Deformations of conically singular Cayley submanifolds

In this article we study the deformation theory of conically singular Cayley submanifolds. In particular, we prove a result on the expected dimension of a moduli space of Cayley deformations of a conically singular Cayley submanifold. Moreover, when the Cayley submanifold is a two-dimensional complex submanifold of a Calabi--Yau four-fold we show by comparing Cayley and complex deformations that in this special case the moduli space is a smooth manifold. We also perform calculations of some of the quantities discussed for some examples.


Introduction
Cayley submanifolds are calibrated submanifolds that arise naturally in manifolds with exceptional holonomy Spin (7). Introduced by Harvey and Lawson [6], calibrated submanifolds are by construction volume minimising, and hence minimal submanifolds. Cayley submanifolds exist in abundance, with the simplest examples being any two-dimensional complex submanifold of a Calabi-Yau four-fold.
The deformation theory of compact calibrated submanifolds in manifolds with special holonomy was studied by McLean [21]. A major obstruction to generalising these results to noncompact manifolds is the failure of an elliptic operator on a noncompact manifold to be Fredholm. However, by introducing a weighted norm on spaces of sections of a given vector bundle on a particular type of noncompact manifold, it is possible to overcome this difficulty, as long as one is careful about the choice of weight. It is therefore possible to study certain moduli spaces of noncompact calibrated submanifolds.
In this article, the noncompact submanifolds that we study are conically singular. Motivated by the SYZ conjecture, an interesting problem is whether a Spin(7)-manifold can be fibred by Cayley submanifolds with some singular fibres. Conically singular Cayley submanifolds are natural candidates for these singular fibres. Deformations of conically singular special Lagrangian submanifolds in Calabi-Yau manifolds are coassociative submanifolds of G 2manifolds have been studied by Joyce [9] and Lotay [17] respectively. Deformations of compact Cayley submanifolds with boundary and asymptotically cylindrical Cayley submanifolds have been studied by Ohst [24,25].
We say that a manifold with a singular point is conically singular if a neighbourhood of the singular point is diffeomorphic to a cone C ∼ = L × (0, ǫ), and moreover the metric approaches the cone metric like r µ−1 as r → 0. We will prove a series of results on the moduli space of Cayley deformations of a Cayley submanifold, conically singular with cone C and rate µ, that also have a conical singularity at the same point with cone C and rate µ.
In Theorem 3.8, we prove that the expected dimension of this moduli space is given by the index of a first order linear elliptic operator acting on smooth normal vector fields that decay like r µ close to the singular point. Motivated by other work of the author [22], we pay special attention to Cayley deformations of a conically singular complex surface N inside a Calabi-Yau four-fold M . In Theorem 3.12, we will show that the elliptic operator in Theorem 3.8 is∂ We will then study the moduli space of complex deformations of N in M that are conically singular at the same point with the same rate and cone as N . We will show in Theorem 3.14 that this moduli space is a smooth manifold, and moreover that there are no infinitesimal Cayley deformations of N that are not infinitesimal complex deformations of N in Corollary 3.16.
In the later sections of this article we will focus on the elliptic operators whose indices we are interested in. In particular, we will characterise the exceptional weights for which these operators are not Fredholm. We will also apply the Atiyah-Patodi-Singer Index Theorem [3] to write down an expression for the index of these operators in terms of topological and spectral invariants of the manifold.
We will conclude this article by performing a series of calculations, including the dimension of the space of infinitesimal Cayley and complex deformations of three complex cones in C 4 , that are themselves cones, motivated by the work of Kawai [13] on deformations of associative submanifolds of the sevensphere.
Layout. In Section 2 we will discuss some background results which the reader may find useful on Cayley submanifolds, conically singular manifolds and Fredholm theory on noncompact manifolds. Section 3 contains the results on the deformation theory of conically singular Cayley and complex submanifolds. In Section 4 we characterise the set D of exceptional weights for which the operators we discuss in this article, before deducing a version of the Atiyah-Patodi-Singer theorem for these operators. In Section 5 we perform calculations of some of the quantities discussed in this article for some examples.
Notation. When M is a complex manifold, we denote by Λ p,q M the bundle of (p, q)-forms Λ p T * 1,0 M ⊗Λ q T * 0,1 M . If N is a submanifold of M , we denote the normal bundle of N in M by ν M (N ). Moreover, if N is a complex submanifold of M then we denote by ν 1,0 M (N ) and ν 0,1 M (N ) the holomorphic and anti-holomorphic normal bundles of N in M respectively. Submanifolds will be taken to be embedded unless otherwise stated.
Definition 2.1. Let (x 1 , . . . , x 8 ) be coordinates on R 8 with the Euclidean metric g 0 = dx 2 1 + · · · + dx 2 8 . Define a four-form on R 8 by Let X be an eight-dimensional oriented manifold. For each p ∈ X define the subset A p X ⊆ Λ 4 T * p X to be the set of four-forms Φ for which there exists an oriented isomorphism T p X → R 8 identifying Φ and Φ 0 given in (2.1), and define the vector bundle AX to be the vector bundle with fibre A p X.
A four-form Φ on X that satisfies Φ| p ∈ A p X for all p ∈ X defines a metric g on X. We call (Φ, g) a Spin(7)-structure on X. If ∇ denotes the Levi-Civita connection of g then say (Φ, g) is a torsion-free Spin (7)-structure on X if ∇Φ = 0.
Then (X, Φ, g) is a Spin (7)-manifold if X is an eight-dimensional oriented manifold and (Φ, g) is a torsion-free Spin(7)-structure on X.
Given a Spin(7)-manifold (X, Φ, g) then Φ is a calibration on X, known as the Cayley calibration. An oriented, four-dimensional submanifold Y of X is said to be Cayley if Φ| Y = vol Y , i.e., Y is calibrated by Φ. We can decompose bundles of forms on Spin(7)-manifolds into irreducible representations of Spin (7). The following proposition is taken from [11,Prop 11.4.4].
Proposition 2.1. Let X be a Spin(7)-manifold. Then the bundle of twoforms M admits the following decomposition into irreducible representations of Spin (7): 2 21 , where Λ k l denotes the irreducible representation of Spin(7) on k-forms of dimension l.
Remark. If Y is a Cayley submanifold of X, then we have that where E is a rank four vector bundle on Y .
The following result allows us to characterise Cayley submanifolds of a Spin(7)-manifold (X, Φ, g) by finding a four-form that vanishes exactly when restricted to a Cayley submanifold of X.
If x, u, v, w are orthogonal, then where π 7 (x ♭ ∧ y ♭ ) = 1 2 (x ♭ ∧ y ♭ + Φ(x, y, ·, ·)) and ♭ denotes the musical isomorphism T X → T * X. Moreover, if e 1 , . . . , e 8 is an orthonormal frame for T X then  (7)-manifold X. The results here are due to McLean [21, §6], although are taken in this form from a paper of the author [22]. We first use the tubular neighbourhood theorem to identify the moduli space of Cayley deformations of Y in X with the kernel of a partial differential operator.

Proposition 2.3 ([21, Thm 6.3]).
Let (X, g, Φ) be a Spin (7)-manifold with compact Cayley submanifold Y . Let exp denote the exponential map and for a normal vector field v define Y v := exp v (Y ). The moduli space of Cayley deformations of Y in X is isomorphic near Y to the kernel of the following partial differential operator (2.6) where τ is defined in Proposition 2.2, V is an open neighbourhood of the zero section in ν X (Y ) and (2.7) Λ 2 7 | Y = Λ 2 + Y ⊕ E, with π : Λ 2 7 | Y → E the projection map.
Moreover, we have that the the linearisation of F at zero is the operator (2.8) where {e 1 , e 2 , e 3 , e 4 } is a frame for T Y with dual coframe {e 1 , e 2 , e 3 , e 4 }, ∇ ⊥ : T Y ⊗ ν X (Y ) → ν X (Y ) denotes the connection on ν X (Y ) induced by the Levi-Civita connection of X and π 7 denotes the projection of two-forms onto Λ 2 7 as in Proposition 2.2.
Proof. We will recap the proof that the linearisation takes the claimed form since McLean's version of this result is presented differently.
We have that where e i = g(e i , · ) and e ijkl := e i ∧ e j ∧ e k ∧ e l . We have that * N L v τ | N = (L v τ )(e 1 , e 2 , e 3 , e 4 ).
Using a formula such as [8,Eqn (4.3.26)], we find that (L v τ )(e 1 , e 2 , e 3 , e 4 ) = (∇ v τ )(e 1 , e 2 , e 3 , e 4 ) + τ (∇ ⊥ e 1 v, e 2 , e 3 , e 4 ) − τ (∇ ⊥ e 2 v, e 1 , e 3 , e 4 ) + τ (∇ ⊥ e 3 v, e 1 , e 2 , e 4 ) − τ (∇ ⊥ e 4 v, e 1 , e 2 , e 3 ), where we have used that τ vanishes on four tangent vectors to a Cayley submanifold. By definition of τ given in Proposition 2.2, we have that and so it remains to show that if Φ is parallel then so is τ . From Equation (2.5), we see that We can see that the second sum in the above expression will vanish when evaluated on e 1 , e 2 , e 3 , e 4 , so it remains to compute we find that Similarly, Using the explicit expression for Φ, we have that Finally, note that since the metric g on X is parallel with respect to the Levi-Civita connection, and so we find that which vanishes since Φ is parallel. Now suppose that M is a four-dimensional Calabi-Yau manifold and N is a two-dimensional complex submanifold of M . We can apply the above results to study the Cayley deformations of N in M , but we will exploit the complex structure of N and M to give these results nicer forms. The following results are due to the author, and proofs can be found in [22].
To begin with, we identify the normal bundle and E with natural vector bundles on N .
where Ω is the holomorphic volume form on M and ♯ : ν * 0,1 M (N ) → ν 1,0 M (N ), and With these isomorphisms in place, we can modify Proposition 2.3.
We will now apply McLean's method to study the complex deformations of N in M . We begin by finding a form which vanishes exactly when restricted to a two-dimensional complex submanifold.
where Ω is the holomorphic volume form of M .
We can now define a partial differential operator whose kernel can be identified with the moduli space of complex deformations of N in M .
where σ was defined in Proposition 2.6 and V is an open neighbourhood of the zero section in ν M (N ).
We can find the linear part of G. Prop 4.4]). Let N be a compact complex surface in a Calabi-Yau four-fold M . Let G be the partial differential operator defined in Proposition 2.6. Then the linearisation of G at zero is equal to the operator where v ∈ C ∞ (ν M (N ) ⊗ C). Therefore v is an infinitesimal complex deformation of N if, and only if, So we can see from this result (in combination with the explicit isomorphism given in Proposition 2.4) that an infinitesimal ) such that∂v +∂ * w = 0 is a complex deformation of N if and only if∂v = 0 =∂ * w. Therefore an infinitesimal Cayley deformation of N in M that is not complex would satisfy∂v = −∂ * w. With a little bit more work, we can prove the following theorem.
. Remark. Comparing this to Kodaira's theorem [14, Thm 1] on the deformation theory of compact complex submanifolds, we see that we agree with the infinitesimal deformation space, but in this special case where the ambient manifold is Calabi-Yau we can integrate all infinitesimal complex deformations to true complex deformations.

Conically singular and asymptotically cylindrical manifolds.
We will now give some facts about closely related conically singular and asymptotically cylindrical manifolds that we will require later.
2.3.1. Conically singular manifolds. Heuristically speaking, a conically singular manifold can be thought of as a compact topological space that is a smooth Riemannian manifold away from a point. If the manifold near this point is diffeomorphic to a product L×(0, ǫ), and the metric on the manifold is close to the cone metric on L × (0, ǫ), then we call the manifold conically singular. This idea is made formal in the following definition, taken from [17, Defn 3.1]. Definition 2.3. Let M be a connected Hausdorff topological space and let x ∈ M . Suppose thatM := M \{x} is a smooth Riemannian manifold with metric g. Then we say that M is conically singular atx with cone C and rate λ if there exist ǫ > 0, λ > 1, a closed Riemannian manifold (L, g L ) of dimension one less than M , an open setx ∈ U ⊆ M and a diffeomorphism Ψ : (0, ǫ) × L → U \{x}, such that where r is the coordinate on (0, ∞) on the cone C = (0, ∞) × L, g C = dr 2 + r 2 g L is the cone metric on C and ∇ C is the Levi-Civita connection of g C .
Definition 2.4. Let M be a conically singular manifold atx with cone (0, ∞) × L. Use the notation of Definition 2.3. We say that a smooth function ρ :M → (0, 1] is a radius function for M if ρ is bounded below by a positive constant on M \U , while on U \{x} there exist constants 0 < c < 1 and C > 1 such that cr < Ψ * ρ < Cr, We will now define weighted Sobolev spaces for conically singular manifolds. The definition given here may be deduced from [15,Defn 4.1]. Definition 2.5. Let M be an m-dimensional conically singular manifold at x with metric g onM := M \{x}. Let ρ be a radius function for M . For a vector bundle E define the weighted Sobolev space L p k,µ (E) to be the set of sections σ ∈ L p k,loc (E) such that Asymptotically cylindrical manifolds. An asymptotically cylindrical manifold is topologically the same as a conically singular manifold, but metrically they are conformally equivalent. Compare the following definition to Definition 2.3.
Definition 2.6. Suppose that (M , g) is a Riemannian manifold. Then we say thatM is asymptotically cylindrical if there exist δ > 0, a closed Riemannian manifold (L, g L ) of dimension one less than M , an open set U ⊆ M and a diffeomorphism Ψ : where t is the coordinate on (0, ∞) on the cylinder C = (0, ∞) × L, g ∞ = dt 2 + g L is the cylindrical metric on C and ∇ ∞ is the Levi-Civita connection of g ∞ .
Notice that if (M , g) is a conically singular manifold with radius function ρ then (M , ρ −2 g) is asymptotically cylindrical.
We have the following weighted spaces on an asymptotically cylindrical manifold.
Definition 2.7. Let (M, g) be an asymptotically cylindrical manifold. For a vector bundle E over M , define the weighted Sobolev spaces W p k,δ (E) to be the space of sections σ ∈ L p k,loc (E) so that where ρ : M → (0, 1] is a smooth function satisfying ce −t ≤ ρ(t) ≤ Ce −t on the cylindrical end on M and is equal to one elsewhere.
We have the following relationship between the weighted spaces W p k,δ and L p k,µ . Let ρ be a radius function for M . Let T q sM be the vector bundle of (s, q)-tensors on M . Denote by W p k,δ (T q sM ) the weighted space of Definition 2.7 with metric ρ −2 g and denote by L p k,µ (T q sM ) the weighted space of Definition 2.5. Then these spaces are isomorphic, with isomorphism given by

Fredholm theory on noncompact manifolds.
A key part in the argument for proving a result on the moduli space of Cayley deformations of compact manifolds is the observation that an elliptic operator on a compact manifold is Fredholm. Unfortunately, this result fails in general when the underlying manifold is not compact, even in the simplest of settings. However, when the noncompact manifold is topologically a compact manifold with a cylindrical end, a theory was developed for certain types of elliptic operators.
Definition 2.8. Let X be a manifold with a cylindrical end L × (0, ∞). Let A : C ∞ 0 (E) → C ∞ 0 (F ), be a differential operator on compactly supported smooth sections of vector bundles. We say that A is translation invariant if it is invariant under the natural R + -action on the cylindrical end L × (0, ∞) of X. If X has an asymptotically cylindrical metric g, then we say that an operator

is asymptotically translation invariant if there exists a translation invariant operator
and δ > 0 such that for all j = 0, . . . , m and k ∈ N ∪ {0} where ∇ is the Levi-Civita connection of g. Here a j , a ∞ j ∈ C ∞ (E * ⊗ F ⊗ (T X) ⊗j ) and '·' denotes tensor product followed by contraction.
The following result may be deduced from the work of Lockhart and McOwen [16,Thm 6.2] in combination with Lemma 2.10.
Proposition 2.11. Let M be a conically singular manifold atx, ρ a radius function for M and T q sM be the vector bundle of (s, q)-tensors onM := M \{x}. Let , be a linear m th -order elliptic differential operator with smooth coefficients such that there exists λ ∈ R so that is asymptotically translation invariant. Then , is a bounded map and there exists a discrete set D A ⊆ R such that (2.16) is Fredholm if, and only if, µ ∈ R\D A .
We will characterise the set D for the operators that feature in this article in Section 4.

Deformations of conically singular Cayley submanifolds
3.1. Conically singular Cayley submanifolds. The following definition gives a preferred choice of coordinates around any given point of X. This definition is analogous to [10,Defn 3.6] and [17,Defn 3.3], which are coordinate systems for almost Calabi-Yau manifolds and G 2 -manifolds respectively.
Call two Spin(7)-coordinate systems χ,χ for X around x equivalent if In particular, when the Spin(7)-manifold X is a four-dimensional Calabi-Yau manifold, we can choose a holomorphic volume form Ω for X so that χ is a biholomorphism and dχ| 0 identifies the Ricci-flat Kähler form ω with ω 0 and Ω with Ω 0 , the Euclidean Kähler form and holomorphic volume form respectively.
Definition 3.2. Let (X, g, Φ) be a Spin(7)-manifold and Y ⊆ X compact and connected such that there existsx ∈ Y such thatŶ := Y \{x} is a smooth submanifold of X. Choose a Spin(7)-coordinate system χ for X aroundx. We say that Y is conically singular (CS) atx with rate µ and cone C if there exist 1 < µ < 2, 0 < ǫ < η, a compact Riemannian submanifold (L, g L ) of S 7 of dimension one less than Y , an open setx ∈ U ⊂ X and a smooth map φ : where ι : (0, ∞) × L → R 8 is the inclusion map given by ι(r, l) = rl, ∇ is the Levi-Civita connection of the cone metric g C = dr 2 + r 2 g L on C, and | · | is computed using g C .
Remark. If the smooth, noncompact submanifoldŶ is a Cayley (complex) submanifold of the Spin(7)-manifold (Calabi-Yau four-fold) X then we say that Y is a CS Cayley (complex) submanifold of X.
Conically singular submanifolds come with a rate 1 < µ < 2. We must have that µ > 1 to guarantee that a conically singular submanifold is a conically singular manifold (in the sense of Definition 2.3). The reason for asking that µ < 2 is so that µ does not depend on the choice of equivalent Spin(7)-coordinate system around the singular point of the conically singular submanifold.
Lemma 3.1. Let Y be a conically singular submanifold atx with rate µ and cone C of a Spin(7)-manifold (X, g, Φ) with Spin(7)-coordinate system χ aroundx. Then Definition 3.2 is independent of choice of equivalent Spin(7)-coordinate system.
Proof. Letχ be another Spin(7)-coordinate system for X aroundx equivalent to χ. Then χ andχ and their differentials agree at zero. Let φ : (0, ǫ) × L → B η (0) be the map from Definition 3.2. We will show that Y is conically singular in X with Spin(7)-coordinate systemχ aroundx. Taking . . , and φ(r, l) = rl +O(r µ ). So we see that Y is conically singular atx with cone C in (X, g, Φ) with Spin(7)-coordinate systemχ, but in order for Y to be CS with rate µ in this case, Equation (3.3) tells us that we must have that µ < 2.
On a Calabi-Yau manifold M we are given a Ricci-flat metric ω that we often have no explicit expression for. The following lemma tells us that Definition 3.2 is independent of choice of Kähler metric on M . Proof. Suppose that N is a CS submanifold of M with respect to ω atx. Choose a Spin(7)-coordinate system for M aroundx, for some η > 0 and open V ⊆ M containingx, so that χ(0) =x and where ω 0 is the standard Euclidean Kähler form on C 4 . Let φ, ǫ, C = (0, ∞) × L, ι and µ be as in Definition 3.2. Now given any other Kähler form ω ′ on M , we can find by [5, pg 107] η ′ > 0, an open set x ∈ V ′ ⊆ M and a diffeomorphism with χ ′ (0) = x and χ ′ * ω ′ = ω 0 + O(|z| 2 ). Since χ and χ ′ are diffeomorphisms, dχ| 0 and dχ ′ | 0 are isomorphisms C 4 → TxM . Then A := (dχ ′ | 0 ) −1 • dχ| 0 is an invertible linear map C 4 → C 4 . We will show that N is conically singular in (M, ω ′ ) (taking χ ′ to be the coordinate system) with cone C ′ = Aι(C) and rate µ.
Firstly note that since A is a linear map, C ′ = Aι(C) = {Av | v ∈ ι(C)} is also a cone. Denote by L ′ the link of C ′ (considered as a Riemannian submanifold of S 7 ), and for any ǫ ′ > 0 write ι ′ : Then this map is well defined (taking ǫ ′ smaller if necessary) and moreover χ ′ • φ ′ is a diffeomorphism onto its image. Moreover, by a similar argument to Lemma 3.1 we have that Finally, we have that and so the tangent cone to N atx is the same with respect to each metric.
Remark. Note that the proof Lemma 3.2 also shows that if N is conically singular with respect to one Spin(7)-coordinate system, it is conically singular with respect to any other Spin(7)-coordinate system, although with a different cone in general, but the same tangent cone.
We can now construct an example of a conically singular complex surface inside a Calabi-Yau four-fold.
Example. We will model our conically singular complex surface on the following complex cone in C 4 . Define C to be the set of (z 1 , z 2 , z 3 , z 4 ) ∈ C 4 satisfying Clearly, if z ∈ C, then also λz ∈ C for any λ ∈ R\{0}, and so C is a cone.
Checking the rank of the matrix , at each point of C, we see that the only singular point of C is zero.
As we will discuss in more detail in Section 4, a complex cone C in C 4 has both a real link L := S 7 ∩ C, and a complex link Σ := π(L), where π : S 7 → CP 3 is the Hopf fibration. We can view the real link of a complex cone as a circle bundle over the complex link of the cone.
In this case, the complex link Σ of C is the Riemannian surface in CP 3 is given by [z 0 : z 1 : z 2 : z 3 ] ∈ CP 3 satisfying We can apply the adjunction formula to find that the canonical bundle of Σ is given by Now consider the Calabi-Yau four-fold M defined by Consider the singular submanifold N of M defined to be the set of all [z 0 : z 1 : z 2 : z 3 : z 4 : z 5 ] ∈ CP 5 satisfying The complex Jacobian matrix of the defining equations of N is given by It can be calculated that there are six singular points on N of the form [ω : 0 : 0 : 0 : 0 : 1], where ω is a 6 th root of −1.
We will now prove that N satisfies Definition 3.2. We will exploit Lemma 3.2 and check the definition using the metric on M induced from the Fubini-Study metric on CP 5 , denoted by ω.
Denote the singular points of N by {p 1 , . . . , p 6 }, where p k = [ω k : 0 : 0 : 0 : 0 : 1] for ω k := e i(2k−1)π/6 . We must construct maps χ k so that there exist η k > 0 and open sets p k ∈ V k ⊆ M and diffeomorphisms with χ k (0) = p k and so that where if a = re iθ for r > 0 and −π < θ ≤ π, we define a 1/6 := r 1/6 e iθ/6 . It is clear that (3.4) is a diffeomorphism onto its image. The induced Fubini-Study metric on M pulls back under χ k to the Euclidean metric on C 4 at each p k = [ω k : 0 : 0 : 0 : 0 : 1]. Taking φ = ι, where ι : C → C 4 is the inclusion map, we see that φ • χ is a diffeomorphism C to N , and so the definition of conically singular is trivially satisfied.
3.2. Tubular neighbourhood theorems. In this section we will prove a tubular neighbourhood theorem for conically singular submanifolds so that we can identify deformations of conically singular submanifolds with normal vector fields. We will do this in two steps. Firstly, in Proposition 3.3 we will construct a tubular neighbourhood of a cone in R n using the well-known tubular neighbourhood theorem for compact submanifolds. We will use this to construct a tubular neighbourhood of a conically singular submanifold in  3.3 (Tubular neighbourhood theorem for cones). Let C be a cone in R n with link L and let g be a Riemannian metric on R n (not necessarily the Euclidean metric). There exists an action of R + on ν R n (C) We can construct open sets V C ⊆ ν R n (C), invariant under (3.5), containing the zero section and T C ⊆ R n , invariant under multiplication by positive scalars, containing C that grow like r and a dilation equivariant diffeomorphism Proof. We will first address the claim that there exists an R + -action on ν R n (C) so that (3.5) holds. First note that points in ν R n (C) take the form where r ∈ R + , l ∈ L and v(r, l) ∈ ν r,l (C). Since any finite-dimensional inner product spaces of the same dimension are isometric, given any r ′ ∈ R + we can think of (r ′ , l, v(r, l)) as another point in ν R n (C) with |v(r, l)| r,l = |v(r, l)| r ′ ,l , where | · | r,l denotes the norm on ν r,l (C) induced from g. Define an action of R + on ν R (C) by (r, l, v(r, l)) → (tr, l, tv(r, l)).
To prove the tubular neighbourhood part of this proposition, we first apply the usual tubular neighbourhood theorem to the compact submanifold L of S n−1 . (Recall that we need a metric on S n−1 to define the exponential map. We take this to be the standard round metric on S n−1 .) This gives us an open set V L ⊆ ν S n−1 (L) containing the zero section and an open set T L ⊆ S 7 containing L and a diffeomorphism so that Ξ L maps the zero section of ν S n−1 (L) to L. Again write points in ν R n (C) as (r, l, v(r, l)), where v ∈ ν r,l (C), and similarly points in ν S n−1 (L) as (l, v(l)) where v ∈ ν l (L) ∼ = ν r,l (C). Then define It is clear that V C is invariant under the R + -action (3.6) by construction of V C and the R + -action. We see that V C grows like r in the sense that if v = (r, l, v(r, l)) ∈ V C then where V is the diameter of the set V . Now define Then it is clear that T C is dilation invariant, in the sense that it is clearly invariant under multiplication by positive scalars, and that C ⊆ T C . We see that T C grows like r in the sense that if t ∈ T , l ∈ L and r ∈ R + then where |T | is the diameter of the set T . Define It is clear that Ξ C is well-defined, bijective and smooth. It is also clear that Finally we have that by definition of Ξ L and so Ξ C maps the zero section of ν R n (C) to C.
We can use this result to prove a tubular neighbourhood theorem for a conically singular submanifold. :V →T , that takes the zero section of ν X (Ŷ ) toŶ . Moreover, we can chooseV and T to grow like ρ as ρ → 0.
Proof. Notice that K := Y \U is a compact submanifold of X. So by the compact tubular neighbourhood theorem we can find open setsV 1 ⊆ ν X (K) containing the zero section andT 1 ⊆ X containing K and a diffeomorphism We will construct a tubular neighbourhood forŶ nearx. Denote C ǫ := C ∩ B ǫ (0). Use the notation of Definition 3.2. Choose φ : C ǫ → R n uniquely by asking that φ(r, l) − ι(r, l) ∈ (T rl ι(C)) ⊥ . Then since |φ − ι| = O(r µ ), for 1 < µ < 2 as r → 0, making ǫ smaller if necessarily, we can guarantee that φ(r, l) lies in the tubular neighbourhood of C given by Proposition 3.3. We can therefore identify φ(C ǫ ) with a normal vector field v φ on C.
Applying Proposition 3.3 gives us V C ⊆ ν R n (C), T C ⊆ R n and a diffeomorphism Denote by V Cǫ the restriction of V C to C ǫ , and define induced from Ψ and ι andT 2 := χ(T φ ). By definition, these sets grow with order ρ as ρ → 0. Then is a diffeomorphism taking the zero section of ν X (Û ) toÛ . DefineV ,T andΞ by interpolating smoothly betweenV 1 andV 2 ,T 1 andT 2 andΞ 1 and Ξ 2 .
3.3. Deformation problem. The moduli space that we will consider will be defined in Definition 3.5 below, and this moduli space will be identified with the kernel of a nonlinear partial differential operator in Proposition 3.5. First we will define a weighted norm on spaces of differentiable sections of a vector bundle.
3.3.1. Weighted norms on spaces of differentiable sections. Let X will be an n-dimensional CS manifold with a radius function ρ, E a vector bundle over X (the nonsingular part of X) with a metric and connection.

Moduli space.
We will now formally define the moduli space of conically singular Cayley deformations of a Cayley submanifold that we will be studying in this article.
Definition 3.5. Let Y be a conically singular Cayley submanifold atx with cone C and rate µ of a Spin(7)-manifold (X, g, Φ) with respect to some Spin(7)-coordinate system χ, and denote the tangent cone of Y atx byĈ. WriteŶ := Y \{x}. Define the moduli space of conically singular (CS) Cayley deformations of Y in X,M µ (Y ), to be the set of CS Cayley submanifolds Y ′ atx with cone C, rate µ and tangent coneĈ of X so that there exists a continuous family of topological embeddings ι t : Y → X with ι 0 (Y ) = Y and ι 1 (Y ) = Y ′ , so that ι t (x) =x for all t ∈ [0, 1] and so that ι t := ι t |Ŷ is a smooth family of embeddingsŶ → X withι 0 (Ŷ ) =Ŷ and We will now end this section by identifying the moduli space of Cayley CS deformations of a CS Cayley submanifold of a Spin (7)-manifold with the kernel of a nonlinear partial differential operator.
Proposition 3.5. Let Y be a CS Cayley submanifold atx with cone C and rate µ ∈ (1, 2) of a Spin (7)-manifold (X, g, Φ). Let τ be the Λ 2 7 -valued four-form defined in Proposition 2.2, π : Λ 2 7 → E be the projection map for the splitting given in (2.4) andV ⊆ ν X (Ŷ ),T ⊆ M andΞ be the open sets and diffeomorphism from the CS tubular neighbourhood theorem 3.4. For Then we can identify the moduli space of CS Cayley deformations of Y in X near Y with the kernel of the following differential operator is a CS submanifold of X atx with cone C and rate µ (with respect to the same Spin(7)-coordinate system as Y ) if, and only if, v ∈ C ∞ µ (V ).
Now we can use Ψ and ι to identify ν X (Û ) with ν Bǫ(0) (ι(C ǫ )), whereÛ := U \{x} and C ǫ : Making ǫ and U smaller if necessary, by the definition of the tubular neighbourhood map in Proposition 3.4, we can define a map φ v : where Ξ φ was defined in the proof of Proposition 3.4, so that χ So we see that for Y v to be a CS submanifold of X with rate µ and cone C we must have that for all j ∈ N as r → 0. Now we can write for j ∈ N as r → 0. But examining the definition of φ v , we see that we can identify φ v − φ with the graph of v C , and so (3.8) holds if, and only if, for j ∈ N as r → 0. But then by definition of v C this is equivalent to So we see that the moduli space of CS Cayley deformations of Y in X can be identified with the kernel of (3.7).

Cayley deformations of a CS Cayley submanifold.
In this section we prove Theorem 3.8 on the expected dimension of the moduli space of CS Cayley deformations of a conically singular Cayley submanifold Y in a Spin (7)-manifold X.
Lemma 3.6. Let Y be a conically singular Cayley submanifold of a Spin (7)manifold X. LetF be the operator defined in Proposition 3.5. Then we can write , is a smooth map of Banach spaces for any 1 < p < ∞ and k ∈ N with k > 1 + 4/p.
Proof. By definition of conically singular, we can split Y into a compact piece K, where we can argue as in the proof of an analogous result for compact Cayley submanifolds (see, for example, [22,Lem 3.4]) that the estimate (3.10) holds, and a piece diffeomorphic to a cone, which is where we must check howF behaves as ρ → 0, where ρ is a radius function forŶ .
Making the compact piece slightly larger, using the definition ofF , we may estimateF by estimatingF C , the operator on the cone defined bŷ where v φ is the normal vector field on C that describes φ(C) as described in the proof of Proposition 3.4, where we are using the notation of Definition 3.2. DefineQ C analogously bŷ By definition ofQ andQ C , we see that and so to estimateQ and its derivatives, it suffices to estimate the right hand side of Equation (3.13). Notice that since for each (r, l) ∈ C we can think ofQ C as a map and so we can make sense of a Taylor expansion ofQ around points of the form (r, l, y, z), for y ∈ ν r,l (C) and z ∈ ν r,l (C) ⊗ T * r,l C. Abusing notation slightly, write ∂Q C ∂y (r, l, y, z), for derivative ofQ C in the y direction at (r, l, y, z), and adopt similar notation for the derivative in the z direction and the higher derivatives. Then we have that for some t ∈ [0, 1]. We would like to estimate the derivatives ofQ C . Sincê Q is smooth in all of its variables, this is possible as long as we restrict the domain ofQ C to a compact set. However, we are working on a cone (with its singular point removed) so this isn't possible. We may, however, fix r = r 0 , for some r 0 ∈ (0, ǫ), perform our estimates, and use the definition ofF C and Q C to study the behaviour of the estimates we find as we let r vary. Recall the action of R + on ν R 8 (C) that was defined in the proof of Proposition 3.3, and indeed the tubular neighbourhood map that we constructed in this proof, which forms part of the operatorF C that we are currently studying. By construction, we can see that where | · | r means that we are taking the norm at the point r. We may deduce that We also have that by construction SinceQ C has no linear parts, we have that by equation (3.14), . Then by taking the supremum over the closed sets l ∈ L and v with |v(r 0 , l)| r 0 ,l + |∇v(r 0 , l)| r 0 ,l ≤ δ, which is possible as long as we take δ sufficiently small, we may bound this expression, as well as the other coefficients of Equation (3.15). Using the scale equivariance properties ofQ C described above, we deduce that as long as v C 1 1 is small, we have that Therefore Finally, we can take k derivatives of Equation (3.15), which will give us a polynomial quadratic in v and its derivatives, whose coefficients depend on between two and k + 2 derivatives ofQ C , the C 1 1 -norm of v φ and δ as above. We can estimate these coefficients as we did above. We will find that and since we may deduce that and so we see that the estimate (3.10) holds.
Finally, as long as µ > 1, we have that C k 2µ−2 (E) ⊆ C k µ−1 (E). We can apply Minkowski's inequality and the smallness of v C 1 1 to (3.10) to deduce (3.11) and that (3.12) is a smooth map of Banach spaces, asQ is smooth. Now that we have described the behaviour of the operatorF close to the singular point of the conically singular manifoldŶ , we will prove a weighted elliptic regularity result for normal vector fields in the kernel ofF .
Proposition 3.7. Let Y be a conically singular Cayley submanifold of a Spin (7)-manifold X. LetF be the map defined in Proposition 3.5. Then Here D is the set of exceptional weights given by applying Proposition 2.11 to the linear part ofF .
Conversely, let v ∈ L p k+1,µ (V ) satisfyF (v) = 0. Here we perform a trick similar to that in [12,Prop 4.6]. Taylor expandingF (v) around zero we can writeF (v) as a polynomial in v and ∇v. Differentiating and gathering terms we can write Consider the second order elliptic linear operator By Sobolev embedding, we know that v ∈ C l µ (V ), for l ≥ 2 by choice of p and k, and therefore the coefficients of the linear operator L v lie in C l−1 loc (V ). Local regularity for linear elliptic operators with coefficients in Hölder spaces (a nice statement is given in [11,Thm 1.4.2], taken from [23, Thm 6.2.5]) tells us that v ∈ C l+1 loc (V ) which is an improvement on the regularity of v, and so bootstrapping we may deduce that v ∈ C ∞ loc (V ). (This is why we must differentiateF (v), to ensure that the coefficients of the linear operator have enough regularity to improve the regularity of v.) Therefore the coefficients of the operator L v are smooth and so we may apply an estimate of Lockhart and McOwen [16,Eq. 2.4] in combination with a change of coordinates which tells us that (3.19) v p,k+2,µ ≤ C( L v v p,k,µ−2 + v p,0,µ ).
SinceF (v) = 0 = ∇F (v), we have that Since E(x, v(x), ∇v(x)) is a polynomial in v and ∇v with coefficients that depend on the C 1 1 -norm of v, and v ∈ C 1 µ (V ) and L p k+1,µ (V ), we have that We may finally deduce the main theorem of this section, on the expected dimension of the moduli space of Cayley CS deformations of a CS Cayley submanifold of a Spin(7)-manifold X. Moreover, the expected dimension ofM µ (Y ) is given by the index of the linear elliptic operator If the cokernel of (3.20) is {0} thenM µ (Y ) is a smooth manifold near Y of the same dimension as the kernel of (3.20). Here D is the set of weights µ ∈ R for which (3.20) is not Fredholm from Proposition 2.11.

3.5.
Cayley deformations of a CS complex surface. In this section we prove Theorem 3.12 which gives the expected dimension of the moduli space of CS Cayley deformations of a two-dimensional conically singular complex submanifold N of a Calabi-Yau four-fold M in terms of the index of the operator∂ +∂ * acting on weighted sections of a vector bundle overN (the nonsingular part of N ).
3.5.1. Deformation problem. We would like to study the moduli space given in Definition 3.5 for the CS Cayley submanifold N that is a complex submanifold of a Calabi-Yau four-fold M . We will now identify this moduli space with the kernel of a nonlinear partial differential operator.
is the image ofV ⊗ C from the tubular neighbourhood theorem under the isomorphism given in Proposition 2.4, and F cx is defined so that the following diagram commutes whereF is the operator defined in Proposition 3.5 and we use the isomorphisms given in Proposition 2.4.
Moreover, the linearisation ofF cx at zero is the operator Proof. By Proposition 3.5 we can identify the moduli space of CS Cayley deformations of N in M with the kernel ofF , which is the same as the kernel ofF cx .
Since the linearisation of the operator ofF is given by the operator D defined in Proposition 2.3, the local argument of Proposition 2.5 still holds, and so we see that the linearisation ofF cx at zero is given by the operator (3.23) as claimed.
3.5.2. Cayley deformations of a CS complex surface. In this section, we will give analogies of the results of Section 3.4, which were on analytic properties of the operatorF defined in Proposition 3.5, for the operatorF cx defined in Proposition 3.9. Due to the relation between the operatorsF andF cx noted in the proof of Proposition 3.9, these results follow immediately from their counterparts.
Lemma 3.10. Let N be a conically singular complex surface inside a Calabi-Yau four-fold M . LetF cx be the operator defined in Proposition 3.9. Then we can write sufficiently small, there exist constants C k > 0 so that and if w ∈ L p k+1,µ (Û ) with w C 1 1 sufficiently small, there exist constants D k > 0 such that , is a smooth map of Banach spaces for any 1 < p < ∞ and k ∈ N with k > 1 + 4/p.
Proof. SinceF cx is defined by composing the operatorF defined in Proposition 3.5 with isomorphisms of vector bundles, the estimates (3.25) and (3.26) follow from the estimates (3.10) and (3.11) respectively since the isomorphisms defined in Proposition 2.4 are isometries.
Moreover, since these isomorphisms are smooth, the claim that (3.27) is a smooth map of Banach spaces follows from the corresponding fact forF from Lemma 3.6.
We may now give a weighted elliptic regularity result forF cx . Proposition 3.11. Let N be a conically singular complex surface inside a Calabi-Yau four-fold M . LetF cx be the map defined in Proposition 3.9. Then , for any µ ∈ (1, 2)\D, 1 < p < ∞ and k ∈ N. Here D is the set of exceptional weights given by applying Proposition 2.11 to the linear part ofF cx .
Proof. This follows from Proposition 3.7 in combination with the fact that the kernels ofF , defined in Proposition 3.5, andF cx are isomorphic by definition, and the isomorphism given in Proposition 2.4 is an isometry.
We deduce the following theorem on the moduli space of CS Cayley deformations of a CS complex surface inside a Calabi-Yau four-fold. This theorem can be proved by an identical argument to the proof of Theorem 3.8, but we will deduce it as a corollary of Theorem 3.8. 3.6. Complex deformations of a CS complex surface. In this section, we will compare the CS complex and Cayley deformations of a CS complex surface inside a four-dimensional Calabi-Yau manifold.
Definition 3.6. Let N be a CS complex surface atx with rate µ and cone C inside a Calabi-Yau manifold M with respect to some Spin(7)-coordinate system χ, and denote byĈ the tangent cone of N . WriteN := N \{x}. Define the moduli space of conically singular (CS) complex deformations of N in M ,M cx µ (N ), to be the set of CS complex surfaces N ′ atx with cone C, rate µ and tangent coneĈ of M so that there exists a continuous family of topological embeddings ι t : N → M with ι 0 (N ) = N and ι 1 (N ) = N ′ , so that ι t (x) =x for all t ∈ [0, 1] and so thatι t := ι t |N is a smooth family of embeddingsN → X withι 0 (N ) =N andι 1 (N ) =N ′ := N ′ \{x}.
We will now identify the moduli space of CS complex deformations of a CS complex surface in a Calabi-Yau manifold M with the kernel of a nonlinear partial differential operator.

29)
where σ was defined in Proposition 2.6. Moreover, the kernel ofĜ is isomorphic to the kernel of its linear part given by the map The kernel of (3.30) is isomorphic to Proof. By definition of σ we see that normal vector fields in the kernel of G correspond to complex deformations ofN , and a similar argument to Proposition 3.5 shows that weighted smooth sections of ν M (N ) ⊗ C give conically singular deformations ofN as required. The linear part ofĜ follows from Proposition 2.8, which was a local argument, and similarly that the kernel ofĜ is equal to the kernel of its linear part follows from the local argument [22,Lem 4.7]. Finally, that the kernel of (3.30) is equal to (3.31) follows from Proposition 2.8, where we proved that where π 1,0 : ν M (N ) ⊗ C → ν 1,0 M (N ) and the isomorphism of Proposition 2.4.
This proposition allows us to prove that the CS complex deformations of a conically singular complex surface are unobstructed. This theorem is a generalisation of Theorem 2.9 to conically singular submanifolds.
Theorem 3.14. Let N be a conically singular complex surface atx with rate µ ∈ (1, 2) and cone C inside a Calabi-Yau four-fold M . The moduli space of CS complex deformations of N in M ,M cx µ (N ) given in Definition 3.6, is a smooth manifold of dimension To compare CS complex and Cayley deformations of a CS complex surface, we require the following result. Proof. Suppose that w ∈ L 2 k+1,µ (ν 1,0 M (N ) ⊕ Λ 0,2N ⊗ ν 1,0 M (N )) satisfies∂w = −∂ * w for µ ∈ (1, 2). Then∂ * ∂ w = 0. We will check whether , that is, whether the integrals on both sides converge. Let ρ be a radius function for N . We have that by Hölder's inequality. This is finite since which again is finite since |ρ µ+4∂ * v| ≤ |ρ 2−µ∂ * v|, for µ ∈ (1, 2). Therefore  14 we see that these spaces must be the same, since any CS complex deformation of N is a Cayley deformation of N .

Index theory
Let Y be a CS Cayley submanifold of a Spin (7)-manifold X with nonsingular partŶ and let N be a CS complex surface inside a four-dimensional Calabi-Yau manifold M with nonsingular partN . In this section, we will be interested in the index of the operators (4.1) D : L p k+1,µ (ν X (Ŷ )) → L p k,µ−1 (E), from Proposition 2.3 on sections with compact support and extended by density to the above spaces, and We will first characterise the set of exceptional weights D for which (4.1) and (4.2) are not Fredholm. We will then explain how we can apply the Atiyah-Patodi-Singer index theorem to operators on conically singular manifolds, before applying this result to the operator (4.2).

4.1.
Finding the exceptional weights for the operators D and∂ +∂ * . In this section we will find the set D of exceptional weights for which the linear elliptic operators (4.1) and (4.2) that appeared in Section 3 are not Fredholm. To do this we will study these operators acting on Cayley and complex cones in R 8 . We will see that the exceptional weights are actually eigenvalues for differential operators on the links of these cones.

4.1.1.
Nearly parallel G 2 structure on S 7 . We can consider R 8 as a cone with link S 7 . Let (Φ 0 , g 0 ) be the Euclidean Spin (7)-structure (as given in Definition 2.1). Define a three-form ϕ on S 7 by the following relation: Then (ϕ, g) is a G 2 -structure on S 7 (here g is the standard round metric on S 7 ). Notice that this G 2 -structure is not torsion-free, however, since Φ 0 is closed we have that
We will now describe the set of exceptional weights for D in terms of an eigenvalue problem on the link of C.
Let L := C ∩S 7 be the link of the cone C, a submanifold of S 7 . Then λ ∈ D D if, and only if, there exists v ∈ C ∞ (ν S 7 (L)) so that where for {e 1 , e 2 , e 3 } an orthonormal frame for T L and ∇ ⊥ the connection on the normal bundle of L in S 7 induced by the Levi-Civita connection of the round metric on S 7 , where × is the cross product on S 7 induced from the nearly parallel G 2structure (ϕ, g) defined by g(u × v, w) = ϕ(u, v, w), for any vector fields u, v, w on S 7 .
Remark. The operator D L can be defined on any associative submanifold of a G 2 -manifold, that is, a manifold with torsion-free G 2 -structure. Normal vector fields in its kernel correspond to infinitesimal associative deformations of the associative submanifold. This can be deduced from the work of McLean [21,, however the operator first appears in this form in [1,Eqn 14]. Infinitesimal associative deformations of an associative submanifold of S 7 with its nearly parallel G 2 -structure, however, satisfy (4.6) with λ = 1 as shown by Kawai [13,Lem 3.5]. Proposition 4.1 can be considered as a different proof of this fact.
Proof. We can apply Proposition 2.11 to the operator D. Suppose that ρ is a radius function for Y . Then since the given Spin(7)-structure on X approaches the Euclidean Spin(7)-structure as we move close to the singular point of Y , ρ −1 Dρ −1 is asymptotic to the translation invariant differential operator where D 0 is defined similarly to D but using the Euclidean Spin(7)-structure pulled back to X by a Spin(7)-coordinate system χ for X aroundx (see Definition 3.1).
By Proposition 2.11 in combination with the discussion in [16, pg 416], we see that λ ∈ D D if, and only if, there exists a normal vector field v ∈ C ∞ (ν L (S 7 )) satisfying where since ν rl,R 8 (C) ∼ = ν l,S 7 (L) for all r > 0 we can consider (r, l) → (r, r λ−1 v(l)) as a normal vector field on the cone. Note also that the induced Euclidean metric on the normal bundle of C in R 8 takes the form r 2 h, where h is the metric on the normal bundle of L in S 7 induced from the round metric on S 7 .
Let {e 1 , e 2 , e 3 } denote an orthonormal frame for T L with dual coframe {e 1 , e 2 , e 3 }, and denote by Φ 0 the Euclidean Cayley form on R 8 and ϕ the nearly parallel G 2 -structure on S 7 defined in (4.3). We compute that as the metric on the normal bundle is of the form r 2 h.
Using the definition of ϕ in (4.3), we find that Now we wish to replace the musical isomorphism ♭ : ν R 8 (C) → ν * R 8 (C) with the musical isomorphism ♭ L : ν S 7 (L) → ν * S 7 (L). Since the metric on ν R 8 (C) is of the form r 2 h, where h is a metric on ν S 7 (L), we find that Notice that E ∼ = ν S 7 (L) via the map Therefore we see that We find that ∂ ∂r Since by definition,

4.1.3.
Exceptional weights for the operator∂ +∂ * . Let N be a CS complex surface with rate µ and cone C inside a Calabi-Yau four-fold M , and writeN for its nonsingular part. In order to prove an analogous result to Proposition 4.1 for the operator ), we will need some preliminary facts about complex cones. The real link of a complex cone C is a circle bundle over the complex link of C. Thinking of L as S 1 × Σ, we can find a globally defined vector field on L that we can think of as being tangent to S 1 in this product. Definition 4.2. Let C be a complex cone in C n+1 , and denote by J the complex structure on C n . The Reeb vector field is defined to be ξ := J r ∂ ∂r .
If p| L : L → Σ is the restriction of the Hopf projection to L, then at each l ∈ L, ξ l spans the kernel of dπ| l : T l L → T p(l) Σ. With these definitions, we may characterise the set of exceptional weights for the operator (4.8) in terms of an eigenproblem on the link of a cone.
where ξ is the Reeb vector field on L. Here ∇ acts on Λ 0,1 h L as the Levi-Civita connection of the metric on L and on ν 1,0 S 7 (L) as the normal part of the Levi-Civita connection on S 7 .
Proof. Similarly to the proof of Proposition 4.1, if ρ is a radius function for N then we can see that∂ +∂ * ρ 2 , onN is asymptotically translation invariant tō ∂ C +∂ * C r 2 , on the cone C where this time we take a metric on ν C 4 (C) that is independent of r. If v ∈ C ∞ (ν S 7 (L) ⊗ C) we can think of r µ v as a complexified normal vector field on C, and moreover the complex structure J on C 4 induces a splitting ν S 7 (L) ⊗ C = ν 1,0 S 7 (L) ⊕ ν 0,1 S 7 (L), of the complexified normal bundle of L in S 7 into holomorphic and antiholomorphic parts (the i and −i eigenspaces of J respectively). Also, by definition of the Reeb vector field, if we take θ ∈ C ∞ (Λ 1 L) to be the dual one-form to ξ we have that dr−irθ is a (0, 1)-form on C. Since Λ 2 C ∼ = Λ 2 L⊕dr∧Λ 1 L, we can see that a (0, 2)-form on C must be of the form where w ∈ C ∞ (Λ 0,1 h L). By Proposition 2.11 in combination with the discussion in [16, pg 416], we deduce that λ ∈ D if, and only if, there exists v ∈ C ∞ (ν S 7 (L)) and w ∈ C ∞ (Λ 0,1 h L ⊗ ν 1,0 S 7 (L)) so that where θ is dual to the Reeb vector field ξ. We can calculate that and therefore (4.12)∂ C (r λ v) = r λ 1 2 We also have that where since w is a horizontal (0, 1)-form we see that any term gained from applying∂ * h to r −1 dr−iθ must be a multiple of w at each point and therefore will vanish under exterior product with w. We have that since ∇ ξ dr = rθ where ∇ is the Levi-Civita connection of the cone metric. We deduce that Equating (4.12) and minus (4.13), we find that λ ∈ D if, and only if, there as claimed.

4.1.4.
An eigenproblem on the complex link. In Proposition 4.2 we characterised the set of exceptional weights D for which the operator (4.9) is not Fredholm in terms of an eigenproblem on the real link of a complex cone C. In this section we will introduce a trick used by Lotay [19, §6] to study an eigenvalue problem on the link of a coassociative cone which is a circle bundle over a complex curve in CP 2 . This will allow us to give an equivalent eigenvalue problem to (4.10)-(4.11) on the real link of C completely in terms of operators and vector bundles on the complex link of C.
Let C be a complex cone in C 4 with real link L ⊆ S 7 and complex link Σ ⊆ CP 3 . Suppose we have a problem of the following form: Find all of the functions f on L that satisfy (4.14) for some m ∈ Z, where ξ is the Reeb vector field on C.
We would like to understand the relationship between the operator∂ h on the real link of C and∂ Σ on the complex link C. Basic functions, forms and vector fields are special because they are in oneone correspondence with functions, forms and vector fields on Σ. It follows from [26, Lem 1] that∂ h acting on basic functions, forms or vector fields on L is equivalent to∂ Σ acting on functions, forms or vector fields on Σ. In Problem (4.14), when m = 0, f is not basic. However, a simple trick allows us to pretend that f is basic.
By the definition of the complex link, we may identify the cone C with the vector bundle O CP 3 (−1)| Σ , that is, the tautological line bundle over CP 3 restricted to Σ. This is then a trivial (real) line bundle over L and therefore has a global section given by the map x → s(x) = x for x ∈ L. It is easy to see that L ξ s = is, and therefore where deg(E) is the degree of the vector bundle E, rk(E) is the rank of the vector bundle and g is the genus of Σ.
We will now apply the trick that we described above to rephrase the eigenvalue problem (4.10)-(4.11) on the real link of a cone as an eigenvalue problem on the complex link on a cone. Proposition 4.4. Let C be a complex cone in C 4 with real link L and complex link Σ. Then given λ ∈ R and m ∈ Z, pairs v ∈ C ∞ (ν 1,0 16) are in a one-one correspondence with pairsṽ ∈ C ∞ (ν 1,0 where ξ is the Reeb vector field, and the eigenvalue problem (4.10)-(4.11).

4.2.
Dimension of the moduli space of complex deformations of a CS complex surface. In this section, we will deduce a version of the Atiyah-Patodi-Singer index theorem for operators on conically singular manifolds. We will then apply this result to prove Theorem 4.8, an index formula for the operator (4.2), which allows us to compare the dimension of the moduli space of CS complex deformations of a conically singular complex surface to what we will think of as the dimension of the moduli space of all complex deformations of a CS complex surface in a Calabi-Yau fourfold based on Kodaira's theorem [14, Thm 1] on deformations of complex submanifolds of complex varieties.

4.2.1.
The Atiyah-Patodi-Singer index theorem for conically singular manifolds. The Atiyah-Patodi-Singer index theorem is predominantly for a certain type of elliptic operator on a manifold with boundary. However, as a corollary to the main theorem, they also prove an index theorem for translation invariant operators on a manifold with a cylindrical end, which we quote here.
Theorem 4.5 ([3, Thm 3.10 & Cor 3.14]). Let be a linear elliptic first order translation invariant differential operator on a manifoldX with a cylindrical end Y × (0, ∞) that takes the special form where u is the inward normal coordinate, σ : E| Y → F | Y is a bundle isomorphism and B is a self adjoint elliptic operator on Y . Then where h, η, α 0 and h ∞ (F ) are defined as follows:  Here we call f an extended L 2 -section of E if f ∈ L 2 loc (E) and on the cylindrical end ofX, for large t, f takes the form f (y, t) = g(y, t) + f ∞ (y), for g ∈ L 2 (E) and f ∞ ∈ Ker B.
We will now explain how we can apply the Atiyah-Patodi-Singer index theorem 4.5 to elliptic operators on conically singular manifolds.
We first give a technical result that relates the adjoint of a differential operator on a conically singular manifold to the adjoint of the related asymptotically translation invariant operator acting on the conformally equivalent manifold with cylindrical end. Lemma 4.6. Let M be an m-dimensional conically singular manifold atx and let ρ be a radius function for M . WriteM := M \{x}, and g for the metric onM . Let be a linear first order differential operator onM and suppose there exists λ ∈ R so thatÃ is an asymptotically translation invariant operator. Then the formal adjoint of the operatorÃ (with respect to the metric ρ −2 g) , is the formal adjoint of A with respect to g. Moreover, using the notation of Definitions 2.7 and 2.5, the kernel of , is isomorphic to the kernel of (4.20) where we have used that A * is the formal adjoint of A with respect to the metric g, which shows that is the formal adjoint ofÃ with respect to the metric ρ −2 g. By Lemma 2.10 , is an isomorphism and so by definition ofÃ * and A * the kernels of (4. 19) and (4.20) are isomorphic.
We may now deduce the following proposition from Theorem 4.5 and Lemma 4.6 to give an index theorem for operators on conically singular submanifolds.
Proposition 4.7. Let M be an m-dimensional conically singular manifold atx with radius function ρ. Let T q sM be the vector bundle of (s, q)-tensors onM := M \{x}. Let , be a first order linear elliptic differential operator so that is asymptotically translation invariant toÃ ∞ for some λ ∈ R. Then for µ ∈ R\D, given in Proposition 2.11, the index of (4.21) A : for ǫ > 0 chosen so that (0, ǫ] ∩ D = ∅ and we use the notation of Theorem 4.5 for the terms on the right hand side of (4.23) (and these terms are defined for the translation invariant operatorÃ ∞ ).
Proof. By Proposition 2.11, we know that A andÃ have the same kernel and cokernel when acting on weighted Sobolev spaces, and moreover, the index of these operators differ from the index ofÃ ∞ by a constant independent of the weight.
SinceÃ ∞ is translation invariant, we can apply Theorem 4.5 toÃ ∞ . Let Ker µÃ∞ and Ker µÃ * ∞ denote the kernels of respectively, whereÃ * ∞ is the formal adjoint ofÃ ∞ with respect to the metric ρ −2 g, where g is the metric onM . Then Theorem 4.5 yields that By definition ofÃ ∞ , Ker 0Ã∞ ∼ = Ker 0 A ∞ , where Ker µ A ∞ denotes the kernel of (4.22), and by Lemma 4.6, Ker 0Ã * ∞ ∼ = Ker λ−m A * ∞ , where A * ∞ is the formal adjoint of A ∞ with respect to the metric g and Ker µ A * ∞ denotes the kernel of So we see that (4.25) dim Denote by D the subset of R for which µ ∈ D if, and only if, (4.22) is not Fredholm. Then we might have a problem equating Then ind ǫ A ∞ is well-defined. Since ǫ > 0, we have that where Ker µ A ∞ denotes the kernel of (4.22). It is claimed that To see this, suppose that α ∈ Ker 0 A ∞ . Then by elliptic regularity, α is smooth, and by definition of weighted norm on L 2 k+1,0 (T q sM ) α must decay to zero as r → 0 and so we must have that α = O(r ǫ ′ ) for some ǫ ′ > 0. Taking ǫ ′ smaller if necessary we can guarantee that D ∩ (0, ǫ ′ ] = ∅. The rate of decay of α allows us to deduce that α ∈ L 2 k+1,ǫ ′′ (T q sM ) where 0 < ǫ ′′ < ǫ ′ . But then we are done, since there is no exceptional weight between ǫ and ǫ ′′ , and so [16,Lem 7.1] says that Ker ǫ A ∞ = Ker ǫ ′ A ∞ . Notice that this tells us that the function µ → dim Ker µ A ∞ is upper semi-continuous at zero.
The above argument also shows that the function µ → dim Ker µ A * ∞ is upper semi-continuous (in particular at µ = λ − m) and so the set Ker −ǫ+λ−m A * ∞ \Ker λ−m A * ∞ , is nonempty, but its elements are exactly the limiting sections of the extended L 2 -sections of T q ′ s ′M . Therefore , exactly the dimension of the space of limiting sections of extended L 2sections of T q ′ s ′M . This allows us to deduce that Applying this to (4.25) we find that (4.26) ind as claimed.
and denote the index of this operator by ind µ (∂ +∂ * ).
Then (4.28) where χ(N, ν 1,0 M (N )) is the holomorphic Euler characteristic of ν 1,0 M (N ), D is the set of λ ∈ R for which (4.15)-(4.16) has a nontrivial solution and then d(λ) is the dimension of the solution space, η is the η-invariant which we can now define to be

Calculations
In this section we will calculate some of the quantities studied in this article for some examples.
In Section 5.1, we will consider deformations of two-dimensional complex cones in C 4 , both as a Cayley submanifold and a complex submanifold of C 4 . In particular, we will consider Cayley deformations of the cone that are themselves cones. The (real) link of such a complex cone is an associative submanifold of S 7 with its nearly parallel G 2 -structure inherited from the Euclidean Spin (7)-structure on C 4 , and so deforming the cone as a complex or Cayley cone in C 4 is equivalent to deforming the link of the cone as an associative submanifold. Homogeneous associative submanifolds of S 7 were classified by Lotay [18], using the classification of homogeneous submanifolds of S 6 of Mashimo [20]. The deformation theory of these submanifolds was studied by Kawai [13], who explicitly calculated the dimension of the space of infinitesimal associative deformations of these explicit examples using techniques from representation theory. Motivated by these calculations, in Section 5.2 we will apply the analysis of the earlier sections to compute the dimension of the space of infinitesimal Cayley conical deformations of the complex cones with these links, and check that these calculations match. We will be able to see explicitly which infinitesimal deformations correspond to complex deformations of the cone and which are Cayley but not complex deformations. In particular we will see that complex infinitesimal deformations and Cayley infinitesimal deformations of a two-dimensional complex submanifold of a Calabi-Yau four-fold are not the same in general. Finally, in Section 5.3 we will compute the η-invariant for a complex cone in C 4 .

Cone deformations.
Let C be a two-dimensional complex cone in C 4 . Let v be a normal vector field on C. If v is sufficiently small, we can apply the tubular neighbourhood theorem for cones 3.3 to identify v with a deformation of C.
where v 1 ∈ C ∞ (ν 1,0 C 4 (C)) and v 2 ∈ C ∞ (ν 0,1 C 4 (C)). We know from Proposition 2.5 that v is an infinitesimal Cayley deformation of C if, and only if, where Ω 0 is the standard holomorphic volume form on C 4 and ♯ denotes the musical isomorphism ν * 0,1 C 4 (C) → ν 1,0 C 4 (C). Moreover by Proposition 2.6 v is an infinitesimal complex deformation of C if, and only if, We would like to know what properties v must have in order for the deformation of C corresponding to v to be a cone itself. By Proposition 3.3, in which we constructed the tubular neighbourhood of a cone, we constructed a map Ξ C : where V C ⊆ ν R 8 (C) contains the zero section and T C ⊆ C 4 contains C. We constructed an action of R + on ν C 4 (C) satisfying |t · v| = t|v|, and the map Ξ C satisfies Ξ C (tr, l, tr · v(r, l)) = tΞ C (r, l, v(r, l)).
Therefore, to guarantee that Ξ C • v is a cone in C 4 , we must have that v(r, l) = r ·v(l), for somev ∈ C ∞ (ν S 7 (L)). In this case, for all r ∈ R + . Choosing a metric on ν C 4 (C) that is independent of r, we see that r ·v(l) = rv(l).
Therefore the dimension of the space of infinitesimal conical Cayley deformations of C is equal to the dimensions of the spaces of solutions to the eigenproblems (4.6) and (4.10)-(4.11) with λ = 1. As remarked after the statement of Proposition 4.1, this particular eigenspace can be identified with the space of infinitesimal associative deformations of the link of the cone in S 7 with its nearly parallel G 2 -structure. This problem was studied by Kawai [13], who computed the dimension of these spaces for a range of examples. In terms of the work done here, this is equivalent to solving the eigenproblem (4.6) when λ = 1. We will study the eigenproblem (4.10)-(4.11) for the three examples of complex cones that were studied by Kawai in his paper. Our analysis will allow us to see directly the difference between the infinitesimal conical Cayley and complex deformations of a cone, and we hope that the complex geometry will make these calculations simpler.
5.1.1. Example 1: L 1 = S 3 . The first example is the simplest, being just a vector subspace (with the zero vector removed). We take where C 1 is the complex cone, L 1 is the real link of C 1 and Σ 1 is the complex link of C 1 .
Our second example is a little less trivial. Take Then it can be shown [13,Ex 6.6] that the link of C 2 , L 2 , is isomorphic to the quotient group SU (2)/Z 2 .

5.1.3.
Example 3: L 3 ∼ = SU (2)/Z 3 . Our third example is the most complicated to state, but is certainly the most interesting.
Define the cone C 3 to be the cone over the submanifold L 3 of S 7 which is defined as follows: consider the following action of SU (2) on C 4 where a, b ∈ C satisfy |a| 2 + |b| 2 = 1. We define L 3 to be the orbit of the above action around the point (1, 0, 0, 0) T , that is, where a, b ∈ C satisfy |a| 2 + |b| 2 = 1. We see that for The complex link of the cone C 3 over L 3 is which is known as the twisted cubic in CP 3 .
This is a particularly interesting example for the following reason [18,Ex 5.8]. Define L 3 (θ) to be the orbit of the above group action around the point (cos θ, 0, 0, sin θ) T . Then L 3 (θ) is associative for θ ∈ [0, π 4 ]. As noted above, L 3 (0) = L 3 is the real link of a complex cone, however, L 3 ( π 4 ) is the link of a special Lagrangian cone. Therefore there exists a family of Cayley cones in C 4 , including both a complex cone and a special Lagrangian cone, that are related by a group action.

5.2.
Calculations. We will now study the eigenvalue problem (4.10)-(4.11) with λ = 1 for C 1 , C 2 and C 3 defined above. Recall that by Proposition 4.4 we can study the eigenproblem (4.15)-(4.16) with λ = 1 on the complex link instead to make our calculations easier. We first explain how to count infinitesimal conical complex deformations and infinitesimal conical Cayley but non complex deformations of a complex cone.
Proposition 5.4. Let C be a complex cone in C 4 with real link L and complex link Σ. Infinitesimal complex conical deformations of C in C 4 are given by holomorphic sections of ν 1,0 CP 3 (Σ). Infinitesimal Cayley conical deformations of C that are not complex are given by v ∈ C ∞ (ν 1,0 where −4 < m < 0.
Proof. We know that infinitesimal complex deformations C will lie in the kernel of∂ C or∂ * C . Recall that these spaces are isomorphic and so we expect them to have the same dimension. Examining the proof of Proposition 4.2 and comparing to Proposition 4.4, we see that infinitesimal complex deformations of C are given by holomorphic sections of ν 1,0 . Since infinitesimal conical deformations of C will correspond to λ = 1 here, we see that infinitesimal complex conical deformations of C correspond to holomorphic sections of ν 1,0 CP 3 (Σ), and antiholomorphic sections of So we see that infinitesimal conical complex deformations of C arise from holomorphic sections of the holomorphic normal bundle of the complex link in CP 3 . The dimension of the space of infinitesimal conical complex deformations of C is then equal to the real dimension (or twice the complex dimension) of the space of holomorphic sections of the holomorphic normal bundle of the complex link.
Finally, we see that any remaining infinitesimal conical Cayley deformations of C must satisfy the eigenproblem (4.15)-(4.16) with λ = 1 and m = 0, −4. Applying∂ * Σ to (4.15) and using (4.16), we see that the remaining infinitesimal conical Cayley deformations of C are given by v ∈ C ∞ (ν 1,0 While we can apply the Hirzebruch-Riemann-Roch theorem 4.3 to count holomorphic sections of holomorphic vector bundles, solving eigenproblems for the Laplacian acting sections of vector bundles such as (5.1) is somewhat more difficult, especially since the degree of the line bundle we consider appears in the eigenvalue itself. Such problems have been studied, however, and we will make use of the following result of López Almorox and Tejero Prieto on eigenvalues of the∂ Σ -Laplacian acting on sections of holomorphic line bundles over CP 1 equipped with a metric of constant scalar curvature.
Theorem 5.5 ([2, Thm 5.1]). Let K be a Hermitian line bundle over Σ, where Σ is CP 1 with metric of constant scalar curvature κ equipped with a unitary harmonic connection ∇ K of curvature F ∇ K = −iBω Σ for some B ∈ R. Then the spectrum of the operator The space of eigensections of 2∂ * Σ∂ Σ with eigenvalue λ q is identified with the space of holomorphic sections of To calculate the dimension of the space of infinitesimal conical Cayley deformations of the cone C 1 = C 2 , which as real link L 1 = S 3 and complex link Σ 1 = CP 1 , we will apply Proposition 5.4. We first calculate the dimension of the space of holomorphic sections of which by the Hirzebruch-Riemann-Roch theorem 4.3 has dimension four. Therefore, the dimension of the space of infinitesimal conical complex deformations of C 1 is eight. Now we study the eigenproblem for v ∈ C ∞ (ν 1,0 and −4 < m < 0. We can apply Theorem 5.5 to solve (5.2) as long as the connection on O CP 3 (m + 1)| Σ ⊕ O CP 3 (m + 1)| Σ takes the form where ∇ i are connections on O CP 3 (m + 1)| Σ . This is the case here, as can be seen from the relation between the connection on the normal bundle of Σ 1 in CP 3 and the connection on the normal bundle of L 1 in S 7 (see [26,Lem 1]) and the fact that the normal bundle of L 1 in S 7 is trivial.
Therefore, by Theorem 5.5, solving (5. We sum this up in a proposition.
Proposition 5.6. The real dimension of the space of infinitesimal conical Cayley deformations of C 1 in C 4 is twelve. The real dimension of the space of infinitesimal conical complex deformations of C 1 in C 4 is eight.
(2)| Σ , which by the Hirzebruch-Riemann-Roch theorem 4.3 has dimension eight, and so we deduce that the space of infinitesimal conical complex deformations of C 2 has dimension sixteen.
Proposition 5.7. The real dimension of the space of infinitesimal conical Cayley deformations of C 2 in C 4 is twenty-two. The real dimension of the space of infinitesimal conical complex deformations of C 2 in C 4 is sixteen.
Remark. The dimension of Spin(7)/SU (4) is six, which implies that the six Cayley but not complex infinitesimal conical deformations of C 2 are just rigid motions induced by the action of Spin(7) on R 8 .

5.2.3.
Example 3: L 3 ∼ = SU (2)/Z 3 . Finally, we compute the dimension of the space of infinitesimal conical Cayley deformations of C 3 in C 4 , which has real link L 3 ∼ = SU (2)/Z 3 and complex link Σ 3 as defined in Section 5.1.3.
The dimension of the space of holomorphic sections of denotes the line bundle of degree n over Σ 3 . By Hirzebruch-Riemann-Roch, Theorem 4.3, this space has dimension twelve, and so the dimension of the space of infinitesimal conical complex deformations of C 3 in C 4 has dimension twenty-four.
Finally, we check the case that m = 4/3. In this case, for the eigenval- we must have a = −2. However, in this case, the eigenvalue is equal to −4, which is negative and therefore not a possible eigenvalue of∂ * Σ 3∂ Σ 3 on sections of O Σ 3 (6).
We have found a total of six infinitesimal conical Cayley deformations of C 3 that are not complex. Remark. Similarly to Proposition 5.7 we have six infinitesimal conical Cayley deformations of C 3 which are not complex, which again implies that these deformations are just rigid motions.

5.3.
Calculating the η-invariant for an example. The final calculation in this article is to compute the η-invariant of the Atiyah-Patodi-Singer index theorem 4.5 for one of the examples we considered in Section 5.1. This will help us to calculate (what we expect to be) the codimension of the space of conically singular complex CS deformations of a CS complex surface N at C with rate µ in a Calabi-Yau manifold M inside the space of all complex deformations of N , for a certain cone C in C 4 , using Theorem 4.8.
It remains to compute the multiplicity of λ as an eigenvalue of where v is a section of O CP 3 (m + 1)| Σ 1 ⊕ O CP 3 (m + 1)| Σ 1 and λ = 1 + m or −3 − m. Theorem 5.5 tells us that this is equivalent to solving the algebraic equation where q is a positive integer.
It can be computed that the multiplicity of integer λ > 0 as an eigenvalue of (5.19) is 2λ(λ + 1) and the multiplicity of integer λ < −2 as an eigenvalue of (5.19) is 2(λ + 2)(λ + 1). So we have that where ζ is the Riemann zeta function.
We have that the multiplicity of the zero eigenvalue in this case in four. So we have found that η(0) + h 2 = 3 2 .
Notice that [α] is the Levi-Civita connection of L and 1 2 ([α − ω] + + [γ] − ) defines the induced connection on the normal bundle of L in S 7 . We have that h defines the second fundamental form II L ∈ C ∞ (S 2 T * L; ν(L)) of L in S 7 , writing II L := h a jk f a ⊗ ω j ω k . Since the associative submanifolds of S 7 that we are considering are S 1bundles over complex curves, we may reduce the structure equations of L.
Proof. This follows from supposing that the complex structure of C 4 acts on C as follows: Jx = e 1 ; Je 2 = e 3 ; Jf 4 = f 5 ; Jf 6 = f 7 .
Acknowledgements. I would like to thank Jason Lotay for his help, guidance and feedback on this project. I would also like to thank Alexei Kovalev, Yng-Ing Lee and Julius Ross for comments on my PhD thesis, from which this work is taken. This work was supported by the UK Engineering and Physical Sciences Research Council (EPSRC) grant EP/H023348/1 for the University of Cambridge Centre for Doctoral Training, the Cambridge Centre for Analysis.