Introduction Local T-duality Global T-duality

T-duality in String Theory

Mark Bugden

Mathematical Sciences Institute Australian National University

Supervisor: Prof. Peter Bouwknegt

July 2014

1 Introduction Local T-duality Global T-duality Acknowledgements

I would like to gratefully acknowledge funding from AMSI and AUSTMS.

2 Introduction Local T-duality Global T-duality Outline

1 Introduction

2 Local T-duality

3 Global T-duality

3 Introduction Local T-duality Global T-duality Outline

1 Introduction

2 Local T-duality

3 Global T-duality

4 Introduction Local T-duality Global T-duality What is T-duality?

T-duality is an equivalence between two physical theories with different spacetime geometries.

These different geometries arise in string theory from the compactification of extra dimensions.

Left image credit: Wikipedia en:User:Jbourjai Right image credit: Wikipedia en:User:Lunch 5 Introduction Local T-duality Global T-duality Why do we care?

T-duality provides a link between different types of string theories.

Type I

Heterotic-O

Heterotic-E

Type II-A

Type II-B

This led to the suggestion that the five distinct string theories are really just limits of one underlying theory, called M-theory.

6 Introduction Local T-duality Global T-duality In Mathematics

T-duality is also interesting from the mathematical point of view.

Mirror Symmetry and the SYZ conjecture Isomorphisms in twisted K-theory Takai duality ...

7 Introduction Local T-duality Global T-duality Outline

1 Introduction

2 Local T-duality

3 Global T-duality

8 Introduction Local T-duality Global T-duality A toy model

The simplest example of T-duality considers a closed Bosonic string on a manifold where one dimension is curled up into a circle.

X 25 ∼ X 25 + 2πmR The integer m is called the winding number.

9 Introduction Local T-duality Global T-duality Equation of motion

The equation of motion for the compactified dimension can be obtained by extremising the Polyakov action. It is

X 25(τ, σ) =q25 + α0p25τ + mRσ r α0 X   +i α25e−ik(τ+σ) +α ˜25e−ik(τ−σ) 2 k k k6=0

25 n where p = R .

10 Introduction Local T-duality Global T-duality Duality

Quantising this, we find that the spectrum of the closed string is invariant under the following transformation:

R α0/R

n m

This invariance is known as T-duality.

11 Introduction Local T-duality Global T-duality A few things...

String theory compactified on small and large circles are equivalent! This duality is not present with point particles!

12 Introduction Local T-duality Global T-duality The B-field

The natural generalisation of this example considers strings coupled to the so-called B-field.

The B-field is the string equivalent to the electromagnetic potential. It is a locally defined 2-form gauge field.

13 Introduction Local T-duality Global T-duality Buscher Rules

Given the non-linear sigma model action

Z √ αβ M N S[X ] = dσdτ hh gMN (X )∂αX ∂βX  αβ M N +ε BMN (X )∂αX ∂βX

and a U(1) isometry, the Buscher rules give a transformation of

the gMN and BMN fields which leave this action invariant.

14 Introduction Local T-duality Global T-duality Buscher Rules

Given the non-linear sigma model action

Z √ αβ M N S[X ] = dσdτ hh gMN (X )∂αX ∂βX  αβ M N +ε BMN (X )∂αX ∂βX

and a U(1) isometry, the Buscher rules give a transformation of

the gMN and BMN fields which leave this action invariant.

14 Introduction Local T-duality Global T-duality Buscher Rules cont.

Explicitly, we have

1 gˆ•• = g•• Bµ• gˆµ• = g•• 1 gˆµν = gµν − (gµ•gν• − Bµ•Bν•) g•• gµ• Bˆµ• = g•• 1 Bˆµν = Bµν − (gµ•Bν• − gν•Bµ•) g••

15 Introduction Local T-duality Global T-duality Outline

1 Introduction

2 Local T-duality

3 Global T-duality

16 Introduction Local T-duality Global T-duality Local vs. Global

The Buscher rules give us a description of T-duality locally, that is, in coordinate patches.

What can we say about T-duality globally?

17 Introduction Local T-duality Global T-duality The global picture

We describe spacetime as a principal S1-bundle

S1 E π

M

where E is the total spacetime, and M is the uncompactified part of spacetime.

Image credit: Niles Johnson 18 Introduction Local T-duality Global T-duality F and H

2 Circle bundles are classified by the first Chern class F ∈ H (M, Z) of the associated line bundle LE = E ×S1 C.

3 Additionally, we include a H-flux, which is a class H ∈ H (E, Z). We have

2 F ∈ H (M, Z) 3 H ∈ H (E, Z)

The Buscher rules suggest that T-duality corresponds to interchanging F and H in some sense.

19 Introduction Local T-duality Global T-duality Gysin Sequence

Theorem (Gysin) Let π : E → M be a principal S1 bundle. Then we have the following long exact sequence

∗ · · · −→ Hk (M) −→π Hk (E) −→π∗ Hk−1(M) −→∧F Hk+1(M) −→ · · ·

20 Fˆ 0

Introduction Local T-duality Global T-duality Defining the dual bundle

For k = 3 we have

π∗ π∗ ∧F ··· H3(M) H3(E) H2(M) H4(M) ···

H

21 0

Introduction Local T-duality Global T-duality Defining the dual bundle

For k = 3 we have

π∗ π∗ ∧F ··· H3(M) H3(E) H2(M) H4(M) ···

H Fˆ

21 Introduction Local T-duality Global T-duality Defining the dual bundle

For k = 3 we have

π∗ π∗ ∧F ··· H3(M) H3(E) H2(M) H4(M) ···

H Fˆ 0

21 Introduction Local T-duality Global T-duality Defining the dual bundle

2 The pushforward of H defines an element Fˆ ∈ H (M, Z), and therefore a new circle bundle

S1 Eˆ πˆ M

What about the H-flux?

22 Hˆ 0

Introduction Local T-duality Global T-duality The dual H-flux

To get the dual H-flux, we note that Fˆ ∧ F = F ∧ Fˆ = 0.

πˆ∗ πˆ∗ ∧Fˆ ··· H3(M) H3(Eˆ ) H2(M) H4(M) ···

F Exactness of the Gysin sequence for the dual bundle then says that 3 there exists some Hˆ ∈ H (Eˆ , Z) such that

F =π ˆ∗Hˆ

23 Hˆ

Introduction Local T-duality Global T-duality The dual H-flux

To get the dual H-flux, we note that Fˆ ∧ F = F ∧ Fˆ = 0.

πˆ∗ πˆ∗ ∧Fˆ ··· H3(M) H3(Eˆ ) H2(M) H4(M) ···

F 0 Exactness of the Gysin sequence for the dual bundle then says that 3 there exists some Hˆ ∈ H (Eˆ , Z) such that

F =π ˆ∗Hˆ

23 Introduction Local T-duality Global T-duality The dual H-flux

To get the dual H-flux, we note that Fˆ ∧ F = F ∧ Fˆ = 0.

πˆ∗ πˆ∗ ∧Fˆ ··· H3(M) H3(Eˆ ) H2(M) H4(M) ···

Hˆ F 0 Exactness of the Gysin sequence for the dual bundle then says that 3 there exists some Hˆ ∈ H (Eˆ , Z) such that

F =π ˆ∗Hˆ

23 Introduction Local T-duality Global T-duality Summary

Theorem (Bouwknegt, Evslin, Mathai) Given a pair (F , H) consisting of a principal circle bundle π : E → M and a H-flux, the T-dual is a pair (Fˆ , Hˆ ) consisting of a principal circle bundle πˆ : Eˆ → M with H-flux.

S1 E S1 Eˆ π πˆ M M

We have

Fˆ = π∗H

F =π ˆ∗Hˆ

24 Introduction Local T-duality Global T-duality Beyond the basics

T-duality for torus bundles Classical duals Non-commutative duals Non-associative duals Spherical T-duality Deligne cohomology Generalised geometry

25 Introduction Local T-duality Global T-duality References

P. Bouwknegt, Quantum Field Theory, Topology and Duality, Geometry and Quantum Field Theory (CareyFest), 20-26 June, 2010 P. Bouwknegt, Lectures on Cohomology, T-duality and Generalized Geometry, 2008 P. Bouwknegt, J. Evslin, V. Mathai, T-duality: Topology change from H-flux, Commun. Math. Phys. 249 (2004) 383-415 P. Bouwknegt, J. Evslin, V. Mathai, On the topology and H-flux of T-dual manifolds, Phys. Rev. Lett. 92 (2004) 181601 T. Buscher, A symnmetry of the string background field equations, Phys. Lett. B194 (1987) 59 W. Gysin, Zur Homologietheorie der Abbildungen und Faserungen von Mannigfaltigkeiten, Commentarii Mathematici Helvetici 14 (1942) 61-122 Jbourjai, Image of Calabi-Yau manifold, http://en.wikipedia.org/wiki/File:Calabi yau.jpg Lunch, Image of Calabi-Yau manifold, http://en.wikipedia.org/wiki/File:Calabi-Yau.png N. Johnson, Image of Hopf fibration, http://www.nilesjohnson.net/hopf.html

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