Part III: Applications of Differential Geometry to

Physics

G. W. Gibbons

D.A.M.T.P.,

Cambridge,

Wilberforce Road,

Cambridge CB3 0WA,

U.K.

January 20, 2011

1 Maps Between Manifolds

In addition to the exterior derivative d, there is another type of derivative onemay define on a manifold possessing no further structure, it is called the Lie

Derivative. To define it we need to a smooth map ψ : M → N from a smoothmanifold M to another smooth manifold N , which need not have the samedimensions, m and n respectively. Of course an interesting special case willcorrespond to taking M = N but for clarity and also with other applicationsin mind, we keep M and N distinct. By smooth map, we mean one givenby smooth functions in every smooth coordinate chart. Thus if xα are localcoordinates for M and yβ for N , then ψ takes a point p to p′ = ψ(p) and ifp has coordinates xα, α = 1, 2, . . .m and p′ has coordinates yβ ,β = 1, 2, . . . n,then yβ = yβ(xα).

Associated with ψ are two maps

Push Forward ψ∗ : Tp(M)→ Tp′(N) (1)

Pull Back ψ∗ : T ∗p (M)← T ∗

p′(N) (2)

The push forward acts on curves vectors and contra-variant tensor, the pullback acts on functions, co-vectors, p-forms and co-variant tensor fields.

1.0.1 The Push Forward Map

Consider a curve in M , i.e a map c : R → M given by xα(λ). The pushedforward a curve c∗ is just obtained by composition, in other words c∗ = ψ c, oryβ(λ) = yβ(xα(λ)). The chain rule now allows us to push forward the tangentvector T

dyβ

dλ=∂yβ

∂xα

dxα

dλ, (3)

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or if ψ∗T =∗ T

∗Tβ(p′) =

∂yα

∂xαTα(p). (4)

Clearly we can extend ψ∗ to arbitrary contra-variant tensor fields.

1.0.2 The Pull Back Map

Consider a function on N , i.e a map N → N given by f(yβ). The pulled backfunction f∗ is just obtained by composition, in other words f∗ = f ψ, orf∗(xα) = f(yβ(xα)). The chain rule now allows us to pull back the gradient co-vector ω = df

∂f∗

∂xα=∂yβ

∂xα

∂f

∂yβ(5)

or if ψ∗ω =∗ ω,

∗ωα(p) =∂yβ

∂xαωβ(p′). (6)

Clearly we can extend ψ∗ to arbitrary co-variant tensor fields.Clearly push forward and pull backs are related. One could define one in

terms of the other by

∗V fp′ = V ∗fp (7)

or〈⋆ω|V 〉p = 〈ω|∗V 〉p′ . (8)

1.0.3 Exterior Derivative commutes with pullback

The formula given above is identical to that used earlier to show that the exteriorderivative takes p-form fields to p+1 -form fields. In the present context thatcalculation shows that d commutes with pull back, i.e.

d(ψ∗ω) = ψ∗(dω). (9)

For example consider the map

ψ : R3 \ z− axis→ S1 , (x, y, z)→ θ = tan−1(

y

x) . (10)

We can pull-bcak the volume form on S1

ψ⋆dθ =−ydx+ xdy

x2 + y2= B dB = 0. (11)

Physically, B is the magnetic field B due to a cuurent flowing along thez-axis and dB = 0⇔ curlB = 0 . Note that

∮

γ

B.dx = 2π =

∮

S1

dθ (12)

where γ is any cloed curve encircling the z-axis once.

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Another example is provided by the map

ψ : S2 → R3s.t.(x, yz) = (sin θ cosφ, sin θ sinφ, cos θ) , (13)

which embeds the 2-sphere into three-dimensional space.Let

ω = zdx ∧ dy + xdy ∧ dz + ydz ∧ dx⇒ dω = 3dx ∧ dy ∧ dz . (14)

We haveψ⋆ω = sin θ ∧ dφ⇒ dψ⋆ω = 0 = ψ⋆dω . (15)

The last equality follows because ψ⋆dω is a 3-form in 2-dimensions.Our last example is related to the Dirac monopole in the same way that the

first was related to the vortex.

ψ : R3 \ 0→ S2 (x, y, z)→ (

x

r,y

r,z

r) (16)

with r =√

x2 + y2 + z2. if

F =zdx ∧ dy + xdy ∧ dz + ydz ∧ dx

r3=

1

2ǫijk

xk

r3dxi ∧ dxk (17)

One has F = ψ⋆η, with η − sin θdθ ∧ dφ the volume form on S2 and so F isclosed, deta = 0⇒ dF = 0. Thus if B is the magnetic field then curlB = 0 and

∫

S

B.dS = 4π, (18)

for any closed surface S which encloses the origin once.This calculation easily generalizes to arbitrary dimnensions and is applied to

the Ramond-Ramond and Neveu-Schwarz charges of various p-brane solutionsin supergravity and sring theory.

1.0.4 Diffeomorphisms :Active versus Passive Viewpoint

In terms of diagrams we are just following the arrows. Analytically we only

need the Jacobian matrix ∂yβ

∂xα , we never need what may not exist, i.e. ∂xβ

∂yβ .However if M = N , and ψ is, at least locally invertible, so that ψ is age local

diffeomorphism, then ∂xβ

∂yβ will exist and what we have been doing is identicalto what we earlier thought of as a local coordinate transformation or change of

chart from φ say to φ′ = φ ψ. This is just a change of viewpoint, initiallyone may have thought of ψ actively as moving the points of the manifold Mwhile we usually think of coordinate transformations passively as relabelingthe same points. The two concepts are essentially interchangeable

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1.0.5 Invariant tensor fields

We can now give a condition that a contra-variant tensor field S of rank k isinvariant under the action of a diffeomorphism ψ, it must equal its push forward

∗S under ψ, i.e.ψ∗S(p) = S(p′), (19)

or in components

Sβ1β2...βk(y) =∂yβ1

∂xα1

∂yβ2

∂xα2 . . .∂yβk

∂xαkSα1α2...αk(x). (20)

Similarly a co-variant second rank tensor field g, for example the metric tensor,is invariant under the action of ψ is that g equals its pull-back,

g(p) = ψ⋆g(p′) (21)

, i.e.

gαβ(x) = gµν(y)∂yµ

∂xα

∂yν

∂xβ, ds2(x) = ds2(y). (22)

Thus infinitesimal lengths are preserved by such diffeomorphisms which arecalled isometries. Later we will discuss infinitesimal isometries and Killing vec-tor fields.

1.1 Immersions and Imbeddings

The derivative map or Jacobian

ψ∗ : Tp(M)→ Tp′(M) (23)

is a linear map, explicitly the matrix ∂yβ

∂xα . Its rank is the number of linearlyindependent rows. Then

ψ∗ is injective if rankψ∗ = dimM , i.e. no vector in Tp(M) is mapped to zero

ψ∗ is surjective if rankψ∗ = dimN , i.e. every vector in Tp′(N) is the image ofsome vector in Tp(M).

The map ψ is called a smooth immersion if locally the inverse ψ−1 exists.The Impliicit Function theorem states that ψ is an immmersion iff ψ∗ is injective.We say that ψ(M) is an immersed submanifold A smooth imbedding1 is a smoothimmersion which is 1-1 on its image and the inverse image of any compact setis compact.

For example an imbedding of S1 into R2 is a circle or an ellipse or any simple

smooth curve which does not intersect itself. An example of an immersion wouldbe a figure of eight curve. For each point on S1 there is a unique tangent vectorbut at two points S1 with the same image in R

2 there are two distinct tangentvectors.

1sometimes called an embedding

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A hypersurface, for example a t = constant surface in spactime is an imbed-ded submanfold of co-dimension one, i.e. dimN = dimM − 1.

In this language, a smooth diifeomorphism is such that ψ is 1-1 and ψ−1

is smooth. The implicit function theorem then asserts that this is true if ψ∗ isboth surjective and injective. Thus

rankψ∗ = dimM = dimN. (24)

It follows that (ψ∗)−1 =and ψ−1

∗ are isomorphisms of Tp(M) and Tp′(N) re-spectively and that the two tensor algebras are also isomorphic.

Thus, in the case of a diffeorphism the distinction between push forward andpull back is to some extent lost. We can, by convention, define the push forwardof a co-variant tensor field Ω by the formula

ψ∗Ω(y) = ψ−1∗Ω(x) (25)

Thus on functions, for example,

(f∗)∗ = f. (26)

2 One-parameter families of diffeomorphims: Lie

Derivatives

Suppose ψt : R×M →M be one parameter family of smooth maps such that

ψt+t′ = ψt ψt′ (27)

and ψ0 is the identity map. Thus

ψ−t = ψ−1. (28)

and we have an action of the additive group R on M . One should have in mindthe example of time translations. The orbit of ψt through a point p ∈ M withcoordinates Xα, ψt(p) is a curve yα(t) with yα(0) = xα, the coordinates of thepoint p. The curves ψt(p) have tangent vectors T (t) and define a vector fieldK(x) on M. Conversely given a vector field K we obtain a one parameter familyof diffeomorphisms ψt by moving the points of M up the integral curves of Kand amount t. Thus in local charts, ψt is given by

ψt : xα → yα(t), (29)

where yα(t) is a solution of

dyα

dt= Kα(yβ(t)) (30)

with yα(0) = xα. Infinitesimally, for small t we have

yα(t) = xα + V α(x)t+O(t2). (31)

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Given ψt we can Lie-Drag curves, and tensor fields along the integral curves,using the push-forward map ψt∗. Thus, for a vector V , ∗V = ψt∗V is given by

∗Vα(yα(t)) =

∂yα(t)

∂xβV β(xγ). (32)

We now define the Lie Derivative £KS of any contra-variant tensor field S sayby

£KS(x) = − d

dtψt∗S(x)

∣

∣

∣

t=0= lim

t→0

(

S(x)− ψt∗S(x))

t. (33)

The origin of the minus sign is as follows. We are evaluating the Lie derive of Sat the point x. To do so we compare S(x) with the value obtained by draggingS from the point ψ−t(x) = ψ−1

t (x) to the point x.As mentioned above, for a (local) diffeomorphism we can extend the push

forward map to co-variant tensor fields and the definition of the Lie Derivativeextends as well. Alternatively one could define it by using the pull back mapbut one now gets a plus sign:

£KΩ(x) = +d

dtψt

∗Ω(x)∣

∣

∣

t=0= lim

t→0

(

ψt∗Ω(x) − Ω(x)

)

t. (34)

2.0.1 Example: Functions

The simplest case to consider is that of functions

£Kf = limt→0

(

f(x)− f(yα(−t))

t(35)

Thus

£Kf = limt→0

(

f(x)− f(xα − tKα(xα)))

t= Kα ∂f

∂xα= Kf. (36)

Doing it the other way we get the same answer

£Kf = limt→0

(

f(xα + tKα(xα)) − f(xα))

t= Kα ∂f

∂xα= Kf. (37)

A, slightly symbolic, but nevertheless illuminating notation makes use ofthe exponential map etK of a vector field. Geometrically this takes points anamount t up the integral curves of the vector field K

etKxα = yα(t). (38)

. On functions therefore we have

etKf = ∗f = ψ∗f. (39)

The push-forward is given by

e−tKf = ∗f = ψ∗f. (40)

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2.0.2 Example: Vector Fields

We need

(£KV )α(xγ) = − d

dt

(

V β(yβ(−t))∂yα(−t)∂xβ

)

. (41)

Since∂yα(−t)∂xβ

(xγ) = δαβ − tKα

,β(xγ) +O(t2), (42)

we obtain

(£KV )α = V α,βK

β −Kα,βV

β = [K,V ]α . (43)

This calculation can also be done using the exponential notation.

(∗V f)(p′) = (V ∗f)(p)⇒ (∗V f)(p) =∗ (V ∗f)(p), (44)

thus

ψt∗V = e−tKV etK ⇒ − d

dtψt∗V

∣

∣

∣

t=0= [K,V ]. (45)

It is straight forward to see how to take the Lie derivative of higher rank con-travariant tensor fields. One gets terms with a partial derivative of K for eachupper index. For example of a second rank contravariant tensor we have

(£KS)αβ = Kγ∂γSαβ − ∂γK

αSγβ − ∂γKβSαγ . (46)

2.0.3 Lie Derivative on covariant tensors

We have

(Lω)α = − d

dt

(

ωβ(yγ(−t))∂yβ(−t)∂xα

)∣

∣

∣

t=0(47)

Using the fact that

∂yβ(−t)∂xα

= δβα − t∂αK

β +O(t2), (48)

one gets

(£Kω)α = Kβ∂βωα + ∂αKβωβ . (49)

For a second rank co-variant tensor, such as the metric tensor one has

(£Kg)αβ = Kγ∂γgαβ + ∂αKγgγβ + ∂βK

γgαγβ. (50)

Note that we have not assumed a particular symmetry for gαβ , but if wehad, that symmetry would be inherited by the Lie derivative.

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2.0.4 Example: Adapted Coordinates, Stationary and Static metrics

From its definition, it is clear that the Lie derivative takes tensor fields to tensorfields. The presence of the terms involving βK

α guarantees that. However, itis always possible locally to introduce adapted coordinates t, xi such that

Kα“ =′′ δαt , ⇔ K ′′ =′′ ∂

∂t. (51)

In these coordinates for any tensor field

£K“ =′′ ∂

∂t. (52)

Thus if K is a Killing vector field, that is

£Kg = 0 (53)

, where g is the metric tensor, then in adapted coordinates the metric is inde-pendent of the coordinate t

ds2 = g00(xk)dt2 + 2g0i(x

k)dtdxi + gij(xk)dxidxj . (54)

The existence of adapted coordinates may be shown as follows. We intro-duce an initial hypersurface nowhere tangent to Σ so that (at least in a localneighbourhood U) the integral curves of K intersect Σ once and only once. Thecoordinates xi are chosen on Σ ∩ U and then Lie dragged along the integralcurves, in other words they are defined to be constant along the integral curves.Points on Σ are assigned the coordinate t = 0. We then assign to each point p inU ⊂M the coordinates t, xi, where xi labels the integral curve passing throughp and t the parameter necessary to reach p starting from Σ.

There is thus clearly some ”gauge freedom” in choosing the hypersurfaceΣ. Changing Σ to some other hypersurface Σ′ will alter the origin of the tparameter along each integral curve. If Σ′ intersects the integral curve labelledby xi at t = τ(xi) then the new coordinate t′ will be given by

t′ = t− τ(xi). (55)

In physical applications, if the Killing vector is timelike, one calls the metricstationary unless there is an additional time reversal symmetry , in which hasone may put g0i = 0. The metric is then said to be static. For example a non-rotating Schwarschild black hole has gives rise to a static metric. A rotatingKerr balck hole has a stationary metric.

2.1 Lie Derivative and Lie Bracket

So far we have not verified explicitly that the Lie derivative £V W is a vectorfield. This may be seen directly by introducing the Bracket or Commutator

[V,W ] = −[W,V ] (56) skew

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of two vector fields V and W by its action on a function f ∈ F(M)

[V,W ]f = V (Wf)−W (V f). (57)

The bracket of two vector fields is itself a vector field. To see this one needsto check that the Leibniz property

[V,W ]fg = g[V,W ]f + f [V,W ]g. (58)

This is a simple, albeit tedious, calculation.Moreover, a short calculation in local coordinates reveals that that

[V,W ]α = (£VW )α. (59)

2.1.1 Jacobi Identity

In the same vein, one easily verifies that the Lie bracket on functions satisfiesthe Jacobi Identity

[U, [V,W ]] + [V, [W,U ]] + [W, [U, V ]] = 0. (60) jac

It follows from (??) and (??) that the set of vector fields X(M) forms aninfinite dimensional Lie Algebra This is sometimes denoted diff(M).

2.1.2 Example: diff(S1) is called the Virasoro Algebra

It crops up in string theory. If θ ∈ (0, 2π] is a coordinate on S1, one defines abasis index by n ∈ Z by

Dn = ieinθ∂θ. (61)

The brackets are[Dn, Dm] = (n−m)Dm+n. (62)

There is a finite sub-algebra spanned by D−1, D0, D−1 whose Lie-algebra is thatof the projective group sl(2,R).

2.2 Lie Bracket and Closure of infinitesimal rectangles

Suppose that we move from O to A an amount t along the integral curves ofU and then from A to B an amount s along the integral curves of V . We maycompare our final position B with what would have happened if we had firstmoved from O to C an amount s along the integral curves of V and then fromC to D an amount t along the integral curves of U . A short calculation in localcoordinates using Taylor’s theorem shows that to lowest no trivial order:

xµB − xµ

D = [U, V ]αst+ . . . . (63)

Thus, if the bracket closes the infinitesimal rectangle DCOAB will close.More generally it may be shown that if the bracket vanishes then the two vector

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fields lie in a 2-surface and conversely. It is not necessary that the bracket vanishfor the vector fields to lie in a 2-surface. It suffices that the bracket [U, V ] isa linear combination of U and V , with coefficients a(x) and b(x) which maydepend on x. This is a special case

2.3 Frobenius’s Theorem

Suppose that k non-vanishing vector fields Ua , a = 1, 2, . . . k satisfy

[Ua, Ub] = −Dac

bUc , (64)

where the functions Dac

b = D(x)ac

b are called structure functions, then thevector fields are tangent to some l-dimensional sub-manifold with 0 < l ≤ k.

Note that if the structure functions are independent of position they becomethe structure constants of some Lie-algebra.

2.3.1 Example: The simplest non-trivial Lie algebra

x∂

∂xand

∂

∂x. (65)

Here k = 2, l = 1. Changing the x ∂∂x to a(x) ∂

∂x would lead to a structurefunction rather than a structure constant , but the two vector fields would stillbe tangent to a one-dimensional manifold.

2.3.2 Example: Two commuting translations

∂

∂xand

∂

∂y. (66)

Here k = 2 and l = 2

2.3.3 Example Kaluza or Torus Reductions

Following the initial idea of Kaluza one could imagine a 4+m dimensional space-time invariant under the action of the abelian torus group Tm = (S1)m.Therewill be m mutually commuting Killing fields Ka, a = 1, . . . µ

[Ka,Kb] = 0 . (67)

By a combination of the Froebenius theorem and the discussion above aboutadapted coordinates, we may introduce coordinates ya such that

Ka =∂

∂ya, (68)

and the metric takes the form

ds2 = gab(xλ)(dya +Aa

µ(xλ)dxµ)(dyb +Abν(xλ)dxν) + gµν(xλ)dxµdxν . (69)

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One may interpret gµν(xλ) as the metric on our lower dimensional spacetimewith coordinates xµ. The co-vector fields Aµ(xλ) may be interpreted as mabelian gauge fields which are called gravi-photons. This works because thecoordinates ym are determined only up to a coordinate transformation of theform

ya → ya = ya + Λa(xλ), (70)

under which the one-forms Aaµ(xλ) undergo a gauge transformation

Aaµ(xλ)→ Aa

µ(xλ)− ∂µΛa(xλ) . (71)

The fields gab(xλ) are invariant under both gauge transformations, and

(u(1)m gauge-transformations. They are scalar fields, called gravi-scalars, whichtake their values in the space of symmetric matrices or m-dimensional metricswhich as we shall see may be identified with the symmetric spaceGL(m,R)/SO(m).It is sometimes convenient to split of the determinant det(gab) and to considerunimodular matrices in which case the remaining scalrs take their values inSL(m,R)/SO(m). Some multiple of the logarithm of det(gab) is often calledthe dilaton .

What has been achieved here is to embed the local (U(1))m gauge group as asubgroup of the the diffeomorphism group of the higher dimensional spacetime.The analogue of time reversal is charge congugation symmetry under whichya → −ya. If it holds for one of the Aa

µ’s then we may set that Aaµ = 0 and the

U(1) ≡ SO(2) contribution to the isometry group is augmented to O(2).

2.3.4 Example: so(3)

Lz = x∂

∂y− y ∂

∂xand cyclically. (72)

Here[Lx, Ly] = −Lz and cyclically, (73)

and we get the Lie algebra of the rotation group so(3). If r2 = x2 + y2 + z2, oneverifies that

Lzr2 = 0, and cyclically, (74)

Thus k = 3 but l = 2, all three vector fields are tangent to 2-spheres centredon the origin of three-dimensional Euclidean space E3. Note that while theLx, Ly, Lz generate rotations in the positive sense in the three orthogonal coor-dinate planes, the structure constants are the negative of the standard structureconstants for so(3), in other words

[Li, Lj ] = −ǫijkLk. (75)

This turns out to be universal sign reversal, as will be explained later.

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2.4 Lie Derivative and Exterior Derivative

It is important that d commutes with Lie, i.e.

LU (dω) = d(LUω), (76)

for any p-form field ω and vector field U . This is no more than the fact thatthe exterior derivative commutes with pull-back, but it may be proved moreconcretely by writing out the relevant expressions in local coordinates. Anotherimportant fact is that one may swap the exterior derivative for the bracket.Thus, for a 1-form ω

〈dω|U, V 〉 = U〈ω|V 〉 − V 〈ω|U〉 − 〈ω|[U, V ]〉. (77)

2.5 Cartan’s formula: Lie derivative and interior product

In the same spirit one has the extremely useful formula

LUω = iUdω + d(iUω), (78)

where ω is a two-form and U a vector field.

2.5.1 Example: electrostatic potentials

Suppose that ω = F , a Maxwell 2-form which by Faraday’s law is closed, dF = 0,and suppose F is invariant under time-translation or some other symmetry,generated by a vector field U , which could be a Killing vector field. We have:

£UF = 0 ⇒ iUF = dΦ, (79)

for some function Φ which plays the role of an an electrostatic or magneto staticpotential. Thus in the Schwarschild solution, for example, if

F = Qdt ∧ drr2

and U =∂

∂t, (80)

then

Φ = −Qr. (81)

This example can readily be generalized to higher rank forms which appear inhigher dimensions in supergravity and superstring theory.

3 Integration on Manifolds: Stokes’ Theorem

The basic idea is that ne can only integrate p-forms over p-chains. Roughlyspeaking, a p-chain is a p-dimensional sub-manifold which may be regarded asthe sum of a set of p-cubes (or alternatively of p-simplices). Each p-cube is theimage in M under some map φ : Rn →M of a standard p-cube in Rn. In other

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words, a p-cube C pushes foward to M to give a curvilinear p-cube C⋆ = ψ⋆C.In order to integrate a p-form ω in M over the p-chain Cstar in M we pullω back to Rn to give ω⋆ = φ⋆ω and integrate it over C using the standardintegration procedure, i.e by means of the definition

∫

C⋆

ω =

∫

C

ω⋆ . (82)

On each p-cube C ∈ Rn we may introduce coordinates 0 ≤ x1 ≤ 1, 0 ≤ x2 ≤1, . . . , 0 ≤ xp ≤ 1.

∫

C

ω⋆ =1

p!

∫

c

ω⋆µ1µ2...µp

dxµ1 ∧ dxµ2 . . . ∧ dxµp (83)

=

∫

. . .

∫

ω12...pdx1dx2 . . . dxp . (84)

A general p-chain is a sum of p-cubes C =∑

iCi. To each p-chain C weasscociate a boundary ∂C =

∑

i ∂Ci. where adjacent cubes contribute with theopposite sign and henc cancel. Thus the boundary of a boundary vanishes:

∂2C = 0 . (85)

By linearity and Because d commutes with pull-back, Stokes’s theorem nowreads

∫

C⋆

dω =

∫

∂C⋆

ω . (86)

In other words we only need only check this formula on a standard p-cube inRn. To check that, recall that

dω⋆µ1µ2...µp+1

=(p+ 1)!

p!∂[µ1

ω⋆µ2...µp+1] . (87)

Thus∫

C

1

(p+ 1)!dω⋆

µ1...µp+1dxµ1 ∧ . . . ∧ dxµp+1 (88)

=

∫

C

(p+ 1)!

(p+ 1)!

1

p!∂[µ1

ω⋆µ2...µp+1] dx

µ1 ∧ . . . ∧ dxµp+1 (89)

=

∫

∂C

1

p!ω⋆

µ1µ2...µpdxµ1 ∧ . . . ∧ dxµp , (90)

where the last line follows from integration by parts.As an example let F =∈ θdθ ∧ dφ.

∫

S2

=

∫ ∫

sin θdθdφ = [− cos θ]π0 [φ]2π0 = 4π . (91)

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As a corollary, we see that F is not exact F 6= dA because if so

∫

FS2 =

∫

S2

dA =

∫

∂S2

= 0 , (92)

because S2 has no boundary ∂S2 = 0.More generally the volume form of any compact manifold is closed but never

exact.

3.0.2 The divergence operator

If we define an inner product on Ω∗(M) by

((α, β)) =

∫

M

(α, β)η =

∫

M

⋆α ∧ β =1

p!

∫

αµ1...µpαm1...µp

√

|g|dn x . (93)

then the formal adjoint 2 of δ of d is defined by

((α, dβ)) = ((δα, β)). (94)

In components

δαµ1...µp= −∇µ0αµ0µ1...µp

. (95)

Now if λ ∈ Λp, ⋆ ⋆ λ = (−1)t(−1)p(n−p)λ and

λ ∧ µ = (⋆, µ)η ,⇒ ⋆λ ∧ µ = (−1)t(−1)p(n−p)(λ, µ)η , (96)

⇒ (α, dβ)η = (−1)t(−1)(p+1)(n−p−1) ⋆ α ∧ dβ , (97)

for α ∈ Λp+1. Moreover

d(⋆α ∧ β) = d ⋆ α ∧ β + (−1)n−1−p) ⋆ α ∧ dβ, (98)

hence up to a boundary term

∫

M

(d ⋆ α) ∧ β = (−1)(n−p)

∫

M

⋆α ∧ dβ . (99)

Thus∫

M

(⋆d⋆, β)η = (−1)(n−p)(−1)t(−1)p(n−p)

∫

m

(α, dβ)η . (100)

Thus up to a sign

δ = ± ⋆ d⋆⇒ δ2 = 0 .

(101)

2It is a formal adjoint because we are not worrying about boundary terms

14

3.1 Brouwer degree

Let φ : M → N be a map between two manifolds of the same dimension. let Nbe equipped with a volume form ηN . The Brouwer degree of the map φ is aninteger given by

deg φ =

∫

Mφ⋆ηN

∫

MηM

. (102)

Intuitively degφ is the number of times the map φ ‘wraps’or ‘winds’themanifold M over N .

It is also the number of inverse images of the map at a generic point x ∈ Ncounted with respect to orientation. That is subtracts the those for which φpreserve orientation from those which reverse it.

3.1.1 Gauss Linking Number

Suppose that γ1 and γ2 are two connected closed curves immersed into E3 andwhich do not intersect each other γ1 ∩ γ2 = ∅. Thus

γ1 : S1 → E3 t→ x1(t) 0 ≤ t < 2π , (103)

γ2 : S1 → E3 s→ x2(s) 0 ≤ s < 2π . (104)

The unit vector

n(t, s)x2(s)− x1(t)

|x2(s)− x1(t)|(105)

gives a map φ : S1×S1 → S2 called the Guass Map. The Gauss-Linking number

Link(γ1, γs) is defined by

Link(γ1, γ2) = deg φ =1

4π

∫

S1×S1

φ⋆η , (106)

where η is the volume form on S2. If x = x2 − x1,

η =zdx ∧ dy + ydz ∧ dx+ zdy ∧ dz

(x2 + y2 + z2)12

(107)

and

Link(γ1, γ2) =1

4π

∮

γ1

∮

γ2

(x2 − x1)× dx1(t).dx2(s)

|x2 − x1|3(108)

=

∮

γ1

B2(x1).dx1 , (109)

where

B2(x) =1

4π

∮

γ2

(x2 − x)× dx2

|x2 − x|3 (110)

is, by the Biot-Savart Law, the magnetic field due to a unit current along γ2.The reader should have no difficulty generalizing this example to the linking

of a p-brane and a q-brane in p+ q + 1 dimensions.

15

3.1.2 Gauss-Bonnet Theorem

Suppose that S is a closed connected orientable 2-surface immersed into E3

given in any local coordinate neighbourhood on Σ by

x = x(u, v), (111)

where u, v are local coordinates on Σ and

n(u, v) =∂ux× ∂vx

|∂ux× ∂vx|(112)

is the unit normal. The normal provides the Gauss map φ : Σ → S2. Acalculation, which the reader is invited to do, shows that

φ⋆ηS2 = KηΣ (113)

where K is the Gauss-curvature of Σ, whose induced metric is

ds2 = ∂ix∂jxdxidxj = gijdx

idxj = Edu2 + 2Fdudv +Gdv2 . (114)

That is the Riemann tensors is

Rijmn = K(gimgjn − gingjm). (115)

.Consideration of the Brouwer degree of φ then leads to an extrinsic version

of the Gauss-Bonnet theorem

1

2χ(Σ) = 1− g(Σ) =

1

4π

∫

K

√

EG− F 2dudv ∈ Z , (116)

where χ is the Euler number and g(Σ) is the genus or number of handles of thesurface Σ.

The reader shold have no dificulty generalizing this argument to an p-branein p+ 1 dimensions.

3.2 A general framework for classical field and brane-theories

Many physical theories can be cast in a,possibly procrustean, framework basedon the space of maps Map(Σ,M) = φ |φ : Σ → M from a manifold Σ toanother manifold M say.

3.2.1 Particles

The simplest example is when Σ = R and we obtain a point particle moving in acurved spacetime M with φ : R → R given by xµ(λ), λ ∈ R being a parameteron the worldline of our particle. Map(R,M) is then just Feynman’s space of

histories of our particle.

16

The action (for massless particles) is

S =1

2

∫

R

N(λ)dλ gµνdxµ

dλ

dxν

dλ. (117)

Variation with respect to the Lagrange mutiplier N(λ) gives the massless con-dition or constraint.

gµνdxµ

dλ

dxν

dλ= 0 . (118)

Mathematically, we may regard N−2dλ2 as a metric on Σ, and N−1dλ is thevolume form or ein-bein on Σ and

gµνdxµ

dλ

dxν

dλ= (φ⋆g)λλ (119)

is the pull-back of the metric gµν on M to Σ.

3.2.2 Strings, Membranes and p-branes

For strings we take Σ to be 2-dimensional. For membranes Σ is three-dimensionalIf dimΣ = p + 1 we get a p-brane3 Now let λi ,i = 1, 2, . . . p + 1 be local coor-dinates on § and γij a metric on §. The pull-back to Σ of the spacetime metricon M is given by

(φ⋆g)ij = g⋆ij = gµν

∂xµ

∂λi

∂xν

∂λj(120)

and

γijg⋆ij = γijgµν

∂xµ

∂λi

∂xν

∂λj(121)

is it’s trace with respect to γ of the pull-back of g The volume form on Σ is

ηγ =√

det(γij) dp+1 λ , (122)

and a suitable action is

S = =1

2

∫

Σ

√γ dp+1 λγij gµν

∂xµ

∂λi

∂xν

∂λj(123)

=1

2

∫

Σ

(Trγφ⋆)ηγ . (124) polyakov

In string theory this is called the Polyakov form of the action. The indepen-dent fields to be varied are the embedding φ, i.e. one varies the world-volumescalar fields xµ(λi), and the world volume metric γij . A different strategy is toto set

γij = g⋆ij = gµν

∂xµ

∂λi

∂xν

∂λj(125)

3p is the spatial dimension of the object and p+1 the dimension of the world-sheet or

world-volume. Sometimes, when no notion of time is involved, a p-brane is taken to be the

same as a p-dimensional immersed sub-manifold

17

That is one identifies the world volume metric with the metric induced on theworld volume from the spacetime metric, via pull-back. The Dirac-Nambu-Goto

Action is now taken to be

S =1

2

∫

§

ηφ⋆g =1

2

∫

§

√

|gµν∂xµ

∂λi

∂xν

∂λj| dp+1 λ . (126)

This is to be varied with respect to the embedding (or immersion) φ, that iswith resepct to the world-volume fields xm(λi. The reader is invited to carry outthat exercise. In the mathematical literature solutions of the Euler-Lagrangeequations are called minimal submanifolds even though they may in fact onlybe stationary points of the Dirac-Nambu-Goto action functional.

3.2.3 Non-linear sigma models

So far we have been thinking of the target M of the map φ as our spacetimeand the domain § as some particle or extended object moving in it. Howeverwe can always change our minds and think of the domain § of the map asour spacetime and the target as some sort of field space or internal space. Thesimplest situation occurs when we think of the metric on § as fixed. For example,could take § to be Minkowski spacetime En−1,1. and we take the Polyakov forof action given above. We then get the action of in physics are called non-

linear sigma models the solutions of whose equations of motion are called inmathematics harmonic maps if the metric on § is positive definite and wave

maps if it is Lorentzian. The terminology arises because if §, γ = En or§, γ = En−1,1 then the equation of motion reduces to the linear Laplaceequation or the wave equation respectively. If the target metric is not flat, thethe equations of motion are non-linear. Historically the name non-linear sigmamodel arose in pion physics. One has four fields πi, σ which are subject (insuitable units) to a constraint

πiπi + σ2 = 1, (127)

The target space may thus be identified with the round 3-sphere S3 withstandard round metric. One may also think of S3 as the group manifold ofSU(2).The high degree of symmetry, SO(4) ≡ SU(2) × SU(2)/Z2, containschiral symmetries and so one also speaks of chiral models or principal chiral

models.Sigma-models, and indeed all geometric theories based on Map, (§,M) admit

in general two types of reparametrization invariance, of § and of M . In the themetrics admit isometries these give rise, via Noether’s theorems, to conservedcurrents on Σ. For example the reader is invited to check that in the case of asigma model,if M admits a Killing vector field K we can convert it to a Killing

1-form

Kg= Kg µ

dxµ = gµνKνdxµ, (128)

pulling back Kgto § gives a conserved current on Σ

JK = K⋆g , d ⋆ J = 0, (129)

18

We can then define a conserved Noether charge

QK =

∫

S

⋆JK , (130)

where S is a suitable spacelike hypersuface in §. Because d ⋆ JK = 0 , then byStokes’s theorem and assuming a suitable behaviour at infinity, the charge QK

will not depend on the choice of the spacelike hypersurface S.

3.2.4 Topological Conservation laws

The conservation of Noether charges depends upon the field equations holding,i.e. the fields must be on shell. Topological conservation laws hold indpendentlyof field equations and resemble the conservation of magnetic charges whoseconservation depends only on Bianchi identies.

Suppose that spacetime which we are regarding as § has dimension n topo-logically takes the form

§ = S × R, (131)

where S is a hypersurfaces and t ∈ R is a time coordinate. The field φ(x, t)gives rise to a one parameter family of maps φt : S →M . If M is equipped witha closed (n− 1)-form ω we can pull it back to § to give a closed (n− 1)-form

ω⋆ = φt⋆ω , dω = 0⇒ d⋆ω = 0 . (132)

The topological charge

Qω =

∫

S

ω⋆, (133)

will, subject to suitable behaviour at infinity, be independent of t and thusconserved. If M is (n− 1)-dimnsional we can choose for ω the volume form ηM ,in which case the conserved cahrge is proprotonal to the Brouwer degree degφt

which is clearly quanitized and thus constant.Consider the Skyrme model. This is a non-linear theory of pions π(x, t)with

higher derivatives. It is based on maps into SU(2) given by x → U(x) =exp(iπiτi), with τi being Pauli matrices. Supppose πi(x) tends to some singlevalue at infinity, independently of direction. Then we may extend the map toS3, the one-point compactification of E3. Thus we get a map S3 → SU(2) = S3

and its Brouwer degree is interpreted as Baryon number in this model.In fact the one-point compactification is a conformal compactification. Con-

sider stereographic projection f : S3 \ north pole→ E3.Explicitly f : (X1, X2, X3, X4) → (r sin θ cosφ, r sin θ sinφ, r cos θ), X4 =

cosχ, sinχ = 2r4+r2 .

We can pull U back to get a map f⋆U : S3 \ north pole → S3, which weextend continously over the north pole to give our map φ : S3 → S3 whosedegree we must calculate. A hedgehog configuration is one for which πi =h(r)(sin θ cosφ, sin θ sinφ, cos θ) with h(0) = 0 and h(∞) = π. It has degreeone.

19

Note that,since stereographic projection preserves angles, the compactifica-tion is conformal. In fact, if ground is the standard metric on S3 and gflat thestandard flat metric on E3, then

f⋆gflat = Ω2ground , (134)

with Ω = 12 (4 + r2).

An analogous construction works in Instanton Physics. The Yang-Mill’sequations in four dimensions are conformally invariant and solutions on E4 andcan be pulled by to S4 by stereographic projection.

3.2.5 Coupling of p-branes to (p+2) forms

To begin with, consider a charged particle, with charge e moving along a worldline § in a manifold M with electromagnetic field F = dA ∈ Ω1(M). One addsto the usual action an interaction term

Sint =

∫

§

eAµdxµ =

∫

§

Aµdxµ

dλdλ =

∫

§

φ⋆A , (135) int

where

φ⋆A = Aµdxµ

dλdλ . (136)

In this notation we can carry over the same expression of a p brane §, withdimΣ = p + 1 moving in the background of a closed p + 2 form F + dA ∈Omegap(M). Moreover Sint is obviously invariant under abelian gauge trans-formations A→ A+ dΛ , Λ ∈ Ωp(M) up tot a surface term because

Sint → Sint +e

∫

Σ

φ⋆dΛ (137)

= Sint +e

∫

Σ

dφdΛ (138)

= Sint +e

∫

§

φ⋆Λ . (139)

The brane equations of motion will thus be gauge-invariant.In local coordinates, and using indices, we get the rather formidable expres-

sion

Sint =e

(p+ 1)!

∫

§

√γ Aµ1...µp+1

∂xµ

∂λi1. . .

∂xµp+1

∂λip+1

ip+1

ηi1...ı1+p dλ1 . . . δλp+1 .

(140)is time-consumimg to type in Latex and and showing that it is invariant up toa boundary term is rather painful, particularly if the details have to be typedin Latex.

The best known case is p = 1, which corresponds to string theory. One hasa closed 3-form

H = dB ∈ Ω3(M) , Hµνλ = ∂µBνλ + ∂λBµν + ∂νBλµ (141)

20

and

Sstring =1

2

∫

§

(gµνγij +Bµνη

ij)∂xµ

∂λi

∂xν

∂λj

√γ d2 λ . (142)

3.2.6 Topological Defects

In so-called mean field theory approaches to condensed matter physics, or incosmology, the space φ in which the field or order parameter φ takes it valuesis typically non-trivial. The simplest case to consider is the global vortex. Forexample in theories involving a field C -valued φ(x, y) in 2 + 1 dimensions theenergy, is

∫

E2

d2 x(1

2|∇φ|2 + V (φ)

)

(143)

and the potential energy V (φ) may be SO(2)-invariant: V (φ) = V (|φ|). Localminima of V define a vacuum manifold N ⊂M = C which will be a collection ofcircles about the origin in C . Suppose there is only one, of finite radius on which

V (φ) attains its absolute lower bound. For example take V (φ) = |φ|4

4 −|φ|2

2 .The vacuum manifold is the unit circle |φ| = 1

On a circle S1∞ ⊂ E2 near infinity, finite energy forces |Φ| = 1, Thus we

get a map S1∞ → S1 whose degree is may be interpreted as a conserved vortex

number. Note that if the vortex number is non-zero, by an argument identicalto the proof of the fundamental theorem of algebra , there must be at least onezero of φ inside S1

∞. Such zeros are interpreted as vortex positions.This example can be trivially generalized to three dimensions with E2,C ≡

R2, SO(2), and S1 replaced by E3,R3, SO(3) and S2, and the words global

vortex replaced by0 the words global monopole.The defects can also be local is the usual gradient operator ∇ is replaced by

a covariant derivative operator D with respect to a U(1) or SU(2) gauge group.

3.2.7 Self-interactions of p-forms

The free action for p-form F = dA is

S = −1

2

1

p!

∫

M

(F, F )η = −1

2

1

p!

∫

M

F ∧ ⋆F (144)

The variation is

δS =(−1)p−1

p!

∫

M

δA ∧ d ⋆ F . (145)

which yields the linear equation of motion

d ⋆ F = 0 . (146)

To get an interacting, i.e. nonlinear theory, we could try an action con-structed out of higher exterior powers of F . It would have the interesting prop-erty of being ‘topological ’independent of any metric on our manifold M . Theproblem is that such a term would either vanish identically, e.g F ∧ F ∧ F for

21

a 3-form in nine dimensions, or or not contribute to the equations of motionbecause it is exact. For example F ∧F ∧F in six dimensions. In all dimensionsin fact we have

F ∧ F ∧ F = d(A ∧ F∧). (147)

However this suggests that in five dimensions

SChern−Simons = cC−S

∫

M

A ∧ F ∧ F , F ∈ Ω2(M) , (148)

where cC−S is a coupling constant, is a possibility. Under a gauge transformationA→ A+ dΛ

A ∧ F ∧ F → A ∧ F ∧ F + d(Λ ∧ F ∧ F ) , (149)

and so the action SChern−Simons changes by a surface term which will not affectthe equations of motion.

Now up to a surface term

δSChern−Simons = 3cC−S

∫

M

δA ∧ F ∧ F . (150)

Thus the non-linear equations of motion are

d ⋆ F − 6cC−SF ∧ F = 0 . (151)

In 3 dimensions the Chern-Simons term gives a mass to the photon. Ineleven dimensional supergravity, which has a 4-form, it gives

d ⋆ F ∝ F ∧ F . (152)

22