**Electro-Magnetic Duality, Magnetic Monopoles and Topological Insulators.**

**ABSTRACT**

The Maxwell equations of electrodynamics acquire an additional symmetry if one assumes the existence of hypothetical particles-magnetic monopoles, carrying a magnetic charge.

The additional internal symmetry is the electro- magnetic duality generated by the rotations in the space of electric and mag- netic charges.

In this project we revise the electromagnetic duality in his global as- pect starting with the celebrated Dirac monopole, a singular solution in a slightly modified Maxwell theory.

We then take account of the new in- sight on the duality in the broken *SO*(3) gauge theory where the magnetic monopoles arose as finite-energy smooth solution (found by ’t Hooft and Polyakov).

The stability of these monopoles is guaranteed by the conserva- tion of topological invariants, i.e., these are topologically protected states.

The spectrum of the gauge theory states enjoys a symmetry between the electrically charged gauge boson and the magnetic monopole, manifesting a quantum electro-magnetic duality which turns out to be a part of larger *SL*(2*, *Z)-group symmetry acting on the 2-dimensional charge lattice.

Recently the idea of magnetic monopoles and dyons was revived by the discovery of new kind of materials known as topological insulators.

The theoretical considerations in the modified axion electrodynamics show that the electric charges on the boundary of a topological insulator induce mirror images carrying magnetic charges.

We consider carefully the mirror images in the case of topological insulator with planar and spherical boundary. We then provide a description of the induced mirror images in a manifestly *SL*(2*, *Z)-covariant form.

**TABLE OF CONTENTS**

- Introduction 2
- Electro-Magnetic Duality in Maxwell Theory 5
- Maxwell Theory with Magnetic Charges . . . . . . . . . . 5

2.2 Dirac Monopole . . . . . .. . . . . . 7

- Angular Momentum of EM field . . . . . . . . . . . . . . 9
- CP-symmetry and Dyon Quantization………………………………… 11

- ’t Hooft-Polyakov monopole 15
- Georgi-Glashow model……………………… 15
- Finite Energy solutions…………………………….. 18
- t’ Hooft-Polyakov ansatz……………………. 18
- Topological Charges…………………………………………………………. 22
- Bogomol’nyi-Prasad-Sommerfeld(BPS) state……………………… 26

- Duality Conjectures 31
- Montonen-Olive Conjecture 31
- The Witten Effect…………………………………………………………… 32
*SL*(2*,*Z)- Duality…………………………………………………………….. 34

- Topological Insulators 37
- Axion Electrodynamics……………………………………………………….. 37
- Topological Insulators with Planar Boundary 38
- Spherical Topological Insulators 41
- Topological insulator and
*SL*(2*,*Z) 48

Bibliography 53

**INTRODUCTION**

This thesis will consider abelian *U *(1) and non-abelian *SO*(3) gauge the- ories allowing for states carrying units of magnetic charge, the so called magnetic monopoles.

The existence of such magnetic monopoles was first suggested by Dirac as a speculation and an outcome of a thought experi- ment [3]. It attracts so much attention because if a monopole with magnetic charge *g *exists in nature, would automatically imply the quantization of the electric charge.

In fact, the requirement that the wave-function solving the Schrodinger in the presence of monopole is single-valued function implies the *Dirac quantization condition **eg *= *nh n *∈ Z where *h *stays for the Plank constant *h *= 2*π*k and then all electric charges are multiples of a minimal electric charge *e *=__ ^{ h}__.

The presence of magnetic charges would restore the broken symmetry between electric and magnetic charges, and the extended Maxwell equa- tions enjoys *electromagnetic duality*, an exchange symmetry of the electric and magnetic components of the electromagnetic field.

The electrodynam- ics with magnetic monopoles becomes highly symmetric, reducing the dif- ference between “electric” and “magnetic” to a matter of convention.

The Dirac monopole suffers from one defect, it is described by a sin- gular potential.

It is only in the ’70 after the advent of the non-abelian gauge theories when ’t Hooft [9] and Polyakov [7] independently discov- ered that in a model with non-abelian gauge group *G *spontaneously broken to a *U *(1) through the Higgs mechanism there exists a non-singular non- perturbative solution with a finite energy which is looking from outside like a Dirac monopole.

BIBLIOGRAPHY

B. Bogomol’nyi, The stability of classical solutions,Soviet J. Nuc. Phys.24(1976), 449-454.

Figueroa-O’Farrill J. M., ”Electromagnetic duality for children.” (1998).

A. M. Dirac, Quantised singularities in the electromagnetic field, Proc. R. Soc.A133(1931), 60-72.

Jeffrey A. Harvey, Magnetic Monopoles, Duality, and Supersymmetry. arXiv:hep-th/9603086

Montonen and D. Olive, Magnetic monopoles as gauge parti- cles,Phys. Lett.72B(1977), 117-120.

Mikio Nakahara, Geormetry, Topology and Physics. IOP, 2003