# How to minimize without knowing how to differentiate (in Riemannian geometry)

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1 How to minimize without knowing how to differentiate (in Riemannian geometry) Spiro Karigiannis Mathematical Institute, University of Oxford Calibrated Geometries and Special Holonomy p. 1/29

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

3 Introduction In this talk I want to consider two simple ideas from calculus and show you some ways in which they arise in Riemannian geometry: Calibrated Geometries and Special Holonomy p. 2/29

4 Introduction In this talk I want to consider two simple ideas from calculus and show you some ways in which they arise in Riemannian geometry: finding some solutions to a second order differential equation by instead solving a first order equation Calibrated Geometries and Special Holonomy p. 2/29

5 Introduction In this talk I want to consider two simple ideas from calculus and show you some ways in which 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 Calibrated Geometries and Special Holonomy p. 2/29

6 Introduction In this talk I want to consider two simple ideas from calculus and show you some ways in which 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 Let us first consider some simple examples. Calibrated Geometries and Special Holonomy p. 2/29

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

8 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 Calibrated Geometries and Special Holonomy p. 3/29

9 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 Then we see that any solution of d dxf = kf will be a solution to our original equation, but not conversely. Calibrated Geometries and Special Holonomy p. 3/29

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

11 We will see two situations in Riemannian geometry where we can obtain some of the solutions to a second order partial 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.) Calibrated Geometries and Special Holonomy p. 4/29

12 We will see two situations in Riemannian geometry where we can obtain some of the solutions to a second order partial 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.) Now let us consider an example of the second idea. Calibrated Geometries and Special Holonomy p. 4/29

13 Example. Suppose we want to find the absolute minima of some fixed function f on a domain without boundary. Calibrated Geometries and Special Holonomy p. 5/29

14 Example. Suppose we want to find the absolute minima of some fixed 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 inflection points. Calibrated Geometries and Special Holonomy p. 5/29

15 Example. Suppose we want to find the absolute minima of some fixed 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 inflection points. In some situations we can completely avoid the need to differentiate and at the same time find the actual minima. Calibrated Geometries and Special Holonomy p. 5/29

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

17 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 obtain the absolute minima of f, if any such points actually exist. Calibrated Geometries and Special Holonomy p. 6/29

18 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 obtain the absolute minima of f, if any such points actually exist. We will see this phenomenon also occur in Riemannian geometry. However, we will be trying to minimize a functional, so the set of points on which it is defined will be a space of functions (or tensors.) Calibrated Geometries and Special Holonomy p. 6/29

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

20 The definition of the functionals will involve derivatives of the arguments, so we will not be able to 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. Calibrated Geometries and Special Holonomy p. 7/29

21 The definition of the functionals will involve derivatives of the arguments, so we will not be able to 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 special situations (in the presence of some additional structure), we can show that solutions to a first order equation are a subset of the solutions to the second order equation, and also that they are actually minima. Calibrated Geometries and Special Holonomy p. 7/29

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

23 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: Calibrated Geometries and Special Holonomy p. 8/29

24 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: Riemann curvature tensor R ijkl Calibrated Geometries and Special Holonomy p. 8/29

25 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: Riemann curvature tensor R ijkl Ricci curvature tensor R ij (a symmetric 2-tensor) Calibrated Geometries and Special Holonomy p. 8/29

26 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: Riemann curvature tensor R ijkl Ricci curvature tensor R ij (a symmetric 2-tensor) Scalar curvature S (a smooth function) Calibrated Geometries and Special Holonomy p. 8/29

27 A natural question to ask is when does M admit a metric g with special curvature properties? For example: Calibrated Geometries and Special Holonomy p. 9/29

28 A natural question to ask is when does M admit a metric g with special curvature properties? For example: R ijkl = 0; a flat metric, M is locally isometric to R n Calibrated Geometries and Special Holonomy p. 9/29

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

30 A natural question to ask is when does M admit a metric g with special curvature properties? For example: R ijkl = 0; a flat metric, M is locally isometric to R n R ij = λg ij ; an Einstein metric R ij = 0; a Ricci-flat metric Calibrated Geometries and Special Holonomy p. 9/29

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

32 A natural question to ask is when does M admit a metric g with special curvature properties? For example: R ijkl = 0; a flat metric, M is locally isometric to R n R ij = λg ij ; an Einstein metric R ij = 0; a Ricci-flat metric S = constant; a constant scalar curvature metric These are all second order, non-linear partial differential equations on M which are in general very difficult to solve. Especially when M is compact, we only expect to find exact solutions in situations with a high degree of symmetry. Calibrated Geometries and Special Holonomy p. 9/29

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

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

35 Sometimes these equations arise naturally as the Euler-Lagrange equations for the critical points of some functional. For example, the Einstein metrics are critical points of F[g] = S g Vol g This is called the Total Scalar Curvature functional. Thus a minimum of F[g] will be an Einstein metric, but the converse need not hold (it could also be a maximum or an inflection point.) M Calibrated Geometries and Special Holonomy p. 10/29

36 Now let N be a k-dimensional immersed submanifold of M. By restriction, g induces a metric g N on N. This determines the second fundamental form h of the immersion, which is a symmetric 2-tensor on N with values in the normal bundle of N in M. Calibrated Geometries and Special Holonomy p. 11/29

37 Now let N be a k-dimensional immersed submanifold of M. By restriction, g induces a metric g N on N. This determines the second fundamental form h of the immersion, which is a symmetric 2-tensor on N with values in the normal bundle of N in M. One again we can consider immersions with special curvature properties. For example: Calibrated Geometries and Special Holonomy p. 11/29

38 Now let N be a k-dimensional immersed submanifold of M. By restriction, g induces a metric g N on N. This determines the second fundamental form h of the immersion, which is a symmetric 2-tensor on N with values in the normal bundle of N in M. One again we can consider immersions with special curvature properties. For example: h ij = 0; a totally geodesic immersion Calibrated Geometries and Special Holonomy p. 11/29

39 Now let N be a k-dimensional immersed submanifold of M. By restriction, g induces a metric g N on N. This determines the second fundamental form h of the immersion, which is a symmetric 2-tensor on N with values in the normal bundle of N in M. One again we can consider immersions with special curvature properties. For example: h ij = 0; a totally geodesic immersion Trace(h ij ) = 0; a zero mean curvature immersion (also called a minimal immersion) Calibrated Geometries and Special Holonomy p. 11/29

40 Now let N be a k-dimensional immersed submanifold of M. By restriction, g induces a metric g N on N. This determines the second fundamental form h of the immersion, which is a symmetric 2-tensor on N with values in the normal bundle of N in M. One again we can consider immersions with special curvature properties. For example: h ij = 0; a totally geodesic immersion Trace(h ij ) = 0; a zero mean curvature immersion (also called a minimal immersion) Trace(h ij ) = constant; a CMC immersion Calibrated Geometries and Special Holonomy p. 11/29

41 These are second order, non-linear partial differential equations on the immersion map. Calibrated Geometries and Special Holonomy p. 12/29

42 These are second order, non-linear partial differential equations on the immersion map. Again, they can be critical points of a functional. The minimal immersions are critical points of the Volume functional Vol N which is why they are called minimal. N Calibrated Geometries and Special Holonomy p. 12/29

43 These are second order, non-linear partial differential equations on the immersion map. Again, they can be critical points of a functional. The minimal immersions are critical points of the Volume functional Vol N which is why they are called minimal. N Note that minimal just means critical point for volume. It could actually be a maximum or inflection point. The actual minima are called minimizing immersions. Calibrated Geometries and Special Holonomy p. 12/29

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

45 Special Holonomy On a Riemannian manifold (M, g), we have a covariant derivative which allows us to differentiate any tensor in the direction of a vector field V. This allows us to define the notion of parallel transport: Calibrated Geometries and Special Holonomy p. 13/29

46 Special Holonomy On a Riemannian manifold (M, g), we have a covariant derivative which allows us to differentiate any tensor in the direction of a vector field V. This allows us to define the notion of 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 Calibrated Geometries and Special Holonomy p. 13/29

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

48 This construction involves solving a linear first order ODE, and hence it can always be done. It is also 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. Calibrated Geometries and Special Holonomy p. 14/29

49 This construction involves solving a linear first order ODE, and hence it can always be done. It is also 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. Suppose M is connected, simply-connected, and oriented. Then one can show that H p = H q for any two points p, q in M. Thus, up to isomorphism, we can talk about the holonomy group H of (M, g), which is a subgroup of GL(n, R). Calibrated Geometries and Special Holonomy p. 14/29

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

51 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. Calibrated Geometries and Special Holonomy p. 15/29

52 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 parallel, the holonomy group is always a subgroup of SO(n). But the holonomy can be a strictly smaller subgroup if there exist other parallel tensors. Calibrated Geometries and Special Holonomy p. 15/29

53 Berger Classification. Let M be irreducible, not locally symmetric. Then the holonomy can only be one of the following seven possibilities: Calibrated Geometries and Special Holonomy p. 16/29

54 Berger Classification. Let M be irreducible, not locally symmetric. Then the holonomy can only be one of the following seven possibilities: H dim Name Parallel 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 Calibrated Geometries and Special Holonomy p. 16/29

55 Some examples: Calibrated Geometries and Special Holonomy p. 17/29

56 Some examples: Projective varieties are all Kähler Calibrated Geometries and Special Holonomy p. 17/29

57 Some examples: Projective varieties are all Kähler By Yau s solution of the Calabi conjecture, any compact Kähler manifold with c 1 = 0 admits a Calabi-Yau metric Calibrated Geometries and Special Holonomy p. 17/29

58 Some examples: Projective varieties are all Kähler By Yau s solution of the Calabi conjecture, any compact Kähler manifold with c 1 = 0 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 ) Calibrated Geometries and Special Holonomy p. 17/29

59 Some examples: Projective varieties are all Kähler By Yau s solution of the Calabi conjecture, any compact Kähler manifold with c 1 = 0 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 Calibrated Geometries and Special Holonomy p. 17/29

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

61 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 tensor. In fact: Calibrated Geometries and Special Holonomy p. 18/29

62 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 tensor. In fact: Calabi-Yau, hyper-kähler, G 2 and Spin(7) manifolds are always Ricci-flat Calibrated Geometries and Special Holonomy p. 18/29

63 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 tensor. In fact: Calabi-Yau, hyper-kähler, G 2 and Spin(7) manifolds are always Ricci-flat Quaternionic-Kähler manifolds are always Einstein Calibrated Geometries and Special Holonomy p. 18/29

64 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 tensor. In fact: Calabi-Yau, hyper-kähler, G 2 and Spin(7) manifolds are always Ricci-flat Quaternionic-Kähler manifolds are always Einstein The holonomy condition is a first order partial differential equation. Thus we see that solutions to a first order equation are automatically solutions to a more complicated second order equation. Calibrated Geometries and Special Holonomy p. 18/29

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

66 Calibrated Geometry (Harvey-Lawson, 1982) Suppose (M, g) is a Riemannian manifold of dimension n. Let 0 < k < n. Calibrated Geometries and Special Holonomy p. 19/29

67 Calibrated Geometry (Harvey-Lawson, 1982) Suppose (M, g) is a Riemannian manifold of dimension n. Let 0 < k < n. Definition. A smooth k-form α on M is called 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 ) Calibrated Geometries and Special Holonomy p. 19/29

68 Calibrated Geometry (Harvey-Lawson, 1982) Suppose (M, g) is a Riemannian manifold of dimension n. Let 0 < k < n. Definition. A smooth k-form α on M is called 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 called calibrated if α p (V p ) = 1 (that is, if the supremum is attained) Calibrated Geometries and Special Holonomy p. 19/29

69 Finally, we say 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 for the immersion of N in M. Calibrated Geometries and Special Holonomy p. 20/29

70 Finally, we say 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 for the immersion of N in M. Fundamental theorem of calibrated geometry: Suppose N is calibrated with respect to α. Then N is absolutely volume minimizing in its homology class. Calibrated Geometries and Special Holonomy p. 20/29

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

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

73 The main examples are: Calibrated Geometries and Special Holonomy p. 22/29

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

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

76 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. Calibrated Geometries and Special Holonomy p. 23/29

77 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. Calibrated Geometries and Special Holonomy p. 23/29

78 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. Many explicit examples of such submanifolds have been found in Euclidean spaces, and some non-compact, non-flat manifolds. Calibrated Geometries and Special Holonomy p. 23/29

79 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. Many explicit examples of such submanifolds have been found in Euclidean spaces, and some non-compact, non-flat manifolds. The only compact examples which have been found so far arise in situations with a high degree of symmetry. Calibrated Geometries and Special Holonomy p. 23/29

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

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

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

83 Analogies between the two subjects Manifolds of special holonomy and calibrated submanifolds both provide first order examples of solutions to second order equations. We replace a quasi-linear second order equation by a fully nonlinear first order equation. Hence it is not necessarily easier to solve. The main examples of calibrations are all defined on manifolds with special holonomy. The reasons are still not very well understood, but seem to be related to spin geometry. (Existence of parallel spinors.) Calibrated Geometries and Special Holonomy p. 24/29

84 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. Calibrated Geometries and Special Holonomy p. 25/29

85 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 has shown that special holonomy metrics are stable critical points of the total scalar curvature functional with respect to some types of deformations. Their arguments use parallel spinors. Calibrated Geometries and Special Holonomy p. 25/29

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

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

88 Mirror Symmetry Modern physics, specifically string theory, 11-dimensional supergravity (M-theory), and F-theory require compact manifolds of dimensions 6, 7, and 8, respectively for their description. Considerations of supersymmetry say that these manifolds have parallel spinors, which implies they are Calabi-Yau, G 2, and Spin(7) manifolds, respectively. Calibrated Geometries and Special Holonomy p. 26/29

89 Mirror Symmetry Modern physics, specifically string theory, 11-dimensional supergravity (M-theory), and F-theory require compact manifolds of dimensions 6, 7, and 8, respectively for their description. Considerations of supersymmetry say that these manifolds have parallel spinors, which implies 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. Calibrated Geometries and Special Holonomy p. 26/29

90 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 suitably compactifying. Calibrated Geometries and Special Holonomy p. 27/29

91 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 suitably 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 others.) Calibrated Geometries and Special Holonomy p. 27/29

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

93 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-Yaus. Calibrated Geometries and Special Holonomy p. 28/29

94 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-Yaus. Similar incomplete conjectures exist for G 2 and Spin(7) manifolds, which involve fibrations by calibrated torii and their dual fibrations. Calibrated Geometries and Special Holonomy p. 28/29

95 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-Yaus. 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 seem to indicate that conical singularities are the most important. Calibrated Geometries and Special Holonomy p. 28/29

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

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

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

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