Electrically Charged Magnetic Monopoles, Superstring Theory

Edward Olszewski

Electromagnetism The Maxwell Theory Compact Notation Minimal Coupling (Quantum Mechanics)

Electrically Charged Magnetic The Dirac Monopole

Monopoles, Superstring Theory, and Grand Uniﬁed Wormholes Theories Magnetic Monopoles and Dyons in Grand Uniﬁed Theories

Edward Olszewski Montonen–Olive Duality

Supersymmetry

Superstring March 14, 2014 Theory Primer Magnetic Monopoles in String Theory

References

Appendix

...... Outline Monopole Edward Olszewski Electromagnetism Electromagnetism The Maxwell Theory The Maxwell Theory Compact Notation Compact Notation Minimal Coupling (Quantum Mechanics) Minimal Coupling (Quantum Mechanics) The Dirac Monopole The Dirac Monopole Grand Unified Grand Unified Theories Theories Magnetic Magnetic Monopoles and Dyons in Grand Unified Monopoles and Dyons in Grand Theories Unified Theories Montonen–Olive Montonen–Olive Duality Duality Supersymmetry Supersymmetry Superstring Superstring Theory Primer Theory Primer Magnetic Monopoles in Magnetic Monopoles in String Theory String Theory References References Appendix Appendix

...... Monopole

The Maxwell Equations Edward Olszewski Electromagnetism The Maxwell Theory Notation: (µ = 0, 1, 2, 3; i = 1, 2, 3), (Minkowski metric Compact Notation Minimal Coupling (Quantum Mechanics) with η00 = −1), (Levi-Cevità symbol: ϵ0123 = ϵ123 = 1), The Dirac (Lorentz-Heaviside units) Monopole

Grand Uniﬁed Theories ∂E Magnetic ∇ · E = ρe ∇ × B − = Je Monopoles and ∂t Dyons in Grand ∂B Uniﬁed Theories ∇ · B = 0 ∇ × E + = 0 Montonen–Olive ∂t Duality Supersymmetry Maxwell - magnetic Potential Superstring Theory Primer

Magnetic The Lorentz Force Equation Monopoles in String Theory

F = qeE + qev × B References Appendix

...... Monopole

Edward Olszewski The Maxwell Equations (magnetic) Electromagnetism The Maxwell Theory Compact Notation Minimal Coupling (Quantum E → B B → −E Mechanics) → → − The Dirac Je Jm Jm Je Monopole

ρe → ρm ρm → −ρe Grand Uniﬁed Theories

Magnetic Monopoles and ∂ Dyons in Grand B Uniﬁed Theories ∇ · B = ρm −∇ × E − = Jm ∂t Montonen–Olive ∂E Duality −∇ · E = 0 ∇ × B − = 0 ∂t Supersymmetry Superstring Theory Primer Maxwell - electric Magnetic Monopoles in The Lorentz Force Equation String Theory References F = qmB − qmv × E Appendix

...... Monopole

The Potential Function Edward Olszewski

Electromagnetism Consequently, The Maxwell Theory Compact Notation Minimal Coupling (Quantum ∇ × A = B since ∇ · (∇ × A) = 0 Mechanics) The Dirac −∇ 0 ∂A ∇ × ∇ 0 Monopole A = E + since A = 0 Grand Uniﬁed ∂t Theories

or Magnetic Monopoles and 0 ∂A Dyons in Grand −∇A − = E . Uniﬁed Theories ∂t Montonen–Olive Duality Maxwell - electric Supersymmetry

Superstring Gauge Transformation Theory Primer Magnetic E and B are invariant under Monopoles in String Theory ∂χ References A0 → A0 + A → A − ∇χ ∂t Appendix

...... Monopole

Edward Olszewski µ i Gauge Potential: A = Aµdx = A0dt + Ai dx Electromagnetism The Maxwell Theory → − µ Compact Notation Gauge Transform: A A dχ(x ) Minimal Coupling (Quantum − Mechanics) Field Strength: F= dA The Dirac Monopole 1 µ ν (gauge invarient = (∂µAν − ∂νAµ)dx ∧ dx Grand Uniﬁed 2 Theories 1 µ ν Magnetic due to ddχ = 0) = Fµνdx ∧ dx Monopoles and 2 Dyons in Grand Uniﬁed Theories k 0 i 1 k i j E and B F =E δki dx ∧ dx + B ϵkij dx ∧ dx Montonen–Olive 2 Duality ∗ ≡ αβ µ ∧ ν Supersymmetry F Fαβ ϵµν dx dx Superstring k 0 i 1 k i j Theory Primer =B δki dx ∧ dx − E ϵkij dx ∧ dx 2 Magnetic ∗ Monopoles in duality : F → F, (Ei → Bi , Bi → −Ei ) String Theory References ∗∗F = −F Appendix

...... Monopole Euler equation (inhomogeneous): δF = −∗d ∗F = j Edward Olszewski Electromagnetism The Maxwell Theory Compact Notation − ∇ · Minimal Coupling (Quantum δF = E dt + Mechanics) − ∇ × · The Dirac + ( ∂t E + B) dx Monopole µ je =jeµ dx = −ρedt + Je · dx Grand Uniﬁed Theories

Magnetic Monopoles and Dyons in Grand Bianchi identity (homogeneous): dF = −ddA = 0 Uniﬁed Theories Montonen–Olive Duality 1 2 3 dF =∇ · Bdx ∧ dx ∧ dx + Supersymmetry Superstring 1 i i j Theory Primer + (∂t B + ∇ × E) ϵijk dt ∧ dx ∧ dx 2 Magnetic Monopoles in =0 String Theory

References

µ Appendix Lorentz force equation: fν = je Fµν

...... Monopole The Action and the Lagrangian Density Edward Olszewski ∫ Electromagnetism 3 L The Maxwell Theory S = dt d x (Aµ, xiµ, uiµ) Compact Notation Minimal Coupling (Quantum ∫ Mechanics) 3 1 µν µ = dt d x (− FµνF − j Aµ + Lm) The Dirac 4 e Monopole Grand Uniﬁed where Theories ∑ 1 µ Magnetic the matter Lagrangian: Lm = mi δ(xi − x)uiµu Monopoles and 2 i Dyons in Grand i Uniﬁed Theories Montonen–Olive and ∑ Duality je = gieδ(xi − x)(−dt + vi · dx) Supersymmetry i Superstring Theory Primer Here g and v are the electric charge and 3-velocity of ie i Magnetic the i-th particle. Monopoles in String Theory

References

Appendix In the Hamiltonian formalism : pµ → pµ − geAµ .

...... Monopole

Edward Olszewski

Electromagnetism The Maxwell Theory Compact Notation Minimal Coupling (Quantum Minimal Coupling Mechanics) The Dirac Monopole pµ = −i∂µ → −i(∂µ − igeAµ) , Grand Uniﬁed Theories where g is the electric charge. e Magnetic Monopoles and Dyons in Grand 1 2 Uniﬁed Theories Schrödinger equation: i∂t ψ = − (∇ − igeA) ψ 2m Montonen–Olive Duality (A, V , χ + geV ψ Supersymmetry

time independent) Superstring Theory Primer

Magnetic Monopoles in String Theory

References

Appendix

...... Monopole Minimal Coupling (continued) Edward Olszewski

Electromagnetism Compactify the range of χ, i.e. χ and χ + 2π correspond The Maxwell Theory e Compact Notation Minimal Coupling (Quantum to the same gauge transformation (e ≡ coupling Mechanics) constant). The Dirac Monopole

Grand Uniﬁed Theories −igeχ Gauge Transformation: ψ → e ψ Magnetic Monopoles and A → A − ∇χ ≡ Dyons in Grand Uniﬁed Theories

Montonen–Olive i igeχ −igeχ ≡ A − e ∇e Duality ge Supersymmetry

Superstring Theory Primer 2π Magnetic Thus ge = 2πn, n ∈ Z Monopoles in e String Theory Charge Quantization ge = ne References Appendix

...... The Dirac Monopole (intuitive construction) Monopole Edward Olszewski

Electromagnetism The Maxwell Theory Compact Notation Minimal Coupling (Quantum Mechanics)

The Dirac Monopole

Grand Uniﬁed Theories

Figure: Given a positive electric charge Magnetic Monopoles and Dyons in Grand Uniﬁed Theories

Montonen–Olive Duality

Supersymmetry

Superstring (a) Solenoid 1 (b) Solenoid 2 Theory Primer Magnetic Monopoles in Figure: Given a north magnetic charge String Theory References

Appendix

...... Wu-Yang construction Monopole Edward Olszewski

Electromagnetism gm eˆϕ The Maxwell Theory A = ∓ (1 ∓ cos θ) Compact Notation N Minimal Coupling (Quantum S 4π r sin θ Mechanics) g = ∓ m (1 ∓ cos θ) dϕ The Dirac 4π Monopole Grand Uniﬁed Theories gmˆr F = −dA → B = Magnetic 4πr 2 Monopoles and Dyons in Grand Uniﬁed Theories

Montonen–Olive Duality

Supersymmetry

Superstring Theory Primer

Magnetic Monopoles in String Theory

References

Appendix

...... Magnetic Charge Quantization Monopole Edward Olszewski

Electromagnetism Since The Maxwell Theory Compact Notation I Minimal Coupling (Quantum − 2π 2πn Mechanics) dχ = χ(2π) χ(0) = = The Dirac I I e ge Monopole Grand Uniﬁed dχ = (AS − AN ) = gm Theories Magnetic 2πn Monopoles and gm = Dyons in Grand ge Uniﬁed Theories Montonen–Olive Duality

Supersymmetry

Superstring Theory Primer the Dirac condition ge gm = 2πn Magnetic 1 qm Monopoles in the magnetic ﬁeld B = eˆ String Theory 4π r 2 r References

Appendix

...... Summary Monopole Edward Olszewski

Electromagnetism I The Maxwell Theory The existence of a single magnetic charge requires Compact Notation Minimal Coupling (Quantum that electric charge be quantized. Mechanics) (−ig χ) The Dirac I The quantities e e are elements of a U(1) group Monopole of gauge transformations. Electric charge is Grand Uniﬁed Theories quantized, since χ and χ + 2π/e ( e being the Magnetic coupling constant) yield the same gauge Monopoles and Dyons in Grand transformation. In the alternative case when charge Uniﬁed Theories 1 Montonen–Olive is not quantized the gauge group is the real line R . Duality

Magnetic monopoles require a compact U(1) gauge Supersymmetry

group. Superstring Theory Primer I Mathematically, we have constructed a non-trivial Magnetic principal ﬁber bundle with base manifold S2 and ﬁber Monopoles in String Theory

U(1). References

Appendix

...... Summary Monopole Edward Olszewski

Magnetic monopoles require a compact U(1) gauge Supersymmetry

group. Superstring Theory Primer I Mathematically, we have constructed a non-trivial Magnetic principal ﬁber bundle with base manifold S2 and ﬁber Monopoles in String Theory

U(1). References

Appendix

...... Summary Monopole Edward Olszewski

Magnetic monopoles require a compact U(1) gauge Supersymmetry

group. Superstring Theory Primer I Mathematically, we have constructed a non-trivial Magnetic principal ﬁber bundle with base manifold S2 and ﬁber Monopoles in String Theory

U(1). References

Appendix

...... Summary Monopole Edward Olszewski

Magnetic monopoles require a compact U(1) gauge Supersymmetry

group. Superstring Theory Primer I Mathematically, we have constructed a non-trivial Magnetic principal ﬁber bundle with base manifold S2 and ﬁber Monopoles in String Theory

U(1). References

Appendix

...... Gauge Transformations and Lie Groups Monopole Edward Olszewski

Electromagnetism The Maxwell Theory Compact Notation Minimal Coupling (Quantum Mechanics)

I The Dirac The gauge groups corresponding to other forces in Monopole nature are obtained by substituting for the gauge Grand Uniﬁed Theories group of electromagnetism another appropriate Lie Magnetic group. Monopoles and Dyons in Grand I Gravity can also be described as a gauge theory Uniﬁed Theories Montonen–Olive based on the equivalence principle. The gauge Duality transformations are the general coordinate Supersymmetry transformations and local Lorentz transformations. Superstring Theory Primer

Magnetic Monopoles in String Theory

References

Appendix

...... Gauge Transformations and Lie Groups Monopole Edward Olszewski

Electromagnetism The Maxwell Theory Compact Notation Minimal Coupling (Quantum Mechanics)

Magnetic Monopoles in String Theory

References

Appendix

...... Gauge Transformations and Lie Groups Monopole Edward Olszewski

Electromagnetism The Maxwell Theory Compact Notation Minimal Coupling (Quantum Mechanics)

Magnetic Monopoles in String Theory

References

Appendix

...... Gauge Transformations (continued) Monopole Edward Olszewski

Electromagnetism The Maxwell Theory Compact Notation Minimal Coupling (Quantum Mechanics)

The Dirac Monopole I Grand Uniﬁed For the remainder of this presentation we shall, for Theories

the most part, restrict our consideration to the gauge Magnetic Monopoles and groups SO(2N) and SU(N), which are relevant to Dyons in Grand Grand Uniﬁcation, because SO(10) −→ SU(5) −→ Uniﬁed Theories × × −→ × Montonen–Olive SUc(3) SUL(2) UY (1) SUc(3) UEM (1). Duality Supersymmetry

Superstring Theory Primer

Magnetic Monopoles in String Theory

References

Appendix

...... Gauge Transformations (continued) Monopole Edward Olszewski

Electromagnetism The Maxwell Theory Compact Notation Minimal Coupling (Quantum Mechanics)

The Dirac Monopole I Grand Uniﬁed For the remainder of this presentation we shall, for Theories

Superstring Theory Primer

Magnetic Monopoles in String Theory

References

Appendix

...... Spontaneous Symmetry Breaking Monopole Edward Olszewski

Electromagnetism scalar ﬁeld The Maxwell Theory Compact Notation Minimal Coupling (Quantum Mechanics) L 1 µ − 2 2 = ∂µA∂ A m A The Dirac 2 Monopole

Grand Unified Theories two coupled scalar fields Magnetic Monopoles and Dyons in Grand Unified Theories 1 µ 1 µ 2 2 L = ∂µA∂ A + ∂µΦ∂ Φ − gΦ A Montonen–Olive 2 2 Duality

Supersymmetry

Superstring scalar ﬁeld coupled to EM Theory Primer Magnetic Monopoles in 1 µν 1 µ String Theory L = − FµνF + DµΦD Φ 4 2 References DµΦ = ∂µΦ − ie AµΦ Appendix

...... Spontaneous Symmetry Breaking Monopole Edward Olszewski

Electromagnetism scalar ﬁeld The Maxwell Theory Compact Notation Minimal Coupling (Quantum Mechanics) L 1 µ − 2 2 = ∂µA∂ A m A The Dirac 2 Monopole

Grand Unified Theories two coupled scalar fields Magnetic Monopoles and Dyons in Grand Unified Theories 1 µ 1 µ 2 2 L = ∂µA∂ A + ∂µΦ∂ Φ − gΦ A Montonen–Olive 2 2 Duality

Supersymmetry

Superstring scalar ﬁeld coupled to EM Theory Primer Magnetic Monopoles in 1 µν 1 µ String Theory L = − FµνF + DµΦD Φ 4 2 References DµΦ = ∂µΦ − ie AµΦ Appendix

...... Spontaneous Symmetry Breaking Monopole Edward Olszewski

Electromagnetism scalar ﬁeld The Maxwell Theory Compact Notation Minimal Coupling (Quantum Mechanics) L 1 µ − 2 2 = ∂µA∂ A m A The Dirac 2 Monopole

Grand Unified Theories two coupled scalar fields Magnetic Monopoles and Dyons in Grand Unified Theories 1 µ 1 µ 2 2 L = ∂µA∂ A + ∂µΦ∂ Φ − gΦ A Montonen–Olive 2 2 Duality

Supersymmetry

Superstring scalar ﬁeld coupled to EM Theory Primer Magnetic Monopoles in 1 µν 1 µ String Theory L = − FµνF + DµΦD Φ 4 2 References DµΦ = ∂µΦ − ie AµΦ Appendix

...... Spontaneous Symmetry Breaking Monopole Edward Olszewski

Electromagnetism scalar ﬁeld The Maxwell Theory Compact Notation Minimal Coupling (Quantum Mechanics) L 1 µ − 2 2 = ∂µA∂ A m A The Dirac 2 Monopole

Grand Unified Theories two coupled scalar fields Magnetic Monopoles and Dyons in Grand Unified Theories 1 µ 1 µ 2 2 L = ∂µA∂ A + ∂µΦ∂ Φ − gΦ A Montonen–Olive 2 2 Duality

Supersymmetry

Superstring scalar ﬁeld coupled to EM Theory Primer Magnetic Monopoles in 1 µν 1 µ String Theory L = − FµνF + DµΦD Φ 4 2 References DµΦ = ∂µΦ − ie AµΦ Appendix

...... Spontaneous Symmetry Breaking Monopole Edward Olszewski (continued) Electromagnetism The Maxwell Theory Compact Notation Minimal Coupling (Quantum Mechanics)

The Dirac ABEGHHK’tH mechanism (the Higgs mechanism) Monopole

Grand Uniﬁed Theories I the lagrangian is invariant under some symmetry, Magnetic Monopoles and e.g. a non-abelian gauge symmetry. Dyons in Grand Uniﬁed Theories I the ground state of the system is continuously Montonen–Olive degenerate under the symmetry. Duality I the system resides in a particular ground state which Supersymmetry Superstring breaks some of the symmetry. An illustrative Theory Primer example is the gauge group SO(3). Magnetic Monopoles in String Theory

References

Appendix

...... Spontaneous Symmetry Breaking Monopole Edward Olszewski (continued) Electromagnetism The Maxwell Theory Compact Notation Minimal Coupling (Quantum Mechanics)

The Dirac ABEGHHK’tH mechanism (the Higgs mechanism) Monopole

References

Appendix

...... Spontaneous Symmetry Breaking Monopole Edward Olszewski (continued) Electromagnetism The Maxwell Theory Compact Notation Minimal Coupling (Quantum Mechanics)

The Dirac ABEGHHK’tH mechanism (the Higgs mechanism) Monopole

References

Appendix

...... Spontaneous Symmetry Breaking Monopole Edward Olszewski (continued) Electromagnetism The Maxwell Theory Compact Notation Minimal Coupling (Quantum Mechanics)

The Dirac ABEGHHK’tH mechanism (the Higgs mechanism) Monopole

References

Appendix

...... Spontaneous Symmetry Breaking Monopole Edward Olszewski (continued) Electromagnetism The Maxwell Theory Compact Notation Minimal Coupling (Quantum Mechanics)

The Dirac ABEGHHK’tH mechanism (the Higgs mechanism) Monopole

References

Appendix

...... The Running Coupling Constant Monopole Edward Olszewski

Electromagnetism The Maxwell Theory Compact Notation Minimal Coupling (Quantum Mechanics)

The Dirac Monopole

Grand Uniﬁed Theories

Magnetic Figure: SU(N) Gauge Theory: (a) Screening due to ﬂavors; (b) Monopoles and Dyons in Grand Antiscreening due to gluons. Uniﬁed Theories

Montonen–Olive Duality 2 ∂α α n 11N Supersymmetry β(µ) ≡ = (+ f − ) , Superstring ∂ log(µ) π 3 6 Theory Primer g2 Magnetic where µ is the relevant energy scale, and α = 4π . Monopoles in String Theory For SU(3), N = 3, and β < 0 requires n < 33 . f 2 References

Appendix

...... The Running Coupling Constant Monopole Edward Olszewski

Electromagnetism The Maxwell Theory Compact Notation Minimal Coupling (Quantum Mechanics)

The Dirac Monopole

Grand Uniﬁed Theories

Magnetic Figure: SU(N) Gauge Theory: (a) Screening due to ﬂavors; (b) Monopoles and Dyons in Grand Antiscreening due to gluons. Uniﬁed Theories

Appendix

...... Running Coupling Constant (continued) Monopole Edward Olszewski

Electromagnetism The Maxwell Theory Compact Notation Minimal Coupling (Quantum Mechanics)

The Dirac Monopole

Grand Uniﬁed Theories

Magnetic Monopoles and Dyons in Grand Uniﬁed Theories

Montonen–Olive Duality ′2 2 U(1)Y α1 = (5/3)g /(4π) = 5α/(3 cos θW ) Supersymmetry 2 2 Superstring SU(2)L α2 = g /(4π) = α/ sin θW Theory Primer 2 Magnetic SU(3)c α3 = gs /(4π) Monopoles in String Theory M (80.4GeV ) α = 0.017, α = 0.034, α = 0.118, Z 1 2 3 References −1 2 α = 129, sin θW = .231 Appendix

...... Running Coupling Constant (continued) Monopole Edward Olszewski

Electromagnetism The Maxwell Theory Compact Notation Minimal Coupling (Quantum Mechanics)

The Dirac Monopole

Grand Uniﬁed Theories

Magnetic Monopoles and Dyons in Grand Uniﬁed Theories

Montonen–Olive Duality

Supersymmetry

Superstring Theory Primer

Magnetic Monopoles in Figure: Theory of Everything. One GeV corresponds to a String Theory −14 length scale of approximately 2 √× 10 cm. The length scale References ′ −33 associated with string theory is α ≈ 10 cm. Appendix

...... Monopole

The Yang–Mills–Higgs Lagrangian Edward Olszewski

1 1 Electromagnetism L − · µν · µ − · The Maxwell Theory = F µν F + DµΦ D Φ V (Φ Φ) , Compact Notation 4 2 Minimal Coupling (Quantum Mechanics)

where The Dirac Monopole

F µν = ∂µAν − ∂νAµ − ig Aµ ∧ Aν . Grand Uniﬁed YM Theories

Magnetic Higgs ﬁeld Φ - scalar transforming in the adjoint Monopoles and Dyons in Grand representation of the gauge group so that Uniﬁed Theories Montonen–Olive Duality DµΦ = ∂µΦ − igYM Aµ ∧ Φ . Supersymmetry

Superstring Theory Primer = a F µν FµνTa Magnetic a Monopoles in Aµ = AµTa String Theory a References Φ = Φ Ta Appendix · ≡ a b a a Note: H G H G 2Tr(TaTb) = H. G...... Monopole

Edward Olszewski · V (Φ Φ) is a potential such that the vacuum expectation Electromagnetism of Φ is non-zero. When a speciﬁc form of V (Φ · Φ) is The Maxwell Theory Compact Notation Minimal Coupling (Quantum required we use Mechanics)

The Dirac λ Monopole V (Φ · Φ) = (Φ · Φ − v 2)2 . Grand Uniﬁed 8 Theories · Magnetic The Lagrangian, when V (Φ Φ) = 0, is scale invariant, Monopoles and · 2 Dyons in Grand but the requirement that Φ Φ = v spontaneously breaks Uniﬁed Theories scale invariance of the vacuum. Montonen–Olive Duality µ µ scale invariance: δx = x δλ; δAµ = −Aµ δλ; Supersymmetry Superstring δΦ = −Φ δλ. Theory Primer Magnetic Monopoles in String Theory

References

Appendix

...... Monopole

References

Appendix

...... Monopole

References

Appendix

...... Monopole How to deﬁne Fµν Edward Olszewski

Electromagnetism The Maxwell Theory Compact Notation Minimal Coupling (Quantum Mechanics) Usual Definition F ≡F · H µν µν The Dirac H ∈ Cartan subalgebra Monopole Grand Unified Theories dF ≠ 0 Fµν =F µν · Φ¯ Magnetic Bianchi identity not satisfied Monopoles and Dyons in Grand Unified Theories t‘Hooft – SO(3) Fµν =F µν · Φ¯− Montonen–Olive 1 Duality D Φ¯ ∧ D Φ¯ · Φ¯ µ ν Supersymmetry igYM Superstring dF = 0 iff Φ¯ ∧ Eα = Eα or 0 Theory Primer Magnetic (Bianchi identity satisfied) Monopoles in String Theory

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

...... Monopole How to deﬁne Fµν Edward Olszewski

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

...... Monopole

Edward Olszewski Make the ansatz that the Higgs ﬁeld Φ and vector Electromagnetism potential A take the form The Maxwell Theory Compact Notation Minimal Coupling (Quantum Mechanics) Φ = (Q(r) α1Tz + α2T⊥) v , The Dirac ge Monopole A = S(r) v α1 Tz dt + Tz (−C) W (r)(1 − cos θ) dϕ , g Grand Uniﬁed Theories

Magnetic where Monopoles and Dyons in Grand W (r), Q(r), S(r) → 0 , as r → 0 ; Uniﬁed Theories → → − g → ∞ Montonen–Olive W (r), Q(r) 1 , S(r) 1 , as r ; Duality √ gm gYM α1 v r g = g2 + g2 . Supersymmetry e m Superstring Theory Primer Here C is an arbitrary constant, and quantities ge and gm Magnetic Monopoles in are the electric and magnetic charges. String Theory

root system – SU(3) References

Appendix

...... Monopole Applying the gauge transformation Edward Olszewski

−iϕTz −iθTy iϕTz χ = e e e Electromagnetism The Maxwell Theory Compact Notation Minimal Coupling (Quantum to A and Φ we obtain Mechanics)

The Dirac 1 Monopole A → χAχ−1 − dχ χ−1 ig Grand Uniﬁed YM Theories g W (r) e − Magnetic = S(r) v α1 Tr dt + ( Tθ sin θ dϕ Tϕ dθ ) . Monopoles and g gYM Dyons in Grand Uniﬁed Theories and Montonen–Olive Duality − Φ → χΦχ 1 Supersymmetry Superstring = v [ α2 T⊥ + Q(r) α1 Tr ] . Theory Primer Magnetic Monopoles in We have used the fact that String Theory

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

...... Monopole

Edward Olszewski

Electromagnetism Dyon charge quantization conditions, SU(N) The Maxwell Theory Compact Notation Minimal Coupling (Quantum Mechanics)

The Dirac Monopole 4π = Grand Uniﬁed magnetic charge gm 2 Theories |α| gYM Magnetic electric charge ge = ne α1gYM Monopoles and √ Dyons in Grand Uniﬁed Theories N = n g , Montonen–Olive e 2(N − 1) YM Duality Supersymmetry

Superstring Theory Primer

Magnetic where ne is an integer. The electric charge quantization is Monopoles in derived from the adjoint representation. String Theory References

Appendix

...... Comparison of the different types of dyons Monopole √ Edward Olszewski

2 N Electromagnetism SU(N), (|α| = 1) gm ge = ne 4π The Maxwell Theory 2(N − 1) Compact Notation Minimal Coupling (Quantum Mechanics) | |2 G2E , ( α = 1/3) gm ge = ne 4π 3 The Dirac Monopole SO(3), t’Hooft/Polykov g g = n 4π m e e Grand Uniﬁed 4π Theories U(1), Dirac gm e = ne Magnetic 2 Monopoles and Dyons in Grand Magnetic Monopole Mass (V = 0) Uniﬁed Theories √ Montonen–Olive | |2 g N Duality SU(N), ( α = 1) M = MX Supersymmetry gYM 2(N − 1) Superstring 2 g Theory Primer G2E , (|α| = 1/3) M = MX 2g Magnetic YM Monopoles in g String Theory SO(3), t’Hooft/Polykov M = M X References gYM U(1), Dirac __ Appendix

...... root system – G2 ...... Monopole

Edward Olszewski

Dyon structure Electromagnetism The Maxwell Theory Compact Notation Minimal Coupling (Quantum Mechanics) 1 gm = ˆ → ∞ The Dirac B 2 er r 4π r Monopole 1 ge Grand Uniﬁed E = eˆ r → ∞ Theories 4π r 2 r Magnetic Φ ≈ r r → 0 . Monopoles and Dyons in Grand Uniﬁed Theories The size of the dyon, i.e. the length scale where massive Montonen–Olive gauge bosons exist, is Duality Supersymmetry g 1 1 1 1 Superstring ≈ ≈ = ≈ ≈ −30 Theory Primer ldyon 16 10 cm gm gYM v α1 gYM v MX 10 GeV Magnetic Monopoles in String Theory At r = 0, Φ = 0, and symmetry is restored. The References monopole can catalyse nucleon decay. Appendix

...... Monopole

Edward Olszewski

...... Monopole

Edward Olszewski

...... Montonen - Olive Conjecture Monopole Edward Olszewski

Electromagnetism Mass (g , g ) Spin The Maxwell Theory e m Compact Notation Minimal Coupling (Quantum Higgs 0 (0, 0) 0 Mechanics) Photon 0 (0, 0) 1 The Dirac Monopole W v e (e, 0) 1 Grand Uniﬁed M vg (0, g) 0 Theories Magnetic Table: The gauge group SO(3). Monopoles and Dyons in Grand Uniﬁed Theories ≡ ≡ 4π Note: MW (MX ) = gYM v; also, e gYM ; g g Montonen–Olive YM Duality

Supersymmetry

Superstring Weak/Strong Duality: E → B; e → g Theory Primer

Magnetic including a relabeling of electric Monopoles in and magnetic states String Theory References

Appendix

...... Monopole Langrangian Density (V = 0) Edward Olszewski

2 2 Electromagnetism θe µν θe i i The Maxwell Theory L = − ∗ · = · Compact Notation Witten Effect: θ 2 Fµν F 2 E B 32π 8π Minimal Coupling (Quantum Mechanics) 1 µν 1 i i i i Traditional: L = − Fµν · F = (E · E − B · B ) The Dirac trad 4 2 Monopole Grand Uniﬁed Theories

Magnetic Lθ causes shifts in allowed values of electric charge in Monopoles and Dyons in Grand the monopole sector. Uniﬁed Theories

Montonen–Olive Consider U(1) gauge transformations about Φ¯: Duality D ¯ Supersymmetry δAµ = µΦ Superstring Theory Primer

Let η be the generator of inﬁnitesimal gauge Magnetic Monopoles in transformations. Since String Theory

References physical quantities require ei2πη = 1 Appendix eigenvalues of η η = n, (n ∈ Z ) ...... Monopole

Edward Olszewski

Electromagnetism The Maxwell Theory Compact Notation Minimal Coupling (Quantum Mechanics)

The Dirac L Monopole ∂ α By Noether’s theorem η = α δAµ Grand Uniﬁed ∂∂0Aµ Theories ge θegm Magnetic Therefore η = + Monopoles and 2 Dyons in Grand e 8π Uniﬁed Theories θ Consequently (n = eg /4π) g = n e − n e Montonen–Olive m m e 2π m Duality Supersymmetry

Superstring Theory Primer

Magnetic Monopoles in String Theory

References

Appendix

...... SL(2, Z) Monopole Define Edward Olszewski θ 4π τ = + Electromagnetism i 2 2π e The Maxwell Theory Compact Notation Minimal Coupling (Quantum θ → θ + 2π τ → τ + 1 Mechanics) 4π 1 The Dirac (e → g ≡ τ → − Monopole m e τ Grand Unified θ = 0) Theories Magnetic Monopoles and Dyons in Grand aτ + b Unified Theories τ → a, b, c, d ∈ Z, ad − bc = 1 cτ + d Montonen–Olive ( ) ( )( ) Duality − n → a b n Supersymmetry − nm c d nm Superstring Theory Primer

Magnetic Monopoles in String Theory

References

Appendix

...... Monopole

Edward Olszewski

Electromagnetism The Maxwell Theory Compact Notation 4π Minimal Coupling (Quantum Magnetic Charge gm = nm Mechanics) e The Dirac θ Monopole g = ne − n e Electric Charge e m Grand Uniﬁed 2π Theories 2 ≥ 2 2 2 Mass of Dyon M v (ge + gm) Magnetic Monopoles and Dyons in Grand 1 θe2 1 Uniﬁed Theories L = µν · − µν · ∗ − DµΦ ·D Φ F Fµν 2 F Fµν µ Montonen–Olive 4 32π 2 Duality 1 µν µν Supersymmetry ≡ − Im(τ)(F + i ∗ F ) · (Fµν + i ∗ Fµν) 32π Superstring Theory Primer 1 µ − D Φ ·DµΦ Magnetic 2 Monopoles in String Theory

References

Appendix

...... Summary Monopole Edward Olszewski I Quantum corrections would be expected to generate Electromagnetism The Maxwell Theory a non-zero potential V (Φ) even if one is absent Compact Notation Minimal Coupling (Quantum classically and should also modify the classical mass Mechanics) formula. Thus there is no reason to think that the The Dirac Monopole

duality of the spectrum should be maintained by Grand Uniﬁed quantum corrections. Theories I Magnetic The W bosons have spin one while the monopoles Monopoles and Dyons in Grand are rotationally invariant indicating that they have Uniﬁed Theories spin zero. Thus even if the mass spectrum is Montonen–Olive Duality invariant under duality, there will not be an exact Supersymmetry number of matching states and quantum numbers. Superstring I The proposed duality symmetry seems impossible to Theory Primer Magnetic test since rather than acting as a symmetry of a Monopoles in single theory it relates two different theories, one of String Theory References which is necessarily at strong coupling where we Appendix have little control of the theory.

...... Summary (continued) Monopole Edward Olszewski

Electromagnetism The Maxwell Theory Compact Notation Minimal Coupling (Quantum Mechanics)

The Dirac Monopole

Grand Unified I The N = 4 Super Yang-Mills theory solves the first Theories Magnetic problem because of exact quantum scale invariance Monopoles and = Dyons in Grand (V 0). It also solves the second problem because Unified Theories

additional particles are added to the spectrum. Montonen–Olive Duality

Supersymmetry

Superstring Theory Primer

Magnetic Monopoles in String Theory

References

Appendix

...... Summary (continued) Monopole Edward Olszewski

Electromagnetism The Maxwell Theory Compact Notation Minimal Coupling (Quantum Mechanics)

The Dirac Monopole

additional particles are added to the spectrum. Montonen–Olive Duality

Supersymmetry

Superstring Theory Primer

Magnetic Monopoles in String Theory

References

Appendix

...... Monopole

Edward Olszewski

I Supersymmetry is an extension of the Poincaré Electromagnetism The Maxwell Theory algebra. Compact Notation Minimal Coupling (Quantum I Every boson(fermion) has a corresponding Mechanics) The Dirac fermionic(bosonic) partner. Monopole Grand Uniﬁed Theories µν 2 Klein Gordon Equation (−η PµPν − m )Φ = 0 Magnetic √ Monopoles and µν µν µ Dyons in Grand Dirac “ η PµPν” = η ΓµPν = PµΓ Uniﬁed Theories µ ν µν where {Γ , Γ } = 2η Montonen–Olive Duality { } − µ supersymmetry algebra Qα, Qβ = 2PµΓαβ Supersymmetry Superstring {Pµ, Qα} = 0 Theory Primer Magnetic Monopoles in String Theory

References

Appendix

...... Monopole

Edward Olszewski

References

Appendix

...... Monopole

Edward Olszewski

References

Appendix

...... Monopole

Edward Olszewski

Do we really need the Higgs field? Electromagnetism The Maxwell Theory Assume an extra dimension Let A4 = Φ. AssumeV = 0. Compact Notation Minimal Coupling (Quantum Mechanics) 1 1 L = − F · F µν + D A · DµA , The Dirac 4 µν 2 µ 4 4 Monopole Grand Unified 1 MN Theories = − F MN · F (M, N = 0 ... 4) 4 Magnetic Monopoles and Dyons in Grand where Unified Theories

Montonen–Olive Duality F µ4 = ∂µA4 − ie Aµ ∧ A4 Supersymmetry

Superstring Theory Primer The Higgs ﬁeld becomes a component of the Yang-Mills Magnetic Monopoles in potential in the extra dimension. String Theory References

Appendix

...... Monopole

Edward Olszewski

Montonen–Olive Duality F µ4 = ∂µA4 − ie Aµ ∧ A4 Supersymmetry

Superstring Theory Primer The Higgs ﬁeld becomes a component of the Yang-Mills Magnetic Monopoles in potential in the extra dimension. String Theory References

Appendix

...... Monopole

Edward Olszewski

Montonen–Olive Duality F µ4 = ∂µA4 − ie Aµ ∧ A4 Supersymmetry

Superstring Theory Primer The Higgs ﬁeld becomes a component of the Yang-Mills Magnetic Monopoles in potential in the extra dimension. String Theory References

Appendix

...... Introduction to superstring theory Monopole Edward Olszewski I Gravity. Every consistent string theory must contain Electromagnetism a massless spin-2 state, whose interactions reduce The Maxwell Theory Compact Notation at low energy to general relativity. Minimal Coupling (Quantum Mechanics) I A consistent theory of quantum gravity, at least in The Dirac perturbation theory. As we have noted, this is in Monopole Grand Unified contrast to all known quantum field theories of Theories gravity. Magnetic Monopoles and I Grand unification. String theories lead to gauge Dyons in Grand Unified Theories groups large enough to include the Standard Model. Montonen–Olive Some of the simplest string theories lead to the Duality same gauge groups and fermion representations that Supersymmetry Superstring arise in the unification of the Standard Model. Theory Primer I Extra dimensions. String theory requires a definite Magnetic Monopoles in number of spacetime dimensions, ten. The field String Theory equations have solutions with four large flat and six References small curved dimensions, with four-dimensional Appendix

physics that resembles the Standard...... Model...... Introduction to superstring theory Monopole Edward Olszewski I Gravity. Every consistent string theory must contain Electromagnetism a massless spin-2 state, whose interactions reduce The Maxwell Theory Compact Notation at low energy to general relativity. Minimal Coupling (Quantum Mechanics) I A consistent theory of quantum gravity, at least in The Dirac perturbation theory. As we have noted, this is in Monopole Grand Unified contrast to all known quantum field theories of Theories gravity. Magnetic Monopoles and I Grand unification. String theories lead to gauge Dyons in Grand Unified Theories groups large enough to include the Standard Model. Montonen–Olive Some of the simplest string theories lead to the Duality same gauge groups and fermion representations that Supersymmetry Superstring arise in the unification of the Standard Model. Theory Primer I Extra dimensions. String theory requires a definite Magnetic Monopoles in number of spacetime dimensions, ten. The field String Theory equations have solutions with four large flat and six References small curved dimensions, with four-dimensional Appendix

physics that resembles the Standard...... Model...... Primer (continued) Monopole Edward Olszewski I Supersymmetry. Consistent string theories require Electromagnetism The Maxwell Theory spacetime super-symmetry, as either a manifest or a Compact Notation Minimal Coupling (Quantum spontaneously broken symmetry. Mechanics) I The Dirac Chiral gauge couplings. The gauge interactions in Monopole

nature are parity asymmetric (chiral). This has been Grand Uniﬁed a stumbling block for a number of previous unifying Theories Magnetic ideas: they required parity symmetric gauge Monopoles and Dyons in Grand couplings. String theory allows chiral gauge Uniﬁed Theories

couplings. Montonen–Olive Duality I No free parameters. String theory has no adjustable Supersymmetry

constants. Superstring Theory Primer I Uniqueness. Not only are there no continuous Magnetic parameters, but there is no discrete freedom Monopoles in String Theory analogous to the choice of gauge group and References

representations in ﬁeld theory: there is a unique Appendix string theory...... Primer (continued) Monopole Edward Olszewski I Supersymmetry. Consistent string theories require Electromagnetism The Maxwell Theory spacetime super-symmetry, as either a manifest or a Compact Notation Minimal Coupling (Quantum spontaneously broken symmetry. Mechanics) I The Dirac Chiral gauge couplings. The gauge interactions in Monopole

couplings. Montonen–Olive Duality I No free parameters. String theory has no adjustable Supersymmetry

representations in ﬁeld theory: there is a unique Appendix string theory...... Primer (continued) Monopole Edward Olszewski

Electromagnetism The Maxwell Theory Compact Notation Minimal Coupling (Quantum Mechanics)

The Dirac Monopole

Grand Unified Figure: Two graviton exchange (field theory) Theories Magnetic Monopoles and Dyons in Grand Unified Theories

Montonen–Olive Duality

Supersymmetry

Superstring Theory Primer

Magnetic Monopoles in Figure: (a) Closed string. (b) Open string. (c) The loop String Theory amplitude in string theory of two graviton exchange, above References Appendix

...... U-duality Monopole Edward Olszewski I SL(2, Z) duality (S duality) i.e. Weak/Strong Duality, Electromagnetism (a consequence of supersymmetry) The Maxwell Theory I Compact Notation O(k, k, Z) T–duality group, including a parity Minimal Coupling (Quantum operation on the right moving open string excitations Mechanics) → ′ α′ The Dirac which implies R R = R Monopole I Closed String Grand Uniﬁed I N wR → ′ Theories pl = R + α′ implies pl pl = pl I N − wR → ′ − Magnetic pr = R α′ implies pr pr = pr Monopoles and I Open String - each end of the open string is Dyons in Grand Uniﬁed Theories constrained to lie on a hypersurface of the space. Montonen–Olive Duality

Supersymmetry

Superstring Theory Primer

Magnetic Monopoles in String Theory Figure: D-branes. Three hyperplanes at different positions, References with various open strings attached. Appendix

Supersymmetry

Superstring Theory Primer

Magnetic Monopoles in String Theory Figure: D-branes. Three hyperplanes at different positions, References with various open strings attached. Appendix

Supersymmetry

Superstring Theory Primer

Magnetic Monopoles in String Theory Figure: D-branes. Three hyperplanes at different positions, References with various open strings attached. Appendix

Supersymmetry

Superstring Theory Primer

Magnetic Monopoles in String Theory Figure: D-branes. Three hyperplanes at different positions, References with various open strings attached. Appendix

Supersymmetry

Superstring Theory Primer

Magnetic Monopoles in String Theory Figure: D-branes. Three hyperplanes at different positions, References with various open strings attached. Appendix

...... Magnetic Monopole Solution Monopole Edward Olszewski I Begin with a Type I SO(32) string theory in ten Electromagnetism The Maxwell Theory dimensions, compactify six of the nine space (the Compact Notation ′ ≫ Minimal Coupling (Quantum compactification radius, R α ), resulting in a six Mechanics) dimensional torus, T 6. The Dirac ′ Monopole I → α Apply T-duality transformations, e.g. ( R R ) to 5 of Grand Unified the 6 compactified space dimensions. The T-duality Theories Magnetic symmetry group is O(5, 5, Z ). This T5 will be the Monopoles and Dyons in Grand internal dimensions of spacetime. Unified Theories I Spacetime now consists of five internal dimensions Montonen–Olive Duality

and a D4-brane comprising a ﬁve dimensional Supersymmetry

spacetime (Type IIA in the bulk). Superstring Theory Primer I Construct a magnetic monople solution on the Magnetic D4-brane, then apply the T-duality transformation, Monopoles in ′ String Theory R → α to the compactiﬁed dimension of the R References

D4-brane, resulting in a D3-brane. The D3-brane is Appendix four dimensional spacetime (Type IIB in the bulk)...... Magnetic Monopole Solution Monopole Edward Olszewski I Begin with a Type I SO(32) string theory in ten Electromagnetism The Maxwell Theory dimensions, compactify six of the nine space (the Compact Notation ′ ≫ Minimal Coupling (Quantum compactification radius, R α ), resulting in a six Mechanics) dimensional torus, T 6. The Dirac ′ Monopole I → α Apply T-duality transformations, e.g. ( R R ) to 5 of Grand Unified the 6 compactified space dimensions. The T-duality Theories Magnetic symmetry group is O(5, 5, Z ). This T5 will be the Monopoles and Dyons in Grand internal dimensions of spacetime. Unified Theories I Spacetime now consists of five internal dimensions Montonen–Olive Duality

and a D4-brane comprising a ﬁve dimensional Supersymmetry

D4-brane, resulting in a D3-brane. The D3-brane is Appendix four dimensional spacetime (Type IIB in the bulk)...... Magnetic Monopole Solution (continued) Monopole Edward Olszewski

The action of the system, S = SDBI + SCS, comprises two Electromagnetism The Maxwell Theory parts: Compact Notation Minimal Coupling (Quantum I The Dirac-Born-Infeld Action - coupling a Dp–brane Mechanics) The Dirac to NS–NS closed string fields Monopole ∫ Grand Unified Theories − { −Φ − SDBI = µp Tr e [ det(Gab+ Magnetic Mp+1 Monopoles and Dyons in Grand ′ 1/2 Unified Theories + Bab + 2πα Fab)] } Montonen–Olive Duality I Chern-Simons Action - coupling a Dp–brane to R–R Supersymmetry

closed string ﬁelds Superstring Theory Primer ∫ [ ] ∑ Magnetic ′ ∧ 2πα F+B Monopoles in SCS = µp C(p+1) Tre String Theory Mp+ 1 p References

Appendix

...... Magnetic Monopole Solution (continued) Monopole Edward Olszewski

closed string ﬁelds Superstring Theory Primer ∫ [ ] ∑ Magnetic ′ ∧ 2πα F+B Monopoles in SCS = µp C(p+1) Tre String Theory Mp+ 1 p References

Appendix

...... Magnetic Monopole Solution (continued) Monopole Edward Olszewski

Electromagnetism The Maxwell Theory Compact Notation Minimal Coupling (Quantum Make the ansatz Mechanics) I p = 4; The Dirac Monopole I 4 C(1) = C4 dx , where C4 is constant and all other Grand Uniﬁed R–R potentials vanish; Theories Magnetic I Monopoles and Dyons in Grand Uniﬁed Theories ge A = S(r) v α1 Tz dt + Montonen–Olive g Duality Supersymmetry + Tz (−C) W (r)(1 − cos θ) dϕ , Superstring Theory Primer A4 = Φ = (Q(r) α1Tz + α2T⊥) v . Magnetic Monopoles in String Theory

References

Appendix

...... Magnetic Monopole Solution (continued) Monopole Edward Olszewski

References

Appendix

...... Magnetic Monopole Solution (continued) Monopole Edward Olszewski

References

Appendix

...... Magnetic Monopole Solution (continued) Monopole Edward Olszewski

References

Appendix

...... Magnetic Monopole Solution (continued) Monopole Edward Olszewski

Electromagnetism The Maxwell Theory Compact Notation Minimal Coupling (Quantum Mechanics)

The Dirac Monopole

Grand Uniﬁed Theories

Magnetic Monopoles and Dyons in Grand Uniﬁed Theories

Montonen–Olive Figure: Wormhole connecting two D3-branes. Duality Supersymmetry

Superstring I There is no event horizon. Theory Primer Magnetic I = Monopoles in The scalar curvature is ﬁnite at r 0. Speciﬁcally, String Theory ≈ 2 ′−1 → R 3 α as r 0. References I There is no singularity at r = 0, since T-duality Appendix

should not create a singularity...... Magnetic Monopole Solution (continued) Monopole Edward Olszewski

Electromagnetism The Maxwell Theory Compact Notation Minimal Coupling (Quantum Mechanics)

The Dirac Monopole

Grand Uniﬁed Theories

Magnetic Monopoles and Dyons in Grand Uniﬁed Theories

Montonen–Olive Figure: Wormhole connecting two D3-branes. Duality Supersymmetry

should not create a singularity...... Magnetic Monopole Solution (continued) Monopole Edward Olszewski

Electromagnetism The Maxwell Theory Compact Notation Minimal Coupling (Quantum Mechanics)

The Dirac Monopole

Grand Uniﬁed Theories

Magnetic Monopoles and Dyons in Grand Uniﬁed Theories

Montonen–Olive Figure: Wormhole connecting two D3-branes. Duality Supersymmetry

should not create a singularity...... Magnetic Monopole Solution (continued) Monopole Edward Olszewski

Electromagnetism The Maxwell Theory Compact Notation Minimal Coupling (Quantum Mechanics)

The Dirac Monopole

Grand Uniﬁed Theories

Magnetic Monopoles and Dyons in Grand Uniﬁed Theories

Montonen–Olive Figure: Wormhole connecting two D3-branes. Duality Supersymmetry

should not create a singularity...... Magnetic Monopole Solution (continued) Monopole Edward Olszewski I The IIB theory transforms into itself under S duality; Electromagnetism The Maxwell Theory i.e. the weakly coupled theory is identical to the Compact Notation Minimal Coupling (Quantum strongly coupled theory with a relabeling of states. In Mechanics) particular, S : F → D1. The Dirac Monopole I Relative strength of couplings: Grand Uniﬁed 2 1 7 2 ′ 4 Theories gravity (10 dim) κ = 2 (2π) g (α ) 1 Magnetic F string tension τF1 = πα′ Monopoles and 2 Dyons in Grand τF1 D1 string tension τD1 = g Uniﬁed Theories Φ Montonen–Olive where g = e , Φ being the constant dilaton Duality

background. Supersymmetry I Relevant length scales (weak coupling, i.e. g ≪ 1): Superstring Theory Primer −1/2 −1/2 −1/4 1/4 τF1 : lg : τD1 = g : 1 : g Magnetic ′ 1/2 Monopoles in I the monopole size, l ≈ (2πα ) String Theory monopole gD3 References where the GUTS Yang-Mills coupling constant is g2 , D3 Appendix 2 where gD3 = 2πg...... Magnetic Monopole Solution (continued) Monopole Edward Olszewski I The IIB theory transforms into itself under S duality; Electromagnetism The Maxwell Theory i.e. the weakly coupled theory is identical to the Compact Notation Minimal Coupling (Quantum strongly coupled theory with a relabeling of states. In Mechanics) particular, S : F → D1. The Dirac Monopole I Relative strength of couplings: Grand Uniﬁed 2 1 7 2 ′ 4 Theories gravity (10 dim) κ = 2 (2π) g (α ) 1 Magnetic F string tension τF1 = πα′ Monopoles and 2 Dyons in Grand τF1 D1 string tension τD1 = g Uniﬁed Theories Φ Montonen–Olive where g = e , Φ being the constant dilaton Duality

Electromagnetism The Maxwell Theory I Compact Notation Steven Weinberg and Edward Witten have shown Minimal Coupling (Quantum that if there is a massless spin-2 particle in the Mechanics) The Dirac spectrum, the the matrix element Monopole Grand Uniﬁed ′ Theories < massless spin 2, k|Tµν| massless spin 2, k > Magnetic Monopoles and Dyons in Grand of the energy momentum tensor (which exists as a Uniﬁed Theories

local observable in the gauge theory) has impossible Montonen–Olive properties. Duality Supersymmetry I Assumption – the graviton bound state moves in the Superstring same spacetime as its gauge boson constituents. Theory Primer I Magnetic Violation of the assumption – the graviton bound Monopoles in state moves in one additional dimension. String Theory References

Appendix

...... Gauge/Gravity Duality Monopole Edward Olszewski

Appendix

...... Gauge/Gravity Duality Monopole Edward Olszewski

Appendix

...... Gauge/Gravity Duality (continued) Monopole Edward Olszewski

Electromagnetism The Maxwell Theory Compact Notation Minimal Coupling (Quantum Mechanics)

The Dirac Monopole

Grand Uniﬁed Theories

Magnetic Monopoles and Dyons in Grand Uniﬁed Theories

Montonen–Olive Duality

Supersymmetry

Figure: Exchange of a closed string between two D-branes. Superstring Equivalently, a vacuum loop of an open string with one end on Theory Primer Magnetic each D-brane. Monopoles in String Theory

References

Appendix

...... References I Monopole Edward Olszewski

Electromagnetism WIKIPEDIA, The Maxwell Theory Compact Notation Magnetic monopole, Minimal Coupling (Quantum Mechanics)

http://en.wikipedia.org/wiki/Magnetic_ The Dirac monopole, 2007, Monopole Grand Uniﬁed [Online; accessed 15-November-2013]. Theories

Magnetic A.RAJANTIE, Monopoles and Dyons in Grand Introduction to Magnetic Monopoles, Uniﬁed Theories

arXiv:1204.3077v1. Montonen–Olive Duality IRAC P. A. M. D , Supersymmetry

Proc. Roy. Soc. A, 60 (1931). Superstring Theory Primer

P. RAMOND, Magnetic Monopoles in Field Theory: A Modern Primer, String Theory

Addison–Wesley Publishing Company, Inc., Boston, References Masssachusetts, 1989. Appendix

...... References II Monopole Edward Olszewski

Electromagnetism The Maxwell Theory L.H.RYDER, Compact Notation Minimal Coupling (Quantum Quantum Field Theory, Mechanics) The Dirac Cambridge University Press, New York, New York, Monopole

1996. Grand Uniﬁed Theories T. EGUCHI, P. B. GILKEY, and A.J.HANSON, Magnetic Physics Reports 66 (1980). Monopoles and Dyons in Grand Uniﬁed Theories J.BAEZ, Montonen–Olive This Week’s Finds in Mathematical Physics (Week Duality 94), Supersymmetry Superstring http: Theory Primer

//math.ucr.edu/home/baez/week94.html, Magnetic Monopoles in 1996, String Theory [Online; accessed 12-March-2014]. References Appendix

...... References III Monopole Edward Olszewski

F. WILCZEK, Electromagnetism The Maxwell Theory Asymptotic Freedom, Compact Notation Minimal Coupling (Quantum hep–th/9609099. Mechanics) The Dirac D. I. KAZAKOV, Monopole Grand Uniﬁed Beyond the Standard Model (In Search of Theories Supersymmetry), Magnetic Monopoles and arXiv:hep-ph/0012288v2. Dyons in Grand Uniﬁed Theories G.HOOFT, Montonen–Olive Nuclear Physics B, 276 (1974). Duality Supersymmetry C.MONTONEN and D. OLIVE, Superstring Theory Primer Phys. Lett. 72B, 117 (1977). Magnetic Monopoles in J.A.HARVEY, String Theory Magnetic Monopoles, Duality, and Supersymmetry, References hep–th/9603086. Appendix

...... References IV Monopole Edward Olszewski

Electromagnetism E.A.OLSZEWSKI, The Maxwell Theory Compact Notation Particle Physics Insights 5, 1 (2012). Minimal Coupling (Quantum Mechanics) OLCHINSKI The Dirac J.P , Monopole

String Theory Volume I, Grand Uniﬁed Cambridge University Press, New York, New York, Theories Magnetic 1998. Monopoles and Dyons in Grand J.POLCHINSKI, Uniﬁed Theories Montonen–Olive String Theory Volume II, Duality

Cambridge University Press, New York, New York, Supersymmetry

1998. Superstring Theory Primer

C. V. JOHNSON, Magnetic Monopoles in D–Branes, String Theory

Cambridge University Press, New York, New York, References

2003. Appendix

...... References V Monopole Edward Olszewski

Electromagnetism The Maxwell Theory Compact Notation Minimal Coupling (Quantum Mechanics)

The Dirac Monopole

Grand Uniﬁed Theories

J.POLCHINSKI, Magnetic Monopoles and Introduction to Gauge/Gravity Duality, Dyons in Grand arXiv:1010.6134. Uniﬁed Theories Montonen–Olive Duality

Supersymmetry

Superstring Theory Primer

Magnetic Monopoles in String Theory

References

Appendix

...... Monopole monopole SU(3) Root System Edward Olszewski H2 Electromagnetism √ √ The Maxwell Theory 1 3 1 3 (− , ) ( , ) Compact Notation 2 2 2 2 Minimal Coupling (Quantum Mechanics)

The Dirac Monopole

Grand Uniﬁed Theories

Magnetic Monopoles and (−1, 0) (1, 0) Dyons in Grand Uniﬁed Theories

H1 Montonen–Olive Duality

Supersymmetry

Superstring Theory Primer

Magnetic √ √ Monopoles in String Theory (− 1 , − 3 ) ( 1 , − 3 ) 2 2 2 2 References

Appendix

...... Monopole monopole G2 Root System Edward Olszewski H √ 2 1 3 ( 2 , 2 ) Electromagnetism The Maxwell Theory Compact Notation Minimal Coupling (Quantum Mechanics) (0, √1 ) 3 The Dirac Monopole

Grand Uniﬁed ( 1 , √1 ) Theories 2 2 3 Magnetic Monopoles and H1 Dyons in Grand Uniﬁed Theories (1, 0) Montonen–Olive Duality

Supersymmetry

Superstring Theory Primer

Magnetic Monopoles in String Theory

References

Appendix

...... References Monopole Edward Olszewski I WIKIPEDIA, Magnetic monopole, Electromagnetism http://en.wikipedia.org/wiki/Magnetic_ The Maxwell Theory Compact Notation monopole, 2007, Minimal Coupling (Quantum Mechanics)

[Online; accessed 15-November-2013] The Dirac Monopole I A.RAJANTIE, Introduction to Magnetic Monopoles, Grand Uniﬁed arXiv:1204.3077v1 Theories Magnetic I P. A. M. DIRAC, Proc. Roy. Soc. A, 60 (1931) Monopoles and Dyons in Grand Uniﬁed Theories I AMOND P. R , Field Theory: A Modern Primer, Montonen–Olive Addison–Wesley Publishing Company, Inc., Boston, Duality Masssachusetts, 1989 Supersymmetry Superstring I L.H.RYDER, Quantum Field Theory, Theory Primer Magnetic Cambridge University Press, New York, New York, Monopoles in 1996 String Theory References I T. EGUCHI, P. B. GILKEY, and A.J.HANSON, Appendix 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 (In Search 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, and Supersymmetry, 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/Gravity Duality, arXiv:1010.6134