electrically charged magnetic monopoles, superstring...

Monopole

Edward Olszewski

ElectromagnetismThe Maxwell Theory

Compact Notation

Minimal Coupling (QuantumMechanics)

The DiracMonopole

Grand UnifiedTheories

MagneticMonopoles andDyons in GrandUnified Theories

Montonen–OliveDuality

Supersymmetry

SuperstringTheory Primer

MagneticMonopoles inString Theory

References

Appendix

Electrically Charged MagneticMonopoles, Superstring Theory, and

Wormholes

Edward Olszewski

March 14, 2014

Monopole

Edward Olszewski

Compact Notation

The DiracMonopole

Supersymmetry

References

Appendix

OutlineElectromagnetism

The Maxwell TheoryCompact NotationMinimal Coupling (Quantum Mechanics)

The Dirac Monopole

Grand Unified TheoriesMagnetic Monopoles and Dyons in Grand UnifiedTheories

Montonen–Olive Duality

Supersymmetry

Superstring Theory Primer

Magnetic Monopoles in String Theory

References

Appendix

Monopole

Edward Olszewski

Compact Notation

The DiracMonopole

Supersymmetry

References

Appendix

The Dirac Monopole

Supersymmetry

References

Appendix

Monopole

Edward Olszewski

Compact Notation

The DiracMonopole

Supersymmetry

References

Appendix

The Dirac Monopole

Supersymmetry

References

Appendix

Monopole

Edward Olszewski

Compact Notation

The DiracMonopole

Supersymmetry

References

Appendix

The Dirac Monopole

Supersymmetry

References

Appendix

Monopole

Edward Olszewski

Compact Notation

The DiracMonopole

Supersymmetry

References

Appendix

The Dirac Monopole

Supersymmetry

References

Appendix

Monopole

Edward Olszewski

Compact Notation

The DiracMonopole

Supersymmetry

References

Appendix

The Dirac Monopole

Supersymmetry

References

Appendix

Monopole

Edward Olszewski

Compact Notation

The DiracMonopole

Supersymmetry

References

Appendix

The Dirac Monopole

Supersymmetry

References

Appendix

Monopole

Edward Olszewski

Compact Notation

The DiracMonopole

Supersymmetry

References

Appendix

The Dirac Monopole

Supersymmetry

References

Appendix

Monopole

Edward Olszewski

Compact Notation

The DiracMonopole

Supersymmetry

References

Appendix

The Dirac Monopole

Supersymmetry

References

Appendix

Monopole

Edward Olszewski

Compact Notation

The DiracMonopole

Supersymmetry

References

Appendix

The Dirac Monopole

Supersymmetry

References

Appendix

Monopole

Edward Olszewski

Compact Notation

The DiracMonopole

Supersymmetry

References

Appendix

The Maxwell Equations

Notation: (µ = 0,1,2,3; i = 1,2,3), (Minkowski metricwith η00 = −1), (Levi-Cevità symbol: ϵ0123 = ϵ123 = 1),

(Lorentz-Heaviside units)

∇ · E = ρe ∇× B − ∂E∂t

∇ · B = 0 ∇× E +∂B∂t

Maxwell - magnetic Potential

The Lorentz Force Equation

F = qeE + qev × B

Monopole

Edward Olszewski

Compact Notation

The DiracMonopole

Supersymmetry

References

Appendix

The Maxwell Equations (magnetic)

E → B B → −EJe → Jm Jm → −Je

ρe → ρm ρm → −ρe

∇ · B = ρm −∇× E − ∂B∂t

−∇ · E = 0 ∇× B − ∂E∂t

Maxwell - electric

The Lorentz Force Equation

F = qmB − qmv × E

Monopole

Edward Olszewski

Compact Notation

The DiracMonopole

Supersymmetry

References

Appendix

The Potential Function

Consequently,

∇× A = B since ∇ · (∇× A) = 0

−∇A0 = E +∂A∂t

since ∇×∇A0 = 0

−∇A0 − ∂A∂t

Maxwell - electric

Gauge Transformation

E and B are invariant under

A0 → A0 +∂χ

∂tA → A −∇χ

Monopole

Edward Olszewski

Compact Notation

The DiracMonopole

Supersymmetry

References

Appendix

Gauge Potential: A = Aµdxµ = A0dt + Aidx i

Gauge Transform: A → A − dχ(xµ)

Field Strength: F= −dA

(gauge invarient =12(∂µAν − ∂νAµ)dxµ ∧ dxν

due to ddχ = 0) =12

Fµνdxµ ∧ dxν

E and B F =Ekδkidx0 ∧ dx i +12

Bkϵkijdx i ∧ dx j

∗F ≡Fαβ ϵαβµν dxµ ∧ dxν

=Bkδkidx0 ∧ dx i − 12

Ekϵkijdx i ∧ dx j

duality : F →∗ F , (E i → Bi , Bi → −E i)∗∗F = −F

Monopole

Edward Olszewski

Compact Notation

The DiracMonopole

Supersymmetry

References

Appendix

Euler equation (inhomogeneous): δF = −∗d∗F = j

δF =−∇ · E dt ++ (−∂tE +∇× B) · dx

je =jeµ dxµ = −ρedt + Je · dx

Bianchi identity (homogeneous): dF = −ddA = 0

dF =∇ · Bdx1 ∧ dx2 ∧ dx3+

+12(∂tB +∇× E)iϵijkdt ∧ dx i ∧ dx j

Lorentz force equation: fν = jµe Fµν

Monopole

Edward Olszewski

Compact Notation

The DiracMonopole

Supersymmetry

References

Appendix

The Action and the Lagrangian Density

∫dt d3x L(Aµ, xiµ,uiµ)

∫dt d3x (−1

4FµνFµν − jµe Aµ + Lm)

the matter Lagrangian: Lm =∑

miδ(xi − x)uiµuµi

andje =

gieδ(xi − x)(−dt + vi · dx)

Here gie and vi are the electric charge and 3-velocity ofthe i-th particle.

In the Hamiltonian formalism : pµ → pµ − geAµ .

Monopole

Edward Olszewski

Compact Notation

The DiracMonopole

Supersymmetry

References

Appendix

Minimal Coupling

pµ = −i∂µ → −i(∂µ − igeAµ) ,

where ge is the electric charge.

Schrödinger equation: i∂tψ =− 12m

(∇− igeA)2ψ

(A,V , χ + geVψtime independent)

Monopole

Edward Olszewski

Compact Notation

The DiracMonopole

Supersymmetry

References

Appendix

Minimal Coupling (continued)

Compactify the range of χ, i.e. χ and χ+ 2πe correspond

to the same gauge transformation (e ≡ couplingconstant).

Gauge Transformation: ψ → e−igeχψ

A → A −∇χ ≡

≡ A − ige

eigeχ∇e−igeχ

Thus ge2πe

= 2πn, n ∈ Z

Charge Quantization ge = ne

Monopole

Edward Olszewski

Compact Notation

The DiracMonopole

Supersymmetry

References

Appendix

The Dirac Monopole (intuitive construction)

Figure: Given a positive electric charge

(a) Solenoid 1 (b) Solenoid 2

Figure: Given a north magnetic charge

Monopole

Edward Olszewski

Compact Notation

The DiracMonopole

Supersymmetry

References

Appendix

Wu-Yang construction

= ∓gm

4π(1 ∓ cos θ)

r sin θ

= ∓gm

4π(1 ∓ cos θ) dϕ

F = −dA → B =gm r4πr2

Monopole

Edward Olszewski

Compact Notation

The DiracMonopole

Supersymmetry

References

Appendix

Magnetic Charge Quantization

Since ∮dχ = χ(2π)− χ(0) =

=2πnge∮

∮(AS − AN) = gm

gm =2πnge

the Dirac condition ge gm = 2πn

the magnetic field B =1

Monopole

Edward Olszewski

Compact Notation

The DiracMonopole

Supersymmetry

References

Appendix

Summary

I The existence of a single magnetic charge requiresthat electric charge be quantized.

I The quantities e(−igeχ) are elements of a U(1) groupof gauge transformations. Electric charge isquantized, since χ and χ+ 2π/e ( e being thecoupling constant) yield the same gaugetransformation. In the alternative case when chargeis not quantized the gauge group is the real line R1.Magnetic monopoles require a compact U(1) gaugegroup.

I Mathematically, we have constructed a non-trivialprincipal fiber bundle with base manifold S2 and fiberU(1).

Monopole

Edward Olszewski

Compact Notation

The DiracMonopole

Supersymmetry

References

Appendix

Summary

Monopole

Edward Olszewski

Compact Notation

The DiracMonopole

Supersymmetry

References

Appendix

Summary

Monopole

Edward Olszewski

Compact Notation

The DiracMonopole

Supersymmetry

References

Appendix

Summary

Monopole

Edward Olszewski

Compact Notation

The DiracMonopole

Supersymmetry

References

Appendix

Gauge Transformations and Lie Groups

I The gauge groups corresponding to other forces innature are obtained by substituting for the gaugegroup of electromagnetism another appropriate Liegroup.

I Gravity can also be described as a gauge theorybased on the equivalence principle. The gaugetransformations are the general coordinatetransformations and local Lorentz transformations.

Monopole

Edward Olszewski

Compact Notation

The DiracMonopole

Supersymmetry

References

Appendix

Monopole

Edward Olszewski

Compact Notation

The DiracMonopole

Supersymmetry

References

Appendix

Monopole

Edward Olszewski

Compact Notation

The DiracMonopole

Supersymmetry

References

Appendix

Gauge Transformations (continued)

I For the remainder of this presentation we shall, forthe most part, restrict our consideration to the gaugegroups SO(2N) and SU(N), which are relevant toGrand Unification, because SO(10) −→ SU(5) −→SUc(3)× SUL(2)× UY (1) −→ SUc(3)× UEM(1).

Monopole

Edward Olszewski

Compact Notation

The DiracMonopole

Supersymmetry

References

Appendix

Gauge Transformations (continued)

I For the remainder of this presentation we shall, forthe most part, restrict our consideration to the gaugegroups SO(2N) and SU(N), which are relevant toGrand Unification, because SO(10) −→ SU(5) −→SUc(3)× SUL(2)× UY (1) −→ SUc(3)× UEM(1).

Monopole

Edward Olszewski

Compact Notation

The DiracMonopole

Supersymmetry

References

Appendix

Spontaneous Symmetry Breaking

scalar field

L =12∂µA∂µA − m2A2

two coupled scalar fields

L =12∂µA∂µA +

12∂µΦ∂

µΦ− gΦ2A2

scalar field coupled to EM

L =− 14

FµνFµν +12

DµΦDµΦ

DµΦ = ∂µΦ− ie AµΦ

Monopole

Edward Olszewski

Compact Notation

The DiracMonopole

Supersymmetry

References

Appendix

scalar field

L =12∂µA∂µA − m2A2

L =12∂µA∂µA +

12∂µΦ∂

µΦ− gΦ2A2

L =− 14

FµνFµν +12

DµΦDµΦ

Monopole

Edward Olszewski

Compact Notation

The DiracMonopole

Supersymmetry

References

Appendix

scalar field

L =12∂µA∂µA − m2A2

L =12∂µA∂µA +

12∂µΦ∂

µΦ− gΦ2A2

L =− 14

FµνFµν +12

DµΦDµΦ

Monopole

Edward Olszewski

Compact Notation

The DiracMonopole

Supersymmetry

References

Appendix

scalar field

L =12∂µA∂µA − m2A2

L =12∂µA∂µA +

12∂µΦ∂

µΦ− gΦ2A2

L =− 14

FµνFµν +12

DµΦDµΦ

Monopole

Edward Olszewski

Compact Notation

The DiracMonopole

Supersymmetry

References

Appendix

Spontaneous Symmetry Breaking(continued)

ABEGHHK’tH mechanism (the Higgs mechanism)

I the lagrangian is invariant under some symmetry,e.g. a non-abelian gauge symmetry.

I the ground state of the system is continuouslydegenerate under the symmetry.

I the system resides in a particular ground state whichbreaks some of the symmetry. An illustrativeexample is the gauge group SO(3).

Monopole

Edward Olszewski

Compact Notation

The DiracMonopole

Supersymmetry

References

Appendix

Monopole

Edward Olszewski

Compact Notation

The DiracMonopole

Supersymmetry

References

Appendix

Monopole

Edward Olszewski

Compact Notation

The DiracMonopole

Supersymmetry

References

Appendix

Monopole

Edward Olszewski

Compact Notation

The DiracMonopole

Supersymmetry

References

Appendix

Monopole

Edward Olszewski

Compact Notation

The DiracMonopole

Supersymmetry

References

Appendix

The Running Coupling Constant

Figure: SU(N) Gauge Theory: (a) Screening due to flavors; (b)Antiscreening due to gluons.

β(µ) ≡ ∂α

∂ log(µ)=α2

3− 11N

where µ is the relevant energy scale, and α = g2

4π .For SU(3), N = 3, and β < 0 requires nf <

Monopole

Edward Olszewski

Compact Notation

The DiracMonopole

Supersymmetry

References

Appendix

The Running Coupling Constant

Figure: SU(N) Gauge Theory: (a) Screening due to flavors; (b)Antiscreening due to gluons.

β(µ) ≡ ∂α

∂ log(µ)=α2

3− 11N

where µ is the relevant energy scale, and α = g2

4π .For SU(3), N = 3, and β < 0 requires nf <

Monopole

Edward Olszewski

Compact Notation

The DiracMonopole

Supersymmetry

References

Appendix

Running Coupling Constant (continued)

U(1)Y α1 = (5/3)g′2/(4π) = 5α/(3 cos2 θW )

SU(2)L α2 = g2/(4π) = α/ sin2 θW

SU(3)c α3 = g2s/(4π)

MZ (80.4GeV ) α1 = 0.017, α2 = 0.034, α3 = 0.118,

α−1 = 129, sin2 θW = .231

Monopole

Edward Olszewski

Compact Notation

The DiracMonopole

Supersymmetry

References

Appendix

Running Coupling Constant (continued)

Figure: Theory of Everything. One GeV corresponds to alength scale of approximately 2 × 10−14 cm. The length scaleassociated with string theory is

√α′ ≈ 10−33 cm.

Monopole

Edward Olszewski

Compact Notation

The DiracMonopole

Supersymmetry

References

Appendix

The Yang–Mills–Higgs Lagrangian

L = −14

Fµν · Fµν +12

DµΦ · DµΦ− V (Φ ·Φ) ,

Fµν = ∂µAν − ∂νAµ − igYM Aµ ∧ Aν .

Higgs field Φ - scalar transforming in the adjointrepresentation of the gauge group so that

DµΦ = ∂µΦ− igYM Aµ ∧Φ .

Fµν = F aµνTa

Aµ = AaµTa

Φ = ΦaTa

Note: H · G ≡ HaGb 2Tr(TaTb) = HaGa

Monopole

Edward Olszewski

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Supersymmetry

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Appendix

V (Φ ·Φ) is a potential such that the vacuum expectationof Φ is non-zero. When a specific form of V (Φ ·Φ) isrequired we use

V (Φ ·Φ) =λ

8(Φ ·Φ− v2)2 .

The Lagrangian, when V (Φ ·Φ) = 0, is scale invariant,but the requirement that Φ ·Φ = v2 spontaneously breaksscale invariance of the vacuum.

scale invariance: δxµ = xµ δλ; δAµ = −Aµ δλ;

δΦ = −Φ δλ.

Monopole

Edward Olszewski

Compact Notation

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Supersymmetry

References

Appendix

V (Φ ·Φ) =λ

8(Φ ·Φ− v2)2 .

δΦ = −Φ δλ.

Monopole

Edward Olszewski

Compact Notation

The DiracMonopole

Supersymmetry

References

Appendix

V (Φ ·Φ) =λ

8(Φ ·Φ− v2)2 .

δΦ = −Φ δλ.

Monopole

Edward Olszewski

Compact Notation

The DiracMonopole

Supersymmetry

References

Appendix

How to define Fµν

Usual Definition Fµν ≡Fµν · HH ∈ Cartan subalgebra

dF = 0 Fµν =Fµν · ΦBianchi identity not satisfied

t‘Hooft – SO(3) Fµν =Fµν · Φ−1

igYMDµΦ ∧ DνΦ · Φ

dF = 0 iff Φ ∧ Eα =± Eα or 0(Bianchi identity satisfied)

Here Eα is any one of the generators of the Lie algebra.

Monopole

Edward Olszewski

Compact Notation

The DiracMonopole

Supersymmetry

References

Appendix

How to define Fµν

Usual Definition Fµν ≡Fµν · HH ∈ Cartan subalgebra

dF = 0 Fµν =Fµν · ΦBianchi identity not satisfied

t‘Hooft – SO(3) Fµν =Fµν · Φ−1

igYMDµΦ ∧ DνΦ · Φ

dF = 0 iff Φ ∧ Eα =± Eα or 0(Bianchi identity satisfied)

Here Eα is any one of the generators of the Lie algebra.

Monopole

Edward Olszewski

Compact Notation

The DiracMonopole

Supersymmetry

References

Appendix

Make the ansatz that the Higgs field Φ and vectorpotential A take the form

Φ = (Q(r) α1Tz + α2T⊥) v ,

gS(r) v α1 Tz dt + Tz (−C) W (r)(1 − cos θ) dϕ ,

W (r),Q(r),S(r) → 0 , as r → 0 ;W (r),Q(r) → 1 , S(r) → 1 − g

gm gYM α1 v r , as r → ∞ ;

g =√

g2e + g2

Here C is an arbitrary constant, and quantities ge and gmare the electric and magnetic charges.

root system – SU(3)

Monopole

Edward Olszewski

Compact Notation

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Supersymmetry

References

Appendix

Applying the gauge transformation

χ = e−iϕTz e−iθTy eiϕTz

to A and Φ we obtain

A → χAχ−1 − 1igYM

dχ χ−1

gS(r) v α1 Tr dt +

W (r)gYM

( Tθ sin θ dϕ − Tϕ dθ ) .

Φ → χΦχ−1

= v [ α2 T⊥ + Q(r) α1 Tr ] .

We have used the fact that

dχ χ−1 = −i [(1 − cos θ) Tr dϕ+ sin θ Tθ dϕ− Tϕ dθ] .

Monopole

Edward Olszewski

Compact Notation

The DiracMonopole

Supersymmetry

References

Appendix

Dyon charge quantization conditions, SU(N)

magnetic charge gm =4π

|α|2gYM

electric charge ge = ne α1gYM

2(N − 1)gYM ,

where ne is an integer. The electric charge quantization isderived from the adjoint representation.

Monopole

Edward Olszewski

Compact Notation

The DiracMonopole

Supersymmetry

References

Appendix

Comparison of the different types of dyons

SU(N), (|α|2 = 1) gm ge = ne 4π

2(N − 1)

G2E , (|α|2 = 1/3) gm ge = ne 4π 3SO(3), t’Hooft/Polykov gm ge = ne 4π

U(1),Dirac gm e = ne4π2

Magnetic Monopole Mass (V = 0)

SU(N), (|α|2 = 1) M =g

2(N − 1)MX

G2E , (|α|2 = 1/3) M =g

2gYMMX

SO(3), t’Hooft/Polykov M =g

U(1),Dirac __root system – G2

Monopole

Edward Olszewski

Compact Notation

The DiracMonopole

Supersymmetry

References

Appendix

Dyon structure

r2 er r → ∞

Φ ≈ r r → 0 .

The size of the dyon, i.e. the length scale where massivegauge bosons exist, is

ldyon ≈ ggm

1gYM v α1

≈ 1gYMv

MX≈ 1

1016 GeV≈ 10−30cm

At r = 0, Φ = 0, and symmetry is restored. Themonopole can catalyse nucleon decay.

Monopole

Edward Olszewski

Compact Notation

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Supersymmetry

References

Appendix

Dyon structure

r2 er r → ∞

Φ ≈ r r → 0 .

ldyon ≈ ggm

1gYM v α1

≈ 1gYMv

MX≈ 1

1016 GeV≈ 10−30cm

Monopole

Edward Olszewski

Compact Notation

The DiracMonopole

Supersymmetry

References

Appendix

Dyon structure

r2 er r → ∞

Φ ≈ r r → 0 .

ldyon ≈ ggm

1gYM v α1

≈ 1gYMv

MX≈ 1

1016 GeV≈ 10−30cm

Monopole

Edward Olszewski

Compact Notation

The DiracMonopole

Supersymmetry

References

Appendix

Montonen - Olive Conjecture

Mass (ge, gm) SpinHiggs 0 (0, 0) 0

Photon 0 (0, 0) 1W± v e (e, 0) 1M vg (0, g) 0

Table: The gauge group SO(3).

Note: MW±(MX ) = gYMv ; also, e ≡ gYM ;g ≡ 4πgYM

Weak/Strong Duality: E → B; e → gincluding a relabeling of electricand magnetic states

Monopole

Edward Olszewski

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Supersymmetry

References

Appendix

Langrangian Density (V = 0)

Witten Effect: Lθ = − θe2

32π2 ∗ Fµν · Fµν =θe2

8π2 Ei · Bi

Traditional: Ltrad = −14

Fµν · Fµν =12(Ei · Ei − Bi · Bi)

Lθ causes shifts in allowed values of electric charge inthe monopole sector.

Consider U(1) gauge transformations about Φ:

δAµ = DµΦ

Let η be the generator of infinitesimal gaugetransformations. Since

physical quantities require ei2πη = 1eigenvalues of η η = n, (n ∈ Z )

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References

Appendix

By Noether’s theorem η =∂L

∂∂0Aαµ

δAαµ

Therefore η =ge

e+θegm

Consequently (nm = egm/4π) ge = n e − θ

2πnm e

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Edward Olszewski

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Supersymmetry

References

Appendix

SL(2,Z )Define

τ =θ

2π+ i

θ → θ + 2π τ → τ + 1

(e → gm ≡ 4πe

τ → −1τ

θ = 0)

τ → aτ + bcτ + d

a, b, c, d ∈ Z , ad − bc = 1(n

(a −bc −d

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Edward Olszewski

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Supersymmetry

References

Appendix

Magnetic Charge gm =4πe

Electric Charge ge = ne − nmθ

Mass of Dyon M2 ≥ v2(g2e + g2

Fµν · Fµν −θe2

32π2 Fµν · ∗Fµν −12DµΦ · DµΦ

≡− 132π

Im(τ)(Fµν + i ∗ Fµν) · (Fµν + i ∗ Fµν)

− 12DµΦ · DµΦ

Monopole

Edward Olszewski

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References

Appendix

Summary

I Quantum corrections would be expected to generatea non-zero potential V (Φ) even if one is absentclassically and should also modify the classical massformula. Thus there is no reason to think that theduality of the spectrum should be maintained byquantum corrections.

I The W bosons have spin one while the monopolesare rotationally invariant indicating that they havespin zero. Thus even if the mass spectrum isinvariant under duality, there will not be an exactnumber of matching states and quantum numbers.

I The proposed duality symmetry seems impossible totest since rather than acting as a symmetry of asingle theory it relates two different theories, one ofwhich is necessarily at strong coupling where wehave little control of the theory.

Monopole

Edward Olszewski

Compact Notation

The DiracMonopole

Supersymmetry

References

Appendix

Summary

Monopole

Edward Olszewski

Compact Notation

The DiracMonopole

Supersymmetry

References

Appendix

Summary

Monopole

Edward Olszewski

Compact Notation

The DiracMonopole

Supersymmetry

References

Appendix

Summary

Monopole

Edward Olszewski

Compact Notation

The DiracMonopole

Supersymmetry

References

Appendix

Summary (continued)

I The N = 4 Super Yang-Mills theory solves the firstproblem because of exact quantum scale invariance(V = 0). It also solves the second problem becauseadditional particles are added to the spectrum.

Monopole

Edward Olszewski

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Appendix

Summary (continued)

I The N = 4 Super Yang-Mills theory solves the firstproblem because of exact quantum scale invariance(V = 0). It also solves the second problem becauseadditional particles are added to the spectrum.

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Edward Olszewski

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Appendix

I Supersymmetry is an extension of the Poincaréalgebra.

I Every boson(fermion) has a correspondingfermionic(bosonic) partner.

Klein Gordon Equation (−ηµνPµPν − m2)Φ = 0

Dirac “√ηµνPµPν” = ηµνΓµPν = PµΓ

where {Γµ, Γν} = 2ηµν

supersymmetry algebra {Qα,Qβ} = −2PµΓµαβ

{Pµ,Qα} = 0

Monopole

Edward Olszewski

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Supersymmetry

References

Appendix

{Pµ,Qα} = 0

Monopole

Edward Olszewski

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Supersymmetry

References

Appendix

{Pµ,Qα} = 0

Monopole

Edward Olszewski

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Supersymmetry

References

Appendix

Do we really need the Higgs field?Assume an extra dimension Let A4 = Φ. AssumeV = 0.

L =− 14

Fµν · Fµν +12

DµA4 · DµA4 ,

= −14

F MN · F MN (M,N = 0 . . .4)

Fµ4 = ∂µA4 − ie Aµ ∧ A4

The Higgs field becomes a component of the Yang-Millspotential in the extra dimension.

Monopole

Edward Olszewski

Compact Notation

The DiracMonopole

Supersymmetry

References

Appendix

L =− 14

Fµν · Fµν +12

DµA4 · DµA4 ,

= −14

F MN · F MN (M,N = 0 . . .4)

Fµ4 = ∂µA4 − ie Aµ ∧ A4

Monopole

Edward Olszewski

Compact Notation

The DiracMonopole

Supersymmetry

References

Appendix

L =− 14

Fµν · Fµν +12

DµA4 · DµA4 ,

= −14

F MN · F MN (M,N = 0 . . .4)

Fµ4 = ∂µA4 − ie Aµ ∧ A4

Monopole

Edward Olszewski

Compact Notation

The DiracMonopole

Supersymmetry

References

Appendix

Introduction to superstring theoryI Gravity. Every consistent string theory must contain

a massless spin-2 state, whose interactions reduceat low energy to general relativity.

I A consistent theory of quantum gravity, at least inperturbation theory. As we have noted, this is incontrast to all known quantum field theories ofgravity.

I Grand unification. String theories lead to gaugegroups large enough to include the Standard Model.Some of the simplest string theories lead to thesame gauge groups and fermion representations thatarise in the unification of the Standard Model.

I Extra dimensions. String theory requires a definitenumber of spacetime dimensions, ten. The fieldequations have solutions with four large flat and sixsmall curved dimensions, with four-dimensionalphysics that resembles the Standard Model.

Monopole

Edward Olszewski

Compact Notation

The DiracMonopole

Supersymmetry

References

Appendix

Monopole

Edward Olszewski

Compact Notation

The DiracMonopole

Supersymmetry

References

Appendix

Monopole

Edward Olszewski

Compact Notation

The DiracMonopole

Supersymmetry

References

Appendix

Monopole

Edward Olszewski

Compact Notation

The DiracMonopole

Supersymmetry

References

Appendix

Monopole

Edward Olszewski

Compact Notation

The DiracMonopole

Supersymmetry

References

Appendix

Primer (continued)I Supersymmetry. Consistent string theories require

spacetime super-symmetry, as either a manifest or aspontaneously broken symmetry.

I Chiral gauge couplings. The gauge interactions innature are parity asymmetric (chiral). This has beena stumbling block for a number of previous unifyingideas: they required parity symmetric gaugecouplings. String theory allows chiral gaugecouplings.

I No free parameters. String theory has no adjustableconstants.

I Uniqueness. Not only are there no continuousparameters, but there is no discrete freedomanalogous to the choice of gauge group andrepresentations in field theory: there is a uniquestring theory.

Monopole

Edward Olszewski

Compact Notation

The DiracMonopole

Supersymmetry

References

Appendix

Monopole

Edward Olszewski

Compact Notation

The DiracMonopole

Supersymmetry

References

Appendix

Monopole

Edward Olszewski

Compact Notation

The DiracMonopole

Supersymmetry

References

Appendix

Monopole

Edward Olszewski

Compact Notation

The DiracMonopole

Supersymmetry

References

Appendix

Primer (continued)

Figure: Two graviton exchange (field theory)

Figure: (a) Closed string. (b) Open string. (c) The loopamplitude in string theory of two graviton exchange, above

Monopole

Edward Olszewski

Compact Notation

The DiracMonopole

Supersymmetry

References

Appendix

U-dualityI SL(2,Z ) duality (S duality) i.e. Weak/Strong Duality,

(a consequence of supersymmetry)I O(k , k ,Z ) T–duality group, including a parity

operation on the right moving open string excitationswhich implies R → R′ = α′

RI Closed String

I pl =NR + wR

α′ implies pl → p′l = pl

I pr =NR − wR

α′ implies pr → p′r = −pr

I Open String - each end of the open string isconstrained to lie on a hypersurface of the space.

Figure: D-branes. Three hyperplanes at different positions,with various open strings attached.

Monopole

Edward Olszewski

Compact Notation

The DiracMonopole

Supersymmetry

References

Appendix

RI Closed String

I pl =NR + wR

I pr =NR − wR

Monopole

Edward Olszewski

Compact Notation

The DiracMonopole

Supersymmetry

References

Appendix

RI Closed String

I pl =NR + wR

I pr =NR − wR

Monopole

Edward Olszewski

Compact Notation

The DiracMonopole

Supersymmetry

References

Appendix

RI Closed String

I pl =NR + wR

I pr =NR − wR

Monopole

Edward Olszewski

Compact Notation

The DiracMonopole

Supersymmetry

References

Appendix

RI Closed String

I pl =NR + wR

I pr =NR − wR

Monopole

Edward Olszewski

Compact Notation

The DiracMonopole

Supersymmetry

References

Appendix

Magnetic Monopole SolutionI Begin with a Type I SO(32) string theory in ten

dimensions, compactify six of the nine space (thecompactification radius, R ≫ α′), resulting in a sixdimensional torus, T6.

I Apply T-duality transformations, e.g. ( R → α′

R ) to 5 ofthe 6 compactified space dimensions. The T-dualitysymmetry group is O(5,5,Z ). This T5 will be theinternal dimensions of spacetime.

I Spacetime now consists of five internal dimensionsand a D4-brane comprising a five dimensionalspacetime (Type IIA in the bulk).

I Construct a magnetic monople solution on theD4-brane, then apply the T-duality transformation,R → α′

R to the compactified dimension of theD4-brane, resulting in a D3-brane. The D3-brane isfour dimensional spacetime (Type IIB in the bulk).

Monopole

Edward Olszewski

Compact Notation

The DiracMonopole

Supersymmetry

References

Appendix

Monopole

Edward Olszewski

Compact Notation

The DiracMonopole

Supersymmetry

References

Appendix

Monopole

Edward Olszewski

Compact Notation

The DiracMonopole

Supersymmetry

References

Appendix

Monopole

Edward Olszewski

Compact Notation

The DiracMonopole

Supersymmetry

References

Appendix

Monopole

Edward Olszewski

Compact Notation

The DiracMonopole

Supersymmetry

References

Appendix

Magnetic Monopole Solution (continued)

The action of the system, S = SDBI + SCS, comprises twoparts:

I The Dirac-Born-Infeld Action - coupling a Dp–braneto NS–NS closed string fields

SDBI = −µp

∫Mp+1

Tr{e−Φ[−det(Gab+

+ Bab + 2πα′Fab)]1/2}

I Chern-Simons Action - coupling a Dp–brane to R–Rclosed string fields

SCS = µp

∫Mp+1

C(p+1)

]∧ Tre2πα′F+B

Monopole

Edward Olszewski

Compact Notation

The DiracMonopole

Supersymmetry

References

Appendix

The action of the system, S = SDBI + SCS, comprises twoparts:

I The Dirac-Born-Infeld Action - coupling a Dp–braneto NS–NS closed string fields

SDBI = −µp

∫Mp+1

Tr{e−Φ[−det(Gab+

+ Bab + 2πα′Fab)]1/2}

I Chern-Simons Action - coupling a Dp–brane to R–Rclosed string fields

SCS = µp

∫Mp+1

C(p+1)

]∧ Tre2πα′F+B

Monopole

Edward Olszewski

Compact Notation

The DiracMonopole

Supersymmetry

References

Appendix

Make the ansatzI p = 4;I C(1) = C4 dx4, where C4 is constant and all other

R–R potentials vanish;I

gS(r) v α1 Tz dt +

+ Tz (−C) W (r)(1 − cos θ) dϕ ,A4 = Φ = (Q(r) α1Tz + α2T⊥) v .

Monopole

Edward Olszewski

Compact Notation

The DiracMonopole

Supersymmetry

References

Appendix

gS(r) v α1 Tz dt +

Monopole

Edward Olszewski

Compact Notation

The DiracMonopole

Supersymmetry

References

Appendix

gS(r) v α1 Tz dt +

Monopole

Edward Olszewski

Compact Notation

The DiracMonopole

Supersymmetry

References

Appendix

gS(r) v α1 Tz dt +

Monopole

Edward Olszewski

Compact Notation

The DiracMonopole

Supersymmetry

References

Appendix

Figure: Wormhole connecting two D3-branes.

I There is no event horizon.I The scalar curvature is finite at r = 0. Specifically,

R ≈ 23α

′−1 as r → 0.I There is no singularity at r = 0, since T-duality

should not create a singularity.

Monopole

Edward Olszewski

Compact Notation

The DiracMonopole

Supersymmetry

References

Appendix

R ≈ 23α

Monopole

Edward Olszewski

Compact Notation

The DiracMonopole

Supersymmetry

References

Appendix

R ≈ 23α

Monopole

Edward Olszewski

Compact Notation

The DiracMonopole

Supersymmetry

References

Appendix

R ≈ 23α

Monopole

Edward Olszewski

Compact Notation

The DiracMonopole

Supersymmetry

References

Appendix

Magnetic Monopole Solution (continued)I The IIB theory transforms into itself under S duality;

i.e. the weakly coupled theory is identical to thestrongly coupled theory with a relabeling of states. Inparticular, S : F → D1.

I Relative strength of couplings:gravity (10 dim) κ2 = 1

2(2π)7g2(α′)4

F string tension τF1 = 12πα′

D1 string tension τD1 = τF1g

where g = eΦ, Φ being the constant dilatonbackground.

I Relevant length scales (weak coupling, i.e. g ≪ 1):τ−1/2F1 : lg : τ

−1/2D1 = g−1/4 : 1 : g1/4

I the monopole size, lmonopole ≈ (2πα′)1/2

where the GUTS Yang-Mills coupling constant is g2D3,

where g2D3 = 2πg.

Monopole

Edward Olszewski

Compact Notation

The DiracMonopole

Supersymmetry

References

Appendix

2(2π)7g2(α′)4

−1/2D1 = g−1/4 : 1 : g1/4

where g2D3 = 2πg.

Monopole

Edward Olszewski

Compact Notation

The DiracMonopole

Supersymmetry

References

Appendix

2(2π)7g2(α′)4

−1/2D1 = g−1/4 : 1 : g1/4

where g2D3 = 2πg.

Monopole

Edward Olszewski

Compact Notation

The DiracMonopole

Supersymmetry

References

Appendix

2(2π)7g2(α′)4

−1/2D1 = g−1/4 : 1 : g1/4

where g2D3 = 2πg.

Monopole

Edward Olszewski

Compact Notation

The DiracMonopole

Supersymmetry

References

Appendix

2(2π)7g2(α′)4

−1/2D1 = g−1/4 : 1 : g1/4

where g2D3 = 2πg.

Monopole

Edward Olszewski

Compact Notation

The DiracMonopole

Supersymmetry

References

Appendix

Gauge/Gravity Duality

I Steven Weinberg and Edward Witten have shownthat if there is a massless spin-2 particle in thespectrum, the the matrix element

< massless spin 2, k |Tµν | massless spin 2, k ′ >

of the energy momentum tensor (which exists as alocal observable in the gauge theory) has impossibleproperties.

I Assumption – the graviton bound state moves in thesame spacetime as its gauge boson constituents.

I Violation of the assumption – the graviton boundstate moves in one additional dimension.

Monopole

Edward Olszewski

Compact Notation

The DiracMonopole

Supersymmetry

References

Appendix

Monopole

Edward Olszewski

Compact Notation

The DiracMonopole

Supersymmetry

References

Appendix

Monopole

Edward Olszewski

Compact Notation

The DiracMonopole

Supersymmetry

References

Appendix

Gauge/Gravity Duality (continued)

Figure: Exchange of a closed string between two D-branes.Equivalently, a vacuum loop of an open string with one end oneach D-brane.

Monopole

Edward Olszewski

Compact Notation

The DiracMonopole

Supersymmetry

References

Appendix

References I

WIKIPEDIA,Magnetic monopole,http://en.wikipedia.org/wiki/Magnetic_monopole, 2007,[Online; accessed 15-November-2013].

A. RAJANTIE,Introduction to Magnetic Monopoles,arXiv:1204.3077v1.

P. A. M. DIRAC,Proc. Roy. Soc. A, 60 (1931).

P. RAMOND,Field Theory: A Modern Primer,Addison–Wesley Publishing Company, Inc., Boston,Masssachusetts, 1989.

Monopole

Edward Olszewski

Compact Notation

The DiracMonopole

Supersymmetry

References

Appendix

References II

L. H. RYDER,Quantum Field Theory,Cambridge University Press, New York, New York,1996.

T. EGUCHI, P. B. GILKEY, and A. J. HANSON,Physics Reports 66 (1980).

J. BAEZ,This Week’s Finds in Mathematical Physics (Week94),http://math.ucr.edu/home/baez/week94.html,1996,[Online; accessed 12-March-2014].

Monopole

Edward Olszewski

Compact Notation

The DiracMonopole

Supersymmetry

References

Appendix

References III

F. WILCZEK,Asymptotic Freedom,hep–th/9609099.

D. I. KAZAKOV,Beyond the Standard Model (In Search ofSupersymmetry),arXiv:hep-ph/0012288v2.

G. HOOFT,Nuclear Physics B, 276 (1974).

C. MONTONEN and D. OLIVE,Phys. Lett. 72B, 117 (1977).

J. A. HARVEY,Magnetic Monopoles, Duality, and Supersymmetry,hep–th/9603086.

Monopole

Edward Olszewski

Compact Notation

The DiracMonopole

Supersymmetry

References

Appendix

References IV

E. A. OLSZEWSKI,Particle Physics Insights 5, 1 (2012).

J. POLCHINSKI,String Theory Volume I,Cambridge University Press, New York, New York,1998.

J. POLCHINSKI,String Theory Volume II,Cambridge University Press, New York, New York,1998.

C. V. JOHNSON,D–Branes,Cambridge University Press, New York, New York,2003.

Monopole

Edward Olszewski

Compact Notation

The DiracMonopole

Supersymmetry

References

Appendix

References V

J. POLCHINSKI,Introduction to Gauge/Gravity Duality,arXiv:1010.6134.

Monopole

Edward Olszewski

Compact Notation

The DiracMonopole

Supersymmetry

References

Appendix

monopole SU(3) Root System

2 )(−12 ,

(12 ,−

2 )(−12 ,−

(1,0)(−1,0)

Monopole

Edward Olszewski

Compact Notation

The DiracMonopole

Supersymmetry

References

Appendix

monopole G2 Root System

2 ,√

(0, 1√3)

Monopole

Edward Olszewski

Compact Notation

The DiracMonopole

Supersymmetry

References

Appendix

ReferencesI WIKIPEDIA, Magnetic monopole,http://en.wikipedia.org/wiki/Magnetic_monopole, 2007,[Online; accessed 15-November-2013]

I A. RAJANTIE, Introduction to Magnetic Monopoles,arXiv:1204.3077v1

I P. A. M. DIRAC, Proc. Roy. Soc. A, 60 (1931)

I P. RAMOND, Field Theory: A Modern Primer,Addison–Wesley Publishing Company, Inc., Boston,Masssachusetts, 1989

I L. H. RYDER, Quantum Field Theory,Cambridge University Press, New York, New York,1996

I T. EGUCHI, P. B. GILKEY, and A. J. HANSON,

Physics Reports 66 (1980)

I J. BAEZ, This Week’s Finds in Mathematical Physics(Week 94),http://math.ucr.edu/home/baez/week94.html,1996,[Online; accessed 12-March-2014]

I F. WILCZEK, Asymptotic Freedom,hep–th/9609099

I D. I. KAZAKOV, Beyond the Standard Model (InSearch of Supersymmetry),arXiv:hep-ph/0012288v2

I G. HOOFT, Nuclear Physics B, 276 (1974)

I C. MONTONEN and D. OLIVE, Phys. Lett. 72B, 117(1977)

I J. A. HARVEY, Magnetic Monopoles, Duality, andSupersymmetry,hep–th/9603086

I E. A. OLSZEWSKI, Particle Physics Insights 5, 1(2012)

I J. POLCHINSKI, String Theory Volume I,Cambridge University Press, New York, New York,1998

I J. POLCHINSKI, String Theory Volume II,Cambridge University Press, New York, New York,1998

I C. V. JOHNSON, D–Branes,Cambridge University Press, New York, New York,2003

I J. POLCHINSKI, Introduction to Gauge/GravityDuality,arXiv:1010.6134

electrically charged magnetic monopoles, superstring...

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