Projective elliptic genera and elliptic pseudodifferential genera
Fei Han, Varghese Mathai

TL;DR
This paper introduces the first construction of projective elliptic genera for compact manifolds with projective vector bundles, establishing their modularity and linking them to elliptic pseudodifferential operators.
Contribution
It develops the concept of projective elliptic genera with topological and analytic interpretations, extending the theory to include elliptic pseudodifferential genera without spin restrictions.
Findings
Constructed projective elliptic genera for manifolds with projective bundles.
Proved modularity properties of these genera.
Linked elliptic pseudodifferential operators to elliptic pseudodifferential genera.
Abstract
In this paper, we construct for the first time the projective elliptic genera for a compact oriented manifold equipped with a projective complex vector bundle. Such projective elliptic genera are rational q-series that have topological definition and also have analytic interpretation via the fractional index theorem in Mathai-Melrose-Singer (2006) without requiring spin condition. We prove the modularity properties of these projective elliptic genera. As an application, we construct elliptic pseudodifferential genera for any elliptic pseudodifferential operator. This suggests the existence of putative rotation-equivariant elliptic pseudodifferential operators on loop space whose equivariant indices are elliptic pseudodifferential genera.
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Projective elliptic genera and elliptic pseudodifferential genera
Fei Han
Department of Mathematics, National University of Singapore, Singapore 119076
and
Varghese Mathai
School of Mathematical Sciences, University of Adelaide, Adelaide 5005, Australia
Abstract.
In this paper, we construct for the first time the projective elliptic genera for a compact oriented manifold equipped with a projective complex vector bundle. Such projective elliptic genera are rational -series that have topological definition and also have analytic interpretation via the fractional index theorem in [25] without requiring spin condition. We prove the modularity properties of these projective elliptic genera. As an application, we construct elliptic pseudodifferential genera for any elliptic pseudodifferential operator. This suggests the existence of putative -equivariant elliptic pseudodifferential operators on loop space whose equivariant indices are elliptic pseudodifferential genera.
Key words and phrases:
Projective elliptic genera, projective elliptic pseudodifferential genera, graded twisted Chern character, modularity, Schur functors
2010 Mathematics Subject Classification:
Primary 58J26, 58J40, Secondary 55P35, 11F55
Contents
Introduction
In 1980’s, Witten studied two-dimensional quantum field theories and the index of Dirac operator in free loop spaces. In [31], Witten argued that the partition function of a type II superstring as a function depending on the modulus of the worldsheet elliptic curve, is an elliptic genus. In [30], Witten derived a series of twisted Dirac operators from the free loop space on a compact spin manifold . The elliptic genera constructed by Landweber-Stong [20] and Ochanine [27] in a topological way turn out to be the indices of these elliptic operators. Motivated by physics, Witten conjectured that these elliptic operators should be rigid. The Witten conjecture was first proved by Taubes [29] and Bott-Taubes [7]. In [21], using the modular invariance property, Liu presented a simple and unified proof of the Witten conjecture. A useful reference in this area is the book by Hirzebruch, Berger and Jung [16]. We also mention the recent generalisation of these genera by the authors in [15] to noncompact manifolds with noncompact almost connected Lie groups acting properly and cocompactly. Let us be more precise as follows.
Let be a dimensional compact smooth spin manifold and be a rank spin vector bundle over . As in [30], let
[TABLE]
where , be the Witten bundle, which is an element in . Construct the bundles
[TABLE]
which are also elements in , and
[TABLE]
which are elements in . Let be the -class of and the spinor bundles of . The bundle twisted elliptic genera are defined to be the integral -series as follows,
[TABLE]
[TABLE]
[TABLE]
[TABLE]
Without the presence of ,
[TABLE]
is the famous Witten genus. By the Atiyah-Singer index theorem, these bundle twisted elliptic genera have analytical interpretation as follows. Let be the spin Dirac operator on . Then
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
One can show that when , these bundle twisted elliptic genera are modular forms of weight over and respectively and when , is a modular form of weight over (see Appendix). Witten showed in [30, 32] that formally can be viewed as the Dirac operator on loop space; , can be viewed as vector bundles over loop space; and
[TABLE]
can be viewed as the Dirac operator on loop space coupled to these bundles. Taubes [29], Bott-Taubes [7] and Liu [21] proved the rigidity of these operators conjectured by Witten. In [22] Liu discovered a profound vanishing theorem for the Witten genus .
In this paper, we construct projective elliptic genera in the case that is a compact oriented manifold not necessarily spin and is a projective vector bundle rather than an ordinary vector bundle on . More precisely, for such and , we construct rational -series
[TABLE]
[TABLE]
which still have both topological definition and analytic interpretation. We also establish the modularity properties of these genera. The key new idea in the topological side is to introduce the graded twisted Chern character for Witten bundles constructed from projective vector bundles (see the definition in (1.15) to (1.18)). For the analytic interpretation, we use the projective spin Dirac operator introduced in [25, 26] and the fractional index theorem proved there. More precisely, let be the projective spin Dirac operator associated to a fixed projective spin structure on the oriented compact manifold . Let be a projective complex vector bundle. The twisted projective spin Dirac operator acts on , where is the projective vector bundle of spinors associated to the Azumaya bundle given by the complex Clifford algebra bundle . The index of has the usual expression in terms of characteristic classes, it is no longer an integer, but only a fraction in general.
In 1983, the physicists Alvarez-Gaumé and Witten [1] discovered the “miraculous cancellation” formula for gravitational anomaly, relating index of the signature operator to indices of twisted Dirac operators in dimension 12. Liu [23] generalised their formula to higher dimension and also allow general bundle twisting rather than the tangent bundle by developing modularities of certain characteristic forms . In this paper, we give a “projective miraculous cancellation” formulae for indices of projective Dirac operators twisted by projective vector bundles (Theorem 2.4) and the 12 dimensional local formula (Theorem 2.5) following Liu’s method.
The Witten genus can be viewed as a morphism from the String bordism ring to the ring of integral modular forms, It has a lift (called ) [2] in homotopy theory where is the deep and powerful theory of topological modular forms, constructed originally by Hopkins and Miller [17], with a new construction due to Lurie [24]. Our projective elliptic genus is a rational modular form over when the first rational Pontryagin classes of the projective bundle and are equal. It seems likely that there is a similar lift in homotopy theory for and a refinement of our projective genera to a version of elliptic cohomology, but we will not address this here.
Other important approaches for construction of Witten genus and elliptic genera include chiral de Rham complex [13],[14], [6], [10] and the application of factorization homology [11]. Our projective genera are twisted version of the usual genera in the presence of a -field. We plan to look at the construction of our projective genera in these approaches in the presence of a -field.
As an interesting application of the projective elliptic genera, we give a construction of the elliptic pseudodifferential genera for any elliptic pseudodifferential operator. More precisely, let be a -dimensional compact oriented manifold. Choose and fix a projective spinc structure on [25, 26]. Let be any elliptic pseudodifferential operator on . We are able to construct elliptic genera type invariants for : and . When is a spinc manifold and is the spinc Dirac operator, and degenerate to the Witten genus of , which is a rational -series on spinc manifold (see Example 3.4). This is similar to looking at the -genus on spinc manifolds in [25]. The key step in our construction is to implicitly use a projective vector bundle coming from by using the projective spinc structure. We also use the Schur functors (c.f. [12]) to understand the Witten bundles of tensor product. Actually we give our construction in a more general setting, namely for projective elliptic pseudodifferential operator which has its own twist.
The paper is organized as follows. In Section 1, we introduce the graded twisted Chern character on the Witten bundles constructed from a projective vector bundle and then construct the projective elliptic genera as well as study their modularities. In Section 2, we first review the index theory for projective elliptic operators in [25, 26] and then give the analytic interpretation of the projective elliptic genera. We also give the “projective miraculous cancellation” formula in this section. As an application, we construct projective elliptic pseudodifferential genera for projective elliptic pseudodifferential operator in Section 3.
Acknowledgements. Fei Han was partially supported by the grant AcRF R-146-000-218-112 from National University of Singapore. He is indebted to Prof. Weiping Zhang for helpful suggestions and discussions. Varghese Mathai was supported by funding from the Australian Research Council, through the Australian Laureate Fellowship FL170100020. He gave a talk based on this paper at the conference, Microlocal methods in Analysis and Geometry (In honor of Richard Melrose’s 70th birthday) CIRM, Luminy, May 6-10 2019, and would like to thank Isadore Singer and Richard Melrose for past collaboration related to this research.
1. Projective elliptic genera
In this section, we give the topological construction of projective elliptic genera and study their modular properties. We will give the the analytic interpretation of them in the next section by using the index theorem of projective elliptic operators in [25, 26].
1.1. Projective vector bundles
Let be a smooth manifold with Riemannian metric and the Levi-Civita connection . Let be a principal bundle over ,
[TABLE]
The Dixmier-Douady invariant of ,
[TABLE]
is the obstruction to lifting the principal -bundle to a principal -bundle (the construction also works for any principal bundle over , together with a central extension of ). Let be the algebra of complex matrices. The associated algebra bundle
[TABLE]
is called the associated Azumaya bundle.
A projective vector bundle on is not a global bundle on , but rather it is a vector bundle , where also satisfies
[TABLE]
where is the primitive line bundle,
[TABLE]
This gives a projective action of on , i.e. an action of on s.t. the center acts as scalars. One can define the twisted Chern character [8], for the projective bundle .
1.2. Projective elliptic genera
Suppose to be closed, oriented and -dimensional. Let be an Hermitian projective vector bundle of rank over , which is a Hermitian vector bundle over with the action in . Let be an Hermitian connection on compatible with the action. Let be a curving of . Let representing the Dixmier-Douady class of such that As the Dixmier-Douady class is torsion element, is exact on .
The condition (1.1) implies that descends to a degree differential form on for all .
Define the first rational projective Pontryagin class of E$$,\mathfrak{p}_{1}(E)\in H^{4}(Z,\mathbb{Q}), such that
[TABLE]
The tensor product satisfies
[TABLE]
Therefore we see that descends to a degree differential form on for all . We can define the twisted Chern character
[TABLE]
As the exterior bundle is a subbundle of , one can also define the twisted Chern character .
Recall that for an indeterminate (c.f. [3]),
[TABLE]
are the total exterior and symmetric powers of respectively.
Let [30]
[TABLE]
be the Witten bundle, which is an element in .
Let be the complex conjugate of , which carries the induced Hermitian metric and connection.
[TABLE]
which are elements in ;
[TABLE]
which are elements in .
In the -expansion of , the coefficient of is integral linear combination of terms of the form
[TABLE]
Pick out the the terms such that and denote their sum by . Then we have the expansion
[TABLE]
is a vector bundle over carrying induced Hermitian metric and connection for each . It is clear that for each fixed , there are only finite many such that is nonzero. satisfies
[TABLE]
Therefore one can define the twisted Chern character .
Similarly in the -expansion of , the coefficient of is integral linear combination of terms of of the form
[TABLE]
Pick out the the terms such that and denote their sum by . Then we have the expansion
[TABLE]
is a vector bundle over carrying induced Hermitian metric and connection for each . For each fixed , there are only finite many such that is nonzero. satisfies
[TABLE]
We can therefore define the twisted Chern character .
One can decompose and in a similar way as
[TABLE]
[TABLE]
and define the twisted Chern characters and as well.
Define the graded twisted Chern character
[TABLE]
[TABLE]
[TABLE]
[TABLE]
Let (compare with (0.1))
[TABLE]
Let be the determinant line bundle of , which carries the induced Hermitian metric and connection. As ther determinant is the highest exterior power, one can define the twisted Chern character .
Define the projective elliptic genera by
[TABLE]
[TABLE]
[TABLE]
[TABLE]
We have the following results.
Theorem 1.1**.**
*(i) If , then is a modular form of weight over .
(ii) If , then is a modular form of weight over , is a modular form of weight over and is a modular form of weight over ; moreover, we have*
[TABLE]
Proof.
Let be a Riemann metric on , be the Levi-Civita connection and be the curvature of . The -form can be expressed as
[TABLE]
By the Chern-Weil theory (c.f. [33]), the Chern-Weil expression of twisted Chern characters ([8]) and the definitions of Jacobi theta functions and the Jacobi identity (see Appendix), we have
[TABLE]
[TABLE]
[TABLE]
[TABLE]
It is known that the generators of are , the generators of are and the generators of are , . Applying the Chern root algorithm on the level of forms (over certain ring extension for each , cf. [19] for details) and the transformation laws of the theta functions, one can see that that when and are cohomologous, the anomalies arising from modular transformation vanish.
∎
2. Analytic definition of projective elliptic genera
In this section, we first briefly review the fractional index theorem of Mathai-Melrose-Singer [25, 26] and then use it to give an analytic construction of the projective elliptic genera.
2.1. Projective elliptic operators
For a compact manifold, and vector bundles and over , the Schwartz kernel theorem gives a 1-1 correspondence,
[TABLE]
[TABLE]
[TABLE]
where is the ‘big’ homomorphism bundle over and is the density bundle from the right factor.
When restricted to pseudodifferential operators, , get an isomorphism with the space of conormal distributions with respect to the diagonal, i.e.
[TABLE]
When further restricted to differential operators (which by definition have the property of being local operators) this becomes an isomorphism with the space of conormal distributions, with respect to the diagonal, supported within the diagonal, . i.e.
[TABLE]
The previous facts motivates our definition of projective differential and pseudodifferential operators when and are only projective vector bundles associated to a fixed finite-dimensional Azumaya bundle
Since a projective vector bundle is not global on , one cannot make sense of sections of , let alone operators acting between sections! However, it still makes sense to talk about Schwartz kernels even in this case, as we explain.
Notice that is a projective bundle on associated to the Azumaya bundle, .
The restriction to the diagonal is an ordinary vector bundle, it is therefore reasonable to expect that also restricts to an ordinary vector bundle in a tubular nbd of the diagonal.
In [25], it is shown that there is a canonical such choice, , such that the composition properties hold.
This allows us to define the space of projective pseudo- differential operators with Schwartz kernels supported in an -neighborhood of the diagonal in , with the space of conormal distributions,
[TABLE]
Despite not being a space of operators, this has precisely the
same local
structure as in the standard case and has similar composition properties
provided supports are restricted to appropriate neighbourhoods of the
diagonal.
The space of *projective * smoothing operators, is defined as the smooth sections,
The space of all projective differential operators, is defined as those conormal distributions that are supported within the diagonal in ,
[TABLE]
In fact, is even a ring when .
Recall that there is a projective bundle of spinors on any even dimensional oriented manifold .
There are natural spin connections on the Clifford algebra bundle and induced from the Levi-Civita connection on
Recall also that has an extension to in a tubular neighbourhood of the diagonal , with an induced connection .
The projective spin Dirac operator is defined as the distributional section
[TABLE]
Here is the connection restricted to the left variables with the contraction given by the Clifford action of on the left.
As in the usual case, the projective spin Dirac operator is elliptic and odd with respect to grading of .
The principal symbol map is well defined for conormal distributions, leading to the globally defined symbol map,
[TABLE]
homogeneous of degree ; here is a globally defined ordinary vector bundle with fibre Thus ellipticity is well defined, as the invertibility of this symbol.
Equivalently, is elliptic if there exists a parametrix and smoothing operators such that
[TABLE]
The trace functional is defined on projective smoothing operators as
[TABLE]
It vanishes on commutators, i.e. if
[TABLE]
which follows from Fubini’s theorem.
The fractional analytic index of the projective elliptic operator is defined in the essentially analytic way as,
[TABLE]
where is a parametrix for and the RHS is the notation for
For , the Guillemin-Wodzicki residue trace is,
[TABLE]
where is an entire family of DOs of complex order which is elliptic and such that The residue trace is independent of the choice of such a family.
- (1)
The residue trace vanishes on all DOs of sufficiently negative order. 2. (2)
The residue trace is also a trace functional, that is,
[TABLE]
for
The regularized trace, is defined to be the residue,
[TABLE]
For general , does depend on the regularizing family . But for smoothing operators it coincides with the standard operator trace,
[TABLE]
Therefore the fractional analytic index is also given by,
[TABLE]
for a projective elliptic operator , and a parametrix for .
The regularized trace is not a trace function, but however it satisfies the trace defect formula,
[TABLE]
where is a derivation acting on the full symbol algebra. It also satisfies the condition of being closed,
[TABLE]
Using the derivation and the trace defect formula, we prove:
- (1)
the homotopy invariance of the index,
[TABLE]
where is a smooth 1-parameter family of projective elliptic DOs; 2. (2)
the multiplicativity property of of the index,
[TABLE]
where for are projective elliptic DOs.
An analogue of the McKean-Singer formula holds,
[TABLE]
where is a globally defined, truncated heat kernel, both in space (in a neighbourhood of the diagonal) and in time. The local index theorem can then be applied, thanks to the McKean-Singer formula, to obtain the index theorem for projective spin Dirac operators.
Theorem 2.1** ([25]).**
The projective spin Dirac operator on an even-dimensional compact oriented manifold , has fractional analytic index,
[TABLE]
Recall that is an oriented but non-spin (actually spinc) manifold such that , e.g.
[TABLE]
2.2. Transversally elliptic operators and projective elliptic operators
Recall that projective half spinor bundles on can be realized as -equivariant honest vector bundles, over the total space of the oriented frame bundle and in which the center, acts as , as follows:
the conormal bundle to the fibres of has vanishing -obstruction, and are just the 1/2 spin bundles of .
One can define the -equivariant transversally elliptic Dirac operator using the Levi-Civita connection on together with the Clifford contraction, where transverse ellipticity means that the principal symbol is invertible when restricted to directions that are conormal to the fibres.
The nullspaces of are infinite dimensional unitary representations of . The transverse ellipticity implies that the characters of these representations are distributions on the group . In particular, the multiplicity of each irreducible unitary representation in these nullspaces is finite, and grows at most polynomially.
The -equivariant index of is defined to be the following distribution on ,
[TABLE]
An alternate, analytic description of the -equivariant index of is: for a function of compact support the action of the group induces a graded operator
[TABLE]
which is smoothing along the fibres. has a microlocal parametrix in the directions that are conormal to the fibres (ie along ). Then for any
[TABLE]
are smoothing operators. The -equivariant index of , evaluated at , is also given by:
[TABLE]
Theorem 2.2** ([26]).**
Let denote the projection. The pushforward map, , maps the Schwartz kernel of the -transversally elliptic Dirac operator to the projective Dirac operator: That is,
[TABLE]
We will next relate these two pictures. An easy argument shows that the support of the equivariant index distribution is contained within the center of .
Theorem 2.3** ([26]).**
Let be such that :
- (1)
* in a neighborhood of , the identity of ;* 2. (2)
. Then
[TABLE]
where is a projective vector bundle associated to on and is the lift of to .
Informally, the fractional analytic index, of the projective Dirac operator , is the coefficient of the delta function (distribution) at the identity in of the -equivariant index for the associated transversally elliptic Dirac operator on
2.3. Analytic definition of projective elliptic genera
In view of the Definition 1.20 to 1.23, Theorem 2.1 and Theorem 2.3, we obtain the following analytic expressions
[TABLE]
where is as in Theorem 2.3.
When is even and is even, the square root line bundle exists and satisfies
[TABLE]
Then we further have
[TABLE]
[TABLE]
where is as in Theorem 2.3.
2.4. Projective miraculous cancellation formula
We have the following “projective miraculous cancellation formula” for projective Dirac operators, generalizing the celebrated Alvarez-Gaumé-Witten “miraculous cancellation formula” [1] in dimension 12 for ordinary Dirac operators.
Let
[TABLE]
where each is a virtual projective bundle over .
The following projective miraculous cancellation can be similarly proved as Theorem 1 in [23] by using the modularities in Theorem 1.1 and the looking at the basis of rings of modular forms over and .
Theorem 2.4**.**
If , is even and is even, then the following equality holds,
[TABLE]
where the virtual projective bundles are canonical integral linear combination of .
When the dimension of is 12, the local formula of the projective miraculous cancellation formula for the top (degree 12) forms reads
Theorem 2.5**.**
[TABLE]
provided
3. Elliptic pseudodifferential genera
In this section, we give an application of the projective elliptic genera by constructing projective elliptic pseudodifferential genera for any projective elliptic pseudodifferential operator. For references on the basics of pseudodifferential operators, see [18, 28]. Our construction of elliptic pseudodifferential genera suggests the existence of putative -equivariant elliptic pseudodifferential operators on loop space that localises to the elliptic pseudodifferential genera, by a formal application of the Atiyah-Segal-Singer localisation theorem, [4, 5]. We also compute the elliptic pseudodifferential genera for some concrete elliptic pseudodifferential operators in this section.
Let be a -dimensional compact oriented smooth manifold. Let be the third integral Stiefel-Whitney class. Fix a projective spinc structure on and let be the complex projective spinc bundle of with twist . Denote by the bundle . Let be the stable complement of the tangent bundle . Let be the projective spinc bundle of with twist . Denote by the bundle . Let be projective Hermitian connections on . Denote by the -graded projective Hermitian connection on .
Let be a projective pseudodifferential elliptic operator with and being projective Hermitian vector bundles over with twist . Let and be projective Hermitian connections on respectively. Denote by the -graded projective Hermitian connection on the bundle . There exists and projective complex vector bundle on with twist such that and . Suppose the rank of is . Define the first rational Pontryagin class of by
[TABLE]
It is clear that the it is well defined.
Let be the Schur functor (c.f. Sec. 6.1 in [12]). They are indexed by Young diagram and are functors from the category of vector spaces to itself . It is not hard to see that the Schur functor is a continuous functor (c.f. [3]) and therefore if is a vector bundle with connection, then applying the Schur functor gives us a vector bundle with connection . If be two vector spaces, the exterior power of a tensor product has the following nice expression via the Schur functors :
[TABLE]
where is the Schur functor with running over all the Young diagram with cells, at most rows, columns, and being the transposed Young diagram. Hence on the projective bundle , there is a projective Hermitian connection
[TABLE]
Denote this connection by .
In the following, when we write , where is certain operations on vector bundles constructed from exterior power, it always means the connections constructed in this way. For instance,
[TABLE]
is a projective Hermitian connection on the -series with virtual projective bundle coefficients,
[TABLE]
Set
[TABLE]
[TABLE]
and for
[TABLE]
Definition 3.1**.**
For the projective elliptic pseudodifferential operator , define the projective elliptic pseudodifferential genera and by the following,
[TABLE]
[TABLE]
[TABLE]
and
[TABLE]
Remark 3.2**.**
*We will see from the next theorem that
(i) the genera for are well defined;
(ii) and *
Theorem 3.3**.**
*(i)
(ii) If , then is a modular form of weight over , is a modular form of weight over , is a modular form of weight over and is a modular form of weight over ; moreover, we have*
[TABLE]
Proof.
(i) By the multiplicativity of the operations and , it is not hard to see that the cohomology class
[TABLE]
is represented by the differential form . Then follows from and . One can similarly use the multiplicativity to prove that
Combining (i) and Theorem 1.1, (ii) is obtained. ∎
In the following, we give two examples of explicit computation of the elliptic pseudodifferential genera.
Example 3.4**.**
Let be a spinc manifold and , the spinc Dirac operator. Then the corresponding to is just the trivial complex line bundle . Hence by (i) in Theorem 3.3, we have
[TABLE]
[TABLE]
[TABLE]
and
[TABLE]
where is the Witten genus of . This is similar to looking at the -genus on spinc manifolds in [25].
Example 3.5**.**
On , take the standard spinc structure from the complex structure. let , where is the canonical complex line bundle and stands for the complex tangent bundle. Let be an elliptic pseudodifferential operator. Here the corresponding to happens to be the honest line bundle rather than a projective one. Let be the generator and By (i) in Theorem 3.3, we have
[TABLE]
due to is an odd function of ;
[TABLE]
[TABLE]
and
[TABLE]
They are all rational -series rather than integral -series.
Appendix. Some number theory preparation
A general reference for this appendix is [9].
Let
[TABLE]
as usual be the modular group. Let
[TABLE]
be the two generators of . Their actions on are given by
[TABLE]
Let
[TABLE]
[TABLE]
[TABLE]
be the three modular subgroups of . It is known that the generators of are , the generators of are and the generators of are , . (cf. [9]).
The four Jacobi theta-functions (c.f. [9]) defined by infinite multiplications are
[TABLE]
[TABLE]
[TABLE]
[TABLE]
where .
They are all holomorphic functions for , where is the complex plane and is the upper half plane.
Let . The Jacobi identity [9],
[TABLE]
holds.
The theta functions satisfy the the following transformation laws (cf. [9]),
[TABLE]
[TABLE]
[TABLE]
[TABLE]
Let be a subgroup of A modular form over is a holomorphic function on such that for any
[TABLE]
the following property holds
[TABLE]
where is a character of and is called the weight of .
The reference list from the paper itself. Each links out to its DOI / PubMed record.
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- 2[2] M. Ando, M.J. Hopkins and N.P.Strickland, Elliptic spectra, the Witten genus and the theorem of the cube. Invent. Math. , 146 (2001) 595-687.
- 3[3] M. F. Atiyah, K − t h e o r y 𝐾 𝑡 ℎ 𝑒 𝑜 𝑟 𝑦 K-theory , Benjamin, New York, 1967.
- 4[4] M. F. Atiyah, G. Segal The index of elliptic operators II, Ann. of Math . (2) 87 1968 531–545.
- 5[5] M. F. Atiyah, I.M. Singer, The index of elliptic operators III, Ann. Math . (2) 87 (1968) 546–604.
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