T -duality for T^{n}-bundles (Oxford, May 2005)

U. Bunke and Th. Schick ^{∗}

May 9, 2005

Contents

1 Twisted K-Theory 1 1.1 Twists . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Twisted K-theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2 T -duality triples 4 2.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.2 The T -duality isomorphism . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.3 Classification of T -duality triples . . . . . . . . . . . . . . . . . . . . . . . 5 2.4 Extension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

3 History 8

1 Twisted K-Theory

1.1 Twists

1.1.1 Some notation:

∗Mathematisches Institut, Universität Göttingen, Bunsenstr. 3-5, 37073 Göttingen, GERMANY, bunke@uni-math.gwdg.de, schick@uni-math.gwdg.de

1 TWISTED K-THEORY 2

(1) P U := U/U(1) - projective unitary group of an infinite-dimensional separable complex Hilbert space

(2) EP U → BP U - its universal bundle

(3) p_{K} : K := EP U ×_{P U} K → BP U - bundle of algebras over BP U, where K are the compact operators on the above Hilbert space, and P U acts by conjugation

1.1.2 Here is a model:

Definition 1.1 A twist of a space X is a map H : X → BP U.

Definition 1.2 A morphism between twists u : H_{0} → H_{1} is a homotopy class of homotopies from H_{0} to H_{1}.

We get a groupoid-valued functor X → T (X).

1.1.3 Note that BP U ∼ K(ℤ, 3).

Corollary 1.3 Isomorphism classes of twists are classified by the transformation

c : T (X) → H^{3}(X, ℤ) .

Corollary 1.4 Hom_{T (X)}(H_{0}, H_{1}) is a torsor over H^{2}(X, ℤ), naturally.

1.2 Twisted K-theory

1.2.1 Consider H ∈ T (X). We define the C^{∗}-algebra

C(X, H) := {f : X → K|p_{K} ∘ f = H} .

Definition 1.5 We define K^{∗}(X, H) := K_{∗}(C(X, H)).

1 TWISTED K-THEORY 3

1.2.2

(1) For f : Y → X we have f^{∗} : K(X, H) → K(Y, f^{∗}H).

(2) For u : H_{1} → H_{0} we have u^{∗} : K(X, H_{0}) → K(X, H_{1}).

(3) K(X, H) is a module over K(X). For a line bundle L → X we have [L]∪x = u^{∗}(x), where u = c_{1}(L) ∈ Aut(H).

1.2.3 Let X be a closed oriented 3-manifold and define H_{n} ∈ T (X) by n =< c(H_{n}), [X] >.

Theorem 1.6 We have

K^{0}(X, H_{n}) ≅ K̃^{0}(X)
K^{1}(X, H_{n}) ≅ ℤ^{rank(K^{1}(X))−1} ⊕ ℤ/nℤ

Proof. Mayer-Vietoris sequence. 1.2.4 Special cases:

K^{0}(S^{3}, H_{n}) ≅ 0 , K^{1}(S^{3}, H_{n}) ≅ ℤ/nℤ .
K^{0}(S^{3}/(ℤ/nℤ), H_{1}) ≅ ℤ/nℤ , K^{0}(S^{3}/(ℤ/nℤ), H_{1}) ≅ 0

This is an example of T -duality for U(1)-bundles. 1.2.5 Consider the diagram

\begin{aligned}& S^{3} ×_{S^{2}} S^{3}/(ℤ/nℤ) \\ p ↙ & & p̂ ↘ \\ H_{n} → S^{3} & & H_{1} → S^{3}/(ℤ/nℤ) \end{aligned} .

There exists an isomorphism

u : p̂^{∗}H_{1} → p^{∗}H_{n}

such that the T -duality transformation

T : p̂_{∗} ∘ u^{∗} ∘ p^{∗} : K(S^{3}, H_{n}) → K(S^{3}/(ℤ/nℤ), H_{1})

is an isomorphism.

2 T -DUALITY TRIPLES 4

2 T -duality triples

2.1 Definition

2.1.1 Data: (1)

\begin{aligned}& & F ×_{B} F̂ \\ & p ↙ & & p̂ ↘ \\ F & & r ↓ & & F̂ \\ & π ↘ & & π̂ ↙ \\ & & B \end{aligned} .

(2) p, p̂ - T^{n}-principal bundles (3)

H ∈ T (F) , Ĥ ∈ T (F̂) , u : p̂^{∗}Ĥ → p^{∗}H

2.1.2 Conditions: (1)

c(H) ∈ F^{2}H^{3}(F, ℤ) , c(Ĥ) ∈ F^{2}H^{3}(F̂, ℤ)

(2) Let c_{i}, ĉ_{i} ∈ H^{2}(B, ℤ) denote the Chern classes of F and F̂

[c(H)]^{2,1} = [∑_{i=1}^{n} y_{i} ∪ ĉ_{i}] ∈ ^{π}E_{2}^{2,1}

[c(Ĥ)]^{2,1} = [∑_{i=1}^{n} ŷ_{i} ∪ c_{i}] ∈ π̂E_{2}^{2,1}

(3) For b ∈ B choose

v̂_{b} : 0 → Ĥ_{|F̂_{b}} , v_{b} : H_{|F_{b}} → 0 .

2 T -DUALITY TRIPLES 5 (4)

u(b) := p_{b}^{∗}b_{b} ∘ u_{|F_{b}×F̂_{b}} ∘ p̂_{b}^{∗}v̂_{b} ∈ Aut_{T (F_{b}×F̂_{b})}(0)

satisfies

[u(b)] = [∑_{i=1}^{n} y_{i} ∪ ŷ_{i}] ∈ H^{2}(F_{b} × F̂_{b}, ℤ)/(im(p_{b}^{∗}) + im(p̂_{b}^{∗})) .

2.1.3 Definition 2.1 A T -duality triple over B is given by

((F, H), (F̂, Ĥ), u) .

satisfying the conditions above. There is a natural notion of an isomorphism between T -duality triples. We obtain a set-valued functor

B ↦ Triple_{n}(B) .

2.2 The T -duality isomorphism

Theorem 2.2 The transformation

T := p̂_{∗} ∘ u^{∗} ∘ p^{∗} : K(F, H) → K(F̂, Ĥ)

is an isomorphism. Proof. True for B a point by explicite calculation, and the in general by Mayer-Vietoris sequence.

2.3 Classification of T -duality triples

2.3.1 Lemma 2.3 The functor Triple_{n} is homotopy invariant.

2 T -DUALITY TRIPLES 6 2.3.2 Consider

q : K(ℤ^{n}, 2) × K(ℤ^{n}, 2) → K(ℤ, 4)

given by

q = ∑_{i=1}^{n} x_{i} ∪ x̂_{i} .

Definition 2.4 Let R_{n} be the homotopy fibre of q. 2.3.3 The maps c, ĉ : R_{n} → K(ℤ^{n}, 2) classify T^{n}-bundle

p_{n}F_{n} → R_{n} , p̂_{n} : F̂_{n} → R_{n} .

Calculation of cohomology:

\begin{matrix}i & 0 & 1 & 2 & 3 \\ H^{i}(F_{n}, ℤ) & ℤ & 0 & ℤ^{n} & ℤ \end{matrix}

Definition 2.5 Define H to be the generator of H^{3}(F_{n}, ℤ). Define Ĥ analogously. 2.3.4 Calculation of cohomology:

\begin{matrix}i & 0 & 1 & 2 & 3 \\ H^{i}(F_{n} ×_{R_{n}} F̂_{n}, ℤ) & ℤ & 0 & 0 & ℤ \end{matrix}

Choose u_{n} : p̂_{n}^{∗}Ĥ_{n} → p_{n}^{∗}H_{n} to be the unique isomorphism. 2.3.5 Proposition 2.6

x_{univ} := ((F_{n}, H_{n}), (F̂_{n}, Ĥ_{n}), u_{n})

is a T -duality triple.

2 T -DUALITY TRIPLES 7

2.3.6

Theorem 2.7 R_{n} is a classifying space for the functor Triple_{n} such that x_{univ} ∈ Triple(R_{n}) is the universal triple.

2.4 Extension

2.4.1 Consider a pair (F, h) of a T^{n}-principal bundle F → B and h ∈ H^{3}(F, ℤ).

Definition 2.8 An extension of (F, h) is a T -duality triple ((F, H), (F̂, Ĥ), u) such that h = c(H). The pair (F̂, c(Ĥ)) is called a T -dual of (F, h).

There is a natural notion of an isomorphism of extensions. Let Ext(F, h) denote the isomorphism classes of extensions.

2.4.2

Theorem 2.9 (1) (F, h) admits a T -dual if and only if h ∈ F^{2}H^{3}(F, ℤ).

(2) Let (F̂, ĥ) be a T -dual of (F, h). Then for any antisymmetric B ∈ Mat(n, n, ℤ) there exists a T -dual (F̂′, ĥ′) of (F, h) with

c_{i}(F̂′) = c_{i}(F̂) + ∑_{i=1}^{n} B_{i,j}c_{j}(F) .

(3) The set of classes [(F, H), (F̂, Ĥ′), u′] ∈ Ext(F, h) with c_{i}(F̂) = c_{i}(F̂′) is an orbit

under the free action of

ker(π^{∗})/im(^{π}d_{2}^{1,1})

with π^{∗} : H^{3}(B, ℤ) → H^{3}(F, ℤ).

3 History

(1) T -duality was found as a duality in string theory. The twist is the H-flux. The surprise was that the H-flux has influence on the topology of the dual. No precise mathematics.

(2) The Raeburn-Rosenberg version (1988) is a translation to the C^{∗}-algebraic context. F is the spectrum of the continuous trace algebra C(F, H). Lift the T^{n}-action on F to an ℝ-action on C(F, H). Then C(F, H) ⋊ ℝ is a again a continuous trace algebra with spectrum F̂, i.e. of the form C(F̂, Ĥ). The T -duality isomorphism is Connes’ Thom isomorphism. The obstruction theory for the ℝ-action lift and the topology of the dual was partially understood.

(3) Topological interpretation of T -duality for U(1)-bundles was obtaind by Bouwknegt-Evslin-Hannabus, Mathai (2003). The described the T -duality transformation using the correspondence. Their approach use forms. Precise definitions and the integral theory was missing.

(4) Bunke, Schick gave an integral version for U(1)-bundles (2004) in the sprit of the present talk. Explicite examples showed, that the torsion part of the twist is important.

(5) In the C^{∗}-algebraic context T -duality for T^{2}-bundles was analysed by Mathai-Rosenberg 2004. Non-commutative torus bundles appear as T -duals if a classical T -dual does not exist.

(6) Bouwknegt-Hannabus-Mathai (2004/05) studied T^{n}-bundles. Precise definitions were missing. Because of the use of forms torsion information was lost.

(7) Mathai-Rosenberg (2004) tried to describe the topology of T -duals for T^{n}-bundles. Mistakes!

(8) Bunke, Schick (2004/05) studied T -duality for T^{n}-bundles from the topological point of view and obtained the definitions and the results presented above.