# A note on the extended superconformal algebras associated with manifolds of exceptional holonomy

###### Abstract

It was observed some time ago by Shatashvili and Vafa that superstring compactification on manifolds of exceptional holonomy gives rise to superconformal field theories with extended chiral algebras. In their paper, free field realisations are given of these extended superconformal algebras inspired by Joyce’s constructions of such manifolds as desingularised toroidal orbifolds. The purpose of this note is to give another realisation of these algebras starting not from free fields, but from the superconformal algebras associated to Calabi–Yau manifolds. These superconformal algebras, originally studied by Odake, are extensions of the Virasoro algebra. For the case of holonomy, our realisation is inspired in the conjectured construction of such manifolds as a desingularisation of , where is a Calabi–Yau 3-fold admitting an antiholomorphic involution. Similarly, for the case of holonomy our realisation suggests a construction of such manifolds as desingularisations of , where is a Calabi-Yau 4-fold admitting an antiholomorphic involution.

^{†}

^{†}thanks: Supported by the EPSRC under contract GR/K57824.

## 1 Introduction

In the context of M-theory [2] and F-theory [13], compactification to four dimensions requires that we do so on manifolds of seven and eight dimensions, respectively. If we require supersymmetry in four-dimensions we are forced to compactify on manifolds admitting parallel spinors. This in turn constraints the holonomy group of the manifold to be contained in the isotropy group of the spinor. For an irreducible riemannian -manifold which is not locally symmetric, the possible holonomy groups are those in Berger’s list. Of the groups in that list, only , , (for ) and (for ) admit parallel spinors (see, for example, [15]). For a 7-dimensional simply-connected manifold to admit the minimum (nonzero) number of parallel spinors, yielding the minimum number of four-dimensional supersymmetries, its holonomy group must be . The same conditions on an 8-dimensional manifold singles out those with holonomy. In both of these cases there is only one parallel spinor.

For traditional superstring compactification from ten to four dimensions, the desired holonomy is . Thanks to Yau’s solution of the Calabi conjecture, any Kähler -fold with vanishing canonical class admits a unique metric of holonomy in the same Kähler class. This provides us with many examples of such manifolds. On the other hand, relatively few examples are known of compact manifolds of exceptional holonomy. It was not until two years ago that the first such manifolds were constructed by Joyce [6, 7] by desingularising toroidal orbifolds.

Such manifolds are not just interesting in their own right, but their study is relevant for superstring phenomenology. Just as in the case of Calabi–Yau 3-folds before them, it is hoped that much can be learned by exploring the superconformal field theories they give rise to. A particularly fruitful approach is to study the orbifold limit, since although the geometry becomes singular, the conformal field theory does not. Moreover, it seems that the orbifold limit captures some of the information on the desingularisation process [10]; for example, different desingularisations seem to be correspond to the different conformal field theories associated with the same orbifold via the process of turning on discrete torsion [14]. This prompted the generalised mirror conjecture in [10], for which more evidence would be welcome.

The superconformal algebras arising from compactification on Calabi–Yau -folds are well known [9]. They correspond to extensions of the Virasoro algebra by a complex field of dimension . For it is an extension of the Virasoro algebra by a complex fermion and a complex boson, whereas for it becomes the (small) Virasoro algebra. This is expected since and all 4-manifolds with holonomy are hyperkähler. These algebras exist for generic values of the Virasoro central charge. For and , the cases of interest in the present note, the Virasoro central charge is fixed to 9 and 12 respectively, corresponding to compactification manifolds of dimensions 6 and 8, respectively.

For compactifications on manifolds of exceptional holonomy, the
resulting superconformal algebras are extensions of the
Virasoro algebra by superfields of weights and for
, and for . These algebras were written down by
Shatashvili and Vafa in [10] for the first time in the present
context; although classical versions of these algebras had appeared in
[5]. The algebra had been discovered previously in a
different context [4].^{1}^{1}1The algebra was also
written down in [3], where the algebra appears for the
first time. The algebra had also appeared in [8]. I
thank Andreas Honecker for reminding me of this work. The
algebra exists only for , corresponding to
7-dimensional compactification manifolds, whereas the
algebra belongs to a one-parameter family of superconformal algebras
[4], of which the point is the interesting one in the
present context.

The algebras in [10] were constructed in a free field realisation appropriate to the study of those manifolds which are presented as desingularised toroidal orbifolds; however other constructions may exist and it is desirable to understand them in the language of conformal field theory. The purpose of this note is to construct new realisations of this algebra which belie constructions of these manifolds starting from Calabi–Yau 3- and 4-folds.

This note is organised as follows. In Section 2 we describe the extended superconformal algebras of interest: the ones associated to manifolds of , , and holonomies. In [6] a construction of compact manifolds with holonomy is conjectured, which consists in desingularising an orbifold , where is a Calabi–Yau 3-fold admitting an antiholomorphic involution, and the generator of the acts as the involution on , and as inversion on the circle. This suggests that there should be a realisation of the superconformal algebra in terms of the superconformal algebra associated to the Calabi–Yau 3-fold , and to the circle. Roughly the geometric induces an automorphism of the superconformal algebra and inside the fixed subalgebra one finds a realisation of the algebra in [10]. This will shown in section 3. There we also show that if we take a Calabi–Yau 4-fold admitting an antiholomorphic involution , then the conformal field theory associated to the orbifold embeds a superconformal subalgebra isomorphic to the algebra. This suggests a construction of compact 8-manifolds with holonomy obtained by desingularising the orbifold. After completion of this work, we became aware of a preprint [1] which mentions the large volume limit of this last embedding.

Throughout we use the notation to denote the residue of the -th order pole in the operator product expansion of the fields and :

and assume familiarity with the axiomatics of these brackets as explained, for example, in [11].

## 2 The superconformal algebras

In this section we write down the superconformal algebras associated with compactifications on manifolds of holonomy and [9], and and [10]. Let be an irreducible manifold with holonomy group in the above list. Every parallel form on gives rise to a generator of the algebra, by pulling the form back to the worldsheet using the fermions. Parallel forms are precisely the singlets under the holonomy group in the representation where is the irreducible representation on tangent vectors. Hence group theory alone tells us all about the parallel forms (indeed tensors) on ; or alternatively we can use the parallel fermions to construct these forms as bispinors.

### 2.1 The superconformal algebra of a Calabi–Yau -fold

For holonomy, the tangent vectors form an irreducible -dimensional representation . Its complexification splits into , where is the fundamental complex representation of and is its complex conjugate. The parallel forms are in one-to-one correspondence with the singlets in . These include the Kähler form and its powers, but also the real 2-dimensional representation , corresponding to the real and imaginary parts of a complex -form . In the large volume limit, we can write down the following generators relative to a complex coordinate basis:

(1) | ||||

which will define an extension of the Virasoro algebra written for the first time in [9].

Let us now write down these algebras. We let and denote the generators of the Virasoro algebra with central charge . We let be a weight one superconformal primary normalised to . Together with its superpartner , they generate the Virasoro algebra. Now let and be superconformal primaries of weight . They are to be understood as the generators corresponding to the real and imaginary parts of the field in (1). They satisfy the following operator product expansion with :

We let and denote their superpartners. Because is actually antiholomorphic, , and are (anti)chiral superfields. This means that they obey the following operator product expansions with the second supersymmetry generator: and . The remaining operator product expansions are given in terms of the ones involving the primary fields and , so we give only these. The others can be reconstructed using the Jacobi-like identities of the brackets, or equivalently the associativity of the operator product expansion. It is here that we must distinguish between and .

#### 2.1.1

The following operator product expansions hold:

where the normal-ordered product is defined by . As it stands, the chiral algebra defined by these brackets is not associative. The Jacobi-like identities are only satisfied modulo the ideal generated by the weight fields

(2) |

These fields are null for this value of the central charge .

#### 2.1.2

Similarly in this case, the following operator product expansions hold:

where the normal-ordered product associates to the left; that is, . Again the above operator product expansions are associative only modulo the ideal generated by the fields given in (2), which now have weight 3 and are null for .

We should remark that it follows from the above brackets that the superconformal algebras associated with a Calabi–Yau -fold have an additional automorphism, corresponding to multiplying the complex fields and by the same phase.

### 2.2 The algebra associated with holonomy

In a 7-dimensional irreducible manifold of holonomy, the tangent vectors are in an irreducible representation of . Computing the singlets in , we find that there is a unique parallel 3-form , a unique parallel 4-form , and their product. In the large volume limit we can write the fields

which generate an extension of the Virasoro algebra.

Let us now write down this algebra. We let and be the generators of the Virasoro algebra with . Let be a superprimary field of weight with operator product expansion and , which defines . The field has weight 2, but it is is not a primary since . In addition, we have

and | ||||||||

It follows from these formulae that , and satisfy an Virasoro algebra with central charge corresponding to the tricritical Ising model. We now define and as the superpartners of and respectively. Because is a superconformal primary, is a Virasoro primary of weight . On the other hand is not a primary because neither is . Instead we have and in addition:

The rest of the relevant operator product expansions are: | ||||||||

and | ||||||||

We can obtain the remaining operator product expansions by using the associativity axiom. For example, to compute we use the fact that and that is an odd derivation over all the brackets:

whence from the above formulae we find:

which corrects a typo in equation (1.8) in the first appendix of [10]. (The notation is the same as in [10] except that here we call what they call , and aside from the above typo, we are in perfect agreement with their results.)

In [10], this algebra was obtained in a free field representation in terms of seven free bosons and seven free fermions. As a consequence, associativity of the operator product expansion is guaranteed. Abstractly, however, the Jacobi-like identities in the above algebra are only satisfied modulo the ideal generated by the weight null field defined by

(3) |

and vanishing identically in the free field realisation of [10].

### 2.3 The algebra associated with holonomy

We finally look at the case of holonomy. Let be an irreducible 8-dimensional manifold with holonomy. The tangent vectors are in the spinorial representation of , which in this context we call . The only singlets in are a self-dual 4-form and its square. In the large volume limit, gives rise to a weight field:

which generates an extension of the Virasoro algebra.

Again let and denote the generators of the Virasoro algebra with central charge . We now let be a weight field which we will choose not to be primary, and be its superpartner of weight but also not primary. The following operator product expansions define the algebra:

Again we can compute all other operator products from these by associativity. Notice that the field obeys a Virasoro algebra with central charge , corresponding to the Ising model. In counterpoint to the algebra, this one obeys associativity abstractly and not modulo an ideal. The reason is that the algebra admits a one-parameter (the central charge) deformation with the same fields. To see this we simply change basis to primary fields and . Then is a superconformal primary of weight , and by the results of [4] there exists a unique such extension of the Virasoro algebra, which exists for generic values of the central charge. In fact, if we further rescale to , and define as its superpartner, then the new algebra satisfied by agrees with the one in the appendix of the second reference in [4] for .

## 3 New realisations

In this section we come to the main results of this note. We will construct new realisations of the and superconformal algebras in sections 2.2 and 2.3 in terms of the algebras of sections 2.1.1 and 2.1.2, respectively.

Let be a Calabi–Yau -fold admitting an antiholomorphic involution . Then on the Kähler form and the (anti)holomorphic -forms and , we have and . At the level of the algebra associated to such a manifold, the involution is represented by an automorphism which fixes , , and , and changes the sign of , , and . It is easy to check that this is an automorphism of the corresponding superconformal algebras. As a result, the subspace of the chiral algebra fixed by this automorphism will be a subalgebra.

In the case we see that this subalgebra contains the superconformal algebra, generated by and together with

All the brackets in section 2.3 are obeyed on the nose, except for (and hence ) which are obeyed only modulo the ideal generated by the null fields in (2). Indeed, if we define , we get that

In the case we need an auxiliary boson-fermion pair normalised to and corresponding to the circle. This algebra has an automorphism given by , which fixes the generators of an Virasoro algebra with central charge :

Let be a Calabi–Yau 3-fold admitting an antiholomorphic involution , and let be the involution on acting by . On the superconformal algebra corresponding to , generated by , , , , , , , , and , the involution is represented by the automorphism which fixes , , and , and changes the signs of the other generators. If we define

then the other fields of the algebra follow:

and computing their brackets, we find those of the superconformal algebra in section 2.2, modulo the ideal generated by the fields in (2). Also, as expected, the null field in (3), belongs to the ideal generated by the fields in (2).

The above embeddings are not unique, of course, since we can still perform the automorphism mentioned at the end of section 2.1, namely and similarly with .

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