Degree in Physics∗ Daniel S. Park April 29, 2013

Abstract We review the mathematical concept of degree. We introduce degree by its integral formula and state its local definition. We also see how it nicely describes properties of gauge bundles on manifolds. Finally, we present some examples of topologically charged objects in physics and see how the concept of degree is useful in understanding them.

1

Introduction

Consider maps from a unit radius circle (S 1 ) to another circle. These maps can be classified by ‘winding number,’ i.e., how much the map winds the initial circle around the final circle. This is a topological number. The concept ‘degree’ is a generalization of this notion for maps between any two manifolds of the same dimension. The concept of degree naturally arises in physics when we consider topologically charged objects [1]. This is because gauge symmetry plays a fundamental role in nature. Topologically charged objects living on some manifold M are described mathematically by a nontrivial gauge bundle whose base is M . For example, an instanton on S 4 is one such object. The bundle is defined by gauge group valued transition functions — or clutching functions — on the overlaps of local coordinate patches. These transition functions are maps from the overlap to the gauge manifold. In certain examples of physical interest, the degree of these transition functions turn out to completely specify the topology of the gauge bundle. In this exposition, we review the mathematical definition of degree and demonstrate its utility in physics through several examples. This exposition is organized in the following way. In section 2 we define degree as an integral of a certain form. We also present how it can be defined locally. In section 3 we see how degree arises in the context of gauge theory. In particular, we relate the degree of certain clutching functions to Chern-Simons forms and Chern characters of gauge bundles. In section 4 we relate the mathematics to physics and see how important topologically charged objects — monopoles, instantons and vortices — can be conveniently described using degree. We also examine how anomalies in gauge theories can be understood from a topological point of view. Concluding remarks are made in section 5. ∗

This is a note based on a term project for MIT course 8.871 “Geometrical and Topological Methods in Physics” taught by Professor Frank Wilczek in 2009.

2 2.1

Definition of Degree Integral Definition

Consider a one-dimensional closed curve C on the plane with a punctured origin. The number of times this curve winds around the origin is a topological invariant. This can be counted by computing the ‘solid angle’ the curve covers, that is, Z −ydx + xdy 1 , (1) n= 2π C r2 where we have used r, θ to denote the polar coordinates of this plane. This formula follows from projecting C onto a unit circle centered at the origin. Note that the projection preserves winding number. Then (1) translates into Z 1 dθ (2) n= 2π C 0 in terms of polar coordinates of the projected curve C 0 . Now let us move up a dimension to two dimensions. Consider a two-dimensional compact surface S in R3 \ {0}. The number of times S ‘wraps’ the origin can be counted by projecting S onto the unit S 2 centered at the origin. The ‘wrapping number’ of S is given by Z 1 dΩ , (3) n= 4π S 0 where S 0 is the projected image of S and dΩ is the volume element of S 2 . The factor of 1/4π is there since the area of the unit sphere is 4π. We can carry on this practice for arbitrary high dimensions. The number of times a D dimensional compact surface S in RD+1 \ {0} wraps around the origin can be counted by projecting S on to the unit S D centered at the origin. Denoting the resulting manifold S 0 the winding number is Z 1 dV , (4) n= Volume(S D ) S 0 where dV is the volume of a unit S D . The topological invariant we are actually counting in each case is how many times the projection map pD : S → S D is wrapping S around S D . We can generalize this notion to any smooth map f : M → N between orientable compact surfaces M and N of the same dimension. We will call the number of times the map f winds M around N the degree of the map f . Taking ω to be the volume form of N , the degree of f is defined to be R f ∗ω R deg f = M , (5) ω N where f ∗ ω denotes the pullback of ω by f [2]. Notice that the degree can be negative — this happens when f flips the orientation of M .

Figure 1: A degree 1 map S 1 → S 1 with ‘folds.’ Notice that the marked three points are mapped to a sigle point in this map.

2.2

Local Definition

Can we define the degree of a map only using local data? Let us imagine we have a map f : M → N that wraps orientable surface M around orientable surface N n times. Let us consider what the inverse image of a generic point1 x on N is. If the map f did not have any ‘folds,’ (see figure 1) that is, if it ‘tightly’ wrapped M around N , the answer would be n. This is, however, not the case in general. In order to obtain the winding number of a smooth map locally, we must cancel contributions coming from ‘folds.’ The way to do this is to take orientation into account. Consider the sign of detdf (the induced map of the tangent spaces) at each point of xi ∈ f −1 (y). One can observe that detdf has the opposite sign for two points x1 , x2 that come from folding, since the map of the volume form of M by dfx1 and dfx2 have opposite orientations. Therefore it follows that the degree of the map is in fact given by [3] X sign detdfxi . (6) xi ∈f −1 (y)

Let us summarize. For a smooth map f : M → N between oriented manifolds M and N of equal dimension, for any regular point y ∈ N of f the number X sign detdfxi (7) xi ∈f −1 (y)

is constant. This number is the degree of the map f as defined in (5).

2.3

A “Nontrivial Calculation” for an Interesting Example

Let us work out an interesting example that makes use of the concept of degree. We can define the linking number of two disjoint loops C1 and C2 embedded in the same three 1

The precise word to use would be ‘regular.’ A regular point in this context means a point y such that for points x ∈ M satisfying f (x) = y, the induced map df between tangent spaces at x and y is not singular.

dimensional space in the following way. Define the map λ : C1 × C2 → S 2 , λ(x, y) =

x−y ||x − y||

(8)

Then the linking number l(l1 , l2 ) — which counts the number of times the two loops are ‘linked’ — is defined to be the degree of λ. From the integral definition of degree (5), it follows that Z Z 1 (x − y)k i l(l1 , l2 ) = dx dy i ijk . (9) 4π l1 ||x − y||3 l2 Now consider two circles C1 : (− sin φ, cos φ, 0) and C2 : (0, cos θ + 1, sin θ) in R3 for φ, θ ∈ [0, 2π). We can explicitly draw the two circles and see that they are linked. (See figure 2.) We now show that the linking number as defined above is in fact −1. λ for these two circles is a map from a torus to a sphere with λ(φ, θ) =

(− sin φ, cos φ − cos θ − 1, − sin θ) p . 1 + 2(1 + cos φ)(1 + cos θ)

(10)

Let us compute the linking number of the two loops by first using the local definition of degree. Let us consider λ−1 (0, 0, 1). Just by examining the geometry we see that λ−1 (0, 0, 1) consists of only one point (φ, θ) = (0, 3π/2). This corresponds to the pair of points (0, 1, 0) and (0, 1, −1) in figure 2. The tangent vector at this points on C1 is (−1, 0, 0) while that on C2 is (0, 1, 0). These two tangent vectors are mapped to a pair of tangent vectors on S 2 as depicted in figure 2. One can see that the orientation of the two vectors are reversed. It therefore follows that l(l1 , l2 ) = −1 . (11) Meanwhile, using the integral equation (9), we see that Z 2π Z 2π (1 − cos φ)(1 + cos θ) − 1 1 dφ dθ p . deg λ = 4π 0 ( 2(1 − cos φ)(1 + cos θ) + 1)3 0

(12)

z 1’ 2’ C1

λ

C2

1

y 2 x x

Figure 2: The map λ takes a pair of points on C1 × C2 to S 2 . The pair of points (0, 1, 0) and (0, 1, 1) is taken to the north pole. dλ maps the two tangent vectors 1, 2 to 10 , 20 . One can observe from the mapping of the vectors that that f is orientation reversing, i.e., detdf < 0 at this point.

We have thus arrived at the nontrivial equality Z 2π Z 2π (1 − cos φ)(1 + cos θ) − 1 dθ p dφ = −4π , ( 2(1 − cos φ)(1 + cos θ) + 1)3 0 0

(13)

which can be verified numerically. In this example, we have found a degree-one mapping from a torus to a sphere. In general, a degree n mapping from a torus to a sphere can be constructed by taking C1 to be a circle and C2 be a closed helix twisting around the circle n times before closing. Specific coordinate constructions would likewise yield nontrivial integral identities.

3

Degree and Gauge Bundles

3.1

Gauge Bundles over S n and Clutching Functions

Consider a gauge theory with gauge group G on a manifold. In this theory, we must consider the possible gauge bundles over the manifold modulo gauge transformations. Bundles are defined in terms of local connections on local patches and gauge group valued transition functions — or clutching functions — on their overlap. For the manifold S n , the situation is exceptionally simple since it can be covered by two open discs — one of which looks like the northern hemisphere extended a bit over the equator and one which looks like the southern hemisphere extended a bit over the equator. S Lie algebra valued gauge fields AN µ and Aµ are defined as a local functions on each patch. At the overlap at equator, which is isomorphic to (−, ) × S n−1 , these two are related by a clutching function g such that −1 ASµ = gAN + ig∂u g −1 . µg

(14)

Note that we have defined Aµ so that the covariant derivative is given by2 ∇µ = ∂µ − iAµ .

3.2

(15)

Topological Charge and Clutching Functions

In the context of topological charge, a more convenient — although less rigorous — way to view a gauge bundle is to think of the two patches of the sphere as the northern and southern hemisphere, which we denote as patch N and S respectively. We then treat the clutching function g as a function g : S n−1 → G. This is possible because we can smoothly deform g to be constant in the direction orthogonal to the equator, i.e., in the (−, ) direction. Since g : S n−1 → G, the classes of clutching functions that are deformable in to each other smoothly is classified by the homotopy group πn−1 (G). In the case that dim G = (n−1) the degree of the clutching function is a topological index — a number that is equivalent for 2

For curved space, we should actually include the metric connection when defining covariant derivatives. When discussing topological charges on S n , however, we can get away with ignoring the curvature, since the Dirac genus of an n-sphere is trivial.

two gauge bundles that can be smoothly deformed into one another. A natural question to ask in this context is whether we can identify this number with another known topological index associated with gauge bundles. We show that we can indeed do so for important examples in physics in section 4. Before we move on, however, let us first examine in detail the topological index of the gauge bundle that we eventually identify the degree of the clutching function with.

3.3

Chern Characters and Chern-Simons Forms

An important topological index that describes the gauge bundle on S 2k is the integral of the kth Chern character. The Chern character is a order-2k differential form defined as  k F 1 (16) chk (F ) = tr − k! 2π where F is the curvature two form of the gauge field. Integrating the differential form over the manifold we obtain Z I= chk (F ) . (17) S 2k

This number counts the difference between the number of positive chirality zero modes and the number of negative chirality zero modes of a Dirac fermion charged under the gauge field.3 Chern characters are closed forms. They therefore can be written on a local patch as chk (F ) = dQ2k−1 (F, A) .

(18)

Q2k−1 (F, A) is called the Chern-Simons form for the Chern character. It follows that Z Z I= chk (FN ) + chk (FN ) S ZN = (Q2k−1 (FN , AN ) − Q2k−1 (FS , AS )) S 2k−1

where S 2k−1 denotes the equator. We note that the respective orientation of the boundaries of patches are accounted for by the relative minus sign. It turns out that the integrand in the last line does not depend on AN and AS but rather only on g up to an exact form. The result of the integral is in fact [4] Z i k k−1 (k − 1)! I = (−1) ( ) tr[(gdg −1 )2k−1 ] , (19) (2k − 1)! 2π 2k−1 S which only depends on the transition function g. This is expected, since the topological non-triviality of the bundle is encoded in g. 3

We discuss this point further in 4.2.

4

Examples in Physics

4.1

Topological Charges of Gauge Bundles on S 2k

In this section, we examine two important objects in physics; the U (1) monopole and the SU (2) instanton. There is a unified way of viewing the two entities: the important physics is described by the topology of gauge bundles over S 2k with a gauge group of dimension (2k − 1). Let us first describe monopoles in terms of a gauge bundle. Consider a sphere wrapping around a monopole. Only caring about the magnetic flux piercing the sphere, we may describe the monopole by a gauge bundle over the sphere with magnetic flux Z F = (magnetic charge) . (20) S2

Now we know that F is proportional to the first Chern character of the bundle Z Z F = −2π ch1 (F ) . S2

(21)

S2

Hence the magnetic charge is a topological charge that is quantized as (magnetic charge) = 2πn,

n ∈ Z.

(22)

Meanwhile, denoting the clutching function at the equator as g : [0, 2π) → U (1),

g(φ) = eiΛ(φ) ,

(23)

we see from (19) that Z i gdg −1 . (24) ch1 (F ) = 2π S 1 S2 We see that the right hand side is precisely the negative of the integral expression for the degree of the map g −1 .4 We can therefore make the identification Z 1 (magnetic charge) = − ch1 (F ) = deg g −1 . (25) 2π S2 Z

Let us carry out an analogous analysis of the SU (2) instanton on S 4 in this way. An SU (2) instanton configuration of instanton number n can be thought of as a gauge bundle with Chern character n. The quantization of the instanton number follows from the fact that the Chern character is an index, i.e., Z Z 1 n= 2 trF ∧ F = ch2 (F ) . (26) 8π S 4 S4 Let us relate this number to the degree of the clutching function of the gauge bundle as we did for the monopole. Denoting the clutching function at the equator as g : S 3 → SU (2) ,

(27)

From now on, g −1 will denote the inverse group element of g, not the inverse function of g. Hence g −1 can also be thought of as a map from S 2k to G. 4

we obtain from (19) that Z ch2 (F ) = S4

1 24π 2

Z

tr[(gdg −1 )(gdg −1 )(gdg −1 )] .

(28)

S3

Interpreting the right hand side needs a bit more work than for the monopole. The idea is that elements of SU (2) can be written as, g(xµ ) = x0 I + i~x · ~σ ,

x20 + ~x2 = 1

(29)

where ~σ are the pauli matrices in vector form. We see that the group manifold of SU (2) parametrized by (x0 , ~x) is just a unit S 3 . By explicit calculation, it can be shown that in this notation the three form of question reduces to tr[(g −1 dg)(g −1 dg)(g −1 dg)] = 12(x0 dx1 dx2 dx3 − dx0 x1 dx2 dx3 + dx0 dx1 x2 dx3 − dx0 dx1 dx2 x3 ) = 12ω , where ω is the volume form of the SU (2) manifold. Therefore (28) becomes Z Z 1 ch2 (F ) = 2 g −1∗ ω = deg g −1 . 2π 4 3 S S Hence we see that

Z n=

ch2 (F ) = deg g −1

(30)

(31)

S3

just as was with the monopole. We have shown that the charges of the U (1) monopole and the SU (2) instanton can be computed as a topological index of a gauge bundle over S 2k with gauge group of dimension (2k − 1). The topological index can be computed by integrating the Chern character of the bundle over the manifold. We have also shown that this index is equivalent (up to sign) to the degree of the clutching function that maps the equator S 2k−1 to the gauge group manifold.

4.2

Anomalies and Zero Modes

In this section we briefly review the statement that the integral of the Chern character of a gauge bundle is an index and comment on its implications. It is a well known fact that the Adler-Bell-Jackiw anomaly [5] for a U (1) axial symmetry of a fermion is given by a Chern character on a 2k dimensional manifold with trivial Dirac genus [6]. As pointed out in [7], the anomaly comes from the fermion measure in the path integral. Taking our 2k dimensional manifold to be M , this means that for the local axial rotation ψ(x) → exp(iγ 2k+1 α(x))ψ(x) (32) the path integral measure for the fermion transforms as Z [Dψ] → [Dψ](1 + i chk F (x)α(x) + O(α2 )) . M

(33)

Note that γ 2k+1 ψ+ = ψ+ for chirality (+) Weyl spinors, and γ 2k+1 ψ− = −ψ− for chirality (−) Weyl spinors. We take a gauge invariant measure for the fermions, where the basis for fermions is given by i/ ∇ψλ = λψλ for the covariant derivative ∇. Now due to the fact that {γ 2k+1 ,∇} / = 0 and 2k+1 2 (γ ) = 1, we see that for λ 6= 0 we may define the basis such that γ 2k+1 ψλ = ψ−λ .

(34)

Meanwhile, for zero eigenstates, γ 2k+1 and i/ ∇ can be simultaneously diagonalized — zero 2k−1 eigenstates have definite helicity. Therefore under constant axial rotation eiαγ the fermion modes ψλ with λ 6= 0 transform as ! ! ! 1 iα ψλ ψλ + O(α2 ) . (35) → iα 1 ψ−λ ψ−λ Meanwhile, the zero modes transform as ψ+ → (1 + iα)ψ+ ,

ψ− → (1 − iα)ψ− .

(36)

We therefore see that the fermion measure transforms as [Dψ] → [Dψ](1 + iα(n+ − n− ) + O(α2 ))

(37)

where n+ and n− are the number of chirality (+) and (−) fermion zero modes in the given gauge field background, respectively. Hence by taking α to be constant in (33) and comparing with (37) we see that Z n+ − n− =

chk F .

(38)

M

We have shown that the integral of the Chern character over a manifold of trivial Dirac genus gives the difference of the number of chirality (+) fermion zero modes and the number of chirality (−) fermion zero modes that couple to the gauge bundle. It follows that the degree of the clutching function of the U (1) monopole on S 2 and the SU (2) instanton on S 4 encodes non-trivial information about the zero modes of fermions coupled to a gauge field with topological charge.

4.3

Vortices and Degree

Aspects of vortices in superconductivity can also be understood using degree. Assume a simple (2 + 1) dimensional system where we have a single complex scalar φ with some charge p — where we have taken the unit charge to be 1 — that brings about superconductivity by condensing due to a potential with U (1) symmetry. We can have ‘vortex solutions’ depending on the system, namely when we have a type II superconductor [1]. The U (1) symmetry is broken to Zp . Hence in order for φ to be well defined, the asymptotics of a vortex solution should behave as φ = vg(θ),

g : S 1 → U (1)/Zp .

(39)

Here, v is the vacuum expectation value for φ and θ parametrizes the angular coordinate at infinity. Meanwhile, the magnetic flux through the vortex is given by Z Z Z i 2π F = A= (40) g −1 dg = −( ) deg g , p S1 p S1 where S 1 is the circle at infinity. The second equality comes from the fact that in order to have finite energy the scalar field must behave as (∂µ − ipAµ )φ → 0

(41)

asymptotically. Therefore the magnetic flux of a vortex in this theory is quantized to be in units of 2π/p. The degree of the map g of (39) counts the number of vortices. Vortices have fractional flux quanta 1/p due to the fact that the U (1) gauge symmetry is not completely broken — it is broken to Zp . For generic p this leads to interesting physical effects in the presence of particles of unit charge still present in the system, as the unit charge particles can ‘feel’ the fractional flux [8].

5

Concluding Remarks

We have mathematically defined degree and have seen how it can be used to describe gauge bundles on manifolds, and hence physical objects with topological charge. In particular, we have considered gauge bundles on S n and have indentified the topological charge of objects of interest in physics with the degree of clutching functions of the bundles. Although a very simple concept, the degree of a map — which was equivalent to the Chern character of a gauge bundle in the examples we have considered — carries non-trivial physical information. In this exposition we have seen this in the context of physical gauge fields, but the same holds even in situations where gauge connections arise in more abstract settings. One such example that has generated much excitement lately is the topological insulator. It was shown recently that the Chern character of the ‘gauge bundle’ — coming from Berry’s phase — over the Brillouin zone of a time reversal invariant insulator has drastic implications on its transport properties. When the degree is even the insulator is an ordinary insulator, while when it is odd it is a topological insulator with novel features [9].

References [1] F. Wilczek, Lecture notes for “8.871 :: Selected Topics in Theoretical Particle Physics,” MIT, Cambridge, MA, 2009. [2] H. Flanders, “Differential Forms with Applications to the Physical Sciences,” Mineola, USA: Dover Publications (1989) 205 p [3] J. W. Milnor, “Topology from the differentiable viewpoint. Based on notes by David W. Weaver,” Princeton, USA: Princeton University Press (1997) 64 p

[4] M. Nakahara, “Geometry, topology and physics,” Boca Raton, USA: Taylor & Francis (2003) 573 p [5] S. L. Adler, “Axial vector vertex in spinor electrodynamics,” Phys. Rev. 177, 2426 (1969). J. S. Bell and R. Jackiw, “A PCAC puzzle: pi0 → gamma gamma in the sigma model,” Nuovo Cim. A 60, 47 (1969). [6] L. Alvarez-Gaume and P. H. Ginsparg, “The Topological Meaning Of Nonabelian Anomalies,” Nucl. Phys. B 243, 449 (1984). [7] K. Fujikawa, “Path Integral For Gauge Theories With Fermions,” Phys. Rev. D 21, 2848 (1980) [Erratum-ibid. D 22, 1499 (1980)], [8] F. Wilczek, “Some global problems in gauge theories (variations on a theme of Aharonov and Bohm),” Invited talk given at Conf. Thirty Years of the Aharonov-Bohm Effect, Columbia, SC, Dec 14-16, 1989. [9] Z. Wang, X. L. Qi, and S. C. Zhang, “Equivalent topological invariants of topological insulators,” arXiv:0910.5954v1 [cond-mat.str-el].