The purpose of this note is to introduce the reader to the notion of Lipschitz structure and its potential applications to the spin geometry required to globally formulate string theory and supergravity on a differentiable manifold.
Spinors are a crucial ingredient in the global formulation of string/M-Theory and its low energy limit supergravity, since these theories incorporate space-time supersymmetry and therefore their complete matter content necessarily contains spinors. Consequently, a complete understanding of spin geometry on the different spaces in which String/M-Theory can be consistently defined is of paramount importance in order to understand how this theory, or rather framework, must be globally formulated. Despite this fundamental and important motivation, the global aspects of spin geometry relevant for string theory are only poorly understood. In fact, the following basic question remains to be properly answered:
What is the appropriate notion of spinor in String Theory?
The main difficulty when trying to answer this question lies in the fact that many constructions in string theory and supergravity are only known at a local level. However, properly understanding the right mathematical description of the fermionic matter-content of string theory is a problem which requires knowing the global behavior and properties of such fields. Local information is simply not enough.
Spinors in string theory and supergravity are tacitly assumed to be given by functions defined on a contractible open set of the space-time manifold and taking values in some representation of the Clifford algebra corresponding to the given dimension and signature. This Clifford-representation is acted upon by local gamma matrices. Although this set up may be convenient to perform local computations, it does not clarify or inform on what is the global structure of these spinors. In addition, although the local structure of Supergravity theories many times requires the corresponding spinors to be real, the physics literature dealing with these situations usually considers spinors as local functions taking values on a complex representation which is then realified, in a non-canonical process that obscures the global properties of the spinors.
It is well-known that string theory can be consistently defined in spaces more general than smooth manifolds, like for example orbifolds 1. However, in order to keep the problem under control, we are going to consider the global formulation of string theory or supergravity on a smooth manifold. We will consider then the problem of understanding the appropriate notion of spinor required to globally formulate string theory or supergravity on a manifold, which will represent the space-time.
Understanding spinors in string theory or supergravity at a global level means fully characterizing them as section of a specific vector bundle over the space-time manifold. The standard approach in the string theory and supergravity literature regarding the global aspects of spin geometry usually consists in assuming the existence of a Spin or a Spinc structure (or in some occasions and in the context of M-theory, a Pin structure 2) , which consists of a principal Spin/Spinc bundle over the space-time manifold satisfying some natural compatibility conditions with respect to the bundle of orientable frames of the space-time. Assuming the existence of Spin or Spinc structure, spinors can be globally understood as sections of a vector bundle associated to the Spin/Spinc structure through the appropriate spinorial representation of the spin group. However, a global spin structure does not have in principle intrinsic physical meaning: what is physically important is not the principal Spin/Spinc bundle itself but the associated vector bundle of spinors S, which is usually called the spinor bundle. The spinor bundle S can be shown to be equipped with Clifford multiplication, a morphism of unital associative algebras from the Clifford bundle to the bundle of endomorphisms of S. Clifford multiplication is the key structure in order to formulate globally physical theories containing spinor (It is also necessary to use a bilinear pairing on S), such as string theory or supergravity. Indeed, it can be seen that Clifford multiplication is the minimal ingredient needed to globally formulate the fermionic sector of any physical theory containing fermions. Therefore, physics reasoning suggests to take Clifford multiplication and not Spin/Spinc structures, or any other fixed spinorial structure for that matter, as the fundamental structure required to describe fermions. In other words, physics reasoning suggests to define spinors not as sections of a vector bundle associated to some Spin/Spinc bundle, but as sections of an abstract vector bundle equipped with Clifford multiplication. We will call such bundle a pinor bundle for reasons that will become apparent later. Notice that a pinor bundle carries a representation of the bundle of Clifford algebras of the underlying manifold, which may be real or complex and not necessarily irreducible. We will intentionally omit specific details about this point since it is beyond the scope of this note. The claim is then the following:
The global formulation of string theory or supergravity on a differentiable pseudo-Riemannian manifold (M,g) is compatible with the spinors or the theory corresponding to sections of a pinor bundle taking values in the appropriate vector bundle.
The reader should notice that this may not be the most general characterization of spinors needed for formulating string theory or supergravity theories on a manifold, but it already captures the most important non-trivial global effects in string theory and supergravity which are missed by the more restricted definition of spinor as a section of a vector bundle associated to a Spin/Spinc structure.
The definition of pinor bundle introduced above poses various natural questions:
- Is every pinor bundle associated to a Spin/Spinc structure? If not, to which principal bundle is associated, if any? We do not expect pinor bundles to be always associated to Spin/Spinc structures, so it is clearly interesting to classify to which sort of spinorial structure they are associated.
- What is the topological obstruction to the existence of a pinor bundle? The existence of a Spin/Spinc structure is obstructed, namely not every manifold admits a Spin/Spinc structure. For the case of a Spin structure, the obstruction is given by the first two Stiefel-Whtiney classes of the tangent bundle. The topological obstruction for the existence of a Spinc structure is slightly different and more general, in the sense that every Spin manifold admits a Spinc structure but the opposite need not to hold. To see this consider for example the complex projective space of complex dimension two. This is a manifold that does not admit a Spin structure but it does admit a Spinc structure, since it is complex. Therefore, we should expect the existence of pinor bundles to be obstructed.
References 34 solved these two questions for the case of some particular types of complex and real pinor bundles: reference  considers bundles of complex faithful Clifford modules over a pseudo-Riemannian manifold whereas reference  considers bundles of real, irreducible, Clifford modules over a pseudo-Riemannian manifold. In other words, in reference  a pinor bundle is a bundle of irreducible real Clifford modules over the bundle of Clifford algebras over the base manifold, which is assumed to be an arbitrary pseudo-Riemannian manifold (M,g).
The results of references [3,4] show that pinor bundles are not necessarily associated to Spin/Spinc structures but to more general spinorial structures, which are generically called Lipschitz structures, following the seminal work of A. Trautman and T. Friedrich . In other words, a pinor bundle of the appropriate type exists over (M,g) if and only if (M,g) admits a Lipschitz structure, which is a principal bundle over M with fiber given by the Lipschitz group and satisfying some further natural conditions that relate it to the bundle of orthonormal frames of (M,g). The particular form of the Lipschitz group depends on the signature and dimension of M, but in general differs from the Spin/Spinc group. Summarizing [3,4]:
A pinor bundle exists on (M,g) if and only if (M,g) admits a Lipschitz structure.
Therefore, the topological obstruction for the existence of pinor a bundle over (M,g) is equivalent to the topological obstruction of a Lipschitz structure. Obtaining the latter is more convenient and that is the way references [3,4] proceeded. References [3,4 ] found that the topological obstruction to the existence of a Lipschitz structure is remarkably more general than the topological obstruction to the existence of a spin structure. This is related to the fact that string theory and supergravity can be defined in manifolds much more general than spin manifolds. In addition, references [3,4] classified the different Lipschitz structures associated to the pinor bundles they consider, giving thus a principal bundle description of spinors, of relevance to further understand spinors in this general sense.
Lipschitz structures generically require the existence of an auxiliary bundle, in the same sense that a Spinc structure requires the existence of a particular U(1) bundle. The auxiliary bundle is obtained from a Lipschitz structure by using the characteristic representation of the Lipschitz group, and therefore it was named in reference  as the characteristic bundle of the corresponding Lipschitz structure. Closer inspection of the characteristic bundles associated to real Lipschitz structures shows an intriguing correspondence between characteristic bundles and the bundles to which supergravity spinors couple according to the local structure of the theory. This provides further evidence that the definition we are using of spinor is indeed the appropriate one for string theory and supergravity, and in particular shows that understanding bundles of irreducible real pinor bundles informs about global aspects of String Theory and Supergravity that would not be accessible otherwise 5.
For the interested reader, we mention now one of the main results proved in reference  regarding the relation between Lipschitz structures and irreducible real pinor bundles:
There exists an equivalence of categories between the grupoid of Lipschitz structures and the groupoid of bundles of irreducible real pinor bundles defined over the same pseudo-Riemannian manifold.
In references [4,5], the authors applied the formalism of pinor bundles and Lipschitz structures to the global spin geometry M-theory, obtaining the following result:
The supersymmetry generator of eleven-dimensional Supergravity in signature (10,1) can be interpreted as a global section of a bundle of irreducible Clifford modules if and only if the space-time manifold is orientable and spin.
It has been known for some time that M-theory is actually consistent on non-orientable manifolds, see for example references , 6. Therefore, the result above shows that this is not always the case and puts constrains on the type of scenarios on which M-theory can be formulated on non-orientable pin manifolds.
References [4,5] are part of a long-term research project aimed at understanding the spin geometry of string theory and supergravity as well as aimed at exploring spin geometry on general pseudo-Riemanninan manifold and its possible applications to the topology and geometry of these spaces.
- Katrin Becker,Melanie Becker and John H. Schwarz, “String Theory and M-Theory: a Modern Introduction”. January 2007. Cambridge University, Cambridge University Press, 2007. ↩
- Edward Witten, “The “Parity” Anomaly On An Unorientable Manifold”, arXiv:1605.02391, 2016. ↩
- Thomas Friedrich and Andrzej Trautman, “Spin spaces, Lipschitz groups, and spinor bundles”, Annals of Global Analysis and Geometry (2000) 18. ↩
- Calin Iuliu Lazaroiu and C. S. Shahbazi, “Real pinor bundles and real Lipschitz structures”, arXiv:1606.07894, 2016. ↩
- Calin Iuliu Lazaroiu and C. S. Shahbazi, “On the spin geometry of supergravity and string theory”, arXiv:1607.02103, 2016. ↩
- Edward Witten, “On flux quantization in M theory and the effective action”, J.Geom.Phys. 22 (1997) 1-13. ↩