Triality and fixed points of Spin-bundles

(1)

Triality and fixed points of Spin-bundles

Madrid, 2007

Alvaro Ant´on Sancho´ CSIC/UCM

Triality and fixed points of Spin-bundles – p. 1/28

(2)

References

(10)

References

1. García-Prada, O.: Involutions of the moduli space of SL(n, C)-Higgs bundles and real forms. Preprint. 2007.

(11)

References

2. García-Prada, O. & Ramanan, S.: Involutions of the moduli space of Higgs bundles. In preparation.

(12)

References

3. Gómez, Tomás L. & Sols, I.: Moduli space of principal sheaves over projective varieties. Ann. of Math. 161, Number 2 (2005), 1033-1088.

(13)

References

4. Ramanan, S.: Orthogonal and spin bundles over hyperelliptic curves. Proc.

Indian Acad. Sci. (Math. Sci.) 90 (1981) 151-166.

(14)

References

5. Ramanathan, A.: Moduli for principal bundles over algebraic curves: I.

Proc. Indian Acad. Sci. (Math. Sci.) 106 (1996), 301-328.

(15)

References

6. Ramanathan, A.: Stable principal bundles on a compact Riemann surface.

Math. Ann. 213 (1975) 129-152.

(16)

References

6. Ramanathan, A.: Stable principal bundles on a compact Riemann surface.

Math. Ann. 213 (1975) 129-152.

(17)

Representations of so (2n, C)

(18)

Representations of so (2n, C)

Let (V, Q) be an m-dimensional C-vector space (m ≥ 2) equipped with a non-degenerate symmetric bilinear form, Q.

(19)

Representations of so (2n, C)

The group G = SO(Q) = SO(n, C) is the group of automorphisms of V

preserving Q and with determinant one. Its Lie algebra is g = so(Q) = so(n, C).

It is simple of dimension m(m − 1)/2.

(20)

Representations of so (2n, C)

Suppose m = 2n.

(21)

Representations of so (2n, C)

Suppose m = 2n.

We can choose a basis in which Q has matrix

M =





0 I_n I_n 0



 .

(22)

Representations of so (2n, C)

Suppose m = 2n.

We can choose a basis in which Q has matrix

M =





0 I_n I_n 0



 .

Let E_ij be the 2n × 2n matrix with 1 in the (i, j) position and 0 in the others.

Then, the matrices

(23)

Representations of so (2n, C)

The adjoint representation

ad : so(2n, C) → gl(so(2n, C))

is irreducible of dimension n(2n − 1) and with maximal weight L₁ + L₂.

(24)

Representations of so (2n, C)

is irreducible of dimension n(2n − 1) and with maximal weight L₁ + L₂. The standard representation

st : so(2n, C) → gl(V ) is irreducible of dimension 2n and maximal weight L₁.

(25)

Representations of so (2n, C)

is irreducible of dimension n(2n − 1) and with maximal weight L₁ + L₂. The standard representation

st : so(2n, C) → gl(V ) is irreducible of dimension 2n and maximal weight L₁. The spin representation of so(2n, C)

∆ : so(2n, C) → gl

•

^ W

! ,

where W is an isotropic n-dimensional subspace of V breaks into the two

half-spin representations Triality and fixed points of Spin-bundles – p. 5/28

(26)

Representations of so (2n, C)

∆⁺ : so(2n, C) → gl

+

^ W

! ,

∆⁻ : so(2n, C) → gl

−

^W

! ,

both irreducible and with maximal weights 1

2 (L₁ + · · · + L_n−1 + L_n) and 1

2 (L₁ + · · · + L_n−1 − L_n) respectively.

(27)

The case of so (8, C)

(28)

The case of so (8, C)

The Dynkin diagram of the algebra so(8, C) is D₄,

(29)

The case of so (8, C)

Theorem. Let g be a simple algebra. The group Out(g) = Aut(g)/ Int(g) is isomorphic to the group of symmetries of the Dynkin diagram of g.

(30)

The case of so (8, C)

From the theorem it follows that Out(so(8, C)) is isomorphic to S₃. In particular, so(8, C) admits two different automorphisms (modulo inner automorphisms) of order 3 (those corresponding to the symmetries of D₄). These automorphisms are mutually inverse.

(31)

The case of so (8, C)

This table sums up the basic information about representations:

(32)

The case of so (8, C)

This table sums up the basic information about representations:

Representation Complex dimension Dominant weight

ad 28 L₁ + L₂

st 8 L₁

∆⁺ 8 ¹₂(L₁ + L₂ + L + L₄)

∆⁻ 8 ¹₂(L₁ + L₂ + L − L₄)

(33)

The case of so (8, C)

The family of weights

{L₁, L₁ + L₂, ¹₂(L₁ + L₂ + L₃ + L₄), ¹₂(L₁ + L₂ + L₃ − L₄)} is a fundamental system of weights of so(8, C).

(34)

The case of so (8, C)

The family of weights

{L₁, L₁ + L₂, ¹₂(L₁ + L₂ + L₃ + L₄), ¹₂(L₁ + L₂ + L₃ − L₄)} is a fundamental system of weights of so(8, C).

In fact, the automorphisms of order three of so(8, C) leaves the standard

representation invariant and moves the other three (ad to ∆⁻, ∆⁻ to ∆⁺ and ∆⁺ to ad), so these automorphisms correspond to the order three symmetries of the Dynkin diagram

(35)

The case of so (8, C)

L₁ L₁+L₂

(L₁+L₂+L₃+L₄)/2

(L₁+L₂+L₃-L₄)/2

(36)

The case of so (8, C)

Let τ and τ⁻¹ be the automorphisms of order three. In Aut(so(8, C)) we consider other equivalence relation, ∼_i, given by conjugation by inner automorphisms, that is, if α, β ∈ Aut(so(8, C)),

α ∼_i β ⇔ ∃θ ∈ Int(so(8, C)) : α = θβθ⁻¹.

(37)

The case of so (8, C)

α ∼_i β ⇔ ∃θ ∈ Int(so(8, C)) : α = θβθ⁻¹.

Let Aut₃(so(8, C)) be the set of automorphisms of order three. In our language, results of Wolf and Gray prove the following propositions.

(38)

The case of so (8, C)

α ∼_i β ⇔ ∃θ ∈ Int(so(8, C)) : α = θβθ⁻¹.

Let Aut₃(so(8, C)) be the set of automorphisms of order three. In our language, results of Wolf and Gray prove the following propositions.

Proposition. Aut₃(so(8, C))/ ∼_i has four elements, the classes of τ and τ⁻¹ and the classes of other two automorphisms of order three, τ^′ and τ^′−1.

Out₃(so(8, C)) is given by the classes of τ y τ⁻¹.

(39)

The case of so (8, C)

Proposition. Via the natural map

Aut₃(so(8, C))/ ∼_i→ Out₃(so(8, C)) ∪ {1},

the classes of τ and τ^′ modulo inner conjugation are sent to the class of τ modulo inner automorphisms and the classes of τ⁻¹ and τ^′−1 modulo inner conjugation are sent to the class of τ⁻¹ modulo inner automorphisms.

(40)

The case of so (8, C)

Aut₃(so(8, C))/ ∼_i→ Out₃(so(8, C)) ∪ {1},

It is relevant for us to study tha algebras of fixed points of τ and τ^′. The same work of Wolf and Gray gives us the key.

(41)

The case of so (8, C)

Aut₃(so(8, C))/ ∼_i→ Out₃(so(8, C)) ∪ {1},

It is relevant for us to study tha algebras of fixed points of τ and τ^′. The same work of Wolf and Gray gives us the key.

Proposition. The algebra of fixed points of τ is isomorphic to the algebra g₂ and the algebra of fixed points of τ^′ is isomorphic to the algebra a₂.

(42)

Triality automorphism

(43)

Triality automorphism

Recall the definition and basic properties of Spin groups.

(44)

Triality automorphism

Let K be a field and (V, Q) a K-vector space equipped with a symmetric bilinear form. In the tensor algebra of V , N V , we consider the ideal

J(V ) = hv ⊗ v − Q(v, v) : v ∈ V i .

(45)

Triality automorphism

J(V ) = hv ⊗ v − Q(v, v) : v ∈ V i .

We define the Clifford algebra associated to V and Q as Cl(Q) = O

V /J(V ).

(46)

Triality automorphism

J(V ) = hv ⊗ v − Q(v, v) : v ∈ V i .

We define the Clifford algebra associated to V and Q as Cl(Q) = O

V /J(V ).

Let Cl(Q)₀ be the subalgebra of the Clifford algebra generated by the products of an even number of elements of V and the automorphism ∗ : Cl(Q) → Cl(Q)

(47)

Triality automorphism

We define the group Spin as

Spin(Q) = {x ∈ Cl(Q)₀ : x is invertible, xx^∗ = 1, xV x^∗ ⊆ V } .

(48)

Triality automorphism

We will consider the complex field and dimension 8.

(49)

Triality automorphism

The following properties of Spin will be relevant for us.

(50)

Triality automorphism

The Lie algebra of Spin(8, C) is so(8, C).

(51)

Triality automorphism

Spin(8, C) is simply connected. So it is the simply connected group of algebra so(8, C).

(52)

Triality automorphism

Spin(8, C) is simply connected. So it is the simply connected group of algebra so(8, C).

The map ρ : Spin(8, C) → SO(8, C), x 7→ ρ(x)(v) = xvx^∗ is a 2 : 1 covering map. So Spin(8, C) is the universal cover of SO(8, C).

(53)

Triality automorphism

We know that there is a bijective correspondence between the automorphisms of a simply connected group and those of its Lie algebra. Moreover, we have the

following result.

(54)

Triality automorphism

following result.

Proposition. Let g be a complex Lie algebra and G the unique connected and

simply connected group with Lie algebra g. Then, there is a natural automorphism of short exact sequences of groups

1 −−−−→ Int(G) −−−−→ Aut(G) −−−−→ Out(G) −−−−→ 1



 y



 y



 y

1 −−−−→ Int(g) −−−−→ Aut(g) −−−−→ Out(g) −−−−→ 1.

(55)

Triality automorphism

following result.

Proposition. Let g be a complex Lie algebra and G the unique connected and

simply connected group with Lie algebra g. Then, there is a natural automorphism of short exact sequences of groups

1 −−−−→ Int(G) −−−−→ Aut(G) −−−−→ Out(G) −−−−→ 1



 y



 y



 y

1 −−−−→ Int(g) −−−−→ Aut(g) −−−−→ Out(g) −−−−→ 1.

We obtain that the automorphism τ lifts uniquely to an automorphism of Spin(8, C). this automorphism is called triality automorphism.

(56)

Triality automorphism

The automorphism τ^′ lifts uniquely to an automorphism of Spin(8, C), too. Let it be j^′.

(57)

Triality automorphism

The automorphism τ^′ lifts uniquely to an automorphism of Spin(8, C), too. Let it be j^′.

From all this we deduce that the group of fixed points of j is isomorphic to G₂ and the group of fixed points of j^′ is isomorphic to SL(3, C).

(58)

Moduli of principal bundles

(59)

Moduli of principal bundles

Definition. Let G be a reductive group. A holomorphic principal G-bundle E is said to be stable (resp. semistable) if for each reduction of the structure group of E to a parabolic subgroup P of G (that is, for each global section σ : X → E/P ), we have that deg σ^∗(T_G/P) > 0 (resp. ≥ 0), where T_G/P is the sub-bundle of

T E/P , tangent along the fibres of E/P .

(60)

Moduli of principal bundles

This notion coincides with the classical notion of Mumford for vector bundles in the case in which G = GL(n, C).

(61)

Moduli of principal bundles

As in the case of vector bundles, we can assign a graded element grE to each semistable G-bundle E, so we have a good notion of S-equivalence.

(62)

Moduli of principal bundles

Definition. Two semistable bundles are said to be S-equivalent if they have isomorphic graded elements.

(63)

Moduli of principal bundles

Definition. Two semistable bundles are said to be S-equivalent if they have isomorphic graded elements.

The moduli of principal G-bundles, M(G), is, then, an algebraic variety that parametrizes classes of S-equivalence of semistable bundles.

(64)

Moduli of principal bundles

This theorem (due to A. Ramanathan) gives some useful properties of the moduli of principal bundles.

(65)

Moduli of principal bundles

Theorem. If G es semisimple, M(G) is an algebraic variety of dimension dim g(g − 1). The number of connected components of M(G) is equal to the number of elements of π₁(G). In particular, if G is simply connected, M(G) is connected and, in fact, irreducible.

(66)

Moduli of principal bundles

From this, M(Spin(8, C)) is a variety of dimension 28(g − 1) and the varieties M(Spin(8, C)), M(G₂) and M(SL(3, C)) are irreducible.

(67)

Moduli of principal bundles

From this, M(Spin(8, C)) is a variety of dimension 28(g − 1) and the varieties M(Spin(8, C)), M(G₂) and M(SL(3, C)) are irreducible.

Recall the notion of simplicity.

Definition. A G-bundle is said to be simple if its unique automorphisms are multiplication by elements of the center of G.

(68)

Moduli of principal bundles

We will study stable and simple fixed points for certain automorphisms of the moduli of principal Spin(8, C)-bundles.

(69)

Moduli of principal bundles

Proposition. If G is semisimple, M(G) is smooth in the open set of stable simple points.

(70)

Moduli of principal bundles

Proposition. If G is semisimple, M(G) is smooth in the open set of stable simple points.

From this proposition we know that we will study smooth fixed points of the moduli.

(71)

Action of Out(G) in M(G)

(72)

Action of Out(G) in M(G)

Let G a complex semisimple Lie group. The group Aut(G) acts on M(G) in this way: if E is a G-bundle and A ∈ Aut(G), A(E) will be equal to E as a variety but equipped with the following action of G,

e♦g = eA⁻¹(g) for e ∈ E and g ∈ G.

(73)

Action of Out(G) in M(G)

If {ψ_ij} are transition functions of E, then {A ◦ ψ_ij} are transition functions of A(E).

(74)

Action of Out(G) in M(G)

If {ψ_ij} are transition functions of E, then {A ◦ ψ_ij} are transition functions of A(E).

With the point of view of transition functions, it is easy to show that if A is an inner automorphism of G, then E is isomorphic to A(E), that is, the action of Aut(G) is trivial on Int(G).

(75)

Action of Out(G) in M(G)

We can assure that the action of Aut(G) defines an action of Out(G), that is, this action of Out(G) in M(G) is well defined: if σ ∈ Out(G), A ∈ Aut(G) is a representant of σ and E ∈ M(G), we define

σ(E) = A(E).

(76)

Action of Out(G) in M(G)

σ(E) = A(E).

Observe that, in order to prove that the preceding action is well defined, it is

necessary to see that, if A ∈ Aut(G), then A(E) is semistable if E is semistable (which is immediate form the definition of semistability) and the following result.

Proposition. If E₁ and E₂ are semistable S-equivalent bundles, then A(E₁) and A(E₂) are S-equivalent.

(77)

Action of Out(G) in M(G)

σ(E) = A(E).

Observe that, in order to prove that the preceding action is well defined, it is

necessary to see that, if A ∈ Aut(G), then A(E) is semistable if E is semistable (which is immediate form the definition of semistability) and the following result.

Proposition. If E₁ and E₂ are semistable S-equivalent bundles, then A(E₁) and A(E₂) are S-equivalent.

The proof of the preceding proposition follows from the definition.

(78)

Fixed points of Spin(8, C)-bundles

(79)

Fixed points of Spin(8, C)-bundles

The main result of this work is the following.

Theorem. Let X be a compact Riemann surface. Let σ ∈ Out(G) with

G = Spin(8, C) an element of order three. Let M^σ(G) be the subset of fixed

points of M(G) for the action of σ and M^σstable,simple(G) be the subset of stable and simple fixed points. Then,

M(G^₂) ∪ M(SL(3, C)) ⊆ M^ ^σ(Spin(8, C)) and

M^σ(Spin(8, C))stable,simple ⊆ M(G^₂) ∪ M(SL(3, C)),^ where, if H is a subgroup of G, M(H) is the image of the map^

M(H) → M(G), E 7→ E ×_H G induced by the inclusion of groups H ֒→ G.

(80)

Fixed points of Spin(8, C)-bundles

Suppose that A is a lifting of σ and E is a simple fixed point. Then E and A(E) are isomorphic. There exists an isomorphism

f : E → A(E).

(81)

Fixed points of Spin(8, C)-bundles

Suppose that A is a lifting of σ and E is a simple fixed point. Then E and A(E) are isomorphic. There exists an isomorphism

f : E → A(E).

A acts on the bundles and on the morphisms. Taking into account that A³ = 1, we have a chain

E → A(E)^f ^{A(f )}→ A²(E) ^A

2(f )

→ E, and, so, an automorphism of E

A²(f ) ◦ A(f ) ◦ f : E → E.

(82)

Fixed points of Spin(8, C)-bundles

From the simplicity of E we deduce the existence of λ ∈ Z(Spin(8, C)) such that A²(f ) ◦ A(f ) ◦ f = λ.

(83)

Fixed points of Spin(8, C)-bundles

In local terms, we obtain that the elements of the group ψf φ⁻¹(x, 1) for x ∈ X are the solutions of the equation

A²(g)A(g)g = λ.

(84)

Fixed points of Spin(8, C)-bundles

In local terms, we obtain that the elements of the group ψf φ⁻¹(x, 1) for x ∈ X are the solutions of the equation

A²(g)A(g)g = λ.

It will be λ ∈ Fix(A). We know that Fix(A) ∼= G2 or Fix(A) ∼= SL(3, C). From this and the fact that λ is in the center, we can deduce that λ = 1. So we have the equation

(85)

Fixed points of Spin(8, C)-bundles

This equation determines a variety H with tangent space at 1 ker dA²

1 + dA|₁ + id .

(86)

Fixed points of Spin(8, C)-bundles

1 + dA|₁ + id .

We consider the map X → E(G/H) defined in the following way. For x ∈ X and (U, φ) a local trivialization of E, the image of x is φ⁻¹(x, 1), H. It is well

defined and defines a global section of E(G/H).

(87)

Fixed points of Spin(8, C)-bundles

1 + dA|₁ + id .

We consider the map X → E(G/H) defined in the following way. For x ∈ X and (U, φ) a local trivialization of E, the image of x is φ⁻¹(x, 1), H. It is well

defined and defines a global section of E(G/H).

Global section of E(G/H) induce global sections of E Gr_k ker dA²

1 + dA|₁ + id → X

(details are due to A. Ramanathan), where k is the dimension of the subalgebra ker dA²

1 + dA|₁ + id.

(88)

Fixed points of Spin(8, C)-bundles

It can be seen that the subalgebras ker dA²

1 + dA|₁ + id and ker ( dA|₁ − id) are semisimple and mutually orthogonal with respect to the Killing form. From this, it is equivalent a global section of

E Gr_k ker dA²

1 + dA|₁ + id → X an a global section of

E (Gr (ker ( dA|₁ − id))) → X

or, what is the same, a global section of E(G/Fix(A)) → X, that is, a reduction of the structure group of E to Fix(A).

(89)

Fixed points of Spin(8, C)-bundles

It can be seen that the subalgebras ker dA²

1 + dA|₁ + id and ker ( dA|₁ − id) are semisimple and mutually orthogonal with respect to the Killing form. From this, it is equivalent a global section of

E Gr_k ker dA²

1 + dA|₁ + id → X an a global section of

E (Gr (ker ( dA|₁ − id))) → X

or, what is the same, a global section of E(G/Fix(A)) → X, that is, a reduction of the structure group of E to Fix(A).

The converse (a bundle that reduces to Fix(A) is a fixed point) is easy.

(90)

Fixed points of Spin(8, C)-bundles

Then, we have proved the desired theorem

Theorem. Let X be a compact Riemann surface. Let σ ∈ Out(G) with

G = Spin(8, C) an element of order three. Let M^σ(G) be the subset of fixed

points of M(G) for the action of σ and M^σstable,simple(G) be the subset of stable and simple fixed points. Then,

M(G^₂) ∪ M(SL(3, C)) ⊆ M^ ^σ(Spin(8, C)) and

M^σ(Spin(8, C))stable,simple ⊆ M(G^₂) ∪ M(SL(3, C)),^ where, if H is a subgroup of G, M(H) is the image of the map^

M(H) → M(G), E 7→ E ×_H G

(91)

Appendix

Finally, the last result.

(92)

Appendix

Theorem.

(93)

Appendix

Theorem. This is the last theorem of the lecture.

(94)

Appendix

Proof.

(95)

Appendix

Proof.

Triality and fixed points of Spin-bundles