. 89 (1983) 341. [6] J.W. Milnor, Enseign. Math. 9 (1963) 198. [7] T. Eguchi, P.B. Gilkey and A,J. Hanson, Phys. Rep. 66 (1980) 213. [8] I R. Porteous, Topological geometry, (Cambridge U.P., London, 1981). [9] A. Trautman, Intern. J. Theor. Phys. 16 (1977) 561.