Preprint typeset in JHEP style - HYPER VERSION

Compendium of useful formulas

Matthew Headrick

Abstract: Almost everything I know. Contents

1. Miscellaneous math 3 1.1 A matrix identity 3 1.2 Campbell-Baker-Hausdorff 3 1.3 Pauli matrices 3 1.4 Level repulsion 3 1.5 The Legendre transform 3 1.6 Gaussian integrals 5 1.7 Saddle-point approximation 5 1.8 Cauchy principal value 6 1.9 A few Fourier transforms 7

2. Differential geometry 7 2.1 Diffeomorphisms vs. coordinate tranformations 7 2.2 Connection and curvatures 8 2.3 Geodesic deviation equation 9 2.4 Forms 10 2.5 Harmonic forms 11 2.6 Vielbein formalism 12 2.7 Lie derivatives and Killing fields 13 2.8 Weyl transformations 14 2.9 Metric perturbations 14 2.10 Warped products 14 2.11 Flat fibrations 15 2.12 Gauss-Bonnet theorems 16 2.13 Hypersurfaces 17 2.14 General submanifolds 21 2.15 K¨ahlergeometry 21 2.16 Poincar´eduality 23 2.17 Coordinates for S2 24 2.18 The Hopf fibration 25 2.19 Unit sphere 27

3. Classical mechanics 27 3.1 Lagrangian and Hamiltonian formulations 27 3.2 Symmetries 29 3.3 Noether’s theorem for higher-order actions 31 3.4 Gauge symmetries 32 3.5 Hamilton-Jacobi theory 34

– 1 – 4. Gauge theories 36 4.1 Generalities 36 4.2 Standard Model 37

5. General relativity 40 5.1 Dimensions 40 5.2 Fundamentals 40 5.3 Spherically symmetric solutions 41 5.4 Cauchy problem 42 5.5 Gravitational waves 44 5.6 Israel junction condition 46 5.7 Boundary term 47 5.8 Dimensional reductions 49

6. String theory & CFT 50 6.1 Complex coordinates 50 6.2 CFT stress tensor 50 6.3 Liouville action 51 6.4 Sigma model 51 6.5 Spacetime action 52 6.6 Dimensional reduction and Buscher duality 53 6.7 More dualities 55 6.8 Orbifolds and quotient groups 56

7. Quantum information theory 56 7.1 Von Neumann entropy 56 7.2 Relative entropy 57 7.3 R´enyi entropy 58 7.4 Mixing-enhancing 59 7.5 Joint systems 60

8. Superselection rules 62 8.1 Joint systems 62 8.2 Joint systems and fermionic operators 63

9. Dimensions of physical quantities 64

– 2 – 1. Miscellaneous math

1.1 A matrix identity

If {gab} is a symmetric matrix, then " #   x vb 1 det = x det gab − wavb = α det{gab}, (1.1) wa gab x −1 " # " c # x vb 1 1 −v = b bc b c , (1.2) wa gab α −w αg + w v  1 −1  1  g − w v = gbc + wbvc (1.3) ab x a b α

bc c bc b bc bc where g is the inverse of gab, v = vbg , w = g wc, and α = x − vbg wc.

1.2 Campbell-Baker-Hausdorff If [A, B] commutes with A and B, then

[A, eB] = [A, B]eB (1.4) eABe−A = B + [A, B] (1.5) eAeB = e[A,B]eBeA (1.6) eA+B = e−[A,B]/2eAeB. (1.7)

1.3 Pauli matrices " # " # " # 0 1 0 −i 1 0 σ = , σ = , σ = . (1.8) 1 1 0 2 i 0 3 0 −1

1.4 Level repulsion In the space of n × n Hermitian matrices, the matrices with two degenerate eigenvalues lie on a (real) co-dimension 3 submanifold. Therefore, a 1-parameter family of matrices H(λ) will not generically intersect the submanifold: as λ is varied, the eigenvalues will move around, but will not cross each other. However, if there is a symmetry, i.e. another Hermitian matrix S that commutes with H(λ) for all λ, then the eigenvalues of H(λ) belonging to different eigenspaces of S are independent, and will generically cross.

1.5 The Legendre transform Let f(x) be a function on a vector space that is bounded below and goes to plus infinity faster than linearly in every direction. The Laplace transform of e−f(x) is eg(k), where g(k) is a function on the dual vector space:

Z i eg(k) = dx ekix −f(x). (1.9)

– 3 – It is easy to show that g(k) is convex. The Legendre transform of f(x) is the saddle-point approximation to g(k). Assume first that f is strictly convex and continuously differentiable. Then for every k the function i f(x) − kix has a unique (local and global) minimum, where

∂f = k . (1.10) ∂xi i

In other words, we have a bijection between x and k. The saddle point approximation to the integral (1.9) is the following function of k.

i g(k) = kix − f(x) ∂f . (1.11) =ki ∂xi

This will also be strictly convex and continuously differentiable. The inverse map is given by i x = ∂g/∂ki, and we have

i i f(x) = kix − g(k) ∂f = kix − g(k) ∂g i . (1.12) =ki =x ∂xi ∂ki

In other words, the Legendre transform is its own inverse. i If f is not convex, then for some values of k, f(x) − kix will have multiple local minima (and saddle points). However, it is the global minimum that dominates the integral (1.9). The x value of this global minimum will jump discontinuously as a function of k. This shows that the Legendre transform g(k) will have a discontinuous first derivative. In fact, the global minimum won’t change at all if we replace f(x) with its convex hull (as in the Maxwell construction). Therefore, two functions with the same convex hull will have the same Legendre transform. If f(x) has a discontinuous first derivative (at a convex point), then the global minimum i of f(x) − kix will be at the same value of x over a range of values of k. Over that range g(k) will be a linear function, and therefore won’t be strictly convex. In summary, on convex functions, having a range over which the function is linear (i.e. not being strictly convex) makes its Legendre transform have a discontinuous first derivative, and vice versa; the Legendre transform continues to be its own inverse. The double Legendre transform of a non-convex function returns its convex hull. If f depends on an auxiliary parameter y, then g will also depend on y. Their derivatives with respect to y obey a simple relation:

∂g ∂f = − . (1.13) ∂y ∂y

– 4 – 1.6 Gaussian integrals

Z 2 dx e−ax /2 = (2π)1/2a−1/2 (1.14)

Z 2 dx x2e−ax /2 = (2π)1/2a−3/2 (1.15)

Z 2 dx x4e−ax /2 = 3(2π)1/2a−5/2 (1.16)

Z 2 dx x6e−ax /2 = 15(2π)1/2a−7/2 (1.17)

   n+1/2 Z 2 1 2 dx x2ne−ax /2 = (2n − 1)!! (2π)1/2a−(n+1/2) = Γ n + (1.18) 2 a

Z i j dmx e−Aij x x = (2π)m/2(det A)−1/2 , (where A is symmetric) (1.19)

1.7 Saddle-point approximation Here we treat only the simplest case of a single real integral of a real function. Let f, g be real analytic functions, with f be bounded below with a single global minimum at x0. Set

00 0 f0 = f(x0) , f2 = f (x0) , g0 = g(x0) , g1 = g (x0) , etc. (1.20)

0 (of course, f (x0) = 0). We will also assume f2, g0 6= 0. Taking  to be a small (positive) parameter, we derive the following approximation, working to the necessary order to get to order  in the final answer: Z dx g(x)e−f(x)/ Z = dx g(x)  1  1 1 1  × exp − f + f (x − x )2 + f (x − x )3 + f (x − x )4 + O(x − x )5  0 2 2 0 6 3 0 24 4 0 0 Z  1  = e−f0/1/2 dy g + 1/2g y + g y2 + O(3/2) 0 1 2 2  1 1 1  × exp − f y2 − 1/2f y3 − f y4 + O(3/2) 2 2 6 3 24 4 Z  g 1 g  −f0/ 1/2 1/2 1 2 2 3/2 = e  g0 dy 1 +  y +  y + O( ) g0 2 g0   1 1 1 2 × 1 − 1/2f y3 − f y4 + f 2y6 + O(3/2) e−f2y /2 6 3 24 4 72 3   2   Z g f g f f 2 −f0/ 1/2 2 2 3 1 4 4 4 3 6 2 −f2y /2 = e  g0 dy 1 +  y − y − y + y + O( ) e 2g0 6g0 24 72 2π 1/2   g f g f 5f 2   −f0/ 1/2 2 3 1 4 3 2 = e  g0 1 +  − 2 − 2 + 3 + O( ) , (1.21) f2 2g0f2 2g0f2 8f2 24f2

– 5 – −1/2 In the second equality we changed the integration variable to y =  (x−x0). In the fourth equality we dropped odd powers of y, since those will integrate to 0. The result is conveniently written as a decreasing series in  for its logarithm:

Z  2   2  −f(x)/ f0 1 2πg0 g2 f3g1 f4 5f3 2 ln dx g(x)e = − + ln +  − 2 − 2 + 3 + O( ) .  2 f2 2g0f2 2g0f2 8f2 24f2 (1.22)

In the case g0 = 0 (but g2 − f3g1/f2 6= 0), one gets instead

Z  π 1/2  f g  −f(x)/ −f0/ 3/2 3 1 dx g(x)e = e  3 g2 − + O() (1.23) 2f2 f2 Z  3    −f(x)/ f0 1 π f3g1 ln dx g(x)e = − + ln 3 + ln g2 − + O() . (1.24)  2 2f2 f2

If there is another local minimum, or if the integration region is bounded (but includes x0 in its interior), then this series is asymptotic, and there are non-perturbative corrections −f(x )/ (f −f(x ))/ to (1.21) that go like e 1 (or to (1.22) like e 0 1 ), where x1 is the local minimum (other than x0) or integration endpoint with the lowest value of f.

1.8 Cauchy principal value

The Cauchy principal value of an integral where the integrand f(x) is singular at b is

Z c Z b− Z c  P dx f(x) := lim dx f(x) + dx f(x) . (1.25) a →0 a b+

For a function f(z) on the complex plane with a singularity at b, the Cauchy principal value of the integral over a contour C containing b is

Z Z P dz f(z) := lim dzf(z) , (1.26) →0 C C\D where D is the disk of radius  about b. In particular, if f is analytic and contains a pole at b, then the principal value is the average of the integral with the contour deformed to go around the pole both ways. 1 If we define [ x ] to be the distribution whose integral against a test function f is

Z  1  Z f(x) dx f(x) := P dx , (1.27) x x then we have 1  1  = − iπδ(x) . (1.28) x + i x

– 6 – 1.9 A few Fourier transforms According to the definition Z 1 Z f˜(k) := dx e−ikxf(x) , f(x) = dk eikxf˜(k) (1.29) 2π of the Fourier transform, we have:

f(x) f˜(k) g(x)∗ g˜(−k)∗ g(−x) g˜(−k) g(−x)∗ g˜(k)∗ g0(x) ikg˜(k) −ixg(x) g˜0(k) g(x − a) e−ikag˜(k) eiaxg(x) g˜(k − a) 1 2πδ(k) ik x (1.30) e 0 2πδ(k − k0) δ(x) 1 −ikx δ(x − x0) e 0 i θ(x) − k−i i θ(−x) k+i 1 x+i −2πiθ(k) 1 x−i 2πiθ(−k) 1 [ x ] πi(θ(−k) − θ(k)) 1 θ(x) − θ(−x) −2i[ k ]

2. Differential geometry

2.1 Diffeomorphisms vs. coordinate tranformations Physicists often use the terms “diffeomorphism” and “change of coordinates” interchangeably. Here I want to make the relationship precise: given a coordinate system on a manifold, its diffeomorphisms are in one-to-one correspondence with other coordinate systems that have the same transition functions on the patch overlaps. Let M be a manifold. A “coordinate system” or “atlas of charts” is a set of open subsets

(“patches”) {Uα ⊂ M} covering M (i.e. ∪αUα = M), together with a one-to-one function (“coordinates”) from each patch to RD (or Cn for a complex manifold)

D xα : Uα → R . (2.1)

Then for every overlap Uα ∩ Uβ, the transition function

−1 D fαβ = xαxβ : xβ(Uα ∩ Uβ) → R , (2.2)

– 7 – is required to be continuous, differentiable, holomorphic, etc. depending on what kind of manifold you want.

Given such a coordinate system {(Uα, xα)}, the diffeomorphisms of M onto itself are in 0 0 one-to-one correspondence with coordinate systems {(Uα, xα)} that have the same images in RD and the same transition functions:

0 0 0 xα(Uα) = xα(Uα), fαβ = fαβ ∀α, β. (2.3) 2.2 Connection and curvatures Antisymmetrization notation: 1 u := (u − u ) (2.4) [ab] 2 ab ba Covariant derivatives: 1 Γa = gad (∂ g + ∂ g − ∂ g ) (2.5) bc 2 b dc c bd d bc a p Γab = ∂b ln |g| (2.6) b b b c ∇av = ∂av + Γacv (2.7) c ∇avb = ∂avb − Γabvc (2.8) a 1 p a ∇av = ∂a( |g|v ) (2.9) p|g| a b a b a b a b a [v, w] = v ∇bw − w ∇bv = v ∂bw − w ∂bv (2.10) Geodesic equation: a a b c x¨ + Γbcx˙ x˙ = 0 (2.11) Riemann tensor:

d d e d Rabc = −2∂[aΓb]c + 2Γc[aΓb]e (2.12)

Rabcd = Rcdab = −Rbacd = −Rabdc (2.13) d R[abc] = 0 (2.14) e ∇[aRbc]d = 0 (2.15) d (∇a∇b − ∇b∇a)vc = Rabc vd (2.16) d d c (∇a∇b − ∇b∇a)v = −Rabc v (2.17) Ricci tensor and scalar:

b Rac = Rabc (2.18) a R = Ra (2.19) Einstein tensor: 1 G = R − Rg (2.20) ab ab 2 ab a ∇ Gab = 0 (2.21)

– 8 – Weyl tensor: 2 2 C = R − (g R − g R ) + Rg g (2.22) abcd abcd D − 2 a[c d]b b[c d]a (D − 1)(D − 2) a[c d]b

Cabcd = Ccdab = −Cbacd = −Cabdc (2.23) d C[abc] = 0 (2.24) a b Cabc = Cabc = 0 (2.25) D = 2: 1 K = R (Gaussian curvature) (2.26) 2 1 R = g (2.27) ab 2 ab Gab = 0 (2.28)

Rabcd = Rga[cgb]d (2.29) (2.30)

D = 3:

Rabcd = 2(ga[cRd]b − gb[cRd]a) − Rga[cgd]b (2.31)

Cabcd = 0 (2.32)  1  Ca = abc∇ R − Rg (Cotton tensor) (2.33) d b cd 4 cd a C a = 0 (2.34)

Cab = Cba (2.35) a ∇aC b = 0 (2.36) 2.3 Geodesic deviation equation

Given a 1-parameter family of curves γs(t), where t is a parameter along each curve and s parametrizes the choice of curve, define tangent vectors T := ∂/∂t, i.e. T a := ∂xa/∂t (velocity) and X := ∂/∂s, i.e. Xa := ∂xa/∂s (deviation vector). The rate of change of the deviation vector with t, or the relative velocity of points on curves γs and γs+ds, is described by the vector field a b a b a v := T ∇bX = X ∇bT , (2.37) where the equality is due to the fact that [X,T ] = 0. The relative acceleration is given by

a b a c b a a c b d a := T ∇bv = X ∇c(T ∇bT ) − Rbcd X T T . (2.38)

b a If the curves are geodesics then T ∇bT = 0 and a a c b d a = −Rbcd X T T (geodesic deviation equation) . (2.39)

a Note that in this case a Ta = 0. In fact, by an s-dependent redefinition of t, we can set a a X Ta = 0 everywhere, so v Ta = 0 as well.

– 9 – 2.4 Forms Let v be a p-form and w a q-form.1

1 v = v dxa1 ∧ · · · ∧ dxap (2.43) p! a1...ap (p + q)! (v ∧ w) = v w (2.44) a1...apb1...bq p!q! [a1...ap b1...bq] v ∧ w = (−1)pqw ∧ v (2.45) a d = dx ∧ ∂a (2.46)

(dv)a1...ap+1 = (p + 1)∂[a1 va2...ap+1] (2.47) Z Z dv = v (M is p + 1 dimensional) (2.48) M ∂M d(v ∧ w) = dv ∧ w + (−1)pv ∧ dw (2.49) 1 (∗v) =  b1...bp v (2.50) a1...aD−p p! a1...aD−p b1...bp  1  (∗v) ∧ v = ∗ v va1...ap = ∗|v|2 (2.51) p! a1...ap Z Z ∗φ = dDxp|g|φ (φ is a scalar) (2.52)

Euclidean signature:

p 1...D = |g| (2.53) 1 1...D = (2.54) p|g| ∗ ∗ v = (−1)p(D−p)v (2.55) 1 d†v = (−1)Dp ∗ d ∗ v = − ∇ va1 dxa2 ∧ · · · ∧ dxap (2.56) (p − 1)! a1 a2...ap

1The definition of the Hodge star depends on who you ask. Above we gave Polchinski’s definition. However, another common one is Wald’s, 1 (∗ v) = v b1...bp , (2.40) W a1...aD−p p! b1...bp a1...aD−p

p(D−p) which is related to Polchinski’s by ∗Wv = (−1) ∗ v. I believe that all of the equations in this subsection continue to hold under ∗ → ∗W, except for (2.51), which becomes

2 v ∧ (∗Wv) = ∗W|v| , (2.41) and (2.56),(2.60), which become

1 d†v = (−1)Dp+D+s ∗ d ∗ v = − ∇ va1 dxa2 ∧ · · · ∧ dxap (2.42) W W (p − 1)! a1 a2...ap

(s = 1 for Euclidean signature, s = 0 for Lorentzian signature).

– 10 – Lorentzian signature: p 0...(D−1) = |g| (2.57) 1 0...(D−1) = − (2.58) p|g| ∗ ∗ v = (−1)p(D−p)+1v (2.59) 1 d†v = (−1)Dp+1 ∗ d ∗ v = − ∇ va1 dxa2 ∧ · · · ∧ dxap (2.60) (p − 1)! a1 a2...ap

2.5 Harmonic forms The natural inner product on forms of the same degree is

1 Z p Z hv, wi := dDx |g|ga1b1 ··· gapbp v w = ∗v ∧ w , (2.61) p! a1···ap b1···bp where v and w are both p-forms. If the manifold is closed, or boundary conditions are imposed so that boundary terms vanish, then

hdv, wi = hv, d†wi (2.62) and hv, dwi = hd†v, wi . (2.63)

v is harmonic if dd†v + d†dv = 0 . (2.64)

For the rest of this subsection we consider a closed Riemannian manifold. A harmonic form v is then both closed, dv = 0, and co-closed, d†v = 0:

dd†v + d†dv = 0 ⇒ hv, dd†v + d†vi = 0 (2.65) ⇒ hd†v, d†vi + hdv, dvi = 0 (2.66) ⇒ hd†v, d†vi = hdv, dvi = 0 (2.67) ⇒ d†v = 0 , dv = 0 (2.68) where in the last two steps we used the fact that the inner product is positive definite. The Hodge decomposition theorem states that any form can be decomposed uniquely into the sum of an exact form dw, a co-exact form d†u, and a harmonic form x:

v = dw + d†u + x . (2.69)

The uniqueness can be seen as follows. First, a form that is both exact and co-exact necessarily vanishes, since if dw = d†u then

hdw, dwi = hdw, d†ui = hw, d†2ui = 0 (2.70)

– 11 – so dw = 0. Second, a form that is both exact and harmonic also vanishes, since if dw is harmonic, then d†dw = 0, which implies 0 = hw, d†dwi = hdw, dwi, hence dw = 0. Finally, a form that is both co-exact and harmonic similarly vanishes. A closed form w can be written as an exact form plus a harmonic form. This follows from the Hodge decomposition theorem (2.69). If dv = 0 then dd†u = 0, which implies

0 = hu, dd†ui = hd†u, d†ui (2.71) so d†u = 0, so v = dw + x . (2.72) (2.72) implies that every cohomology class contains a harmonic representative. Further- more, this representative is unique, since, as explained above, a form that is both harmonic and exact must vanish. The harmonic representative is distinguished by the fact that it is co-closed, d†v = 0. It also minimizes the norm hv, vi within the class. This can be seen as follows. The functional hv, vi is convex, so an extremum is necessarily a minimum. At the extremum we have, for an arbitrary p − 1 form w and to first order in ,

0 = hv + dw, v + dwi − hv, vi = (hv, dwi + hdw, vi) = (hd†v, wi + hw, d†vi) (2.73)

Setting w = d†v, we see that d†v = 0, so v is harmonic.

2.6 Vielbein formalism Let µ, ν be indices in the vector representation of the appropriate Lorentz group (raised and lowered with ηµν), and a, b be the usual tangent and co-tangent space indices. The vielbein 1-forms eµ are defined by µ (eµ)a(e )b = gab (2.74) which is equivalent to a (eµ) (eν)a = ηµν. (2.75)

µ The connection 1-forms ων are valued in the Lie algebra of the Lorentz group (i.e. ωµν = −ωνµ), and defined by µ deσ = eµ ∧ ωσ (2.76) or in components b (ωµν)a = (eµ) ∇a(eν)b. (2.77) ν (Wald writes this ωaµν.) The Riemann 2-forms Rµ are also valued in the Lie algebra of the Lorentz group, and are defined by

ν c ν d (Rµ )ab = (eµ) (e )dRabc . (2.78)

They are given by ν ν α ν Rµ = dωµ + ωµ ∧ ωα . (2.79)

– 12 – In order to calculate the connection 1-forms from the vielbein, one may do it either by inspection from (2.76), or, more algorithmically, as follows. First compute

a Kµσν ≡ (eσ)a[eµ, eν] , (2.80) which equals ωµσν − ωνσµ, and from that compute 1 ω = (K + K + K ) . (2.81) µσν 2 µσν σµν σνµ

The scalars ωµσν are called Ricci rotation coefficients. The connection 1-forms can be com- puted from them: µ (ωσν)a = (e )aωµσν . (2.82) (Note that it is the first index on the Ricci rotation coefficients that becomes the 1-form index on the connection.) The Riemann tensor can then be computed either using (2.78) or directly from the Ricci rotation coefficients:

a a αβ Rρσµν = (eρ) ∂aωσµν − (eσ) ∂aωρµν − η (ωρβµωσαν − ωσβµωραν + ωρβσωαµν − ωσβρωαµν) . (2.83) The Christoffel symbols can also be computed from the connection 1-forms:

c µν c  αβ  Γab = η (eµ) ∂a(eν)b − η (eα)b(ωβν)a . (2.84)

2.7 Lie derivatives and Killing fields Let va be a vector field. Then

a a b a b a b a b a £vw = [v, w] = v ∂bw − w ∂bv = v ∇bw − w ∇bv (2.85) b b b b £vwa = v ∂bwa + wb∂av = v ∇bwa + wb∇av (2.86)

£vgab = ∇avb + ∇bva (2.87)

b If in some coordinate system the components of v are constant, ∂av = 0, then in that coordinate system the Lie derivative coincides with the ordinary derivative:

a £v = v ∂a. (2.88)

Under a small diffeomorphism xa = x0a + va, (2.89) a tensor T will transform into 0 T = T + £vT. (2.90) So this diffeomorphism is an isometry if

£vgab = 0 (2.91)

(Killing’s equation).

– 13 – 2.8 Weyl transformations Let 2ω g˜ab = e gab. (2.92) Then

p|g˜| = eDωp|g| (2.93) g˜ab = e−2ωgab (2.94) ˜a a a a a Γbc = Γbc + ∂bωδc + ∂cωδb − ∂ ωgbc (2.95) ˜ d d d

d d e Rabc = Rabc + 2δ[a ∇b] + ∂b]ω ∂cω − 2gc[a ∇b] + ∂b]ω ∂ ω − 2gc[aδb]∂eω∂ ω (2.96) 2  d  R˜ac = Rac − gac∇ ω + (D − 2) −∇a∂cω + ∂aω∂cω − gac∂dω∂ ω (2.97) −2ω 2 a
R˜ = e R − 2(D − 1)∇ ω − (D − 2)(D − 1)∂aω∂ ω (2.98) d d C˜abc = Cabc (2.99) −ω C˜ab = e Cab (D = 3) (2.100) In D = 2, all metrics are locally conformally flat. In D = 3, the metric is locally conformally flat if and only if Cab = 0. In D ≥ 4, the metric is locally conformally flat if and only if Cabcd = 0.

2.9 Metric perturbations

Let us perturb the metric gab by δgab = hab. (2.101)

Then, to first order in hab (and always using gab to raise and lower indices), 1 δp|g| = p|g|h (2.102) 2 δgab = −hab (2.103) 1 δΓa = ∇ h a − ∇ah (2.104) bc (b c) 2 bc d d δRabc = −2∇[aδΓb]c (2.105) 1 1 δR = ∇ ∇ h c − ∇ ∂ h − ∇2h = ∆ h − ∇ v , (2.106) ab c (a b) 2 a b 2 ab L ab (a b) 1 1 ∆ h ≡ − ∇2h − R c dh + R ch , v ≡ ∂ h − ∇ hb (2.107) L ab 2 ab a b cd (a b)c a 2 a b a ab 2 ab δR = ∇a∇bh − ∇ h − h Rab (2.108) 2.10 Warped products Consider the warped product geometry

2 2 2τ(x(1)) 2 ds = ds(1) + e ds(2) (2.109) m m n 2τ(xm) µ µ ν = gmn(x )dx dx + e gµν(x )dx dx , (2.110)

– 14 – m µ 2 where x(1) = x and x(2) = x . Let k be the dimensionality of the space ds(2). The Christoffel symbols are:

µ (2)µ Γνρ = Γνρ (2.111) µ µ Γνp = ∂pτδν (2.112) m mn 2τ Γνρ = −g ∂nτe gνρ (2.113) m µ Γnρ = Γnp = 0 (2.114) m (1)m Γnp = Γnp (2.115)

The Riemann tensor is:

σ (2) σ 2τ mn σ σ
Rµνρ = Rµνρ + e g ∂mτ∂nτ gνρδµ − gµρδν (2.116) s σ σ σ Rµνρ = Rµνp = Rmνρ = Rµnρ = 0 (2.117) σ s Rmnρ = Rµνp = 0 (2.118) σ  (1)  σ Rmνp = − ∇m ∂pτ + ∂mτ∂pτ δν (2.119)

σ  (1)  σ Rµnp = ∇n ∂pτ + ∂nτ∂pτ δµ (2.120)

s sa  (1)  2τ Rmνρ = g ∇m ∂aτ + ∂mτ∂aτ e gνρ (2.121)

s sa  (1)  2τ Rµnρ = −g ∇n ∂aτ + ∂nτ∂aτ e gµρ (2.122) σ s s s Rmnp = Rµnp = Rmνp = Rmnρ = 0 (2.123) s (1) s Rmnp = Rmnp (2.124)

The Ricci tensor is:

(2)  2 mn  2τ Rµρ = Rµρ − ∇(1)τ + kg ∂mτ∂nτ e gµρ (2.125)

Rµp = 0 (2.126) (1)  (1)  Rmp = Rmp − k ∇m ∂pτ + ∂mτ∂pτ (2.127)

The Ricci scalar is:

−2τ 2 mn R = R(1) + e R(2) − 2k∇(1)τ − k(k + 1)g ∂mτ∂nτ (2.128)

2.11 Flat fibrations

Consider the following metric ansatz:

2 m n µ ν ds = gmn(x)dx dx + hµν(x)dy dy . (2.129)

– 15 – 2 m n We denote quantities derived from the base metric dsˆ = gmndx dx by hats. The Christoffel symbols are (here and below we write only non-zero components):

l ˆl Γmn = Γmn (2.130) 1 Γm = − gmn∂ h (2.131) µν 2 n µν 1 Γµ = Γµ = hµλ∂ h . (2.132) mν νm 2 m λν The Riemann tensor is:

Rmnpl = Rˆmnpl (2.133) 1 R = R = hµν (∂ h ∂ h − ∂ h ∂ h ) (2.134) mnρλ ρλmn 4 m λν n µρ n λν m µρ 1 1 R = −R = −R = R = − ∇ˆ ∂ h + hνλ∂ h ∂ h (2.135) µnρl nµρl µnlρ nµlρ 2 n l µρ 4 n λρ l µν 1 R = gmn (∂ h ∂ h − ∂ h ∂ h ) (2.136) λµνρ 4 n µν m ρλ n νλ m µρ The Ricci tensor is: 1 1 R = Rˆ − hµρ∇ˆ ∂ h − ∂ hµρ∂ h (2.137) mn mn 2 m n µρ 4 m n µρ 1 1 1 √ R = − ∇ˆ 2h + gmnhρλ∂ h ∂ h − gmn∂ h ∂ ln h. (2.138) µν 2 µν 2 n λµ m ρν 2 m µν n The Ricci scalar is: 3 √ √ R = Rˆ − hµρ∇ˆ 2h − gmn∂ hµρ∂ h − gmn∂ ln h∂ ln h. (2.139) µρ 4 m n µρ m n 2.12 Gauss-Bonnet theorems Here we discuss only closed manifolds. In odd dimensions, the Euler number always vanishes. In zero dimensions, it is the number of connected components (i.e. points). In two dimensions it is given by: 1 Z √ χ = d2x gR. (2.140) 4π In four dimensions it is given by 1 Z √   χ = d4x g R2 − 4R Rab + R Rabcd . (2.141) 32π2 ab abcd For an even-dimensional hypersurface in flat space, Z 2 D √ j χ = d x γ det{Ki } , (2.142) ΩD where ΩD is the volume of the unit D-sphere (see 2.19), and γij and Kij are the induced metric and extrinsic curvature respectively (see 2.13).

– 16 – 2.13 Hypersurfaces Let’s begin with the most covariant description of a hypersurface in a manifold, in which the hypersurface S and the manifold M have independent coordinate systems: M has D coordinates xµ while S has D − 1 coordinates yi. The embedding of S in M is described by the functions xµ(y), whose first derivatives give a D × (D − 1) matrix:

∂xµ M µ = . (2.143) i ∂yi

This matrix can be used to pull a co-vector back from M onto S,

µ vi = Mi vµ, (2.144) or to push a vector forward from S into M,

µ µ i v = Mi v . (2.145)

A vector vµ in M is said to be tangent to S if there exists a vector vi in the tangent space to S such that 2.145 is satisfied. The notion of being tangent to S does not exist for co-vectors (in the absence of a metric), which instead have the notion of being normal to S: a co-vector vµ in M is said to be normal to S if it pulls back to a vanishing co-vector in S. In fact, up to rescaling, there is a unique normal co-vector. For example, if f(x) is a scalar field on M that vanishes on S, f(x(y)) = 0 for all y, then its gradient vµ = ∂µf is normal to S. Given a foliation by hypersurfaces and a normal co-vector field vµ, it is easy to show that

v ∧ dv = 0 , (2.146)

µ where v = vµdx is the corresponding one-form. Frobenius’s theorem asserts that (2.146) is also a sufficient condition for the existence of a foliation by hypersurfaces normal to vµ. None of the above depends on the metric. If M is endowed with a metric gµν, then several structures follow from it. First of all, we can pull it back to a metric on S:

µ ν γij = Mi Mj gµν (2.147)

We’ll always use gµν to raise and lower µ, ν indices, and γij to raise and lower i, j indices. Now that we don’t have to distinguish so carefully between vectors and co-vectors, we can make statements like: The pull-back of the push-forward of a vector vi is itself. The push- forward of the pull-back of a vector vµ lies tangent to S; this defines an orthogonal projection operator: µ µ i P ν = Mi Mν. (2.148)

Unless the hypersurface is null, we can define a unit normal co-vector nµ by

µ µ Mi nµ = 0 , nµn = σ (2.149)

– 17 – where σ = ±1, which makes nµ unique up to a sign. It’s clear that

µ µ µ P ν = δν − σn nν, (2.150) since both 2.148 and 2.150 are 0 acting on nµ and 1 on vectors tangent to S.

The induced metric γij defines the intrinsic geometry of S, and one can use it to calculate covariant derivatives and curvature tensors. The curvature of the embedding of S in M is captured by the (covariant) derivative of the unit normal along S,

ρ Kiν = Mi ∇ρnν. (2.151)

Of course, this tensor is a bit awkward, having one index in M and the other in S. However, ν since the ν index is tangent to S (n Kiν = 0), we can pull it back to S without losing any information: ν ρ ρ ν Kij = Mj Mi ∇ρnν = −nνMi ∇ρMj . (2.152) Alternatively, we can push the i index forward to M:

ρ Kµν = P µ∇ρnν. (2.153)

One can show that both Kij and Kµν are symmetric. They are both known as the extrinsic curvature. µ µ µ Recall that the normal vector n was unique up to a sign. Under n → −n , Kij and Kµν change sign. The mean curvature is the trace of the extrinsic curvature:

µν ij K = g Kµν = γ Kij . (2.154)

If M is foliated by a family of hypersurfaces, then their normal vectors nµ will define a vector field on M. In this case there are a few other useful formulas for the extrinsic curvature and its trace: 1 1 1 K = £ P = P α P β £ g = (∇ n + ∇ n − σnα∇ (n n )) , (2.155) µν 2 n µν 2 µ ν n αβ 2 µ ν ν µ α µ ν µ K = ∇µn . (2.156)

If the coordinates on M are {λ, yi}, with S being a surface of constant λ, and we use the coordinates yi on S, then we have

µ λ j j Mi : Mi = 0 ,Mi = δi , (2.157)

γij = gij , (2.158) λλ −1/2 nλ = σ|g | , ni = 0 , (2.159) λ Kij = −nλΓij . (2.160)

(The coefficient σ in nµ is put in to agree with (2.161) below.)

– 18 – Gaussian normal coordinates is a special case of the coordinate system of the last para- graph that is adapted to metric, with the coordinates for M fixed (at least in a neighborhood of S) in terms of those for S. From each point on S, send out a geodesic with initial velocity nµ. Sufficiently close to S, each point p of M will have a single such geodesic passing through it. Call the point on S from which that geodesic emanates q(p). The point p can then be labelled by the coordinates yi of q(p) together with the affine parameter λ of the geodesic at p. Obviously, on S (which is the λ = 0 surface) the unit normal vector in Gaussian normal coordinates is

nλ = 1, ni = 0. (2.161)

Less obvious, but straightforward to show (see section 3.3 of Wald), is that the vector 2.161 is the unit normal vector to the surfaces of constant λ even away from S. Therefore in the neighborhood of S where the Gaussian normal coordinates are defined, the metric may be written

2 2 i j ds = σdλ + γij(y, λ)dy dy . (2.162)

In these coordinates the extrinsic curvature is

1 K = ∂ γ . (2.163) ij 2 λ ij

The formulas below give useful quantities for M in Gaussian normal coordinates in terms of those for the surfaces of constant λ. We write K˙ ij = ∂λKij, and a prime indicates a quantity derived from γij rather than gµν, i.e. describing the intrinsic geometry of the surfaces of constant λ:

gλλ = σ (2.164) gi0 = 0 (2.165) gij = γij (2.166) p|g| = p|γ| (2.167) i 0i Γjk = Γjk (2.168) λ Γij = −σKij (2.169) i i Γλj = K j (2.170) i λ λ Γλλ = Γiλ = Γλλ = 0 (2.171)

– 19 – 0 Rijkl = Rijkl − σ (KikKjl − KjkKil) (2.172) 0 0 Rijkλ = ∇iKjk − ∇jKik (2.173) m Riλjλ = −K˙ ij + Ki Kmj (2.174)

Rijλλ = Riλλλ = Rλλλ = 0 (2.175) 0  ˙ k  Rij = Rij − σ Kij + KKij − 2KikK j (2.176) 0 j Riλ = ∇jK i − ∂iK (2.177) ij ij Rλλ = −γ K˙ ij + K Kij (2.178) 0  ij 2 ij  R = R − σ 2γ K˙ ij + K − 3K Kij (2.179)

Combining (2.178) and (2.179) gives the following equation, valid in any coordinate sys- tem: 0 2 ij µ ν
R = R + σ K − K Kij − 2n n Rµν (2.180)

Under a small change in the metric gµν → gµν +hµν, the tensors defined above will change. Here we write their changes in the Gaussian normal coordinates of the original metric gµν:

µ δMi = 0 (2.181)

δγij = hij (2.182) 1 δn = h , δn = 0 (2.183) λ 2 λλ i 1 δnλ = − σh , δni = −hi (2.184) 2 λλ λ λ λ i i i δP λ = δP i = δP j = 0 , δP λ = h λ (2.185) 1 1 δK = −∇0 h − σh K + h˙ (2.186) ij (i j)λ 2 λλ ij 2 ij

(note that hiλ is a co-vector field from the point of view of S). Under a Weyl transformation, 2ω g˜µν = e gµν , (2.187) the tensors mentioned above transform as follows:

˜ µ µ Mi = Mi (2.188) 2ω γ˜ij = e γij (2.189) µ µ P˜ ν = P ν (2.190) ω n˜µ = e nµ (2.191) n˜µ = e−ωnµ (2.192) ω ρ K˜ij = e (Kij + γijn ∂ρω) (2.193) ω ρ K˜µν = e (Kµν + Pµνn ∂ρω) (2.194) −ω ρ K˜ = e (K + (D − 1)n ∂ρω) (2.195)

– 20 – 2.14 General submanifolds The extrinsic curvature generalizes easily to a codimension-k submanifold. We work in the most covariant description (the first one used in the previous subsection). As before, we have µ µ i µ ν µ µ i the matrix Mi = ∂x /∂y , the induced metric γij = Mi Mj gµν, and the matrix P ν = Mi Mν which projects vectors onto S. The normal vector is generalized into an orthonormal frame µ for the normal bundle, a set of k vectors na satisfying

µ na nµb = ηab , (2.196) where ηab is diagonal with eigenvalues ±1. The frame indices a, b are raised and lowered with ηab. There is an SO(k) (or one of its Lorentzian generalizations) gauge (i.e. y-dependent) µ ambiguity in the choice of na . We now have, instead of (2.150),

µ µ µ a P ν = δν − na nν . (2.197)

The extrinsic curvature now carries an index in the normal bundle:

a ν ρ a a ρ ν Kij = Mj Mi ∇ρnν = −nνMi ∇ρMj . (2.198)

Note that this transforms covariantly under y-dependent SO(k) transformations of the basis µ na . Any or all of these indices can be put into embedding-coordinate indices; for example, we can write λ λ ν ρ a Kµκ = naP κP µ∇ρnν . (2.199) The mean curvature is the normal vector obtained by tracing on the tangent indices:

λ µκ λ K = g Kµκ . (2.200)

2ω Under a Weyl transformation,g ˜µν = e gµν, the extrinsic and mean curvatures transform as follows:

˜ λ λ λα Kµκ = Kµκ + PµκQ ∂αω , (2.201) λ −2ω  λ λα  K˜ = e K + (D − k)Q ∂αω , (2.202) where Qλα = gλα − P λα . (2.203)

2.15 K¨ahler geometry A point to keep in mind is that the complexified tangent space of a manifold is twice as big (in real dimension) as its real tangent space—you have the same number of basis vectors (D, the real dimension of the manifold), but they can have complex coefficients. So it is possible to impose a reality condition on a vector, v¯ı = (vi)∗. Of course, the holomorphic and anti-holomorphic parts of the tangent space each have complex dimension D/2.

– 21 – b An almost complex structure is a (real) tensor Ja satisfying

b c c Ja Jb = −δa . (2.204)

i b Complex coordinates z adapted to Ja are ones in which it takes the form

j j ¯ ¯ j ¯ Ji = iδi ,J¯ı = −iδ¯ı ,J¯ı = Ji = 0. (2.205)

For an almost complex structure to be a complex structure its Niejenhuis tensor must vanish (see Candelas). The complex structure is preserved by holomorphic changes of coordinates.

A metric gab is hermitian (with respect to a given complex structure) if it satisfies any of the following conditions, which are all equivalent:

c d gij = g¯ı¯ = 0, gab = Ja Jb gbd,Jab = −Jba. (2.206)

Thus metric and complex structure together define a 2-form, the K¨ahlerform J, whose components in complex coordinates are given by

Ji¯ = −Ji¯ = igi¯,Jij = J¯ı¯ = 0. (2.207)

The measure is p | det{gab}| = det{gi¯}, (2.208) and the volume form is 1 J n, (2.209) n! where n = D/2 is the complex dimension. If the K¨ahlerform is closed, dJ = 0, then the metric is called K¨ahler. By dividing dJ into (1,2) and (2,1) components, we find

∂kgi¯ = ∂igk¯, ∂k¯g¯ıj = ∂¯ıgkj¯ . (2.210)

If follows that the mixed components of the Christoffel symbols vanish, and the pure ones are given by i i¯ı ¯ı ¯ıi Γjk = g ∂jgk¯ı, Γ¯k¯ = g ∂¯gki¯ . (2.211) It then follows that the only non-zero components of the Riemann tensor are

¯l ¯l ¯l l l l −Ri¯k¯ = Ri¯ k¯ = ∂iΓ¯k¯, −R¯ıjk = Rj¯ık = ∂¯ıΓjk, (2.212) and of the Ricci tensor are p Ri¯ = Ri¯ = −∂i∂¯ ln |g|. (2.213) (Be careful: I think that Candelas uses the opposite sign convention for the curvature tensors.) It also follows from K¨ahlerity that the metric can locally be expressed in terms of a K¨ahlerpotential K:

gi¯ = ∂i∂¯K (2.214)

– 22 – or in other words J = i∂∂K.¯ (2.215)

The K¨ahlerpotential is a scalar under (holomorphic) diffeomorphisms, but it has an additional gauge freedom: the metric is invariant under K¨ahlertransformations

K → K + f(z) + f¯(¯z). (2.216)

In fact, the K¨ahlerpotential may not be globally well-defined, and it may be necessary to do K¨ahlertransformations in going from patch to patch. Example: CP n is defined by

(z1, . . . , zn+1) ∈ Cn+1 \{0}, (z1, . . . , zn+1) ∼ λ(z1, . . . , zn+1)(λ ∈ C \{0}). (2.217)

m j j m m On the patch where z 6= 0, we can define coordinates ζm = z /z (of course ζm = 1 is not a coordinate). The Fubini-Study metric is defined by the K¨ahlerpotential

n+1  X j 2 Km = ln  |ζm|  . (2.218) j=1

A simple calculation shows that

ln p|g| = −(n + 1)K, (2.219) and therefore

Ri¯ = (n + 1)gi¯. (2.220)

2.16 Poincar´eduality Let M be a D-dimensional real compact orientable manifold without boundary. For a given p (0 ≤ p ≤ D, p 6= D/2) consider the following four vector spaces: the pth and (D − p)th de Rham cohomologies Hp(R) and HD−p(R), and the pth and (D − p)th homologies Hp(R) and HD−p(R). If we arrange these four spaces at the vertices of a square, then they are connected along each edge by a natural bilinear form:

p R D−p H (R) → M α ∧ β ← H (R) ↓ ↓ R R a α b β (2.221) ↑ ↑

Hp(R) → #(a, b) ← HD−p(R)

p D−p Here α, β, a, and b are elements of H (R), H (R), Hp(R), and HD−p(R) respectively, and #(a, b) is the intersection number of the cycles a and b. It is important that these bilinear forms are independent of the choice of representative in each homology or cohomology class.

– 23 – Poincar´eduality is an isomorphism between the diagonally opposite vector spaces of 2.221 under which all four bilinear forms are equal. Given a p-cycle a ∈ Hp(R), the Poincar´edual is an element of HD−p(R) for which we can construct a representative form β as follows. In each patch choose coordinates xi (i = 1, . . . , p), ym (m = 1,...,D − p) such that the submanifold a is the y = 0 locus. Then

β = δD−p(y)dy1 ∧ · · · ∧ dyD−p (2.222)

D−p is a closed (D − p)-form. It is possible to show that this map from Hp(R) to H (R) is bijective. Since under Poincar´eduality all four bilinear forms in 2.221 are equal, we really have just two distinct vector spaces connected by a single bilinear form. Furthermore, de Rham’s theorems show that this form is non-degenerate, in other words it is represented by an invertible matrix. This makes each vector space canonically isomorphic to the dual vector space (in the usual sense) of the other. Their dimension is called the pth Betti number bp (= bD−p) of M. Poincar´eduality is actually stronger than this, as it is an isomorphism between the integral homology and cohomology groups. Consider the lattice of integral homology Hp(Z) ⊂ p p Hp(R). H (Z) ⊂ H (R) is defined as the dual lattice in the dual vector space:  Z  p p H (Z) = α ∈ H (Z): ∀a ∈ Hp(Z), α ∈ Z . (2.223) a

Since the intersection number of two elements of Hp(Z) is always an integer, the image of p Hp(Z) under Poincar´eduality sits inside H (Z). The non-trivial statement (see section 0.4 of Griffiths and Harris) is that Poincar´eduality is in fact an isomorphism between Hp(Z) and Hp(Z). Let us now deal with the case p = D/2. We now start with just two vector spaces, Hp(R) and Hp(R). These are both equipped with inner products: on Hp(R) by the intersection p R number and on H (R) by M α ∧ β. These inner products are symmetric or antisymmetric p depending on whether p is even or odd. Between Hp(R) and H (R) we also have a bilinear R p form given by a α. Poincar´eduality is an isomorphism between Hp(R) and H (R) under which the two inner products and the bilinear form all agree. So we effectively have a single vector space equipped with a (symmetric or antisymmetric) inner product. Furthermore,

Poincar´eduality asserts that, under that inner product, Hp(Z) ⊂ Hp(R) is a self-dual lattice.

2.17 Coordinates for S2 There are three standard sets of coordinates on S2, namely polar coordinates:

(θ, φ) : 0 ≤ θ ≤ π, φ ∼ φ + 2π; (2.224) embedding coordinates:

3 2 2 2 (x1, x2, x3) ∈ R : x1 + x2 + x3 = 1; (2.225)

– 24 – and stereographic coordinates: y ∈ C. (2.226) They are related by:

2y 1 − |y|2 x1 + ix2 = sin θ eiφ = , x3 = cos θ = , (2.227) 1 + |y|2 1 + |y|2

θ |y| = tan , arg y = φ. (2.228) 2 The unit round metric is 4dy dy¯ dΩ2 = dθ2 + sin2 θ dφ2 = dx2 + dx2 + dx2 = . (2.229) 2 1 2 3 (1 + |y|2)2

The metric is K¨ahlerwith respect to the y coordinate, with K¨ahlerpotential

K = 2 ln(1 + |y|2). (2.230)

2.18 The Hopf fibration The Hopf fibration is a fiber bundle of S1 over S2 that is topologically S3. Define S3 as the 2 set (z1, z2) ∈ C satisfying 2 2 |z1| + |z2| = 1, (2.231) 2 3 and S as the set (x1, x2, x3) ∈ R satisfying

2 2 2 x1 + x2 + x3 = 1. (2.232)

The Hopf fibration is defined by the following map from S3 to S2:

∗ 2 2 x1 + ix2 = 2z1z2, x3 = |z1| − |z2| . (2.233)

2 1 The inverse image of each point in S is the S defined by simultaneous phase rotations of z1 3 and z2 (note that z1 and z2 can never simultaneously vanish). If we parametrize S by three angles as follows: θ i θ i z = cos exp (ψ + φ), z = sin exp (ψ − φ), (2.234) 1 2 2 2 2 2 where 0 ≤ θ ≤ π, (φ, ψ) ∼ (φ + 2π, ψ + 2π) ∼ (φ + 2π, ψ − 2π) (2.235) then θ and φ parametrize the base S2 (in the usual way), and ψ parametrizes the fiber. It is useful to define the following three one-forms on S3, known as the “left-invariant one-forms” due to their role in SU(2): 1 1 σ + iσ = − eiψ(idθ + sin θdφ), σ = (cos θdφ + dψ). (2.236) x y 2 z 2

– 25 – They satisfy

dσx = 2σy ∧ σz (2.237) and cyclic permutations. The unit round metric on S3 is given by 1 dΩ2 = σ2 + σ2 + σ2 = dθ2 + dφ2 + 2 cos θdφdψ + dψ2
(2.238) 3 x y z 4 The volume form is given by 1 −σ ∧ σ ∧ σ = sin θdθ ∧ dφ ∧ dψ. (2.239) x y z 8 The cross term in the metric 2.238 shows that the fibration is not trivial. In fact it has the form of the Kaluza-Klein ansatz,

1 2 dΩ2 = ds2 + e2σ dψ + A , (2.240) 3 2 2 KK

1 (We put 2 dψ because ψ has periodicity 4π rather than 2π as is conventional in the Kaluza- Klein ansatz.) Here 1 ds2 = σ2 + σ2 = dΩ2 (2.241) 2 x y 4 2 is the metric on the base, which is a round S2. Then we have

1  eσ dψ + A = σ , (2.242) 2 KK z so that the Kaluza-Klein scalar σ vanishes while the Kaluza-Klein gauge field, 1 A = cos θdφ, (2.243) KK 2 is the same as the gauge field on an S2 surrounding a magnetic monopole in three spatial dimensions. Taub-NUT is a four-dimensional Ricci-flat metric that fills in the radial direction of the Kaluza-Klein monopole: r + m r − m ds2 = dr2 + 4(r2 − m2)(σ2 + σ2) + 16m2 σ2. (2.244) r − m x y r + m z

At long distances the base is asymptotically R3 while the fiber goes to a constant size. 3 The free Zn action on S ,

2πi/n 2πi/n (z1, z2) 7→ (e z1, e z2), (2.245) is rather convenient to describe in terms of the Hopf fibration, since it acts only on the fiber and not on the base: 4π (θ, φ, ψ) 7→ (θ, φ, ψ + ). (2.246) n

– 26 – The metric obtained by modding out by this symmetry can still be described by the Kaluza- Klein ansatz, but we must adjust some of the coefficients since ψ now has periodicity 4π/n: n  eσ dψ + A = σ (2.247) 2 KK z now yields

σ = − ln n, AKK = n cos θdφ. (2.248) The sphere therefore now carries n units of magnetic flux. Corresponding four-dimensional multi-Taub-NUT solutions, with the individual monopoles either coincident or separated in three dimensions, are also known (see Eguchi-Gilkey-Hanson). There is a more general Hopf fibration of S2n+1 over CP n. Here we consider S2n+1 as non-zero points in R2n+2 =∼ Cn+1, identified under multiplication by positive real numbers. These are mapped to points in CP N simply by quotienting further by phases, which amounts to quotienting points in Cn+1 by arbitrary non-zero complex numbers. It’s clear that the fiber over every point in CP n is a circle.

2.19 Unit sphere The unit SD has the following curvature tensors: D R = g g −g g ,R = (D−1)g ,R = D(D−1) ,G = (D−1)(1− )g . abcd ac bd ad bc ab ab ab 2 ab (2.249) Its total volume is 2π(D+1)/2 ΩD = D+1 . (2.250) Γ( 2 ) For example, D 0 1 2 3 4 5 6 7 2 2 3 3 4 . (2.251) ΩD 2 2π 4π 2π 8π /3 π 16π /15 π /3 Its Euler character χ is 2 when D is even and 0 when D is odd. D+1 The unit sphere embedded in R has extrinsic curvature Kij = γij (in the notation of subsection 2.13).

3. Classical mechanics

3.1 Lagrangian and Hamiltonian formulations To define Lagrangian mechanics we need a configuration space, which is a manifold, and a Lagrangian, which is a function L(q, v) on its tangent bundle. (We are assuming that Lagrangian is first-order and not explicitly time-dependent.) A path is a map qi(t) from time (parametrized by the real line or a segment thereof) to the configuration space, and the action is a functional of the path: Z S[q(t)] = dt L(q(t), q˙(t)). (3.1)

– 27 – Extremizing this action with respect to the path gives the Euler-Lagrange equation, which is the equation of motion: δS d ∂L ∂L 0 = = − + . (3.2) δqi dt ∂q˙i ∂qi In Hamiltonian mechanics we start with a phase space, which is a symplectic manifold, and a Hamiltonian, which is a function H(x) on it. The equation of motion is Hamilton’s equation, a ab x˙ = ω ∂bH, (3.3)

ab where ω is the inverse of the symplectic form ωab. It follows that the time dependence of any function on phase space is given by

a f˙ = ∂af x˙ = {f, H}, (3.4) where the Poisson bracket {f, g} is defined by

ab {f, g} = ∂afω ∂bg. (3.5)

Given a Lagrangian system, it is straightforward to map it to a Hamiltonian system with the same dynamics. The phase space is taken to be the cotangent bundle of the configuration i space, parametrized by coordinates q and momenta pi, with symplectic structure defined by the Poisson brackets i j i i {q , q } = {pi, pj} = 0, {q , pj} = δj. (3.6) The Hamiltonian is taken to be the Legendre transform of the Lagrangian with respect to the tangent vector vi, at fixed coordinate qi:

i H(q, p) = piv − L(q, v) ∂L . (3.7) =pi ∂vi

A history qi(t) is mapped into phase space using the map from the tangent space to the cotangent space implicit in the Legendre transform:

∂L pi(t) = i . (3.8) ∂v vi=q ˙i(t)

The inverse of this map is ∂H q˙i = , (3.9) ∂pi which is one of Hamilton’s equations. The other one is the translation of the Euler-Lagrange equation (3.2) into the Hamiltonian language, using (1.13) and (3.8):

∂H p˙ = − . (3.10) i ∂qi

– 28 – 3.2 Symmetries Symmetries are treated somewhat differently in the Lagrangian and Hamiltonian formula- tions. However, just as a Lagrangian system can be mapped into a Hamiltonian one, any symmetry of the former can be mapped into a symmetry of the latter. For the rest of this subsection, we will deal only with continuous symmetries, but first we would like to make a comment that applies to discrete or continuous symmetries. We will need the following lemma: Suppose that a (differentiable) function S on some space is invariant under the action of a group G, and let I be the G-invariant subset of P . At any point x ∈ I, the gradient of S must be G-invariant. Therefore if x extremizes S within I then it extremizes it within the full space. In Lagrangian mechanics, a symmetry is a group G acting on the space of paths which leaves the action S invariant. (We will make a further restriction below for continuous sym- metries.) As a consequence of the above lemma, for G-invariant paths, it suffices to check that they extremize the action within the space of G-invariant paths. In other words, one can work consistently within the space of G-invariant paths, and any solution to the equations of motion derived there will solve the equations of motion of the full system. For example, for a static path of a time translation-invariant system, one just needs to check that the path extremizes the action within the space of static paths, in other words that the potential energy is extremized. We now specialize to the case of continuous symmetries. Again, a symmetry of a La- grangian system is a transformation on the space of paths under which the action is invariant. To be more specific, we require that the change in the Lagrangian at each point on the path be expressible as the time derivative of a locally defined quantity. It follows that paths obeying the equations of motion are mapped onto other such paths, i.e. that the equations of motion are invariant. We thus have ∂L ∂L δL = δqi + δq˙i = K,˙ (3.11) ∂qi ∂q˙i where δq is the generator of the symmetry. When the equations of motion are satisfied, the following quantity (“charge”) is conserved: ∂L Q = δqi − K. (3.12) ∂q˙i This is Noether’s theorem. If (as is usually the case) δq and K are functions only of q andq ˙, then obviously the charge Q is also. In the Hamiltonian formulation, symmetries are special cases of canonical transforma- tions, which are tranformations on the phase space which leave the symplectic form invariant. The generator δxa of a canonical transformation must obey

a a a a a 0 = £δxa ωbc = δx ∂aωbc + ωac∂bδx + ωba∂cδx = ∂b(ωacδx ) − ∂c(ωabδx ), (3.13)

a where in the last equality we used the fact that the symplectic form is closed. Since ωabδx is closed, it must be locally exact, in other words there must be a function f (possibly multiple-

– 29 – valued if the phase space is multiply connected) such that

a ab δx = ω ∂bf. (3.14)

The function f is also sometimes called the generator of the canonical transformation δxa.A symmetry is a canonical tranformation which preserves the Hamiltonian (and therefore the equations of motion), as well as the symplectic form:

a δx ∂aH = 0. (3.15)

In other words, its generator commutes with the Hamiltonian,

{f, H} = 0, (3.16) and therefore by (3.4) is conserved. Since the Hamiltonian obviously commutes with itself, by Hamilton’s equation time evolution acts by a symmetry. Since f is conserved, it is consistent to restrict the dynamics to a subspace of the phase space with f = f0 for some constant f0. However, the symplectic form restricted to that subspace will be degenerate (as it must be, since the subspace is odd-dimensional): for any a a b vector v lying in the subspace, v ωabδx = 0. Hence, to obtain a sensible restricted phase space, it is necessary to quotient the surface f = f0 by the action of the symmetry, i.e. a by the vector field δx . Thus, one dimension is lost by the constraint f = f0 and another by the quotient; the resulting restricted phase space has a well-defined symplectic form and a Hamiltonian (since δx ∂aH = 0). An application of this construction is to the global part of a gauge symmetry: as we will show below, the Noether charge associated to this symmetry is constrained to vanish, and the phase space should be quotiented by it to obtain the physical phase space. As noted above, given a symmetry of a Lagrangian system such that δq and K depend only on q andq ˙, the same goes for corresponding conserved charge Q, which can therefore equally well be considered a function on the phase space of the corresponding Hamiltonian system. Furthermore, since it’s conserved, it must commute with the Hamiltonian, and therefore must generate a symmetry in the Hamiltonian sense. In fact, it’s straightforward to show that the transformation on the phase space that it generates is the original symmetry i i transformation, i.e. that ∂Q/∂pi = δq and ∂Q/∂q = −δpi (where δp is the image of δq i and δq˙ under the usual map pi = ∂L/∂q˙ ). To prove the first identity, we use the fact that δL − K˙ should vanish for a general pathq ˙ (not necessarily one satisfying the equations of i motion). Using pi = ∂L/∂q˙ , such a path can be mapped into a path (q(t), p(t)) in phase i space, which will satisfy one of Hamilton’s equations,q ˙ = ∂H/∂pi, but not necessarily the i other,p ˙i = −∂H/∂q . We have ∂L ∂L δL − K˙ = δqj + δq˙j − K˙ (3.17) ∂qj ∂q˙j  j   j  ∂H j ∂δq ∂K i ∂δq ∂K = − j δq + pj i − i q˙ + pj − p˙i. (3.18) ∂q ∂q ∂q ∂pi ∂pi

– 30 – Since this must vanish for arbitraryp ˙ (independent of q and p), we find the relation

∂δqj ∂K pj − = 0. (3.19) ∂pi ∂pi Using this relation, the equation ∂Q = δqi (3.20) ∂pi i follows immediately from the definition of Q. To show that ∂Q/∂q = −δpi, we will need to use the equation of motion. This is because δp depends on δq˙, which (if δq depends on q˙) depends onq ¨. So the equation of motion is necessary to write δp as a function of q and p, which ∂Q/∂pi is. We won’t calculate δp directly, but will instead use a trick, which is to notice that δq˙ can be written as a function of q and p in two different ways. We have:

δq˙i = δ(q ˙i) ∂H  = δ ∂pi 2 2 ∂ H j ∂ H = j δq + δpj. (3.21) ∂pi∂q ∂pi∂pj On the other hand we have d δq˙i = δqi dt ∂Q  = ,H ∂pi  ∂H  = − Q, ∂pi ∂Q ∂2H ∂Q ∂2H = − j + j . (3.22) ∂q ∂pi∂pj ∂pj ∂pi∂q The first term of (3.21) and the second term of (3.22) are equal in view of the identity 2 i (3.20). Furthermore, the matrix ∂ H/∂pi∂pj = ∂q˙ /∂pj is invertible in view of the assumed i invertibility of the map pi = ∂L/∂q˙ . So we have, as promised, ∂Q = −δp . (3.23) ∂qi i 3.3 Noether’s theorem for higher-order actions We will work in field theory notation. The treatment of Noether’s theorem in most textbooks assumes a first-order Lagrangian. We will be more general, but will still assume that the action is the integral of a local Lagrangian L. We define a (continuous) symmetry to be an infinitesimal variation δφi of the fields φi, which is a local function of the fields and their derivatives, under which the change in the Lagrangian is a total divergence,

a δL = ∂aK , (3.24)

– 31 – where Ka is also a local function of the fields and their derivatives. (Here  is a constant infinitesimal parameter.) This is slightly weaker than demanding that the action be invariant, but stronger than demanding that the equations of motion be unchanged. Now consider letting  be an arbitrary function of spacetime. Then we can expand the change in the Lagrangian in derivatives of :

a a ab δL = ∂aK + ∂aJ(1) + ∂a∂bJ(2) + ··· (3.25) a = −∂aj + total divergence, (3.26) where a a a ab j = −K + J(1) − ∂bJ(2) + ··· . (3.27) When the fields satisfy the equations of motion, the change in the action must vanish for any  that vanishes (along with enough of its derivatives) on the boundary. The equations of motion must therefore imply a ∂aj = 0. (3.28) It is straightforward to derive explicit expressions for all the terms in the current (except Ka), order by order in derivatives of the fields. For example, up to second derivatives of the fields we have   a ab ∂L ∂L ∂L J(1) − ∂bJ(2) = − ∂b + ··· δφ + ∂bδφ + ··· (3.29) ∂(∂aφ) ∂(∂a∂bφ) ∂(∂a∂bφ) (sum over fields implied).

3.4 Gauge symmetries A symmetry is called a gauge symmetry if its action on the system depends on a parameter which is a function of spacetime. This implies that histories (or paths) that are related by the symmetry are experimentally indistinguishable and should therefore be identified in the true space of histories. To see this, imagine that we are studying the system with an experimental apparatus which we imagine to be separated from it in space or time. If we act on the system but not the apparatus with a global symmetry, this is not a symmetry of the full system- plus-apparatus ¨uber-system; therefore the results reported by the apparatus may be different. On the other hand, acting only on the system with a gauge symmetry is a symmetry of the ¨uber-system, and the results of the apparatus will be unchanged. Another way to see that histories related by a gauge symmetry must be identified is that to retain them as physically distinct would entail a massive loss of determinism. Given initial data specified at some initial time, and a history with those initial conditions satisfying the equations of motion, any other history related by a gauge transformation whose parameter vanishes at the initial time will have the same initial conditions and will also satisfy the equations of motion. This plethora of solutions for the same initial conditions is physically sensible only if we declare all of them to be physically the same. Thus we see that a gauge symmetry necessarily reveals a redundancy in our description of the physics.

– 32 – A gauge symmetry necessarily includes a global part, in which the gauge parameter is constant. The global part will have an associated conserved Noether charge. Let us show that this charge in fact vanishes on the equations of motion. We work in the notation of a mechanical (0 + 1 dimensional) system, with positions qi and Lagrangian L(q, q˙) (we assume a first-order action). The gauge transformation will be parametrized by a function (t). We assume that the corresponding change in qi is local in time, but can depend on both  and its time derivative: i i i δq = δ0q +δ ˙ 1q . (3.30) Hence we have i i i i i δq˙ = δ0q˙ +δ ˙ 0q +δ ˙ 1q˙ +δ ¨ 1q . (3.31) We now assume that under this transformation the Lagrangian changes by a total derivative: ∂L ∂L δL = δqi + δq˙i = K˙ ; (3.32) ∂qi ∂q˙i this guarantees that the equations of motion are unchanged. K will itself depend on  and ˙:

K = K0 +K ˙ 1. (3.33)

Equating coefficients of , ˙, and ¨ in 3.32 gives us ∂L ∂L δ qi + δ q˙i = K˙ (3.34) ∂qi 0 ∂q˙i 0 0 ∂L ∂L ∂L δ qi + δ qi + δ q˙i = K + K˙ (3.35) ∂qi 1 ∂q˙i 0 ∂q˙i 1 0 1 ∂L δ qi = K . (3.36) ∂q˙i 1 1 The first of these is Noether’s theorem: defining the charge ∂L Q = δ qi − K , (3.37) ∂q˙i 0 0 it asserts dQ =∼ 0, (3.38) dt where =∼ means “equal on the equations of motion”. The last, 3.36, constrains the canonical momenta (without using the equations of motion); it asserts that the phase space is smaller than one would naively expect from the configuration space. The middle equation, 3.35, is equal to the time derivative of the last one plus

Q =∼ 0. (3.39)

Since Q can be calculated from the initial data, this equation asserts that the equations of motion not only dictate the time evolution but also constrain the initial data, making the physical phase space yet smaller.

– 33 – Let us now generalize this story to multiple dimensions, with fields φ and Lagrangian a density L(φ, ∂aφ). Naively one might guess that now the Noether current j will vanish. However, we will see that the story is more complicated.

The field variations δ0φ and δ1φ are a scalar and a vector respectively:

a δφ =  δ0φ + ∂a δ1φ . (3.40)

As before, we will assume that the change in the Lagrangian density is a total divergence:

a a a ab δL = ∂aK ,K = K0 + ∂bK1 . (3.41)

The Noether current associated to the global part of the symmetry is

a ∂L a j = δ0φ − K0 . (3.42) ∂(∂aφ) It is also useful to define the following tensor:

ab ∂L a ba F = δ1φ − K1 . (3.43) ∂(∂bφ) The generalizations of equations 3.34, 3.35, 3.36 above are

a ∼ ∂aj = 0 (3.44) a ab ∼ j + ∂bF = 0 (3.45) F (ab) = 0. (3.46)

The key difference is the last one, which is weaker than the naive generalization of 3.36, ab F = 0. This occurs because it is the coefficient in δL of ∂a∂b, which is necessarily a symmetric tensor, so that only the symmetric part of F ab is constrained.

3.5 Hamilton-Jacobi theory Hamilton’s equations of motion are first-order equations in phase space:

i ∂H ∂H q˙ = , p˙i = − i . (3.47) ∂pi ∂q The Hamilton-Jacobi formalism is a trick for writing them as first-order equations on configu- ration space (the space of positions qi) alone. Let S(q, t) be a function satisfying the following differential equation: ∂S = −H| ∂S . (3.48) pi= ∂t ∂qi Then if we let the time evolution of the positions be governed by the first of Hamilton’s equations 3.47, and at every time set the momenta equal to the gradient of S with respect to the positions, ∂S p = , (3.49) i ∂qi

– 34 – then they will automatically satisfy the second of Hamilton’s equations. An example of a function S that satisfies 3.48 (as well as the motivation for the name S) is provided by the action evaluated on trajectories that satisfy the equations of motion. Fix i i i an initial time t0 and position q0 = q (t0). Any q1 at any final time t1 then uniquely deter- i mines a classical trajectory qc(t), pc,i(t). The action evaluated along this classical trajectory, i considered as a function of q1 and t1,

Z t1 i
Sq0,t0 (q1, t1) = dt pc,iq˙c − H(qc, pc, t) , (3.50) t0 can be shown to satisfy

∂Sq0,t0 i = pc,i(t1), (3.51) ∂q1 as well as 3.48. i In a saddle-point approximation to the path integral, the wave function at position q1 at i time t1 of a particle that started at position q0 at time t0, is exp(iSq0,t0 (q1, t1)/~) (times a factor depending on the second functional derivative of the action about the classical path). More generally, 3.48 is the ~ → 0 limit of the Schr¨odingerequation for the wave function ψ(q, t) = eiS(q,t)/~. Consider now a situation where the Hamiltonian has no explicit time dependence. Then there exists a special class of solutions to 3.48 of the following form:

S(q, t) = S0(q) − Et, (3.52) where E is a fixed constant and S0 satisfies the following equation:

H| ∂S0 = E. (3.53) pi= ∂qi

S0 is called Hamilton’s principal function. For such an S, all trajectories have the same energy E. This is the classical equivalent of an energy eigenstate. Still assuming time-translation symmetry, we can also note an interesting property of the on-shell action Sq0,t0 (q1, t1). Fixing q0, q1, we write it S(∆t) (where ∆t := t1 − t0); from 3.48, we then have dS(∆t) = −E. (3.54) d∆t The Legendre transform of S(∆t) is the area under the phase-space curve (q(t), p(t)) as a function of E: Z t1 Z q1 i E∆t + S(∆t) = dt(E + L) = dq pi =: A(E) (3.55) t0 q0 It follows from properties of the Legendre transform that dA(E) = ∆t , S(∆t) = A(E) − E∆t . (3.56) dE The first equation can easily be checked for the simple case H = p2/(2m) + V (q).

– 35 – 4. Gauge theories

4.1 Generalities

Consider a gauge group G. The connection (or gauge field) Aµ(x) is a Lie-algebra valued one-form that transforms under gauge transformations U(x) ∈ G like

−1 −1 Aµ → UAµU + i(∂µU)U . (4.1)

The field strength Fµν is a Lie-algebra valued 2-form:

Fµν = ∂µAν − ∂νAµ + i[Aµ,Aν]. (4.2)

It transforms homogeneously under gauge transformations:

−1 Fµν → UFµνU . (4.3)

When G is non-abelian, the Lagrangian for Aµ is conventionally

1 L = − tr F F µν, (4.4) YM 2g2 µν where the trace is taken in the fundamental representation of G. When it is U(1), the Lagrangian is conventionally 1 L = − F F µν. (4.5) em 4g2 µν

If G is not simple, its different simple parts can have different couplings g. For a matter field φ transforming in the representation r of G,

φ → Urφ, (4.6) the covariant derivative r r Dµ = ∂µ + iAµ (4.7) transforms homogeneously: r r Dµφ → UrDµφ. (4.8)

The field strength in this representation is simply

r r r Fµν = −i[Dµ,Dν]. (4.9)

– 36 – Another convention is obtained by the replacements Aµ → gAµ, Fµν → gFµν in the above formulas:

i A → UA U −1 + (∂ U)U −1 (4.10) µ µ g µ

Fµν = ∂µAν − ∂νAµ + ig[Aµ,Aν] (4.11) −1 Fµν → UFµνU (4.12) 1 L = − tr F F µν (4.13) YM 2 µν 1 L = − F F µν (4.14) em 4 µν φ → Urφ (4.15) r r Dµ = ∂µ + igAµ (4.16) r r Dµφ → UrDµφ (4.17) i F r = − [Dr ,Dr ]. (4.18) µν g µ ν

4.2 Standard Model

Here is the field content before symmetry breaking (i = 1, 2, 3 labels generation):

field Lorentz SU(3)c SU(2)L U(1)Y 1 1 g ( 2 , 2 ) 8 1 0 1 1 W ( 2 , 2 ) 1 3 0 1 1 B ( 2 , 2 ) 1 1 0 q ( 1 , 0) 3 2 1 i 2 6 (4.19) c 1 ¯ 2 ui ( 2 , 0) 3 1 − 3 c 1 ¯ 1 di ( 2 , 0) 3 1 3 1 1 li ( 2 , 0) 1 2 − 2 c 1 ei ( 2 , 0) 1 1 1 1 H (0, 0) 1 2 − 2

The gauge bosons are g for SU(3)c with coupling g3, W for SU(2)L with coupling g2, and B for U(1)Y with coupling g1. In the following, we write gµ and Wµ as matrices in the a a a 1 fundamental representations of SU(3) and SU(2) respectively: gµ = gµT (where T are 2 a b 1 ab i i i 1 the 8 Gell-Mann matrices, satisfying tr T T = 2 δ ), Wµ = Wµt (t = 2 σi). We also use

– 37 – the second normalization convention of the previous subsection. The Lagrangian is:

LSM = Lgauge + Lfermions + LHiggs + LYukawa (4.20) 1 1 1 L = − tr g gµν − tr W W µν − B Bµν (4.21) gauge 2 µν 2 µν 4 µν † µ c† µ c c† µ c † µ c† µ c Lfermions = qi iσ¯ Dµqi + ui iσ¯ Dµui + di iσ¯ Dµdi + li iσ¯ Dµli + ei iσ¯ Dµei (4.22) λ v2 L = −(D H)†DµH − (H†H − ) (4.23) Higgs µ 2 2 † u c AB d c AB e c LYukawa = −H qiλijuj − HA qBiλijdj − HA lBiλijej + h.c., (4.24)

µ whereσ ¯ = (1, −~σ), † conjugates spinors and transposes SU(2)L and SU(3)c indices, and 12 A, B = 1, 2 are indices of the fundamental representation of SU(2)L with  = 1. † 2 In the vacuum H H = v /2, and by making an appropriate choice of SU(2)L × U(1)Y gauge we can always put H in the form " # 1 v + h(x) H = √ (4.25) 2 0 where h(x) is a canonically normalized real scalar. This is called unitarity gauge. The unbroken SU(2)L × U(1)Y generator is

Q = t3 + Y, (4.26) which is therefore the generator of the surviving U(1)em gauge group. In unitarity gauge the Higgs Lagrangian is (keeping only quadratic terms)

1 v2 λv2 L = − ∂ h∂µh − (g W 3 − g B )2 + (g W 1)2 + (g W 2)2
− h2 (4.27) Higgs 2 µ 8 2 µ 1 µ 2 µ 2 µ 2 1 1 1 = − ∂ h∂µh − m2 h2 − m2 Z Zµ − m2 W +W −µ, (4.28) 2 µ 2 h 2 Z µ W µ where the mass of the physical Higgs boson is √ mh = λv, (4.29) and we have defined the fields 1 Z = (g W 3 − g B ), (4.30) µ p 2 2 2 µ 1 µ g1 + g2 1 W ± = √ (W 1 ± iW 2), (4.31) µ 2 µ µ with masses 1q 1 m = g2 + g2 v, m = g v. (4.32) Z 2 1 2 W 2 2

– 38 – 3 The linear combination of Wµ and Bµ that is orthogonal to Zµ, 1 A = (g W 3 + g B ), (4.33) µ p 2 2 1 µ 2 µ g1 + g2 remains massless, and this is the gauge field for the unbroken U(1)em, as we can see explicitly by expanding the covariant derivatives for a field in a general SU(2)L and U(1)Y representa- i tion (here t are the representation matrices for SU(2)L in a general representation):

i i Dµ = ∂µ + ig2Wµt + ig1YBµ (4.34) g i = ∂ + i√2 (W +t+ + W −t−) + ieQA + (g2t3 − g Y )Z , (4.35) µ µ µ µ p 2 2 2 1 µ 2 g1 + g2 where t± = t1 ± it2, g g e = 1 2 (4.36) p 2 2 g1 + g2 3 is the coupling of the U(1)em gauge field, and Q = t + Y as above. This goes for the vector ± fields Wµ as well, so they have electric charges Q = ±1. Defining the mixing angle θW by g g cos θ = 2 , sin θ = 1 , (4.37) W p 2 2 W p 2 2 g1 + g2 g1 + g2 we have

3 3 Zµ = cos θW Wµ − sin θW Bµ,Aµ = sin θW Wµ + cos θW Bµ (4.38) 3 Wµ = cos θW Zµ + sin θW Aµ,Bµ = − sin θW Zµ + cos θW Aµ, (4.39) e e mW = cos θW mZ , g1 = g2 = . (4.40) cosθW sin θW Inserting the Higgs vev into the Yukawa terms in the Lagrangian gives some of the fermions (Dirac) masses. Expand the SU(2)L doublet fermions out into components: " # " # ui νi qi = , li = . (4.41) di ei

The Yukawa Lagrangian becomes (again, keeping only quadratic terms):

v u c v d c v e c LYukawa = −√ uiλ u − √ diλ d − √ eiλ e + h.c. (4.42) 2 ij j 2 ij j 2 ij j

Note that the neutrinos remain massless. By global U(3) rotations on the fields ei and c e u ei , it is possible to make the matrix λij diagonal, real, and positive. Similarly for λij and d λij. However, the necessary U(3) rotations on ui and di will be different. Since these two fields are components of the same SU(2)L doublet, making different rotations on them will complicate their couplings to the W bosons, introducing the CKM matrix there and leading to generation-changing weak-interaction processes.

– 39 – After symmetry breaking, the fields and their quantum numbers are as follows:

field Lorentz massive SU(3)c Q g real vector no 8 0 A real vector no 1 0 W complex vector yes 1 1 Z real vector yes 1 0 2 (4.43) ui Dirac yes 3 3 1 di Dirac yes 3 − 3 ei Dirac yes 1 −1 νi Weyl no 1 0 h real scalar yes 1 0

5. General relativity

5.1 Dimensions

We set c = 1, but not GN or ~. We are thus left with two units, mass (M), and length (L), in terms of which

[ds2] = L2 (5.1) [dDxp|g|] = LD (5.2) [S] = [~] = LM (5.3) 2 D−3 −1 [GN] = [κ ] = L M (5.4) [R] = [Λ] = L−2 (5.5) a 1−D [Lm] = [ρv] = [Ta ] = L M (5.6)

5.2 Fundamentals The action for gravity coupled to matter is Z 1 D p S = d x |g| (R − 2Λ) + Sm, (5.7) 16πGN which gives rise to the Einstein equation 1 R − Rg + Λg = 8πG T , (5.8) ab 2 ab ab N ab where ab 2 δSm T = p . (5.9) |g| δgab

If Z D p Sm = d x |g|Lm, (5.10)

– 40 – then the stress tensor is given in terms of Lm by

ab δLm ab T = 2 + Lmg . (5.11) δgab Having a cosmological constant is equivalent to having a vacuum energy Λ ρv = , (5.12) 8πGN which means having a term in Lm equal to −ρv, and thus a term in Tab equal to −ρvgab. Sometimes in place of GN one uses 2 κ = 8πGN. (5.13)

One can also define the Planck mass mP, D−3 D−2 ~ mP = . (5.14) GN

In the weak-field limit, gab = ηab + hab, |hab|  1, massive particles moving non- relativistically feel a Newtonian gravitational potential equal to −h00/2. Two masses M1 and M2 separated by a distance r feel a gravitational attraction of magnitude

(D − 3)8πGNM1M2 F = D−3 . (5.15) (D − 2)ΩD−2r In our world, 2 −27 −1 8πGN = κ = 1.9 × 10 cm g , (5.16) and −5 19 mP = 2.2 × 10 g = 1.3 × 10 GeV. (5.17) A top-form field strength makes a positive contribution to the cosmological constant.

5.3 Spherically symmetric solutions We work in D ≥ 3 and look for SO(D − 1) invariant vacuum solutions. Birkhoff’s theorem states that any such solution is static. Given D and Λ, there is a one-parameter family of such metrics: 2Λr2 µ ds2 = −f(r)dt2 + f(r)−1dr2 + r2dΩ2 , f(r) = − + 1 − . (5.18) D−2 (D − 2)(D − 1) rD−3 The coordinate r ranges over the region where r ≥ 0 and f(r) ≥ 0. Requiring that such a region exist, and that the solution be free of naked singularities, leads to a complicated classification (and nomenclature). µ = 0 is allowed for any D and for any value of Λ; these are called de Sitter, Minkowski, and anti-de Sitter space for Λ positive, zero, and negative respectively. The curvature radius

R(A)dS of (anti-)de Sitter space is related to Λ by (D − 1)(D − 2) Λ = ± 2 . (5.19) 2R(A)dS

– 41 – For D = 3 the only other non-nakedly singular solutions have Λ < 0 and µ > 1; they are called BTZ. For D ≥ 4 and Λ > 0, we can have any value of µ in the range

2 (D − 3)(D − 2)(D−3)/2 0 < µ < ; (5.20) D − 1 2Λ these are called dS-Schwarzschild. (In the limit that µ goes to its upper bound, the solution D−2 becomes dS2 × S .) For D ≥ 4 and Λ = 0 we can have any µ > 0; these are called Schwarzschild (for D > 4 also sometimes called Schwarzschild-Tangherlini). Similarly, for D ≥ 4 and Λ < 0 we can have any µ > 0, and these are called AdS-Schwarzschild. In the solutions with Λ ≤ 0 the coordinate r is unbounded above, so they are asymptot- ically flat (Λ = 0) or AdS (Λ < 0). The ADM masses of these solutions (relative to AdS or Minkowski space) are (D − 2)Ω µ M = D−2 . (5.21) 16πGN 5.4 Cauchy problem Let’s first review the situation for electromagnetism (in flat spacetime, without sources). The equation of motion for the gauge field Aa is

2 b ∂ Aa − ∂a∂ Ab = 0, (5.22) subject to the gauge freedom

Aa → Aa + ∂aλ. (5.23) In order to obtain a deterministic time evolution, we will obviously need to fix this gauge freedom. Also, we must choose a time slicing of spacetime, breaking the manifest relativistic invariance of the theory. Therefore a natural choice of gauge is the Coulomb gauge:

A0 = 0. (5.24)

(Note that to impose this gauge condition it is not necessary to be on-shell.) The remaining degrees of freedom are the spacelike components Ai. These are still subject to the residual gauge freedom

Ai → Ai + ∂iλ, (5.25) where now the gauge parameter λ is required to be time-independent, i.e. this gauge freedom operates only on the initial value surface, not throughout the spacetime. We denote the time derivatives of the Ai, the momenta, by Ei:

Ei = A˙ i. (5.26)

There are D −1 “positions” (values of Ai) to specify at each point on the initial value surface, but due to the residual gauge freedom 5.25, only D − 2 are physically meaningful. Meanwhile we appear still to have D−1 momenta Ei, one too many. Worse, we seem to have D equations

– 42 – of motion 5.22, two too many. On closer inspection, however, one of the equations of motion, the time component of 5.22, turns out not to contain second time derivatives of Ai:

∂iEi = 0. (5.27)

It is a constraint equation (the Gauss law), reducing the number of independent momenta in the initial conditions to D − 2. The dynamical equations of motion are the D − 1 spacelike components of 5.22,

−E˙ i + ∂j(∂jAi − ∂iAj) = 0. (5.28)

In fact, the divergence of this equation merely serves to conserve the constraint 5.27, so there are actually only D − 2 dynamical equations of motion, which is the correct number. The Cauchy problem for general relativity is treated in painful detail in section 10.2 of Wald. Here we give a shortcut to extract some intuition and the most important equations. It will provide a local (in time) formulation of the Cauchy problem. As in the case of electromagnetism, in order to have a deterministic time evolution we need to eliminate the gauge freedom, in this case local reparametrization invariance. An ideal gauge for treating the Cauchy problem is provided by Gaussian normal coordinates, discussed in subsection 2.13 above. Given a spacelike hypersurface S of a pseudo-Riemannian manifold, and coordinates xi on S, there is a unique set of coordinates {x0, xi} (on some neighborhood of S) such that the x0 = 0 surface is S, and such that the metric takes the form

2 0 2 i j ds = −(dx ) + γijdx dx . (5.29)

(Note that in making this statement we are not using the equations of motion—this gauge choice is possible even off-shell.) Requiring the metric to be of the form 5.29 fixes all of the local diffeomorphism freedom, except the possibility to choose a different surface S or to reparametrize it (in total, D functions on S, compared to the usual D functions on the full spacetime for general diffeomorphisms.) Of course, for generic choices of S the region covered by the normal coordinates will not be particularly large, meaning that the Cauchy evolution dictated by the equations derived below will typically develop a coordinate singularity well before any real singularity or Cauchy horizon forms in the spacetime—this is why it’s local in time. We’ll treat the vacuum Einstein equation, but it’s trivial to include a stress tensor or cosmological constant. By our gauge choice, we have gotten rid of all the superfluous com- ponents of the metric; the dynamical variables are the components γij, which describe the intrinsic geometry of each x0 = constant surface. Their time derivatives define the extrinsic curvature of the surface: 1 K = ∂ γ . (5.30) ij 2 0 ij

However, we are not free to choose all of the components of γij and Kij independently on the initial value surface S. For each gauge condition we have imposed, one of the equations

– 43 – of motion will turn into a constraint equation. These are the 00 and i0 components of the

Einstein equation, which do not contain second time derivatives of γij:

0 2 ij 2G00 = R + K − KijK = 0 (5.31) 0 j Gi0 = ∇jK i − ∂iK = 0 (5.32)

0 0 (where R and ∇ are derived from γij). The initial value data thus consists of a total of D(D − 2) independent functions (although, given the residual gauge freedom, only D(D − 3) of these are meaningful in the sense of giving rise to different spacetimes). The true dynamical equation of motion is Gij = 0. This is a bit complicated, but is equivalent to the simpler Rij equation if one assumes that the above constraints are satisfied:

0 k Rij = ∂0Kij + Rij + KKij − 2KikK j = 0. (5.33)

Together with 5.30, this equation determines the evolution of Kij and γij. Although I haven’t checked it myself, it should be the case that if the constraints 5.31 are satisfied in the initial conditions, then they will automatically continue to be satisfied.

5.5 Gravitational waves Let us first recall the situation for electromagnetic waves (in flat space). We wish to solve the equation of motion 2 b ∂ Aa − ∂a∂ Ab = 0 (5.34) modulo the gauge freedom

Aa → Aa + ∂aλ. (5.35) It is easy to show using the equation of motion that any wave with spacelike or timelike momentum must have a polarization parallel to its momentum and must therefore be pure gauge. Therefore, on-shell one may impose the gauge condition

2 ∂ Aa = 0, (5.36) which is equivalent (on-shell) to a ∂ Aa = 0 (5.37)

(assuming Aa is nicely behaved at infinity). This condition, which partially fixes the gauge, is known as Lorenz gauge. (Off-shell, neither 5.36 nor 5.37 can be imposed.) Since 5.36 and 5.37 together imply the equation of motion, one may take them together as equation of motion-cum-gauge condition.

Working in Lorenz gauge, we can fix a null momentum ka and choose light-cone coordi- nates x±, xm such that

ds2 = −2dx+dx− + dxmdxm (5.38) − m k = k+ = 0, k = km = 0. (5.39)

– 44 – In these coordinates the component A− is pure gauge, and the equation of motion is

A+ = 0. (5.40)

The on-shell physical degrees of freedom are the components Am, which form a vector of SO(D − 2). We can fix the remaining gauge freedom by setting A = 0. If we have chosen light-cone √ − coordinates such that (x+ + x−)/ 2 coincides with the original time coordinate x0, then this condition is equivalent to A0 = 0 (Coulomb gauge). Now let us turn to gravitational waves in flat space. The equation of motion for a metric

fluctuation hab is 1 1 δR = ∂ ∂ ha − ∂ ∂ h − ∂2h = 0. (5.41) cd a (c d) 2 c d 2 cd This must be solved modulo the gauge freedom

hab → hab + ∂(aξb). (5.42)

The analysis is similar to the case of electromagnetism. Again, waves with spacelike or timelike momentum are easily seen using the equation of motion to be pure gauge. Thus on-shell we may partially fix the gauge by imposing the condition

2 ∂ hab = 0, (5.43) or equivalently 1 ∂ah − ∂ h = 0. (5.44) ab 2 b (Again, neither condition may be imposed off-shell, and again, together they imply the equa- tion of motion and may therefore be taken together as a combined equation of motion and gauge fixing condition.) This is the analog of Lorenz gauge, and it is sometimes called de

Donder gauge. In the coordinate system 5.38, the components h−a, ha− are pure gauge, and the equation of motion is

h++ = hm+ = h+m = hmm = 0. (5.45)

So the on-shell physical degrees of freedom are hmn, with hmm = 0, i.e. a symmetric traceless tensor of SO(D − 2).

We can fix the gauge a little more, but still not completely, by also setting h+− = h−+ = 0. In position space this amounts to setting

a 2 h = 0, ∂ hab = 0, ∂ hab = 0. (5.46)

This is transverse traceless gauge. (Among the four equations 5.41, 5.46, any three imply the fourth with suitably nice behavior at infinity.)

Finally, we can completely fix the gauge by also setting h−a = ha− = 0, or (in the original coordinate system, in position space) h0a = ha0 = 0.

– 45 – The transverse traceless gauge can be generalized to curved backgrounds as follows (see section 7.5 of Wald):

a 2 c d h = 0, ∇ hab = 0, ∇ hab + 2Ra b hcd = 0. (5.47)

The wave operator acting on hab in the last equation of 5.47 is known as the Lichnerowicz operator. 1 R p The quadratic approximation to the Einstein-Hilbert action S = 2κ2 |g|R for a trans- verse traceless fluctuation hab about a solution, is

1 Z 1  S = dDxp|g| hab ∇2h + R c dh + O(h3) 4κ2 2 ab a b cd (5.48) 1 Z 1  = − dDxp|g| ∇chab ∇ h − ∇ h + O(h3). 4κ2 2 c ab a bc

5.6 Israel junction condition

Suppose that the metric is continuous but has components with discontinuous first derivatives across a timelike hypersurface S. Work in Gaussian normal coordinates (see subsection 2.13), so the metric is 2 i j 2 ds = γij(y, λ)dy dy + dλ , (5.49) with S at λ = 0. The extrinsic curvature of the constant-λ hypersurfaces is

1 ∂γ K = ij . (5.50) ij 2 ∂λ

We assume that γij is continuous but has a discontinuous first derivative at λ = 0, so Kij is double-valued there: ± ± Kij := Kij(λ = 0 ) . (5.51)

− Note that Kij is the extrinsic curvature with respect to the normal pointing into S on the + λ < 0 side, while Kij is the extrinsic curvature with respect to the normal pointing out of S on the λ > 0 side. Let ∆Kij be their difference:

+ − ∆Kij := Kij − Kij . (5.52)

From (2.176)–(2.179), we see that the Einstein tensor has a delta-function with components tangent to S:

Gij = (−∆Kij + ∆Kγij) δ(λ) + finite (5.53)

Giλ = finite (5.54)

Gλλ = finite . (5.55)

– 46 – For the spacetime to be a solution to the Einstein equation requires the stress tensor to include a piece localized on S with components tangents to it:

Tij = tijδ(λ) + finite (5.56)

Tiλ = finite (5.57)

Tλλ = finite , (5.58) where 1 tij = (−∆Kij + ∆Kγij) . (5.59) 8πGN R 1/2 Such a localized stress tensor would result from an action localized on S: SS = S |γ| LS, where Lm depends on the metric only through the induced metric on S. The finite components of the stress tensor need not be continuous across S. The continuity ab + − equation ∇aT = 0 relates the discontinuity ∆Tab := Tab(λ = 0 ) − Tab(λ = 0 ) to tij:

0 i ij ∇it j = −∆Tjλ ,Kijt = ∆Tλλ , (5.60)

0 ave 1 + − where ∇ is the covariant derivative with respect to γij, and Kij := 2 (Kij + Kij ). (This can be derived by thickening S into a thin slab.) The first equation in (5.60) allows S to exchange energy and momentum with the surroundings, while the second relates a pressure difference on the two sides of S to a bending or acceleration of a brane. If S is spacelike, then all the above equations still hold except (5.59), which is replaced by 1 tij = − (−∆Kij + ∆Kγij) , (5.61) 8πGN and the second equation in (5.60), which becomes

ij Kijt = ∆Tλλ . (5.62)

However, such a stress tensor can never obey the null energy condition.

5.7 Boundary term Let us consider the Einstein-Hilbert action (for pure gravity): 1 Z S = dDxp|g|(R − 2Λ). (5.63) bulk 2κ2 If the manifold on which we define this action has a boundary ∂, then the variation of the action under a small change in the metric δgab = hab is

δSbulk = Z   Z 1 D p ab 1 ab ab 1 D−1 p a  b  − 2 d x |g| R − Rg + Λg hab + 2 d x |γ|n ∇ hab − ∂ah , 2κ 2 2κ ∂ (5.64)

– 47 – a where γab is the induced metric on the boundary and n is the unit normal vector to the boundary. In Gaussian normal coordinates (see subsection 2.13) the integrand of the surface term is

a  b  0 i ij ij ˙ n ∇ hab − ∂ah = ∇ih λ + Kijh + σKhλλ − γ hij (5.65) ij ij 0 i = K hij − 2γ δKij − ∇ih λ , (5.66) where i, j index directions along the boundary, hij = δγij, and Kij is the extrinsic curvature of the boundary.2 The extrinsic curvature is roughly speaking the normal derivative of the tangential components of the metric (see (2.163)). Its appearance here can be traced to the fact that the action (5.63) contains second derivatives of the metric. Since the last term i in (5.66) is the divergence of the boundary vector field h λ, its integral over the boundary vanishes, so the surface term can be written as Z 1 D−1 p ij ij
2 d x |γ| K hij − 2γ δKij , (5.67) 2κ ∂

If we wish to have a well-posed variational problem, we will need to impose some boundary conditions on the metric. Let us first impose Dirichlet boundary conditions, i.e. fix γij. This implies hij = 0, so the first term in 5.67 vanishes. The second term may be cancelled by adding the following boundary term to the action:3 Z 1 D−1 p Sboundary = 2 d x |γ|K. (5.69) κ ∂

This boundary term may be justified for another reason, namely that it makes the action extensive. Specifically, suppose we divide the spacetime volume V on which we are evaluating the action into two pieces V1 and V2 along the hypersurface S. If we demand that the metric be continuous across S, but not its first derivative (i.e. S may have different a extrinsic curvature on the V1 side than on the V2 side), then the Ricci scalar may have a delta-function singularity on S (see 2.179). Without the boundary term, the Einstein-Hilbert action evaluated on V1 alone will miss this contribution, as will the action evaluated on V2 alone. Hence the action evaluated on all of V , which includes this contribution, will not be the sum of the actions evaluated on V1 and V2. Inclusion of the boundary term remedies this defect.

2 To get the sign of Kij right: For a timelike boundary Kij is defined as the outward-directed derivative of γij (so that, using 2.163, λ < 0 inside the region being integrated). For a spacelike boundary Kij is the inward-directed derivative of γij (the usual time derivative for a past boundary, minus the usual one for a future boundary; using 2.163, λ > 0 inside the region being integrated). 3In Euclidean signature, the Einstein-Hilbert action, including boundary term, is

1 Z √ 1 Z √ S = − dDx gR − dD−1x γK , (5.68) 2κ2 κ2 R √ where Kij is the outward-directed derivative of γij at the boundary. (For example, for a flat disk, γK = 2π.)

– 48 – Now let us consider the effect of matter. If the matter Lagrangian depends on the first a derivative of the metric (e.g. on Γbc), then the surface term will pick up additional contribu- tions proportional to hij, which will vanish with Dirichlet boundary conditions. Therefore we need not include any further boundary term in the action.

One can also consider other consistent boundary conditions, such as Neuman, Kij = 0, or mixed (or umbilic), Kij = αγij where α is a constant. For pure gravity, the former does not require a boundary term, as (5.67) vanishes identically, while the latter requires the boundary term4 Z α D−1 p Sboundary = 2 d x |γ| . (5.70) κ ∂ In the presence of matter, additional boundary terms may be required for these boundary conditions.

5.8 Dimensional reductions

2 If we dimensionally reduce d + k dimensions down to d dimensions on a manifold dsk of 2 vanishing Ricci scalar, then the following ansatz relates the (Einstein-frame) metric dsd+k of 2 the higher-dimensional manifold to the (Einstein-frame) metric dsd of the lower-dimensional one: 2 −2kτ/(d−2) 2 2τ 2 dsd+k = e dsd + e dsk. (5.71) The Einstein-Hilbert action is Z Z     1 d+k p 1 d p k m 2 d x |gd+k|Rd+k = 2 d x |gd| Rd − k + 1 ∂mτ∂ τ , (5.72) 2κd+k 2κd d − 2

2 2 where κd = κd+k/Vk. This is a consistent truncation of (d + k)-dimensional gravity only if 2 the metric dsk is fixed (up to the overall scale) by symmetries, for example for a circle or a square torus. In that case, the equations of motion derived from (5.72) are equivalent to the full d + k dimensional Einstein equation. (See the discussion near the beginning of subsection 3.2.) More generally, any parameters of the compact space which are not determined by symmetries (for example, the moduli of a general torus) must be included. We can obtain dilaton gravity by dimensionally reducing pure Einstein gravity on a circle, Z Z d+1 p d p −2Φ m d x |gd+1|Rd+1 = 2πr d x |g|e (R + 4∂mΦ∂ Φ) , (5.73)

4 One way to impose the boundary condition Kij = αγij is to double the manifold along the boundary, 2 impose a Z2 symmetry on the metric, and insert a brane of tension −2(D − 2)α/κ at the (former) boundary. 2 In the notation of subsection 5.6, the localized stress tensor due to the brane is tij = −2(D − 2)αγij /κ , and the Israel junction condition (??) requires (in view of the Z2 symmetry) Kij = −∆Kij /2 = αγij . In this setup the boundary condition thus becomes an equation of motion. Imposing this equation, there is, in addition to 2 R p 2 R p the brane action −2(D − 2)α/κ ∂ |γ|, a localized piece of the Einstein-Hilbert action 2α(D − 1)/κ ∂ |γ| (see (2.179)). Adding them, we obtain twice the boundary action (5.70) (recall that we’ve doubled the bulk action).

– 49 – with the ansatz  4   4  ds2 = exp √ Φ dy2 + exp √ Φ ds2, (5.74) d+1 d − 1 d − 1 − (d − 1) where y is the coordinate on the circle (with coordinate radius r) and ds2 is the d-dimensional (string frame) metric. The case d = 10 gives the usual reduction of M-theory to 10- dimensional supergravity.

6. String theory & CFT

6.1 Complex coordinates On a Euclidean worldsheet we can use real coordinates σ1, σ2 or complex coordinates (Polchin- ski’s conventions) z = σ1 + iσ2, z¯ = σ1 − iσ2. (6.1) In these coordinates we have 1 1 ∂ = ∂ = (∂ − i∂ ), ∂¯ = ∂ = (∂ + i∂ ), (6.2) z 2 1 2 z¯ 2 1 2 and for a general vector va 1 1 vz = v1 + iv2, vz¯ = v1 − iv2, v = (v − iv ) v = (v + iv ). (6.3) z 2 1 2 z¯ 2 1 2 Note that 1 d2σ = d2z, (6.4) 2 so we define the delta function 1 δ2(z, z¯) = δ2(σ1, σ2). (6.5) 2

If the worldsheet is flat, gab = δab, then 1 g = g = , g = g = 0, gzz¯ = gzz¯ = 2, gzz = gz¯z¯ = 0, (6.6) zz¯ zz¯ 2 zz z¯z¯ and i  = − = , zz¯ = −zz¯ = −2i. (6.7) zz¯ zz¯ 2 The divergence theorem is Z I 2 z z¯ z z¯ d z(∂zv + ∂z¯v ) = i (dzv¯ − dzv ). (6.8) R ∂R 6.2 CFT stress tensor By convention in 2d CFT, the stress tensor is defined with an extra factor of 2π compared to the standard definition (5.9) in GR: 4π δS T ab := √ . (6.9) g δgab

– 50 – 6.3 Liouville action The stress tensor of a CFT is traceless on a flat world-sheet, but the Weyl anomaly implies the following value for the trace of the stress tensor (where c is the central charge): c T a = − R. (6.10) a 12 Note that, whereas the other stress tensor components are operators, the trace is a c-number. 2ω Writing the metric on a closed surface as e gab, where gab is a fixed fiducial metric, we then have the following functional differential equation for the partition function as a functional of the metric: 2π δ ln Z[e2ωg ] c c − √ ab = hT a i = T a = − R[e2ωg ] = − e−2ω R − 2∇2ω
(6.11) e2ω g δω a a 12 ab 12

(where R := R[gab]), in other words

δ ln Z[e2ωg ] c ab = R − 2∇2ω
. (6.12) δω 24π The solution to (6.12) is c ln Z[e2ωg ] = ln Z[g ] + S [ω, g ] , (6.13) ab ab 6 L ab where SL is the Liouville action: 1 Z √   S [ω, g ] := d2σ g gab∂ ω∂ ω + Rω . (6.14) L ab 4π a b The Liouville action obeys the following composition rule, as it must for consistency of (6.13):

2ω1 SL[ω1 + ω2, gab] = SL[ω1, gab] + SL[ω2, e gab] . (6.15)

If ω is constant then

SL[ω, gab] = χω , (6.16) where χ is the Euler character of the surface.

6.4 Sigma model The bosonic sigma model action is (for a Euclidean worldsheet) 1 Z √ h  i S = d2σ g gabG (X) + iabB (X) ∂ Xµ∂ Xν + α0R Φ(X) . (6.17) σ 4πα0 µν µν a b (2)

2ω In conformal gauge, gab = e δab, we can use complex coordinates, in which case the action becomes: 1 Z S = d2z (G (X) + B (X)) ∂Xµ∂X¯ ν − 2α0Φ(X)∂∂ω¯  . (6.18) σ 2πα0 µν µν

– 51 – The trace of the worldsheet stress tensor for this model is 1   1 T a = − gabβG + iabβB ∂ Xµ∂ Xν − βΦR , (6.19) a 2α0 µν µν a b 2 (2) where the beta functions are 1 1 βG = R + 2∇ ∂ Φ − H H λξ + O(α0) (6.20) α0 µν µν µ ν 4 µλξ ν 1 1 βB = − ∇ξH + ∂ξΦH + O(α0) (6.21) α0 µν 2 ξµν ξµν 1 D + c0 1 1 βΦ = − ∇2Φ + ∂ Φ∂ξΦ − H Hµνλ + O(α0). (6.22) α0 6α0 2 ξ 24 µνλ

Here c0 is the total central charge of any other fields on the world-sheet, which are assumed to be CFTs (for example −26 for the ghosts). To this order in α0, the beta functions obey the following identities:5

Bµν Bµν ∇µβ − 2∂µΦβ = 0 (6.23) Gµ Gµ Gµ Bµν Φ −2∇µβ α + ∂αβ µ + 4∂µΦβ α + Hµναβ − 4∂αβ = 0 . (6.24)

The sigma model defines a CFT as long as G, B, and Φ satisfy βG = βB = 0.6 Under these conditions, the Wess-Zumino consistency condition requires βΦ to be a c-number rather than an operator, i.e. to be constant in the target space; this can also be seen from (6.24) at the corresponding order in α0. This constant is, up to a factor of 6, the central charge:

c = 6βΦ. (6.25)

Thus βΦ = 0 is the further condition for having a good string background.

6.5 Spacetime action

The equations βG = βB = βΦ = 0 for G, B, and Φ can be derived as the equations of motion of the following spacetime action:7

1 Z  D + c0 1  S = dDxp|G|e−2Φ −2 + R + 4∂ Φ∂µΦ − H Hµνλ + O(α0) , (6.26) 2κ2 3α0 µ 12 µνλ

5As we will discuss in the next subsection, the beta functions can be derived as equations of motion from a target-space action. (6.23) follows from the fact that this action is invariant under B-field gauge transformations, and (using (6.23)) (6.24) follows from the fact that it is invariant under diffeomorphisms. Since the derivability of the beta functions from a gauge- and diffeomorphism-invariant action is believed to hold to all orders in α0, the identities (6.23), (6.24), possibly with explicit α0-corrections, should also hold to all orders. 6Note that it is necessary to specify Φ to define the sigma model even if one is working on a flat worldsheet, because it enters into the definition of the stress tensor. 7 1 R D−1 p −2Φ The boundary term for this action is κ2 ∂ d x |γ|e K.

– 52 – Specifically,

δS = 1 Z  1 1  − dDxp|G|e−2Φ δG βGµν + δB βBµν − 8(δΦ − GµνδG )(βΦ − βGξ) . 2κ2α0 µν µν 4 µν 4 ξ (6.27)

The RHS of this equation defines a metric on the space of fluctuations of the fields G, B, Φ. In other words, using I,J to index both the field and the spacetime point, we have

I J δS = −δφ gIJ β . (6.28)

It follows that RG flow is gradient flow (albeit with an indefinite metric) with respect to a potential given by the spacetime action

I dφ I IJ δS − = −β = g J . (6.29) d ln Λren δφ

(Note that gIJ is not the Zamolodchikov metric for the vertex operators corresponding to the fluctuations, since it does not vanish for pure gauge fluctuations. However, perhaps there is some ambiguity in its definition, and one could choose a metric which vanishes on pure gauge fluctuations and is positive definite on physical ones. This would then presumably equal the Zamolodchikov metric. If this is the case, it would be interesting to consider whether it has any implications for RG flows.) The action (6.26) is also the NS-NS part of the action for the superstring in D = 10, with the first (α0−1) term absent and 2κ2 = (2π)7α04. (6.30)

6.6 Dimensional reduction and Buscher duality We consider configurations of the metric, B-field, and dilaton that have a U(1) invariance. Call the invariant direction y, the other directions xµ, and all of them xM . The coordinate radius of y is R. The Buscher rules are as follows: α0 R0 = (6.31) R 0 −1 −1 Gµν = Gµν − Gyy GyµGyν + Gyy ByµByν (6.32) 0 −1 −1 Bµν = Bµν − Gyy GyµByν + Gyy ByµGyν (6.33) 0 −1 Gyµ = Gyy Byµ (6.34) 0 −1 Byµ = Gyy Gyµ (6.35) 0 −1 Gyy = Gyy (6.36) 0 2Φ0 α 2Φ e = 2 e (6.37) GyyR

– 53 – These can be more usefully organized in terms of the fields of the dimensionally reduced theory. These are: a metric gµν; a two-form bµν; two gauge fields Aµ and A˜µ; and two scalars σ and φ. The fields of the original theory are given in terms of these as follows (we set the √ coordinate radius R equal to α0):

2 M N µ ν 2σ µ 2 ds = GMN dx dx = gµνdx dx + e (dy + Aµdx ) (6.38) B = b + (A + 2dy) ∧ A˜ (6.39) 1 Φ = φ + σ. (6.40) 2 Other useful formulas include:

2σ Gµν = gµν + e AµAν (6.41) 2σ Gyµ = e Aµ (6.42) 2σ Gyy = e (6.43) Gµν = gµν (6.44) Gµy = −Aµ (6.45) Gyy = e−2σ + A2 (6.46) √ √ G = eσ g (6.47) 1 R(G) = R(g) − 2e−σ∇2eσ − F F µν (6.48) 4 µν ˜ Bµν = bµν + A[µAν] (6.49)

Byµ = A˜µ. (6.50)

The spacetime action is

1 Z  1  S = p|G|e−2Φ R(G) + 4∂ Φ∂M Φ − H HMNL (6.51) 2κ2 M 12 MNL √ 2π α0 Z = p|g|e−2φ (6.52) 2κ2  1 1 1  × R(g) − ∂ σ∂µσ + 4∂ φ∂µφ − e2σF F µν − e−2σF˜ F˜µν − h hµνλ . (6.53) µ µ 4 µν 4 µν 12 µνλ

We raise and lower M,N with GMN , and µ, ν with gµν. The field strengths F, F˜ are defined in the usual way (F = dA, F˜ = dA˜), while h also has Chern-Simons contributions:

1 1 h = db + A ∧ F˜ + A˜ ∧ F. (6.54) 2 2 This action has several gauge invariances (the higher dimensional origin of each is indicated in parentheses):

• Diffeomorphisms (diffeomorphisms x0µ = x0µ(xµ), y0 = y).

– 54 – • Gauge transformations on b: b0 = b + dλ (B-field transformations where the gauge parameter does not have a y component).

0 0 1 ˜ 0µ µ • Gauge transformations on A: A = A + dλ, b = b + 2 dλ ∧ A (diffeomorphisms x = x , y0 = y + λ(xµ)).

˜ ˜0 ˜ 0 1 • Gauge transformations on A: A = A + dλ, b = b − 2 A ∧ dλ (B-field transformations where the gauge parameter is proportional to dy).

0 0 0 • The Z2 transformation σ = −σ, A = A˜, A˜ = A (T-duality).

The definitions of the lower-dimensional fields have been carefully chosen so as to give the simplest possible transformation rules. In particular, gµν is defined the way it is in order to be invariant under the gauge transformations of A; as a bonus it also ends up being invariant under T-duality. φ and b are defined as they are in order to be invariant under T-duality. Note that the 3-form field strength h is invariant under all of these transformations (actually, covariant under diffeomorphisms). The metric (6.27) on the space of field configurations becomes, in terms of these variables,

1 Z  1 2 dS2 = p|G|e−2Φ dG2 + dB2 − dG M − dΦ
(6.55) 2κ2α0 MN MN 2 M Z  2 2π p −2φ 2  ˜ ˜  2σ 2 −2σ ˜2 = √ |g|e dg + dbµν − A[µdAν] − A[µdAν] + 2e dA + 2e dA 2κ2 α0 µν µ µ 1  + 4dσ2 − (dg µ − 4dφ)2 (6.56) 2 µ

This is manifestly invariant under T-duality. It follows that the beta functions are also invariant under T-duality. This is because T-duality is an equivalence between sigma models, irrespective of whether they are conformal.

6.7 More dualities

The NS-NS 2-form potential B has mass dimension −2, and the R-R p-form potentials, denoted Cp, have dimension −p. All metrics are in string frame. In the first two dualities below, all fields are assumed independent of the coordinate y, on which the dualities are performed. M-theory–IIA:

2 2 µ ν 2σ µ 2 M11dsM = Gµνdx dx + e (dy + Aµdx ) , y ∼ y + 2πR (6.57) m (6.58)

1 2 σ µ ν Φ σ 3/2 1 1 µ dsIIA = Re Gµνdx dx , e = (Re ) , √ C1 = Aµdx (6.59) α0 α0 R

– 55 – T-duality:

2 µ ν 2σ µ 2 dsIIA = Gµνdx dx + e (dy + Aµdx ) , y ∼ y + 2πR, (6.60) 1 B = B dxµ ∧ dxν + b dy∧dxµ, eΦIIA ,C = C dxµ + C dy (6.61) IIA 2 µν µ 1 µ y m (6.62) 0 2 µ ν −2σ µ 2 α dsIIB = Gµνdx dx + e (dy + bµdx ) , y ∼ y + 2π , (6.63) R√ 1 α0 B = B dxµ ∧ dxν + b A dxµ∧dxν + A dy ∧ dxµ, eΦIIB = eΦIIA , (6.64) IIB 2 µν µ ν µ Reσ R µ R C2 = √ Cµdx ∧ dy, C0 = √ Cy (6.65) α0 α0

S-duality:

02 −Φ 2 0 0 0 ds = e ds , Φ = −Φ,B = C2,C2 = −B (6.66)

6.8 Orbifolds and quotient groups

Let G be a group acting on a space (or a CFT), and H a normal subgroup of G. Then the space (or CFT) obtained by orbifolding by G is the same as that obtained by orbifolding first by H and then by G/H.

7. Quantum information theory

In the following, ρ, σ denote density matrices and {λi} is a probability distribution (i.e. λi ≥ 0, P i λi = 1). The Shannon entropy is X H({λi}) := − λi ln λi , (7.1) i where 0 ln 0 is defined to equal 0.

7.1 Von Neumann entropy

Definition:

S(ρ) := − Tr ρ ln ρ = h− ln ρiρ = H({pa}) , (7.2) where {pa} is the spectrum of ρ.

Positivity: S(ρ) ≥ 0 (7.3) with equality if and only if ρ is pure.

– 56 – Maximal value: S(ρ) ≤ ln d (7.4) (d is the dimension of the Hilbert space). For d finite, there is equality if and only if ρ = I/d. For d infinite, S(ρ) can be arbitrarily large or infinite; e.g. if the nth eigenvalue of ρ (in decreasing order) goes like 1/(n(ln n)2), then S(ρ) = ∞.

Invariance: S(UρU −1) = S(ρ) (7.5) (U unitary).

Expansibility: S(ρ ⊕ 0) = S(ρ) (7.6)

Concavity: S (λρ + (1 − λ)σ) ≥ λS(ρ) + (1 − λ)S(σ) (7.7) for 0 ≤ λ ≤ 1, with equality if and only if λ = 0, λ = 1, ρ = σ, S(ρ) = ∞, or S(σ) = ∞. For P a general convex combination ρ = i λiρi, X X λiS(ρi) ≤ S(ρ) ≤ λiS(ρi) + H({λi}) . (7.8) i i

The term H({λi}) is called the mixing entropy. The difference between S(ρ) and the left-hand side is called the Holevo information: X χ({λi, ρi}) := S(ρ) − λiS(ρi) . (7.9) i So

0 ≤ χ({λi, ρi}) ≤ H({λi}) . (7.10)

If the ρi are pairwise orthogonal then the second inequality is saturated: X S(ρ) = λiS(ρi) + H({λi}) , χ({λi, ρi}) = H({λi}) . (7.11) i Canonical ensemble: Given a Hamiltonian H and energy E, S(ρ) is maximized within the set of density matrices satisfying hHiρ = E by

−βH −βH ρβ = e /Z , Z = Tr e , (7.12) where β is chosen to satisfy hHiρ = E. For fixed H, S(ρβ) is a decreasing function of β.

7.2 Relative entropy Definition:

S(σ|ρ) := Tr ρ (ln ρ − ln σ) = h− ln σiρ − S(ρ) (7.13)

– 57 – Positivity:

S(σ|ρ) ≥ 0 , (7.14) with equality if and only if σ = ρ.

Norm bound: 1 S(σ|ρ) ≥ kσ − ρk2 (7.15) 2 1

Relative entropy in terms of entropy: Define

Sλ(σ|ρ) = S(λσ + (1 − λ)ρ) − λS(σ) − (1 − λ)S(ρ) (0 ≤ λ ≤ 1) . (7.16)

Sλ(σ|ρ) is concave in λ and satisfies

0 ≤ Sλ(σ|ρ) ≤ −λ ln λ − (1 − λ) ln(1 − λ) (7.17)

(so in particular S0(σ|ρ) = S1(σ|ρ) = 0). Then

d S(σ|ρ) = Sλ(σ|ρ) . (7.18) dλ λ=0

Joint convexity:

S(σ|ρ) ≤ λS(σ1|ρ1) + (1 − λ)S(σ2|ρ2) (7.19) where σ = λσ1 + (1 − λ)σ2, ρ = λρ1 + (1 − λ)ρ2.

7.3 R´enyi entropy

The R´enyi entropy Sα(ρ) is defined for 0 ≤ α ≤ ∞:

1 S (ρ) = ln Tr ρα (α 6= 0, 1, ∞) (7.20) α 1 − α

S0(ρ) = ln rank ρ = lim Sα(ρ) (7.21) α→0

S1(ρ) = S(ρ) = lim Sα(ρ) (7.22) α→1

S∞(ρ) = − ln p1 = lim Sα(ρ) (7.23) α→∞

– 58 – where p1 is the largest eigenvalue of ρ. S0 is also called the Hartley entropy. Properties (for fixed ρ):

Sα ≥ 0 (= 0 iff ρ is pure) (7.24) d S ≤ 0 (= 0 iff ρ ∝ identity on its image) (7.25) dα α d2 S ≥ 0 (7.26) dα2 α α S ≤ S (α > 1) (7.27) α α − 1 ∞ d α − 1  S ≥ 0 (7.28) dα α α d ((α − 1)S ) ≥ 0 (7.29) dα α d2 ((α − 1)S ) ≤ 0 (7.30) dα2 α 7.4 Mixing-enhancing

0 Let {pa} be the eigenvalues of ρ, arranged in decreasing order, and similarly {pa} for σ. If, for all n less than or equal to the smaller of their dimensions,

n n X X 0 pa ≤ pa (7.31) a=1 a=1 then we write ρ σ and say that ρ is more mixed than σ and that σ is purer than ρ. If ρ σ then S(ρ) ≥ S(σ) . (7.32) More generally Tr f(ρ) ≥ Tr f(σ) (7.33) for any concave f, and

Sα(ρ) ≥ Sα(σ) (7.34) for all α. A transformation F on density matrices such that F (ρ) ρ for all ρ is called mixing- enhancing. Examples:

1. f(ρ) ρ 7→ (7.35) Tr f(ρ) for f concave, positive on (0, 1], and with f(0) = 0.

2. X ρ 7→ PiρPi , (7.36) i

– 59 – P where {Pi} is a set of orthogonal projectors such that i Pi = I (representing possible outcomes of a measurement, for example). If the Pi are rank 1, then this corresponds to deleting the off-diagonal elements in the matrix representing ρ in the basis corresponding

to the Pi; more generally it corresponds to deleting the off-block-diagonal elements.

3. X −1 ρ 7→ λiUi ρUi , (7.37) i

where the Ui are unitary operators.

7.5 Joint systems

A, B, C denote disjoint systems. AB denotes A ∪ B, etc. Partial traces are implied by subscripts, e.g. ρA = TrB ρAB. S(A) denotes S(ρA). Note that ln(ρA ⊗ ρB) = ln ρA ⊗ IB + IA ⊗ ln ρB.

Separable and entangled states: A separable state on AB is one that can be written in the form X (i) (i) ρAB = λiρA ⊗ ρB , (7.38) i

(i) where ρA,B are density operators. A state that is not separable is called entangled.

Additivity:

S(ρA ⊗ ρB) = S(ρA) + S(ρB) . (7.39)

More generally,

Sα(ρA ⊗ ρB) = Sα(ρA) + Sα(ρB) . (7.40)

Subadditivity: S(AB) ≤ S(A) + S(B) (7.41) with equality if and only if ρAB = ρA ⊗ ρB. Among the R´enyi entropies, only the Hartley is also subadditive: S0(AB) ≤ S0(A) + S0(B). This is true even for classical distributions. A simple counterexample for the α > 1 R´enyis is the following distribution on two bits, p00 = 1 − 2x, p01 = p10 = x, p11 = 0. This violates subadditivity for x in the range 0 ≤ x ≤ xα (where 0 < xα < 1/2). S and S0 are the only functions of ρ obeying invariance, additivity, and subadditivity.

Araki-Lieb (or triangle): |S(A) − S(B)| ≤ S(AB) (7.42)

S(A) − S(B) = S(AB) if and only if A can be decomposed as A = AL ⊗ AR such that

ρAB = ρAL ⊗ ρARB where ρARB is pure (see arXiv:1105.2993).

– 60 – Strong subadditivity:

S(ABC) + S(B) ≤ S(AB) + S(BC) (7.43) S(A) + S(C) ≤ S(AB) + S(BC) (7.44)

L The first inequality if saturated if and only if B can be decomposed as B = i BLi ⊗ BRi such that M ρ = λ ρi ⊗ ρi (7.45) ABC i ABLi BRiC i (see arXiv:quant-ph/0304007).

Additivity of relative entropy:

S(σA ⊗ σB|ρA ⊗ ρB) = S(σA|ρA) + S(σB|ρB) (7.46)

Monotonicity of relative entropy:

S(σAB|ρAB) ≥ S(σA|ρA) (7.47)

Mutual information:

I(A : B) := S(A) + S(B) − S(AB) = S(ρA ⊗ ρB|ρAB) (7.48)

Properties of mutual information:

I(A : B) = I(B : A) (7.49)

I(A : B) ≥ 0 (with equality if and only if ρAB = ρA ⊗ ρB) (7.50) I(A : B) + 2S(C) ≥ I(A : BC) ≥ I(A : B) (7.51)

Bound on correlators: Let MA,MB be bounded operators in A, B respectively. Then

1 hM M i − hM ihM i2 A B A B ≤ I(A : B) (7.52) 2 kMAkkMBk

Conditional entropy:

H(A|B) := S(AB) − S(B) = ln dB − S(ρA ⊗ IB/dB|ρAB) (7.53) where dB is the dimension of the B Hilbert space. Concavity of conditional entropy:

H(A|B)λρAB +(1−λ)σAB ≥ λH(A|B)ρAB + (1 − λ)H(A|B)σAB (7.54)

– 61 – Purification: A purification of ρA, is a Hilbert space B and a pure joint density matrix ρAB such that ρA = TrB ρAB. Then S(B) = S(A). In general there exist many purifications of a given density matrix. One construction, sometimes called the canonical purification, is as follows. Let pa, |aiA be the eigenvalues and corresponding eigenvectors of ρA, so ρA = P a pa|aiAAha|. Let the B Hilbert space have dimension equal to the dimension of A, and let |aiB be an orthonormal basis for it. Then let

ρAB = |ψihψ| (7.55) where |ψi is the following state on AB:

X 1/2 |ψi = pa |aiA|aiB . (7.56) a

8. Superselection rules

See arXiv:0710.1516 for a review of superselection rules. Here we will be brief. A superselection rule is defined by some observable R, for example electric charge. The rule says that we consider only eigenstates of R, i.e. we do not allow superpositions of states with different eigenvalues. However, we do allow mixtures of states with different eigenvalues. This is equivalent to the requirement that the density matrix ρ commutes with R. We also consider only operators that carry a definite R quantum number, i.e.

[R,M] = rM M (8.1)

F where rM is a c-number. (Alternatively, if R is unitary, for example R = (−1) , where F is fermion number, then we would require

−1 R MR = rM M.) (8.2)

This guarantees that M preserves the condition of being an eigenstate of R. Only the opera- tors that commute with R (rM = 0 in (8.1) or rM = 1 in (8.2)) represent physical observables (i.e. have non-zero expectation values on physical states, and have eigenvectors that are phys- ical states); these form a subalgebra of the full algebra of operators.

8.1 Joint systems On a joint system AB, the superselection rule will typically be defined by an observable of the form

R = RA ⊗ IB + IA ⊗ RB (8.3) F F F (or, in the unitary case, R = RA ⊗ RB, for example (−1) = (−1) A ⊗ (−1) B ). Two interesting points arise. The first is that superselection rules are preserved by restriction to a subsystem. For example, we can consider a state with R = 1 that is a superposition of an RA = 1, RB = 0

– 62 – state and an RA = 0, R=1 state. But when we trace over B, we will get a mixture of states on A with RA = 0 and 1. More generally, if ρ commutes with R, then ρA commutes with RA. Furthermore, if an operator or observable is allowed by R and localized on A, i.e. is of the form MA ⊗ IB, then MA is allowed by RA. On the other hand, an observable on AB may be formed by tensoring operators on A and

B that are not themselves observables, as long as their charges cancel. Thus M = MA ⊗ MB is an observable if rM = rMA +rMB = 0, even if rMA 6= 0. Hence the algebra of observables on the full system is larger than the tensor product of observables on the subsystems, something that does not happen in the absence of superselection rules.

8.2 Joint systems and fermionic operators If two subsystems A, B contain fermionic operators (e.g. if A, B are spatial regions in a field theory with fermionic fields), then in the joint systems those operators must anticommute.

This is confusing, since in the tensor product HAB = HA ⊗ HB, operators on A would seem to always commute with operators on B. Is the joint system really described by the tensor product, or by some more exotic structure? The solution to the puzzle is simple. The joint system is indeed described by the usual tensor product. However, the statement above about operators on HA and HB was too fast. Given an operator MA on A, we need a way to associate to it an operator on AB. This mapping TA should be an algebra homomorphism and should preserve the expectation value for an arbitrary product state, i.e.

hT (M )i = hM i ∀ ρ , ρ . (8.4) A A ρA⊗ρB A ρA A B

The simplest option (which is usually assumed implicitly) is TA(MA) = MA ⊗ IB. In fact, in the absence of superselection rules, this is the unique homomorphism obeying (8.4). However, if we do the same for the B operators, TB(MB) = IA ⊗MB, then indeed TA(MA) and TB(MB) will always commute. So we need to find a different homomorphism.

For bosonic operators, the mapping TA is fixed by the requirement that the expectation value in an arbitrary product physical state should be preserved, i.e. (8.4). However, the fact that the expectation value of a fermionic operator vanishes in any physical state gives us more freedom in choosing the homomorphism. Let us denote the statistics of an operator by s(M), where s(M) = 0 for bosonic operators and s(M) = 1 for fermionic ones. Then it is easy to see that the following is a homomorphism:

s(MA)FB TA(MA) = MA ⊗ (−1) . (8.5)

Furthermore, if we map the B operators to AB in the usual way, TB(MB) = IA ⊗ MB, then TA(MA) and TB(MB) will anticommute when MA and MB are both fermionic and commute otherwise, which is what we wanted. Another option is to put the cocyle on TB, i.e.

s(MB )FA TA(MA) = MA ⊗ IB .TB(MB) = (−1) ⊗ MB . (8.6)

– 63 – (We can put cocycles on both TA and TB, but then the fermionic operators will again com- mute.) These two choices are equivalent, since they are related by conjugation by (−1)FAFB .

For multiple systems A1,...,An, we can for example take

s(M )F Ai Ai+1 s(MAi )FAn TAi (MAi ) = IA1 ⊗ · · · ⊗ IAi−1 ⊗ MAi ⊗ (−1) ⊗ · · · ⊗ (−1) . (8.7)

It is easy to see that this prescription reproduces the usual action of the creation and anni- hilation operators on the Fock space of a free fermionic field theory.

The freedom in choosing the homomorphism presumably extends to the case with more than two superselection sectors (e.g. with anyons). It might even be possible for the cocycles to be non-abelian, although I haven’t thought through that case.

9. Dimensions of physical quantities

The tables below follow the SI system. I have chosen charge, mass, length, and time as the basic dimensions. This is obviously an arbitrary choice but seems to be conventional, at least at the undergraduate level. In particular, in undergraduate physics, mass is taken as a fundamental concept, with concepts like energy and action derived; whereas from a more advanced viewpoint (based on QFT for example), action is fundamental and mass is derived. One interesting observation from the table is that time almost always appears with a negative power.

– 64 – quantity dimensions unit constant value −5 1/2 −44 time T s (G~c ) 5.4 × 10 frequency, angular velocity, δ(t), ˙ T −1 Hz angular acceleration T −2 −3 1/2 −35 length L m (G~c ) 1.6 × 10 2 −3 −70 area L G~c 2.6 × 10 volume L3 wave number/vector, δ(x), ∇ L−1 number density, δ3(r) L−3 spacetime volume L3 T speed, velocity LT −1 c 3.0 × 108 acceleration LT −2 diffusion constant, kinematic viscosity L2 T −1 probability/number current/flux L−2 T −1 −1 1/2 −8 mass M kg (G ~c) 2.2 × 10 moment of inertia ML2 mass density ML−3 momentum MLT −1 2 −2 −1 5 1/2 9 energy, torque ML T J (G ~c ) 2.0 × 10 force MLT −2 N power ML2 T −3 W G−1c5 3.6 × 1052 energy flux MT −3 2 −1 −34 angular momentum, action ML T ~ 1.1 × 10 Young/shear/bulk modulus, energy density, pressure, stress ML−1 T −2 Pa momentum density ML−2 T −1 dynamic viscosity ML−1 T −1 surface tension, spring constant MT −2 M −1 L3 T −2 G 6.7 × 10−11

– 65 – quantity dimensions unit constant value 1/2 −19 charge Q C e = (α4π0~c) 1.6 × 10 electric dipole moment QL electric 2n-pole moment QLn charge density QL−3 current QT −1 A current density QL−2 T −1 electric field Q−1 MLT −2 electric potential Q−1 ML2 T −2 V electric flux Q−1 ML3 T −2 2 −1 −3 2 −12 electric permittivity Q M L T 0 8.9 × 10 −2 3 −2 −1 9 Q ML T (4π0) 9.0 × 10 capacitance Q2 M −1 L−2 T 2 F resistance, impedance Q−2 ML2 T −1 Ω resistivity Q−2 ML3 T −1 conductivity Q2 M −1 L−3 T displacement D, polarization QL−2 magnetic dipole moment QL2 T −1 magnetic field B Q−1 MT −1 T −1 2 −1 −1 −15 magnetic flux Q ML T Wb Φ0 = π~e 2.1 × 10 −2 2 −1 −6 magnetic permeability Q ML µ0 = (0c ) 1.3 × 10 inductance Q−2 ML2 H vector potential Q−1 MLT −1 H, magnetization QL−1 T −1

– 66 –