Pin groups in general relativity

There are eight possible Pin groups that can be used to describe the transformation behavior of fermions under parity and time reversal. We show that only two of these are compatible with general relativity, in the sense that the configuration space of fermions coupled to gravity transforms appropriately under the space-time diffeomorphism group.


I. INTRODUCTION
For bosons, the space-time transformation behavior is governed by the Lorentz group Oð3; 1Þ, which comprises four connected components. Rotations and boosts are contained in the connected component of unity, the proper orthochronous Lorentz group SO ↑ ð3; 1Þ. Parity (P) and time reversal (T) are encoded in the other three connected components of the Lorentz group, the translates of SO ↑ ð3; 1Þ by P, T and PT.
For fermions, the space-time transformation behavior is governed by a double cover of Oð3; 1Þ. Rotations and boosts are described by the unique simply connected double cover of SO ↑ ð3; 1Þ, the spin group Spin ↑ ð3; 1Þ. However, in order to account for parity and time reversal, one needs to extend this cover from SO ↑ ð3; 1Þ to the full Lorentz group Oð3; 1Þ.
This extension is by no means unique. There are no less than eight distinct double covers of Oð3; 1Þ that agree with Spin ↑ ð3; 1Þ over SO ↑ ð3; 1Þ. They are the Pin groups Pin abc , characterized by the property that the elements Λ P and Λ T covering P and T satisfy Λ 2 P ¼ −a, Λ 2 T ¼ b and ðΛ P Λ T Þ 2 ¼ −c, where a, b and c are either 1 or −1 (cf. [1,2]).
In this paper, we show that the consistent description of fermions in the presence of general relativity (GR) imposes severe restrictions on the choice of Pin group. In fact, we find that only two of the eight Pin groups are admissible: the group Pin þ ¼ Pin þþ− and the group Pin − ¼ Pin −−− . The source of these restrictions is the double cover of the frame bundle, which, in the context of GR, is needed in order to obtain an infinitesimal action of the space-time diffeomorphism group on the configuration space of fermions coupled to gravity.
We derive these restrictions in the "universal spinor bundle approach" for fermions coupled to gravity, as developed in [3][4][5] for the Riemannian and in [6][7][8][9] for the Lorentzian case. However, our results remain valid in other formulations that are covariant under infinitesimal space-time diffeomorphisms, such as the "global" approach of [2,[10][11][12]. To underline this point, we highlight the role of the space-time diffeomorphism group in restricting the admissible Pin groups.
Selecting the correct Pin groups is important from a fundamental point of view-it determines the transformation behavior of fermionic fields under reflections-but also because the Pin group can affect observable quantities such as currents [13][14][15]. Due to their transparent definition in terms of Clifford algebras, the "Cliffordian" Pin groups Pinð3; 1Þ ¼ Pin þ−þ and Pinð1; 3Þ ¼ Pin −þþ have attracted much attention [13,[16][17][18][19]. Remarkably, the two Pin groups Pin þ and Pin − that are compatible with GR are not the widely used Cliffordian Pin groups Pinð3; 1Þ and Pinð1; 3Þ.

II. THE LORENTZIAN METRIC
In order to establish notation, we briefly recall the frame or vierbein formalism for a Lorentzian metric g on a fourdimensional space-time manifold M.
A frame e x based at x is a basis e μ a ∂ μ of the tangent space T x M, with basis vectors labeled by a ¼ 0, 1, 2, 3. The space FðMÞ of all frames (with arbitrary x) is called the frame bundle, and we denote by F x ðMÞ the set of frames with base point x. Note that the group Glð4; RÞ of invertible 4 × 4 matrices A a b acts from the right on F x ðMÞ, sending e x to the frame e 0 x ¼ e x A with e 0 μ a ¼ e μ b A b a . This action is free and transitive; any two frames e x and e 0 x over the same point x are related by e 0 x ¼ e x A for a unique matrix A a b . For a given Lorentzian metric g, the orthonormal frame bundle O g ðMÞ ⊂ FðMÞ is the space of all orthonormal frames e μ a , satisfying g μν e μ a e ν b ¼ η ab . Since two orthonormal frames e x and e 0 x over the same point x differ by a Lorentz transformation Λ, e 0 x ¼ e x Λ, the Lorentz group Oð3; 1Þ acts freely and transitively on the set O g x ðMÞ ⊂ F x ðMÞ of orthonormal frames based at x.
Specifying a metric g at x is equivalent to specifying the set O g x ðMÞ of orthonormal frames. Since O g x ðMÞ ⊂ F x ðMÞ is an orbit under the action of the Lorentz group Oð3; 1Þ on F x ðMÞ, specifying the metric at x is equivalent to picking a point in the orbit space R x ðMÞ ¼ F x ðMÞ=Oð3; 1Þ. This is the set of equivalence classes ½e x of frames at x, where two frames e x and e 0 x are deemed equivalent if they differ by a Lorentz transformation Λ, e 0 x ¼ e x Λ. We denote the bundle of all equivalence classes ½e x (with arbitrary x) by RðMÞ.
To describe fermions in the presence of GR, it will be convenient to view a metric g on M as a section of RðMÞ; a smooth map g∶M → RðMÞ that takes a point x to an equivalence class ½e x of frames at x. The configuration space [20] of general relativity can thus be seen as the space ΓðRðMÞÞ of sections of the bundle RðMÞ.

III. FERMIONIC FIELDS IN A FIXED BACKGROUND
We start by describing fermionic fields on M in the presence of a fixed background metric g. In order to do this, a number of choices have to be made, especially if we wish to keep track of the transformation behavior of spinors under parity and time reversal.
The local transformation behavior is fixed by choosing one out of the eight possible Pin groups Pin abc , together with a (not necessarily C-linear) representation V that extends the spinor representation of Spin ↑ ð3; 1Þ ⊂ Pin abc . For example, V consists of n copies of C 4 in the case of n Dirac fermions, and it consists of m copies of C 2 in the case of m Majorana fermions [21].
Once a Pin group has been selected, the second choice one has to make is a choice of Pin structure. A Pin structure is a twofold cover u∶ Q g → O g ðMÞ of the orthonormal frame bundle, equipped with a Pin abc -action that is compatible with the action of the Lorentz group on O g ðMÞ. The compatibility entails that ifΛ ∈ Pin abc covers Λ ∈ Oð3; 1Þ, then uðq xΛ Þ ¼ uðq x ÞΛ for all pin frames q x in Q g . A pin frame q x is based at the same point as its image, the frame uðq x Þ. We denote by Q g x the set of pin frames based at x. For a given manifold M and a given Pin group Pin abc , a Pin structure may or may not exist, and if it does, it need not be unique. The obstruction theory for this problem has been completely solved for the Cliffordian Pin groups in [22], and for the general case in [1].
Once a Pin structure Q g has been chosen, one can construct the associated bundle S g ¼ ðQ g × VÞ=Pin abc of spinors. A spinor ψ x ¼ ½q x ; ⃗ v at x is thus an equivalence class of a pin frame q x ∈ Q g x and a vector ⃗ v ∈ V, where ðq xΛ ; ⃗ vÞ is identified with ðq x ;Λ ⃗ vÞ for every elementΛ of the Pin group Pin abc . For a given background metric g, the fermionic fields are then described by sections of the spinor bundle S g , that is, by smooth maps ψ∶M → S g that assign to each space-time point x a spinor ψ x based at x. The configuration space for the fermionic fields at a fixed metric g is thus the space ΓðS g Þ of sections of the spinor bundle S g .

IV. FERMIONIC FIELDS COUPLED TO GR
We now wish to describe the configuration space for fermionic fields coupled to gravity. This is not simply the product of the configuration space of general relativity and that of a fermionic field; the main difficulty here is that the very space S g where the spinor field ψ takes values depends on the metric g. A solution to this problem was proposed in [3,4] for the Riemannian case, and in [6][7][8][9] for metrics of Lorentzian signature. In order to handle reflections, we need to adapt this procedure as follows.
First, we choose a twofold cover of Glð4; RÞ that agrees with the universal coverGl þ ð4; RÞ over Gl þ ð4; RÞ. In Sec. V we show that there are only two such covers, which, for want of a better name, we will call Gin þ and Gin − . Having made our choice of Gin AE , we choose what one may call a Gin structure; a twofold cover u∶Q → FðMÞ with a Gin AE -action that is compatible with the Glð4; RÞ-action on FðMÞ. Corresponding to every (not necessarily orthogonal) frame e x , there are thus two gin framesq x andq 0 x . If A ∈ Gin AE covers A ∈ Glð4; RÞ, then the two gin frames corresponding to e x A areq xÃ andq 0 xÃ . We denote by Pin AE the twofold cover of Oð3; 1Þ inside Gin AE . Choosing a Gin structureQ for the group Gin AE is equivalent to choosing a Pin structure Q g for the group Pin AE . Indeed, for every Gin AE structureQ, the preimage Q g ⊂Q of O g ðMÞ ⊂ FðMÞ under the map u∶Q → FðMÞ is a Pin AEstructure, since the restriction u g ∶ Q g → O g ðMÞ of u to Q g intertwines the Pin AE -action on Q g with the action of the Lorentz group Oð3; 1Þ on O g ðMÞ. Conversely, every Pin AEstructure u∶ Q g → O g ðMÞ gives rise to the associated Gin AE -structureQ ¼ ðQ g × Gin AE Þ=Pin AE . This is the space of equivalence classes ½q x ;Ã, where ðq xΛ ;ÃÞ is identified with ðq x ;ΛÃÞ for everyΛ in Pin AE . The obstruction theory for Gin AE -structures therefore reduces to the obstruction theory for Pin AE -structures, which has been worked out in [1].
Using the Gin structureQ, one constructs the universal spinor bundle Σ ¼ ðQ × VÞ=Pin AE in analogy with [9]. A universal spinor Ψ x ¼ ½q x ; ⃗ v at x is an equivalence class of a gin frameq x ∈Q x and a vector ⃗ v ∈ V, where ðq xΛ ; ⃗ vÞ is identified with ðq x ;Λ ⃗ vÞ for everyΛ in Pin AE . Note that a universal spinor Ψ x in Σ ¼ ðQ × VÞ=Pin AE defines a metric g μν at x, together with a spinor ψ x in the spinor bundle S g ¼ ðQ g × VÞ=Pin AE that corresponds with the metric g μν induced by Ψ x .
Indeed, since the covering map u∶Q → FðMÞ intertwines the Pin AE -action onQ with the Glð4; RÞ-action on FðMÞ, it identifies the quotient ofQ by Pin AE with the quotient of FðMÞ by Oð3; 1Þ, which is the orbit space RðMÞ. From a universal spinor Ψ x ¼ ½q x ; ⃗ v at x, we thus obtain an equivalence class ½uðq x Þ in R x ðMÞ, and hence a metric g μν at the point x.
To obtain not only the metric g μν but also the spinor ψ x , recall that the Pin structure Q g corresponding to g μν is the preimage of O g ðMÞ under the double cover u∶Q → FðMÞ. Since Q g ⊂Q contains the gin frameq x , the equivalence class Ψ x ¼ ½q x ; ⃗ v in Σ ¼ ðQ × VÞ=Pin AE yields an equivalence class ψ x ¼ ½q x ; ⃗ v in the spinor bundle S g ¼ ðQ g × VÞ=Pin AE by setting q x ¼q x . Here, S g is the spinor bundle derived from the metric g μν that is induced by Ψ.
We conclude that both the metric g and the fermionic field ψ are described by a single section Ψ∶M → Σ, a smooth map assigning to each point x of space-time a universal spinor Ψ x based at x. The configuration space of fermionic fields coupled to gravity is thus the space ΓðΣÞ of sections of the universal spinor bundle Σ.

V. COVERING GROUPS
Out of the eight Pin groups covering Oð3; 1Þ, the only two that are compatible with this formalism are the twofold cover Pin þ of Oð3; 1Þ inside Gin þ , and the twofold cover Pin − of Oð3; 1Þ inside Gin − . We show that their coefficients in the sense of Sec. I are ða; b; cÞ ¼ ðþ; þ; −Þ and ða; b; cÞ ¼ ð−; −; −Þ.
First we show that there are only two double covers of Glð4; RÞ that reduce to the universal cover over Gl þ ð4; RÞ. Assume that G is such a cover. If Λ T is an element of G that covers the time reversal operator T ∈ Glð4; RÞ, then the automorphism Ad Λ T ðÃÞ ≔ Λ TÃ Λ −1 T ofGl þ ð4; RÞ covers the automorphism Ad T ðAÞ ≔ TAT −1 of Gl þ ð4; RÞ. By the universal covering property, Λ TÃ Λ −1 T is uniquely determined byÃ, and it depends neither on the choice of G, nor on the choice of Λ T inside G. Since every element of G can be written as eitherÃ orBΛ T , there are four types of products, namely those of the formÃÃ 0 ,ÃðBΛ T Þ, ðBΛ T ÞÃ and ðBΛ T ÞðB 0 Λ T Þ, whereÃ;Ã 0 ;B;B 0 are inGl þ ð4; RÞ. Products of the first 2 types are determined by the group structure onGl þ ð4; RÞ. This is true for the third type as well, since ðBΛ T ÞÃ ¼BðΛ TÃ Λ −1 T ÞΛ T , and T , the only choice in the product structure on G lies in the sign of Λ 2 T ¼ AE1, yielding the two groups Gin AE . The twofold cover Pin þ of Oð3; 1Þ inside Gin þ thus has b ¼ þ1, whereas the twofold cover Pin − inside Gin − has b ¼ −1.
In particular, we conclude that the two Pin groups Pin AE compatible with GR are not the widely used Cliffordian Pin groups Pinð3; 1Þ and Pinð1; 3Þ.

VI. TRANSFORMATION UNDER DIFFEOMORPHISMS
In the above derivation of the two admissible Pin groups, a crucial role is played by the continuous covering map u∶Q → FðMÞ. This map has physical significance, since it induces an infinitesimal action of the space-time diffeomorphism group DiffðMÞ on the configuration space of fermions coupled to gravity (cf. [3,8]). This allows one to formulate a theory which is (up to sign) covariant under general coordinate transformations (cf. [3,11]), and to construct a stress-energy-momentum tensor via Noether's theorem (cf. [23,24], and cf. [ [6], Sec. 6] for an approach using variation of the metric).
To construct the infinitesimal action, note that DiffðMÞ acts by automorphisms on the frame bundle FðMÞ, a diffeomorphism ϕ maps e x ∈ F x ðMÞ to Dϕðe x Þ ≔ ∂μϕ μ e¯μ a in F ϕðxÞ ðMÞ. A one-parameter group ϕ ε of diffeomorphisms thus yields a one-parameter group Dϕ ε of automorphisms of FðMÞ. Since u∶Q → FðMÞ is a double cover, this lifts to a unique one-parameter group Dφ ε of automorphisms ofQ. On the universal spinor bundle Σ ¼ ðQ × VÞ=Pin AE , we define the lift by Dφ ε ½q x ; ⃗ v ¼ ½Dφ ε ðq x Þ; ⃗ v. For the infinitesimal variation of the universal spinor field Ψ∶ M → Σ along ϕ ε , this yields δΨ x ¼ d dε j 0 Dφ ε ðΨ ϕ −1 ε ðxÞ Þ.

VII. THE ROLE OF DIFFEOMORPHISMS IN RESTRICTING THE PIN GROUPS
We stress that the above restrictions on the Pin groups are not needed to construct the configuration space for fermions coupled to gravity, but to ensure that it transforms appropriately under space-time diffeomorphisms.
Indeed, to construct the configuration space, one could simply choose any principal Pin abc -bundle P → RðMÞ (for example the trivial one), and construct the universal spinor bundle Σ ¼ ðP × VÞ=Pin abc as in Sec. IV. Its sections Ψ ∈ ΓðΣÞ can be interpreted as a fermionic field ψ together with a metric g, so ΓðΣÞ may serve as a configuration space. This requires no restrictions on the Pin groups, nor on the topology of M.
However, this simple construction leaves the space-time transformation behavior undetermined. We show that the restrictions on the Pin groups are recovered by imposing appropriate transformation behavior on ΓðΣÞ. Compatibility with the Lorentz group leads to the familiar restrictions on the topology of M, compatibility with infinitesimal diffeomorphisms leads to Pin groups with c ¼ −1, and compatibility with a double cover of the diffeomorphism group requires Pin groups with a ¼ b as well as c ¼ −1.

A. Lorentz transformations
The pullback of a bundle E → Y along a map f∶ X → Y is the bundle f Ã E → X with ðf Ã EÞ x ≔ E fðxÞ . Starting from the principal bundle P → RðMÞ, one thus obtains for every metric g∶ M → RðMÞ a principal Pin abc -bundle g Ã P → M. Its fibre g Ã P x at x is the fibre P gðxÞ of P at gðxÞ ∈ RðMÞ. The bundle g Ã P is not quite a Pin structure, since the action of Pin abc on g Ã P is as yet unrelated to the action of Oð3; 1Þ on O g ðMÞ. To define the transformation behavior of Ψ under infinitesimal isometries, we need to choose a Pin structure on each of the bundles g Ã P. That is, for any possible metric g ∈ ΓðRðMÞÞ, we need to choose a double cover u g ∶ g Ã P → O g ðMÞ that intertwines the action of Pin abc on g Ã P with the action of Oð3; 1Þ on O g ðMÞ. This is where the restrictions on the topology of M arise: if the conditions in [1] are met, then it is possible to endow every single bundle g Ã P → M with a double covering map u g ∶ g Ã P → O g ðMÞ, making it into a Pin structure.

B. Infinitesimal diffeomorphisms
The problem is that, in general, these covering maps u g do not depend continuously on the metric g. If we require this to be the case, then we recover the infinitesimal action of the diffeomorphism group on the configuration space, as well as the restriction c ¼ −1 on the Pin groups. This already excludes the "Cliffordian" Pin groups Pinð3; 1Þ and Pinð1; 3Þ.
It consists of all pairs ðp; gÞ ∈ P × ΓðRðMÞÞ where p lies in g Ã P. The maps u g for the different metrics g ∈ ΓðRðMÞÞ then combine to a single map u∶ ev Ã P → FðMÞ, defined by uðp; gÞ ≔ u g ðpÞ. We say that u g depends continuously on g if the map u∶ ev Ã P → FðMÞ is continuous.
If u g depends continuously on g, then we obtain an infinitesimal action of DiffðMÞ on the configuration space ΓðΣÞ of fermions coupled to gravity. Since the (left) action of DiffðMÞ on FðMÞ commutes with the (right) action of Glð4; RÞ, we have an action of DiffðMÞ on RðMÞ, yielding the usual space-time transformation behavior g x ↦ Dϕg ϕ −1 ðxÞ on the space ΓðRðMÞÞ of metrics. To obtain the transformation behavior of spinors coupled to gravity, note that since u∶ ev Ã P → FðMÞ is continuous, it induces a double cover from ev Ã P to ev Ã FðMÞ, the space of all pairs ðe x ; gÞ ∈ FðMÞ × ΓðRðMÞÞ with e x ∈ O g ðMÞ. Since DiffðMÞ acts on ev Ã FðMÞ, it has an infinitesimal action on the double cover ev Ã P. This yields an infinitesimal action on ev Ã Σ → M × ΓðRðMÞÞ, the space of all pairs ð½q x ; v; gÞ ∈ Σ × ΓðRðMÞÞ whereq x is in g Ã P. This yields an infinitesimal action on ΓðΣÞ, since a section Ψ ∈ ΓðΣÞ can be viewed as a map from M to ev Ã Σ, sending x ∈ M to the pair ðΨ x ; gÞ, where g is the metric obtained from the section Ψ.

C. Double cover of the diffeomorphism group
In the above line of reasoning, the group structure on P þ 0 stems from its identification with the universal cover of the connected Lie group Glð4; RÞ þ . Since we lack a group structure on the disconnected space P 0 , we cannot directly infer that a ¼ b. This does, however, follow from the slightly stronger assumption that the DiffðMÞ-action on ev Ã FðMÞ lifts to an action by automorphisms of a double cover d DiffðMÞ on ev Ã P. This yields an action of d DiffðMÞ on ev Ã Σ, and by identifying Ψ ∈ ΓðΣÞ with a map from M to ev Ã Σ as before, one obtains an action of d DiffðMÞ on ΓðΣÞ. Explicitly, ϕ ∈ DiffðMÞ acts on ev Ã FðMÞ by taking ðe x ; gÞ to ðDϕðe x Þ; Dϕ ∘ g ∘ ϕ −1 Þ. If this lifts to an automorphism Dφ of ev Ã Σ, then Dφ maps Ψ ∈ ΓðΣÞ to the unique Ψ 0 ∈ ΓðΣÞ with ðΨ 0 To see that this yields the restriction a ¼ b, consider the case M ¼ R 4 . Then Glð4; RÞ is a subgroup of DiffðR 4 Þ, and its preimage in d DiffðR 4 Þ is one of the two Gin groups Gin AE . The (left) action of Gin AE by automorphisms on ev Ã P covers the (left) action of Glð4; RÞ by automorphisms on ev Ã FðMÞ, so in particular, the (left) action of Gin AE on σ Ã ev Ã P ¼ P 0 covers the (left) action of Glð4; RÞ on σ Ã ev Ã FðR 4 Þ ¼ F 0 ðR 4 Þ. This intertwines the (right) action of Pin abc on P 0 with the (right) action of Oð3; 1Þ on F 0 ðR 4 Þ. Since all these actions are free, we can identify Pin abc with a subgroup of Gin AE that covers the Lorentz group Oð3; 1Þ. Following the line of reasoning in Sec. V, we thus find a ¼ b as well as c ¼ −1.
We conclude that although an infinitesimal action of the space-time diffeomorphism group on the configuration space of fermions coupled to gravity requires c ¼ −1, an action of a double cover of the diffeomorphism group can only be achieved if the Pin group additionally satisfies the relation a ¼ b.

VIII. DISCUSSION
The conclusion that only two of the eight Pin groups are compatible with general relativity, appears to be quite robust. It is based on the elementary observation that the twofold spin cover of the orthonormal frame bundle O g ðMÞ is compatible with a twofold cover of the full frame bundle FðMÞ. Although we derived this from the setting outlined in Sec. IV (going back to [3][4][5] in the Riemannian and [6][7][8][9] in the Lorentzian case), the use of double covers of the full frame bundle-and hence our conclusion that only two Pin groups are admissible-is common to many other approaches, such as the more "global" formalism developed in [2,[10][11][12]. In fact, the restrictions on the Pin groups are closely linked to the transformation behavior of fermions coupled to gravity under space-time diffeomorphisms.
Since any principal bundle with an infinitesimal action of the space-time diffeomorphism group is associated to a discrete cover of a (higher order) frame bundle [7,8], we expect that our restrictions on the Pin group are not an artefact of the particular description that we have adopted.

ACKNOWLEDGMENTS
I would like to thank Edward Witten for several valuable comments. This research is supported by the NWO Grant No. 639.032.734 "Cohomology and representation theory of infinite dimensional Lie groups".