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8 Brief introduction to differential forms

“ cannot be understood without differential forms”

V. I. Arnold

In this chapter we give a very brief introduction to differential forms fol- lowing the chapter 7 of Ref.[26]. We refer the reader to [26, 9, 8] for more detailed (and more precise) introductions.

8.1 Exterior forms 8.1.1 Definitions Let Rn be an n-dimensional real . Definition An exterior algebraic form of degree k (or k-form) is a function of k vectors which is k-linear and antisymmetric. Namely:

ω(λ1ξ1! + λ2ξ1!!,ξ2,...,ξk)=λ1ω(ξ1! ,ξ2,...,ξk)+λ2ω(ξ1!!,ξ2,...,ξk), and ω(ξ ,...,ξ )=( 1)νω(ξ ,...,ξ ) i1 ik − 1 k with ( 1)ν = 1 ( 1) if permutation (i , . . . , i ) is even (odd). Here ξ Rn − − 1 k i ∈ and λ R. i ∈ The set of all k-forms in Rn form a real vector space with

(ω + ω )(ξ)=ω (ξ)+ω (ξ),ξ = ξ ,...,ξ 1 2 1 2 { 1 k} and (λω)(ξ)=λω(ξ).

Let us suppose that we have a system of linear coordinates x1, . . . , xn on n R . We can think of these coordinates as of 1-forms so that xi(ξ)=ξi - the i-th coordinate of vector ξ. These coordinates form a of 1-forms in the

65 66 Brief introduction to differential forms

n 1-form vector space (which is also called the (R )∗). Any 1-form can be written as a of basis 1-forms

ω1 = a1x1 + ... + anxn.

8.1.2 Exterior products

Definition An exterior product of k 1-forms ω1,ω2,...,ωk is a k-form defined by (ω ... ω )(ξ ,...,ξ ) = det ω (ξ ). 1 ∧ ∧ k 1 k i j Exterior products of basis 1-forms x ... x with i < . . . < i form a i1 ∧ ∧ ik 1 k basis in the space of k-forms. The of the latter space is obviously k Cn. A general k-form can be written as ωk = a x ... x , i1...ik i1 ∧ ∧ ik 1 i1<...

(ωk ωl)(ξ ,...,ξ )= ( 1)νωk(ξ ,...,ξ )ωl(ξ ,...,ξ ), ∧ 1 k+l − i1 ik j1 jl ! where i1 < . . . < ik and j1, . . . , jl and the sum is taken over all permutations (i , . . . , i ,j , . . . , j ) with ( 1)ν being +1( 1) for even (odd) permutations. 1 k 1 l − − One can check that this definition is consistent with the exterior product of 1-forms defined above and that the exterior product is distributive, asso- ciative, and skew-commutative. The latter means ωk ωl =( 1)klωl ωk. ∧ − ∧ If ω is a 1-form (or a form of an odd degree) one can easily show that ω ω = 0. ∧

8.1.3 Behavior under mappings Let f : Rm Rn be a , and ωk an exterior k-form on Rn. Then → k m we can define a k-form (f ∗ω ) on R by

k k (f ∗ω )(ξ1,...,ξk)=ω (fξ1,...,fξk).

Notice that the obtained mapping of forms f ∗ acts in “opposite” direction to f. Namely, f :Ω (Rn) Ω (Rm), where Ω (Rn) is a vector space of ∗ k → k k k-forms on Rn. 8.2 Differential forms 67 8.2 Differential forms Definition A differential k-form ωk at a point x of a M is an |x exterior k-form on the space TMx to M at x, i.e., a k-linear skew- symmetric function of k vectors ξ1,...,ξk tangent to M at x. If such a form is given at every point x of M and if it is differentiable, we say that we are given a k-form ωk on the manifold M. We introduce the coordinate basis 1-forms dxi with i =1, . . . , n (n- dimension of M). The notation dxi instead of xi used for exterior forms emphasizes that these basis forms act on TMx at a given point x of M. In the neighborhood of the point x one can always write the general differential k-form as

ωk = a (x) dx ... dx , (8.1) i1...ik i1 ∧ ∧ ik i1<...

n ∂f df = dxk, ∂xk x k=1 " ! " " where dxk are basis differential 1-forms. The value of this 1-form on a vector ξ TM is given by ∈ x n ∂f df (ξ)= ξk. ∂xk x k=1 " ! " "

8.2.1 Behavior under mappings Let f : M N be a differentiable map of a smooth manifold M to a → smooth manifold N, and let ω be a differential k-form on N. The mapping f induces the mapping f : TMx TMf(x) of tangent spaces. The latter ∗ → mapping f is called the differential of the map f. The mapping f is a ∗ ∗ mapping of linear spaces and gives rise to the mapping of forms defined on corresponding tangent spaces (see Sec. 8.1.3). As a result a well-defined differential k-form (f ∗ω) exists on M:

(f ∗ω)(ξ1,...,ξk)=ω(f ξ1, . . . , f ξk). (8.2) ∗ ∗ 68 Brief introduction to differential forms 8.3 Integration of differential forms over chains 8.3.1 Integration of k-form over k-dimensional cell k Let D be a bounded convex polyhedron in R and x1, . . . , xk an oriented on Rk. Any differential k-form on Rk can be written as ωk = φ(x) dx ... dx , where φ(x) is a differentiable function on Rk. We 1 ∧ ∧ k define the of the form ωk over D as the integral of the function φ:

k ω = φ(x) dx1dx2 . . . dxk. (8.3) #D #D Let us define a k-dimensional cell σ of n-dimensional manifold M as a polyhedron D in Rk defined above with a differentiable map f : D M. → One can think about σ as of “curvilinear polyhedron” - the image of D on M. If ω is a differentiable k-form on M we define the integral of a form ω over the cell σ as

ω = f ∗ω, (8.4) #σ #D where f ∗ is a mapping of k-forms induced by f (see Sec. 8.2.1). The cell σ inherits an orientation from the orientation of Rk. The k- dimensional cell which differs from σ only by the choice of orientation is called the negative of σ and is denoted by σ or by ( 1)σ. One can show − − that under a change of orientation the integral changes sign:

ω = ω. (8.5) σ − σ #− #

8.3.2 Chains and boundary operator It is convenient to generalize our definition of the integral of a form over cell to the integral over chain. Definition A chain of dimension k on a manifold M consists of a fi- nite collection of k-dimensional oriented cells σ1,...,σr in M and integers m1, . . . , mr, called multiplicities. A chain is denoted

ck = m1σ1 + ... + mrσr. (8.6) One can introduce the structure of a commutative group on a set of k-chains on M with natural definitions of addition of chains ck + bk. The boundary of a convex oriented k-polyhedron D on Rk is the (k 1)- − chain ∂D on Rk defined as

∂D = σi, !i 8.4 Exterior differentiation and Stokes formula 69 where the cells σ the (k 1)-dimensional faces of D with orientations in- i − herited from the orientation of Rk (see [26]). One can easily extend this definition to the definition of the boundary of a cell ∂σ on M and then to the boundary of a chain (8.6) as

∂ck = m1∂σ1 + ... + mr∂σr. (8.7) ∂c is a (k 1)-chain on M. Additionally, we define a 0-chain is a collection k − of points with multiplicities and the boundary of an oriented interval AB' is B A. The boundary of a point is empty. − One can show that the boundary of the boundary of the cell is zero ∂(∂σ) = 0 and therefore

∂(∂ck)=0 for any k-chain ck. We write this property as ∂∂ =0. (8.8)

8.3.3 Integration of k-form over k-chains An integral of a k-form over k-chain (8.6) is defined as

k k ω = mi ω . (8.9) ck σi # ! #

8.4 Exterior differentiation and Stokes formula An exterior of the differential k-form ωk is a (k+1)-form Ω= dωk. One can define it using a choice of coordinates on M as Ω= dωk = da dx ... dx (8.10) i1...ik ∧ i1 ∧ ∧ ik if ωk is given by (8.1). Here!da is a 1-form of a differential of a function a(x). One can show that the definition (8.10) does not actually depend on the choice of coordinates (see [26] for more rigorous discussion). We think of the 1-form given by the differential of a scalar function (see Sec. 8.2) as of an external derivative of 0-form (scalar function). It is easy to see from the definition (8.10) that

d(dxi)=0. Using this property one can show that generally d(dω)=0, 70 Brief introduction to differential forms where ω is a general differential k-form. We write the latter property as

dd =0. (8.11)

One can easily prove the formula for differentiating an exterior product of forms d(ωk ωl)=dωk ωl +( 1)kωk dωl. (8.12) ∧ ∧ − ∧ If f : M N is a smooth map and ω is a k-form on N we have → f ∗(dω)=d(f ∗ω). (8.13)

8.4.1 Stokes’ formula One of the most important formulae in differential geometry is the Newton- Leibniz-Gauss-Green-Ostrogradskii-Stokes-Poincar´eformula:

ω = dω, (8.14) #∂c #c where c is any (k + 1)-chain on a manifold M and ω is any k-form on M. The formula is proved similarly to Green’s formula in .

In particular case when the boundary ∂c = 0 we have c dω = 0 (integra- tion of a complete derivative over closed ). $

8.5 Homologies and Cohomologies 8.5.1 Closed and exact forms Definition A differential form ω on a manifold M is closed if dω = 0. In particular, on a 3d we have dω2 =( A')ω3 =0 A$ ∇· is equivalent to ( A') = 0, i.e. the corresponding vector field is divergenless ∇· (we used (8.30)). Using Stokes formula (8.14) we have for a closed form

ωk =0, if dωk =0. (8.15) #∂ck+1 Definition A differential form ω on a manifold M is exact if ω = dµ, where µ is another differential form. By (8.11) all exact forms are closed. However, there are closed forms which are not exact. For example on a circle S1 parametrized by an angle φ [0, 2π] one can introduce the 1-form ω1 defined by ω1(∂ γ)=∂ γ, where ∈ t t ∂tγ is the “velocity” along the path φ = γ(t) which belongs to the tangent 8.5 Homologies and Cohomologies 71 space to S1 at φ. The form is obviously closed (dω1 is a 2-form which is automatically zero on S1 - one-dimensional manifold). However, T 1 ω = dt∂tγ =2π. 1 #S #0 Although ∂S1 = 0, the integral is not zero, and, therefore, ω1 is not exact. We notice that locally the introduced 1-form can be written as ω1 = dφ but this form is not valid at φ = 0. This is the general situation. According to Poincar`e’slemma any closed form is locally exact. The existence of locally but not globally exact closed form is related to topological properties of M.

8.5.2 Cycles and boundaries Definition A cycle on a manifold M is a chain whose boundary is equal to zero. Using Stokes formula (8.14) we have

k dω =0, if ∂ck+1 =0. (8.16) #ck+1 The chains which can be considered as boundaries of some other chains are called boundaries. By (8.8) all boundaries are cycles. However, generally, not all cycles are boundaries. The existence of cycles which are not boundaries is related to topological properties of M. For example, let us take one of the cycles of torus. It is the cycle (its boundary is zero). However it does not bound any chain on torus.

8.5.3 Homologies and Cohomologies The set of all k-forms on M is a vector space, the closed k-forms a sub- space, and the differentials of (k 1)-forms (exact k-forms) a subspace of − the subspace of closed forms. The quotient space (closed forms) = Hk(M,R) (8.17) (exact forms) is called the k-th group of the manifold M. An element of this group is a class of closed forms differing from one another only by an exact form. For a circle S1 we have H1(S1,R)=R. The dimension of Hk is called the k-th Betti number of M. The first Betti number of S1 is 1. The cohomology groups of M are important topological properties of M. 72 Brief introduction to differential forms Let us consider two k-cycles a and b such that their difference is a bound- ary of a (k+1)-chain, i.e., a b = ∂c (such cycles are called homologous). − k+1 If dωk = 0 we have from (8.15)

ωk = ωk, (8.18) #a #b that is cycles can be replaces one by another. The quotient group (cycles) = H (M) (8.19) (boundaries) k is called the k-th group of M. An element of this group is a class of cycles homologous to one another. The rank of this group is also equal to k-th Betty number of M.

8.5.4 Homologies and There are important relations between homology and groups of the topological space M (see e.g., [8]). Let us suppose that π1(M) and H1(M) are fundamental group and first homology group of M respectively. Then, H1(M)=π1(M)/[π1,π1], where [π1,π1] is the commutator of the group π1(M). In particular, if π1(M) is Abelian, then π1(M)=H1(M). For higher homotopy groups there is Gurevich theorem. Gurevich theorem If πk(M) = 0 for all k < n. Then

πn(M)=Hn(M). (8.20) Homology (and cohomology) groups are usually easier to calculate than homotopy groups.

8.6 Differential forms in 8.6.1 U(1) Gauge field n Let us suppose that we have a gauge field Aµ(x) in euclidian space R . It is very useful to think of the gauge field as of 1-form ν A = Aν dx . (8.21) The exterior differential of this form ∂A 1 F = dA = ν dxµ dxν = (∂ A ∂ A ) dxµ dxν ∂xµ ∧ 2 µ ν − ν µ ∧ 1 = F dxµ dxν (8.22) 2 µν ∧ 8.6 Differential forms in physics 73 is a 2-form which has the field Fµν as coefficients. If 1-form A is exact, i.e., A = df with some scalar function f we call A pure gauge. In components Aµ = ∂µf. In this case F = 0 as F = dA = d(df ) = 0 because of (8.11). On euclidian space Rn the inverse is also true. If F = dA = 0, i.e, A is closed, the form A is exact and there exists f such that A = df . This happens because Rn is a simply-connected space. Indeed, in a simply-connected space every 1-cycle c1 is a boundary c1 = ∂c2 (homologous to zero). Therefore, having F = 0 we have by Stokes theorem A = F = 0. c1 c2 $ Let us$ now assume that the 1-form A is given on a manifold M which is not simply connected π (M) = 0. Then, there are cycles which are not 1 & boundaries and the integral of A over such cycles might not vanish. It is possible to have, therefore, closed but not exact 1-forms A. In classical mechanics this is not essential as only 2-form F is physical but for closed 1- forms A we have F = dA = 0. In quantum mechanics, however the integral A = 0 is a phase which is picked up by quantum particle moving around c1 & $the cycle c1. One can observe the interference due to this phase even in cases where particle is never in the presence of “magnetic field” (F =0 everywhere). This phenomenon is called Aharonov-Bohm effect. The integral Φ= A = 0 can be interpreted as a magnetic flux through c1 & the “holes” of a multiply$ connected space M. For example, the flux through the ring for the particle on the ring or two different fluxes through the non- trivial basic cycles of 2d torus etc.

8.6.2 Dirac monopole

We have seen that the 2-form F corresponding to the field tensor Fµν is closed and this statement dF = 0 is equivalent to a pair of Maxwell’s equa- tions (Bianchi identity). On Rn this means that F = dA where A is some 1-form given globally on Rn and F is necessarily exact. Let us consider now the closed 2-form F on a non-trivial manifold M (think of M = S2 as an example). For a 2-dimensional surface c2 with boundary c1 = ∂c2 we have F - the flux through the surface c . According to Poincare’s lemma we c2 2 $can write F = dA with some 1-form A defined in the vicinity of c2 (as it can be contracted to a point). Then we have F = A. If the integral of F c2 c1 over the other surface c2! with the same boundary$ $∂c2! = c1 is different then the 1-form A can not be defined globally on M. Indeed

0 = F = (A A!). & c2 c c1 − # − 2! # 74 Brief introduction to differential forms It means that if the flux through some 2-cycle is not zero, the vector potential 1-form does not exist globally. As an example we consider the particle moving on a in a magnetic field of a point magnetic charge g placed in the center of the sphere B = g/R2, where R is the radius of the sphere. The flux of magnetic field through 2 the sphere Bd x =4πg and considering c2 and c2! as northern and southern hemispheres$ we have 1 4πg F = (A A!)= (∆φ ∆φ!), c2 c c1 − e − # − 2! # where e is the electric charge of the particle and ∆φ = e A is the phase c1 picked up by quantum particle when moved around the$ equator c1 of the sphere. For the consistency of quantum mechanics one should require that ∆φ ∆φ =2π integer. Therefore, we have the famous Dirac monopole − ! × charge quantization condition 2ge = integer. (8.23)

8.6.3 Theta and WZW terms for Let us consider the sphere Sn. We parameterize this manifold by “embed- ding coordinates” 'π =(π1,...,πn+1) with 'π 2 = 1. We have Hk(Sn,R)=R for k =0,n and 0 for k =0,n. The normalized n-form from Hn(Sn,R) is a & form on Sn which can be explicitly written as 1 ωn = *a1...an+1 πan+1 dπa1 ... dπan . (8.24) n! Area (Sn) ∧ ∧ Let us use this forms to write down the expressions for theta and WZW terms with spheres as target spaces.

8.6.3.1 Theta terms Consider D-dimensional spacetime SD and a target space M = SD. We introduce D Q[π(x)] = π∗ω , (8.25) D #S D where the integral is taken over the spacetime and π∗ω is the of the ωD existing on the target space M = SD. The value of (8.25) is the degree of mapping π : SD SD = M. → Notice, that this is a particular example where Gurevich theorem (8.20) is applicable and there is a correspondence between homotopies (degree of mapping) to cohomologies (integral of D-form of volume). 8.7 Exercises 75 The integer value Q is a topological invariant of the mapping π(x) and can be used to write down the topological theta-term as

Sθ[π]=iθQ[π], (8.26) where θ is a coupling constant.

8.6.3.2 WZW terms Assume now that the spacetime is SD but the target space is M = SD+1. The field configuration is given by the mapping π : SD SD+1 = M. We → extend this mapping continuously toπ ˜ : DD+1 SD+1 = M, where the → physical spacetime is the boundary of the disk SD = ∂DD+1. We write

D+1 WD[π]= π˜∗ω , (8.27) D+1 #D D+1 whereπ ˜∗ω is a pullback of the volume form on the target space to the auxiliary disk DD+1. The expression (8.27) is multiply-valued. Indeed, D+1 suppose that we have another extension of π(x) toπ ˜! on D !. We have

D+1 D+1 D+1 π˜∗ω π˜∗!ω = π˜∗ω D+1 − D+1 D+1 #D #D ! #S = ωD = integer. (8.28) D+1 #M=S We see that WD[π] is defined modulo integer and the action

SWZW [π]=i2πkWD[π] (8.29) is a well defined object provided the coupling constant k is integer. This is WZW term. To generalize this construction to arbitrary target space M one needs the existence of closed but not exact form ωD+1 on M, i.e., HD+1(M,R) = 0. &

8.7 Exercises

Exercise 1: Forms on R3 Consider the oriented euclidian space R3 (with a given scalar product). Every vector A' R3 determines a 1-form by ω1 (ξ')=(A' ξ') and a 2-form by ω2 (ξ' , ξ' )= ∈ A! · A! 1 2 (A' [ξ'1 ξ'2]). Show· × that

a) ω1 ω1 = ω2 , A! B! [A! B! ] ∧ × 76 Brief introduction to differential forms

b) ω1 ω2 =(A' B' )x x x . A! ∧ B! · 1 ∧ 2 ∧ 3

Exercise 2: on 3-dimensional manifold In a 3-dimensional oriented riemannian space M, every vector field A'(x) corre- sponds to a 1-form ω1 and ω2 defined similarly to exterior forms in a previous A! A! exercise. a) Show that

df = ω1 , dω1 = ω2 , dω2 =( A')ω3, (8.30) f A! A! A! ∇ ∇× ∇· where f is a 0-form (scalar function) and ω3 is a volume 3-form on M. b) Identify the Stoke’s formula applied to each equation in (8.30) with known formulae of vector calculus.

Exercise 3: Vector calculus on 3-dimensional manifold Using differential forms and the results of previous exercises show that

a) [A' B' ]=( A') B' ( B' ) A,' ∇· × ∇× · − ∇× · b) (aA')=(a) A' + a( A'), ∇× ∇ × ∇× c) (aA')=(a) A' + a( A'). (8.31) ∇· ∇ · ∇· Hint: Use the formula for the derivative of the product of forms. References

V. I. Arnold, Mathematical Methods of Classical Mechanics, Springer; 2nd edition (1997). Mikio Nakahara, Geometry, , and Physics, 2nd edition, Taylor & Francis, New York, 2003. M. Monastyrsky, O. Efimov (Translator), “Topology of Gauge Fields and Con- densed Matter” Plenum Pub Corp; March 1993; ISBN: 0306443368

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