# Two simple ideas from calculus applied to Riemannian geometry

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1 Calibrated Geometries and Special Holonomy p. 1/29 Two simple ideas from calculus applied to Riemannian geometry Spiro Karigiannis Department of Pure Mathematics, University of Waterloo

2 Introduction Calibrated Geometries and Special Holonomy p. 2/29

3 Calibrated Geometries and Special Holonomy p. 2/29 Introduction We will consider two simple ideas from calculus and see how they arise in Riemannian geometry:

4 Calibrated Geometries and Special Holonomy p. 2/29 Introduction We will consider two simple ideas from calculus and see how they arise in Riemannian geometry: finding some solutions to a second order differential equation by instead solving a first order equation

5 Calibrated Geometries and Special Holonomy p. 2/29 Introduction We will consider two simple ideas from calculus and see how they arise in Riemannian geometry: finding some solutions to a second order differential equation by instead solving a first order equation in the presence of additional structure, finding the absolute minima of some function (and not just the critical points) without needing to take derivatives

6 Calibrated Geometries and Special Holonomy p. 3/29 Introduction Example. Consider the differential equation d 2 dx 2f = k2 f

7 Calibrated Geometries and Special Holonomy p. 3/29 Introduction Example. Consider the differential equation d 2 dx 2f = k2 f We can rewrite this equation as ( ) ( ) d d dx + k dx k f = 0

8 Calibrated Geometries and Special Holonomy p. 3/29 Introduction Example. Consider the differential equation d 2 dx 2f = k2 f We can rewrite this equation as ( ) ( ) d d dx + k dx k f = 0 d Thus any solution of dxf = kf will be a solution to our original equation, but not conversely.

9 Calibrated Geometries and Special Holonomy p. 4/29 Introduction We will see two situations in Riemannian geometry where we can find some of the solutions to a second order differential equation, by instead solving a first order equation.

10 Calibrated Geometries and Special Holonomy p. 4/29 Introduction We will see two situations in Riemannian geometry where we can find some of the solutions to a second order differential equation, by instead solving a first order equation. It will not be the case in general that the second order equation factors into the composition of two first order differential operators. (Although in some cases this can be shown to happen.)

11 Calibrated Geometries and Special Holonomy p. 5/29 Introduction Example. Suppose we want the absolute minima of a function f on a domain without boundary.

12 Calibrated Geometries and Special Holonomy p. 5/29 Introduction Example. Suppose we want the absolute minima of a function f on a domain without boundary. By solving the algebraic equation d dx f = 0 we can find all possible candidate points, but they might be maxima or saddle points.

13 Calibrated Geometries and Special Holonomy p. 5/29 Introduction Example. Suppose we want the absolute minima of a function f on a domain without boundary. By solving the algebraic equation d dx f = 0 we can find all possible candidate points, but they might be maxima or saddle points. Sometimes we can avoid the need to differentiate and at the same time find the actual minima.

14 Calibrated Geometries and Special Holonomy p. 6/29 Introduction For example, suppose that for some reason we know f(x) 1 for all x in our domain.

15 Calibrated Geometries and Special Holonomy p. 6/29 Introduction For example, suppose that for some reason we know f(x) 1 for all x in our domain. Then by solving the algebraic equation f(x) = 1 we get absolute minima of f, if such points exist.

16 Calibrated Geometries and Special Holonomy p. 6/29 Introduction For example, suppose that for some reason we know f(x) 1 for all x in our domain. Then by solving the algebraic equation f(x) = 1 we get absolute minima of f, if such points exist. This phenomenon also occurs in Riemannian geometry. However, we will be trying to minimize a functional, so the points on which it is defined will be a space of functions (or tensors.)

17 Calibrated Geometries and Special Holonomy p. 7/29 Introduction The definition of the functionals will involve derivatives, so we will not completely avoid taking derivatives. But we will be able to take one less derivative than we would normally need.

18 Calibrated Geometries and Special Holonomy p. 7/29 Introduction The definition of the functionals will involve derivatives, so we will not completely avoid taking derivatives. But we will be able to take one less derivative than we would normally need. More precisely, we will have functionals whose critical points (potential minima) are given by solutions to certain second order equations.

19 Calibrated Geometries and Special Holonomy p. 7/29 Introduction The definition of the functionals will involve derivatives, so we will not completely avoid taking derivatives. But we will be able to take one less derivative than we would normally need. More precisely, we will have functionals whose critical points (potential minima) are given by solutions to certain second order equations. In some cases, solutions to a first order equation will be a subset of the solutions to the second order equation, and they will be actual minima.

20 Riemannian geometry Calibrated Geometries and Special Holonomy p. 8/29

21 Calibrated Geometries and Special Holonomy p. 8/29 Riemannian geometry Let M be a smooth manifold with Riemannian metric g. This is a smoothly varying, positive definite inner product g p on each tangent space T p M of M, for each point p in M. The metric g determines some curvature tensors on M:

22 Calibrated Geometries and Special Holonomy p. 8/29 Riemannian geometry Let M be a smooth manifold with Riemannian metric g. This is a smoothly varying, positive definite inner product g p on each tangent space T p M of M, for each point p in M. The metric g determines some curvature tensors on M: The Riemann curvature R ijkl

23 Calibrated Geometries and Special Holonomy p. 8/29 Riemannian geometry Let M be a smooth manifold with Riemannian metric g. This is a smoothly varying, positive definite inner product g p on each tangent space T p M of M, for each point p in M. The metric g determines some curvature tensors on M: The Riemann curvature R ijkl The Ricci curvature R ij (a symmetric tensor)

24 Calibrated Geometries and Special Holonomy p. 8/29 Riemannian geometry Let M be a smooth manifold with Riemannian metric g. This is a smoothly varying, positive definite inner product g p on each tangent space T p M of M, for each point p in M. The metric g determines some curvature tensors on M: The Riemann curvature R ijkl The Ricci curvature R ij (a symmetric tensor) The scalar curvature S (a smooth function)

25 Calibrated Geometries and Special Holonomy p. 9/29 Riemannian geometry Natural question: When does M admit a metric g with special curvature properties? For example:

26 Calibrated Geometries and Special Holonomy p. 9/29 Riemannian geometry Natural question: When does M admit a metric g with special curvature properties? For example: R ijkl = 0, a flat metric; locally isometric to R n

27 Calibrated Geometries and Special Holonomy p. 9/29 Riemannian geometry Natural question: When does M admit a metric g with special curvature properties? For example: R ijkl = 0, a flat metric; locally isometric to R n R ij = λg ij, an Einstein metric

28 Calibrated Geometries and Special Holonomy p. 9/29 Riemannian geometry Natural question: When does M admit a metric g with special curvature properties? For example: R ijkl = 0, a flat metric; locally isometric to R n R ij = λg ij, an Einstein metric R ij = 0, a Ricci-flat metric

29 Calibrated Geometries and Special Holonomy p. 9/29 Riemannian geometry Natural question: When does M admit a metric g with special curvature properties? For example: R ijkl = 0, a flat metric; locally isometric to R n R ij = λg ij, an Einstein metric R ij = 0, a Ricci-flat metric S = C, a constant scalar curvature metric

30 Calibrated Geometries and Special Holonomy p. 9/29 Riemannian geometry Natural question: When does M admit a metric g with special curvature properties? For example: R ijkl = 0, a flat metric; locally isometric to R n R ij = λg ij, an Einstein metric R ij = 0, a Ricci-flat metric S = C, a constant scalar curvature metric These are all second order, non-linear PDE s on M, in general very difficult to solve. When M is compact, we only expect to find exact solutions in situations with a high degree of symmetry.

31 Calibrated Geometries and Special Holonomy p. 10/29 Riemannian geometry Sometimes these equations arise naturally as the Euler-Lagrange equations for the critical points of some functional.

32 Calibrated Geometries and Special Holonomy p. 10/29 Riemannian geometry Sometimes these equations arise naturally as the Euler-Lagrange equations for the critical points of some functional. Example: Einstein metrics are critical points of F[g] = S g Vol g This is the Total Scalar Curvature functional. M

33 Calibrated Geometries and Special Holonomy p. 10/29 Riemannian geometry Sometimes these equations arise naturally as the Euler-Lagrange equations for the critical points of some functional. Example: Einstein metrics are critical points of F[g] = S g Vol g This is the Total Scalar Curvature functional. Thus a minimum of F[g] is an Einstein metric, but not conversely (could be a max or a saddle.) M

34 Calibrated Geometries and Special Holonomy p. 11/29 Riemannian geometry Let N be a k-dimensional submanifold of M. There is a second fundamental form h, which is a symmetric tensor on N with values in the normal bundle of N in M, measuring a kind of curvature.

35 Calibrated Geometries and Special Holonomy p. 11/29 Riemannian geometry Let N be a k-dimensional submanifold of M. There is a second fundamental form h, which is a symmetric tensor on N with values in the normal bundle of N in M, measuring a kind of curvature. Again, one can consider submanifolds with special curvature properties. For example:

36 Calibrated Geometries and Special Holonomy p. 11/29 Riemannian geometry Let N be a k-dimensional submanifold of M. There is a second fundamental form h, which is a symmetric tensor on N with values in the normal bundle of N in M, measuring a kind of curvature. Again, one can consider submanifolds with special curvature properties. For example: h ij = 0, a totally geodesic submanifold

37 Calibrated Geometries and Special Holonomy p. 11/29 Riemannian geometry Let N be a k-dimensional submanifold of M. There is a second fundamental form h, which is a symmetric tensor on N with values in the normal bundle of N in M, measuring a kind of curvature. Again, one can consider submanifolds with special curvature properties. For example: h ij = 0, a totally geodesic submanifold Trace(h ij ) = 0, a zero mean curvature submanifold; also called minimal

38 Calibrated Geometries and Special Holonomy p. 11/29 Riemannian geometry Let N be a k-dimensional submanifold of M. There is a second fundamental form h, which is a symmetric tensor on N with values in the normal bundle of N in M, measuring a kind of curvature. Again, one can consider submanifolds with special curvature properties. For example: h ij = 0, a totally geodesic submanifold Trace(h ij ) = 0, a zero mean curvature submanifold; also called minimal Trace(h ij ) = C, a CMC submanifold

39 Calibrated Geometries and Special Holonomy p. 12/29 Riemannian geometry These are second order, non-linear PDE s.

40 Calibrated Geometries and Special Holonomy p. 12/29 Riemannian geometry These are second order, non-linear PDE s. Again, they can be critical points of a functional. Minimal submanifolds are critical points of the Volume functional Vol N which is why they are called minimal. N

41 Calibrated Geometries and Special Holonomy p. 12/29 Riemannian geometry These are second order, non-linear PDE s. Again, they can be critical points of a functional. Minimal submanifolds are critical points of the Volume functional Vol N which is why they are called minimal. N Minimal just means critical. It could be a max or a saddle. Actual minima are called minimizing.

42 Special Holonomy Calibrated Geometries and Special Holonomy p. 13/29

43 Calibrated Geometries and Special Holonomy p. 13/29 Special Holonomy Let (M,g) be a Riemannian manifold. There is a covariant derivative which enables us to differentiate any tensor in the direction of a vector field V. Then we can define parallel transport:

44 Calibrated Geometries and Special Holonomy p. 13/29 Special Holonomy Let (M,g) be a Riemannian manifold. There is a covariant derivative which enables us to differentiate any tensor in the direction of a vector field V. Then we can define parallel transport: Given a tangent vector X p at a point p on M, and a closed loop γ : [0, 1] M at p, define τ γ,p : T p M T p M X p X γ(1) where X γ(t) is in T γ(t) M, and γ (t)x γ(t) = 0 0 t 1

45 Calibrated Geometries and Special Holonomy p. 14/29 Special Holonomy This involves solving a linear first order ODE, and hence can always be done. It is easy to see that the map τ γ,p is in GL(T p M).

46 Calibrated Geometries and Special Holonomy p. 14/29 Special Holonomy This involves solving a linear first order ODE, and hence can always be done. It is easy to see that the map τ γ,p is in GL(T p M). In fact, because τ γ1 γ 2 = τ γ1 τ γ2 and τ γ 1 = τ 1 γ, the set of all such τ γ s at a point p is a subgroup H p of GL(T p M), called the holonomy group at p.

47 Calibrated Geometries and Special Holonomy p. 14/29 Special Holonomy This involves solving a linear first order ODE, and hence can always be done. It is easy to see that the map τ γ,p is in GL(T p M). In fact, because τ γ1 γ 2 = τ γ1 τ γ2 and τ γ 1 = τ 1 γ, the set of all such τ γ s at a point p is a subgroup H p of GL(T p M), called the holonomy group at p. If M is connected, simply-connected, and oriented, then H p = Hq for any p, q in M. Hence, up to isomorphism, we can define the holonomy group H of (M,g). It is a subgroup of GL(n, R).

48 Calibrated Geometries and Special Holonomy p. 15/29 Special Holonomy A tensor T on M is called parallel if V T = 0 for all vector fields V.

49 Calibrated Geometries and Special Holonomy p. 15/29 Special Holonomy A tensor T on M is called parallel if V T = 0 for all vector fields V. Theorem. The holonomy group H is the subgroup of GL(n, R) preserving all parallel tensors on M.

50 Calibrated Geometries and Special Holonomy p. 15/29 Special Holonomy A tensor T on M is called parallel if V T = 0 for all vector fields V. Theorem. The holonomy group H is the subgroup of GL(n, R) preserving all parallel tensors on M. Since the metric g and volume form Vol are always parallel, the holonomy group is in fact a subgroup of SO(n). But the holonomy can be a strictly smaller subgroup if there exist other parallel tensors.

51 Calibrated Geometries and Special Holonomy p. 16/29 Special Holonomy Berger Classification. Let M be irreducible, and not locally symmetric. The holonomy can only be:

52 Calibrated Geometries and Special Holonomy p. 16/29 Special Holonomy Berger Classification. Let M be irreducible, and not locally symmetric. The holonomy can only be: H dim Name Tensors SO(n) n Riemannian U(m) 2m Kähler ω SU(m) 2m Calabi-Yau ω, Ω Sp(m) 4m hyper-kähler ω I, ω J, ω K Sp(m) Sp(1) 4m quaternionic-kähler σ G 2 7 G 2 manifold ϕ, ψ Spin(7) 8 Spin(7) manifold Φ

53 Calibrated Geometries and Special Holonomy p. 17/29 Special Holonomy Some examples of special holonomy manifolds:

54 Calibrated Geometries and Special Holonomy p. 17/29 Special Holonomy Some examples of special holonomy manifolds: Projective varieties are all Kähler manifolds

55 Calibrated Geometries and Special Holonomy p. 17/29 Special Holonomy Some examples of special holonomy manifolds: Projective varieties are all Kähler manifolds By Yau s solution of the Calabi conjecture, any compact Kähler manifold with vanishing first Chern class admits a Calabi-Yau metric

56 Calibrated Geometries and Special Holonomy p. 17/29 Special Holonomy Some examples of special holonomy manifolds: Projective varieties are all Kähler manifolds By Yau s solution of the Calabi conjecture, any compact Kähler manifold with vanishing first Chern class admits a Calabi-Yau metric (Bryant-Salamon, 1989) Non-compact G 2 and Spin(7) manifolds: Λ 2 (S 4 ), Λ 2 (CP 2 ), /S(S 4 )

57 Calibrated Geometries and Special Holonomy p. 17/29 Special Holonomy Some examples of special holonomy manifolds: Projective varieties are all Kähler manifolds By Yau s solution of the Calabi conjecture, any compact Kähler manifold with vanishing first Chern class admits a Calabi-Yau metric (Bryant-Salamon, 1989) Non-compact G 2 and Spin(7) manifolds: Λ 2 (S 4 ), Λ 2 (CP 2 ), /S(S 4 ) (Joyce, 1994) Compact G 2 and Spin(7) manifolds: obtained by resolving orbifolds

58 Calibrated Geometries and Special Holonomy p. 18/29 Special Holonomy Ambrose-Singer Theorem. The Lie algebra h of H is generated by the Riemann curvature tensor R ijkl.

59 Calibrated Geometries and Special Holonomy p. 18/29 Special Holonomy Ambrose-Singer Theorem. The Lie algebra h of H is generated by the Riemann curvature tensor R ijkl. Thus, if the holonomy group H is small, this puts restrictions on the Riemann curvature. In fact:

60 Calibrated Geometries and Special Holonomy p. 18/29 Special Holonomy Ambrose-Singer Theorem. The Lie algebra h of H is generated by the Riemann curvature tensor R ijkl. Thus, if the holonomy group H is small, this puts restrictions on the Riemann curvature. In fact: Calabi-Yau, hyper-kähler, G 2 and Spin(7) manifolds are always Ricci-flat

61 Calibrated Geometries and Special Holonomy p. 18/29 Special Holonomy Ambrose-Singer Theorem. The Lie algebra h of H is generated by the Riemann curvature tensor R ijkl. Thus, if the holonomy group H is small, this puts restrictions on the Riemann curvature. In fact: Calabi-Yau, hyper-kähler, G 2 and Spin(7) manifolds are always Ricci-flat Quaternionic-Kähler manifolds are Einstein

62 Calibrated Geometries and Special Holonomy p. 18/29 Special Holonomy Ambrose-Singer Theorem. The Lie algebra h of H is generated by the Riemann curvature tensor R ijkl. Thus, if the holonomy group H is small, this puts restrictions on the Riemann curvature. In fact: Calabi-Yau, hyper-kähler, G 2 and Spin(7) manifolds are always Ricci-flat Quaternionic-Kähler manifolds are Einstein The holonomy condition is a first order PDE. Hence the solutions to a first order equation are automatically solutions to a more complicated second order equation.

63 Calibrated Geometry Calibrated Geometries and Special Holonomy p. 19/29

64 Calibrated Geometries and Special Holonomy p. 19/29 Calibrated Geometry Suppose (M,g) is a Riemannian manifold of dimension n. Let 0 < k < n.

65 Calibrated Geometries and Special Holonomy p. 19/29 Calibrated Geometry Suppose (M,g) is a Riemannian manifold of dimension n. Let 0 < k < n. Definition. A k-form α on M is a calibration if dα = 0 and whenever {e 1,...,e k } is an oriented orthonormal basis for a tangent k-plane in T p M, α p (e 1,...,e k ) 1 = Vol p (e 1,...,e k )

66 Calibrated Geometries and Special Holonomy p. 19/29 Calibrated Geometry Suppose (M,g) is a Riemannian manifold of dimension n. Let 0 < k < n. Definition. A k-form α on M is a calibration if dα = 0 and whenever {e 1,...,e k } is an oriented orthonormal basis for a tangent k-plane in T p M, α p (e 1,...,e k ) 1 = Vol p (e 1,...,e k ) An oriented k-plane V p in T p M is calibrated if α p (V p ) = 1 (that is, if the upper bound is attained)

67 Calibrated Geometries and Special Holonomy p. 20/29 Calibrated Geometry An oriented k-dimensional submanifold N k is a calibrated submanifold if each oriented tangent space to N is calibrated. This is equivalent to α N = Vol N which is a first order PDE.

68 Calibrated Geometries and Special Holonomy p. 20/29 Calibrated Geometry An oriented k-dimensional submanifold N k is a calibrated submanifold if each oriented tangent space to N is calibrated. This is equivalent to α N = Vol N which is a first order PDE. Fundamental theorem of calibrated geometry: Suppose N is calibrated with respect to α. Then N is absolutely minimizing in its homology class.

69 Calibrated Geometries and Special Holonomy p. 21/29 Calibrated Geometry Proof. Suppose [N] = [N ], so N N = L. Then Vol(N) = Vol N = α = α + α N N N L = α + dα Vol N = Vol(N ) N L N

70 Calibrated Geometries and Special Holonomy p. 21/29 Calibrated Geometry Proof. Suppose [N] = [N ], so N N = L. Then Vol(N) = Vol N = α = α + α N N N L = α + dα Vol N = Vol(N ) N L N Thus, calibrated submanifolds are solutions to a first order differential equation which are automatically solutions to the second order equation for minimal submanifolds. That is, they have zero mean curvature.

71 Calibrated Geometries and Special Holonomy p. 22/29 Calibrated Geometry The main examples of calibrated geometries are:

72 Calibrated Geometries and Special Holonomy p. 22/29 Calibrated Geometry The main examples of calibrated geometries are: Let M be a Kähler manifold, with Kähler form ω. Then α = 1 p! ωp is a calibration, and the calibrated submanifolds are 2p-dimensional complex submanifolds.

73 Calibrated Geometries and Special Holonomy p. 22/29 Calibrated Geometry The main examples of calibrated geometries are: Let M be a Kähler manifold, with Kähler form ω. Then α = 1 p! ωp is a calibration, and the calibrated submanifolds are 2p-dimensional complex submanifolds. Let M be Calabi-Yau, with Kähler form ω and holomorphic volume form Ω. Then α = Re(Ω) is a calibration. Calibrated submanifolds are special Lagrangian. They are Lagrangian with respect to the symplectic form ω, and satisfy an additional metric condition.

74 Calibrated Geometries and Special Holonomy p. 23/29 Calibrated Geometry Let M 7 be a G 2 manifold. The 3-form ϕ and 4-form ψ are both calibrations. Calibrated submanifolds are called associative and coassociative, respectively.

75 Calibrated Geometries and Special Holonomy p. 23/29 Calibrated Geometry Let M 7 be a G 2 manifold. The 3-form ϕ and 4-form ψ are both calibrations. Calibrated submanifolds are called associative and coassociative, respectively. Let M 8 be a Spin(7) manifold. The 4-form Φ is a calibration. Calibrated submanifolds are called Cayley.

76 Calibrated Geometries and Special Holonomy p. 23/29 Calibrated Geometry Let M 7 be a G 2 manifold. The 3-form ϕ and 4-form ψ are both calibrations. Calibrated submanifolds are called associative and coassociative, respectively. Let M 8 be a Spin(7) manifold. The 4-form Φ is a calibration. Calibrated submanifolds are called Cayley. Explicit examples have been found in R n, and in some non-compact, non-flat manifolds.

77 Calibrated Geometries and Special Holonomy p. 23/29 Calibrated Geometry Let M 7 be a G 2 manifold. The 3-form ϕ and 4-form ψ are both calibrations. Calibrated submanifolds are called associative and coassociative, respectively. Let M 8 be a Spin(7) manifold. The 4-form Φ is a calibration. Calibrated submanifolds are called Cayley. Explicit examples have been found in R n, and in some non-compact, non-flat manifolds. The only compact examples known arise in situations with a high degree of symmetry.

78 Analogies between the two subjects Calibrated Geometries and Special Holonomy p. 24/29

79 Calibrated Geometries and Special Holonomy p. 24/29 Analogies between the two subjects Manifolds of special holonomy and calibrated submanifolds both give first order examples of solutions to second order equations.

80 Calibrated Geometries and Special Holonomy p. 24/29 Analogies between the two subjects Manifolds of special holonomy and calibrated submanifolds both give first order examples of solutions to second order equations. We replace a quasi-linear 2nd order equation by a fully nonlinear 1st order equation, so it is not necessarily easier to solve.

81 Calibrated Geometries and Special Holonomy p. 24/29 Analogies between the two subjects Manifolds of special holonomy and calibrated submanifolds both give first order examples of solutions to second order equations. We replace a quasi-linear 2nd order equation by a fully nonlinear 1st order equation, so it is not necessarily easier to solve. The main examples of calibrations are all defined on manifolds with special holonomy. The reasons are not very well understood, but are related to existence of parallel spinors.

82 Calibrated Geometries and Special Holonomy p. 25/29 Analogies between the two subjects We are missing a classification theorem of calibrations. Is there some kind of analogue of the Ambrose-Singer theorem? This appears to be much more complicated.

83 Calibrated Geometries and Special Holonomy p. 25/29 Analogies between the two subjects We are missing a classification theorem of calibrations. Is there some kind of analogue of the Ambrose-Singer theorem? This appears to be much more complicated. Is there a simple analogue to the fundamental theorem of calibrated geometry for metrics with special holonomy? Work of M.Wang and Dai-Wang-Wei shows that special holonomy metrics are stable critical points of the total scalar curvature functional with respect to some types of deformations.

84 Mirror Symmetry Calibrated Geometries and Special Holonomy p. 26/29

85 Calibrated Geometries and Special Holonomy p. 26/29 Mirror Symmetry Theories of physics, specifically string theory, 11-dimensional supergravity (M-theory), and F-theory require for their description compact manifolds of dimensions 6, 7, and 8, respectively.

86 Calibrated Geometries and Special Holonomy p. 26/29 Mirror Symmetry Theories of physics, specifically string theory, 11-dimensional supergravity (M-theory), and F-theory require for their description compact manifolds of dimensions 6, 7, and 8, respectively. Supersymmetry forces these manifolds to have parallel spinors, which says they are Calabi-Yau, G 2, and Spin(7) manifolds, respectively.

87 Calibrated Geometries and Special Holonomy p. 26/29 Mirror Symmetry Theories of physics, specifically string theory, 11-dimensional supergravity (M-theory), and F-theory require for their description compact manifolds of dimensions 6, 7, and 8, respectively. Supersymmetry forces these manifolds to have parallel spinors, which says they are Calabi-Yau, G 2, and Spin(7) manifolds, respectively. Physics also predicts there sometimes exist dual manifolds which are different but actually describe the same physics. These are called mirror manifolds.

88 Calibrated Geometries and Special Holonomy p. 27/29 Mirror Symmetry Strominger-Yau-Zaslow Conjecture, Let M 6 be a compact Calabi-Yau 3-fold which admits a mirror. There should exist a fibration f : M 6 L 3. The generic (non-singular) fibres of f are special Lagrangian torii in M, and the mirror M is obtained from M by taking the dual torus over each non-singular fibre, and compactifying.

89 Calibrated Geometries and Special Holonomy p. 27/29 Mirror Symmetry Strominger-Yau-Zaslow Conjecture, Let M 6 be a compact Calabi-Yau 3-fold which admits a mirror. There should exist a fibration f : M 6 L 3. The generic (non-singular) fibres of f are special Lagrangian torii in M, and the mirror M is obtained from M by taking the dual torus over each non-singular fibre, and compactifying. This conjecture is still not formulated precisely. Some progress has been made, but much remains to be done. (D. Joyce, M. Gross, N.C. Leung, W.D. Ruan, S.T. Yau, E. Zaslow, and many others.)

90 Calibrated Geometries and Special Holonomy p. 28/29 Mirror Symmetry We know there must exist singular fibres.

91 Calibrated Geometries and Special Holonomy p. 28/29 Mirror Symmetry We know there must exist singular fibres. The SYZ conjecture is probably only true near a large complex structure limit point in the moduli space of Calabi-Yau manifolds.

92 Calibrated Geometries and Special Holonomy p. 28/29 Mirror Symmetry We know there must exist singular fibres. The SYZ conjecture is probably only true near a large complex structure limit point in the moduli space of Calabi-Yau manifolds. Similar incomplete conjectures exist for G 2 and Spin(7) manifolds, which involve fibrations by calibrated torii and their dual fibrations.

93 Calibrated Geometries and Special Holonomy p. 28/29 Mirror Symmetry We know there must exist singular fibres. The SYZ conjecture is probably only true near a large complex structure limit point in the moduli space of Calabi-Yau manifolds. Similar incomplete conjectures exist for G 2 and Spin(7) manifolds, which involve fibrations by calibrated torii and their dual fibrations. We need to understand the types of singularities that can arise in calibrated submanifolds, to understand the possible singular fibres. Predictions from physics indicate conical singularities are important.

94 Calibrated Geometries and Special Holonomy p. 29/29 Mirror Symmetry Mirror Symmetry for Calabi-Yau manifolds also involves Gromov-Witten invariants. These are invariants which count holomorphic curves.

95 Calibrated Geometries and Special Holonomy p. 29/29 Mirror Symmetry Mirror Symmetry for Calabi-Yau manifolds also involves Gromov-Witten invariants. These are invariants which count holomorphic curves. Similar theories in the G 2 and Spin(7) cases (invariants which count calibrated submanifolds) are still far from being properly formulated. We expect the analytic difficulty to be much greater.

96 Calibrated Geometries and Special Holonomy p. 29/29 Mirror Symmetry Mirror Symmetry for Calabi-Yau manifolds also involves Gromov-Witten invariants. These are invariants which count holomorphic curves. Similar theories in the G 2 and Spin(7) cases (invariants which count calibrated submanifolds) are still far from being properly formulated. We expect the analytic difficulty to be much greater. Thank you for your attention.

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