EElectrostatics and Riemann Surfaces
Spencer Tamagni ∗ and Costas Efthimiou † Department of Physics, University of Central Florida, Orlando, FL 32816May 31, 2020
Abstract
Using techniques from geometry and complex analysis in their simplest form, we presenta derivation of electric fields on surfaces with non-trivial topology. A byproduct of thisanalysis is an intuitive visualization of elliptic functions when their argument is complex-valued. The underlying connections between these techniques and the theory of Riemannsurfaces are also explained. Our goal is to provide students and instructors a quick referencearticle for an extraordinary topic that is not included in the standard books.
Concepts from topology and geometry are ubiquitous in physics. In the era of the topologicalinsulator, this is quite evident. Moreover, geometrical and physical intuition are intimatelylinked: each aids in elucidating the key concepts of the other.Unfortunately, modern geometry is somewhat dense, and is inaccessible at the undergraduatelevel. In spite of this, the underlying ideas are concrete and intuitive. In addition, geometricalconcepts are usually introduced to physicists in relatively sophisticated physical contexts, typ-ically involving general relativity or quantum field theory. We aim to (partially) remedy thisissue by considering a 19th century physics problem which illustrates geometrical concepts —the formulation and solution of electrostatics on compact surfaces. In particular, we focus onthe simplest cases of the sphere and torus.We believe that working out this problem in detail is of pedagogical value because electro-statics is a standard subject which students of physics are held accountable for knowing well.Therefore, we may use it as a laboratory for exploring more unfamiliar geometrical ideas, whichturn out to involve the theory of Riemann surfaces. This also establishes that geometricalconcepts appear even at the introductory level in physics, although we tend to overlook them.We have structured this paper as follows: in the first section, we provide a brief reviewof some background mathematics from complex analysis. In section 2, we explore some topo-logical subtleties of electrostatics on compact spaces. In section 3, we explain how to use thestereographic projection to obtain electric fields on the sphere from familiar fields on the plane.Section 4 concerns fields on the torus, where we use geometry to explain a beautiful connectionbetween electrostatics and the theory of elliptic functions.In order to remain concise, we avoid exhaustive and general mathematical constructions,choosing instead to develop all concepts by example. In the concluding section, we direct the ∗ [email protected] † [email protected] The exact mathematical definition of a compact surface is not necessary for our purposes. The reader mayimagine such a surface as a finite surface (i.e. one that does not go to infinity) and without boundary (i.e. withouta borderline that belongs to it). a r X i v : . [ phy s i c s . g e n - ph ] J un nterested reader to the relevant literature to learn the general formulation of the concepts weillustrate. Our most essential tool will be the concept of a holomorphic function. One typically encounterssuch functions in a course on Mathematical Methods in physics and, of course, on ComplexAnalysis in mathematics. We will review the fundamental results that we will need in thissection.If we introduce the complex coordinate z = x + iy on the plane, we have that x = z + z , y = z − z i , where z = x − iy is the complex conjugate. Due to these formulas, a general function f ( x, y )will depend on both z and z when expressed in complex coordinates, f ( z, z ).The functions for which f ( z, z ) is in fact independent of z are known as holomorphic , writtenas f ( z ), and satisfy the trivial Cauchy-Riemann condition ∂f∂z = 0 . (1)We may write f = u + iv and then we have u = u ( x, y ); (2a) v = v ( x, y ) . (2b)Taking the real and imaginary parts of (1) gives the more conventional statement of the Cauchy-Riemann conditions, ∂u∂x = + ∂v∂y , (3a) ∂u∂y = − ∂v∂x , (3b)a simple exercise we encourage for the reader.One useful property of holomorphic functions is the following: regarded as a transformation,equations (2) give a mapping from the xy -plane to the uv -plane. This mapping is conformal , orangle-preserving. Precisely, if two curves C and C (cid:48) in the xy -plane meet at some point P withangle θ , the corresponding curves f ( C ), f ( C (cid:48) ) meet at f ( P ) in the uv -plane at the same angle θ . Another useful property of holomorphic functions is that the families of curves u ( x, y ) =const. and v ( x, y ) = const. are orthogonal, that is, the curves always meet at right angles.Again, the reader is encouraged to prove this, using the Cauchy-Riemann conditions.A useful tool for both analysis and visualization of two-dimensional fields is the so-called“complex potential”, which we proceed to define and explain.The main equation for electrostatics is Gauss’ law. In differential form and for points thathave no charge, the law is ∇ · (cid:126)E = 0. We traditionally introduce the scalar potential ϕ by (cid:126)E = − ∇ ϕ . By direct substitution in Gauss’ law, this results in Laplace’s equation ∇ ϕ = 0 . In two dimensions, or in a three dimensional situation with translation symmetry in one direc-tion, using the complex coordinate z = x + iy , the Laplacian takes the form ∇ = ∂ ∂x + ∂ ∂y = (cid:16) ∂∂x − i ∂∂y (cid:17)(cid:16) ∂∂x + i ∂∂y (cid:17) = 4 ∂ z ∂ z . ∂ z ∂ z ϕ = 0 , with the general solution ϕ ( z, z ) = F ( z )+ G ( z ), where F ( z ) , G ( z ) are any arbitrary holomorphicfunctions . Since the potential must be real, ϕ ( z, z ) = f ( z ) + f ( z )2 . In other words, ϕ is the real part of the holomorphic function f ( z ) which is known as the complex potential . It will be most useful for us to deal with f directly instead of ϕ .Note that the equipotential surfaces are simply the solutions to Re f ( z ) = const. The electricfield is of course normal to the equipotential surfaces. It then follows that that the solutions toIm f ( z ) = const. — which form a family of curves orthogonal to the equipotential surfaces —are the electric field lines. Hence, knowing the complex potential f ( z ) allows one to reconstructthe entire picture of the field.To gain some intuition for this, consider the case f ( z ) = V ln( z/R ), where V , R are twoconstants with dimensions of potential and length, respectively. In terms of polar coordinates z = ρ e iφ , f ( z ) = V ln( ρ/R ) + iV φ . The equipotential lines are ρ = const. and the field linesare φ = const. — i.e. the field is directed radially. We recognize this field — it is simply theplanar cross section of the field due to an infinite line of charge in three dimensions. This is asexpected, because a two-dimensional problem is equivalent to a three-dimensional one with asymmetry in the third direction. In two dimensions, this is the field due to a point charge. Theparameter V determines the strength of the linear charge density (in three dimensions) or thecharge (in two dimensions) and R is a parameter related to the reference point for the potential.The reader is encouraged to find f ( z ) for some other familiar fields. The essential point is thattwo-dimensional electrostatics is most naturally formulated in terms of holomorphic quantities. A consequence of the existence of f ( z ) is that electrostatics is conformally invariant. Sincethe composition of two holomorphic functions is again holomorphic, this means that under aconformal transformation, an electric field gets mapped into an electric field. In fact, the fullset of Maxwell equations are conformally invariant, but this is subtler to show.To gain familiarity with the complex potential, we will work this out explicitly and relatetwo different fields by a conformal transformation.Let us consider the example of a constant electric field E pointing in the + x direction inthe plane. This could be sourced, for example, by a line of charge parallel to the y -axis (ora sheet of charge in three dimensions which is perpendicular to the xy -plane and contains the y -axis), far away from the region of interest. The complex potential is given by f ( z ) = E z. (4)Consider now the conformal mapping w = e z . Let us consider the image of the field in the w -plane. Since w = e z = e x + iy = e x e iy , we see that the lines x = C are mapped to circles ofradius e C in the w -plane, and lines of y = C are mapped to rays at an angle C with the positive x -axis. In particular, the field lines point out radially. The complex potential in the w plane is g ( w ) = f ( z ( w )) = E ln w. (5) This ensures that G ( z ) is antiholomorphic, that is, only depends on z . To be precise in terms of dimensional analysis, we should write w/L = exp( z/R ), where
L, R are two lengths.Without loss of generality, we can always assume that L = R = 1 or that the coordinates z, w are really thedimensionless ratios z/R , w/L . E = V .Something peculiar has happened: we started with a field sourced by a line of charge intwo dimensions and finished with a point charge sitting at the origin. Where did this chargecome from? To understand this subtlety, we may consider placing the the line of charge alongsome line x = x in the z -plane. That is, we take the line to be at a finite distance, instead ofinfinitely far away. The image of this in the w -plane is a circle of radius e x . The field in the w -plane corresponds precisely to the field due to this circle of charge . Consider now the limit x → −∞ , when the source is infinitely far away in the z -plane. In the w -plane, this is thezero-radius limit of the circle, which is a point charge.The “ghost charge” at the origin in the w -plane seems to spontaneously appear, but as wesaw from the physics, it is merely a limit. It is the necessary charge density to source the fieldunder consideration. We can state this mathematically as follows. The exponential map w = e z does not send the z -plane to the w -plane surjectively. Its image is C (cid:114) { } , the w -plane minusthe origin. When we formally add back the point in w = 0 as a limit point, a ghost chargeappears there.We aim to describe electric fields on more general domains, namely compact surfaces. Forcompact surfaces, it turns out that ghost charges are required by topology — on a genericsurface, one cannot have a sourceless field. It would be beyond our scope to prove this, but thereader may hopefully develop some intuition from our treatment of the examples.Let us develop some results which will be useful in later sections. Recall that electromag-netism is linear. We may therefore use the superposition principle, and add solutions togetherto find other solutions. Given a positive charge at z = a and a negative charge at z = − a ofequal strength, the complex potential f ( z ) = V ln( z − a ) − V ln( z + a ) = V ln z − az + a describes an electric dipole created from two charges separated by length 2 a . To find the complexpotential of a fundamental dipole, we take the limit a → V → V (2 a ) = p remains finite. At this limit f ( z ) = − pz . Similarly, we can take two opposite dipoles at z = ± a : f ( z ) = − p (cid:18) z − a − z + a (cid:19) = − a p ( z − a ) ( z + a ) . In the limit a → p → ap = Q remains finite, we find the quadrupole complexpotential: f ( x ) = − Qz . The negative sign in the previous potentials is not important; it relates to the relative orientationof the charges. If we interchange the position of the charges or dipoles, a (cid:55)→ − a , then theopposite sign appears. We may continue in this fashion to find higher-order multipole fields. Inparticular, it is easy to verify that multipoles come in the form f ( x ) = M n z n , with n = 1 , , . . . and M n a constant that defines the strength of the (2 n )-pole. We will call M n the moment of the (2 n )-pole; for the dipole, M = p is the dipole moment and for the This equation seems to violate dimensional analysis, but we must recall that we have chosen units by settinga length scale to unity, as discussed in the previous footnote. In three dimensions, this would represent the cross-sectional view of a cylindrical configuration. M = Q is the quadrupole moment. We have discovered our key rule: specifyingthe order of a pole in the complex function f ( z ) is equivalent to specifying a multipole sourcefor the field. A function which is holomorphic everywhere, with the exception of a finite set ofpoles, is called meromorphic . Liouville’s theorem in complex analysis states that a meromorphicfunction is uniquely fixed by its limiting behavior at its poles and zeroes. Finally, recall thata solution to an electrostatics problem with given boundary conditions is unique. It does notmatter how we discover a solution; once we do, we know that it is the only solution. This,together with Liouville’s theorem, will allow us to completely determine the fields. Now that we have gained some experience with our basic principle — that conformal maps takefields to fields, and limit points imply ghost charges — we can explain how to do electrostaticson the sphere. We begin by explaining how to describe the sphere in complex coordinates. Thekey is a geometric construction known as stereographic projection.Before continuing, we address a question that the alert reader may have. Electrostaticson the sphere could just as well be studied by using the ambient three-dimensional space,parameterized in spherical polar coordinates ( r, θ, φ ), and keeping r constant. We would thenobtain the potential as a function of the angles ( θ, φ ). Although such an approach is possible, itis analytically complicated and ignores some fundamental aspects of the topic. The geometricalapproach we present is much more efficient. We will still use the ambient space, but in a subtlerway — it will allow us to obtain a more convenient parametric description of the sphere. Thereader will understand these kind of subtleties in more detail if he/she studies the general theoryof manifolds.Consider a sphere with radius R in three dimensions, X + Y + Z = R .X, Y, Z, are three-dimensional Cartesian coordinates. We use capital letters to avoid confusionwith the complex coordinate z = X + iY on the XY -plane. Without loss of generality, we willset R = 1. The north pole, in Cartesian coordinates ( X, Y, Z ), is N = (0 , , r, θ, φ ) bethe standard spherical coordinates. Hence the points of the sphere are parametrized as (1 , θ, φ )or simply ( θ, φ ); two parameters are sufficient since the sphere is a 2-dimensional surface.Given any point P on the sphere, draw the line NP. Denote the point of intersection of NPwith the XY -plane as P (cid:48) . This point P (cid:48) may be described in polar coordinates ( ρ, φ ) in the XY -plane. Taking a cross-section, by elementary trigonometry and similar triangles one deducesthat the distance in the XY -plane from the origin to P (cid:48) is ρ = cot( θ/ φ in the plane is the same as the 3-dimensional azimuthal angle φ . Now, we regard the XY -plane as the complex plane, with coordinate z = ρe iφ , so that we can write the stereographicprojection as z = e iφ cot (cid:16) θ (cid:17) . (6)Hence, given the point on the sphere with angular coordinates ( θ, φ ), the correspondingpoint in the XY -plane, regarded as the complex plane, has coordinate z given by the aboveformula. This equation is valid for every point on the sphere, except for the north pole — aswe approach θ = 0, | z | → ∞ . The stereographic projection has ‘unwrapped’ the sphere minusa point to the plane. Turning this around, we may consider the sphere as the plane, togetherwith “the point at infinity”, namely the north pole. This is analogous to what we did in theprevious section, but in that case we added the origin as a limit point. Since the sphere iscompact, mathematicians say that the sphere gives a one-point compactification of the plane.The theory of manifolds provides a concrete set-up which specifies how to deal with surfacesthat have points which escape from our mapping, which in this case is the local coordinate5SO P P (cid:48) X Y φθ Figure 1: Stereographic projection of the sphere. z . Without going into the details, when this happens, a second choice of local coordinate isnecessary in a neighborhood of such points. If the local coordinates we have introduced do notcover the surface, we add a third map, and so on. For the stereographic projection of the sphere,a second map is all we need. That is, we would like to have another coordinate description ofthe sphere which includes the north pole. This can be accomplished if we stereographicallyproject from the south pole S = (0 , , − XY -plane at some P (cid:48)(cid:48) . If we use a complex coordinate w inthe plane to denote the coordinates P (cid:48)(cid:48) , supposing P has angular coordinates ( φ, θ ), we obtainthat w = e − iφ tan (cid:16) θ (cid:17) . (7)The sign on φ must be reversed in order to have a consistent orientation on the sphere.Notice that any point P of the sphere which is not the north or the south pole can bedescribed with any of the two complex coordinates z or w , such that z, w (cid:54) = 0, w = 1 z . This means that our two coordinates are in fact holomorphic functions of one another: math-ematicians would say that we have given the sphere the structure of a complex manifold —we have covered it with two open neighborhoods, each with a choice of complex coordinate(commonly referred to as the z -patch and w -patch), such that the two choices of coordinate arerelated by a holomorphic change of variable. The sphere, when described in terms of complexcoordinates, is known as the Riemann sphere . It gives us a precise way to talk about infinity inthe complex plane — simply work in the w -patch and consider w = 0.How does this construction relate to physics? The key is that the stereographic projection isin fact a conformal map. To verify this, recall that the infinitesimal distance ds between pointson a sphere is given by ds = dθ + sin θ dφ . By relating the differentials dθ and dφ to dz and dz one obtains, after some algebra: ds = 4(1 + | z | ) dzdz. (8)From this expression, we can see that the stereographic projection is conformal — the infinites-imal distance squared on the plane is simply dzdz = dX + dY , and on the sphere we findthe same infinitesimal distances, simply rescaled by a position-dependent scale factor. Since6escaling preserves angles, this local rescaling preserves angles at every point, so the mappingis conformal.Conformal invariance then implies that the electrostatic equation takes the remarkably sim-ple form on the Riemann sphere ∂ z ∂ z ϕ = 0 . (9)In particular, it is identical to what we found on the complex plane. We may therefore use thecomplex potential. This means that given any field line pattern on the plane, we simply invertthe stereographic projection to obtain the corresponding field line pattern on the sphere. Sincestereographic projection is conformal, we know that this procedure will produce a good electricfield on the sphere.Let us do some examples. First, we take a point charge + q sitting at the south pole, z = 0.The complex potential is f ( z ) = q π(cid:15) ln z. (10)We have used a more standard choice of parameters for this physical example–in the notationof the previous sections, V = q/ π(cid:15) . From the stereographic projection, we see that the fieldlines run along the lines of longitude of the sphere. Since we are on the sphere, we have to checkthe behavior at infinity, that is, in the w -patch near w = 0. We see that f ( z ( w )) = q π(cid:15) ln (cid:16) w (cid:17) = − q π(cid:15) ln w. (11)We see another point charge, this time of charge − q , a sink in the field. This is a ghost chargethat has appeared at our limit point w = 0. Its appearance is topologically necessary for anon-singular field on the sphere. We conclude that it is impossible to have an isolated pointcharge on a sphere — topology demands that we obtain a dipole. In fact, one of the results ofthe general theory of Riemann surfaces implies that it is impossible to have an isolated pointcharge on any surface at all. The charges must always sum to zero. Intuitively this statement isobvious: imagine some positive charges on the surface. These are sources of electric field lines.The latter must terminate somewhere on the surface. But Riemann surfaces (like the Riemannsphere) are compact, in particular finite. Therefore, the lines cannot go out to infinity. Theremust necessarily be the right amount of sinks (negative charges) to allow the lines to terminate.As a second example, let’s take the field given by f ( z ) = E z on the plane, which weconsidered in the last section and check its behavior at infinity on the Riemann sphere. We seethat f ( z ( w )) = E w . (12)This is the complex potential for a dipole. Thus, the constant field induces a ghost dipole atinfinity. Continuing in this fashion, it is easy to see that by considering potentials f ( z ) ∼ z n , weend up with ghost multipole fields. Hence, the procedure to solve electrostatics on the Riemannsphere is equivalent to specifying the zeroes and poles of the function f ( z ).In this section, we have sketched a physical explanation of the fact that the poles and zeroesof the function correspond to the sources of the electric field. And we know that, given thecharge distribution, the electric field is fixed uniquely. We draw some examples in Figure 2 sothat the reader can have a visual representation of the mathematical statements. The study of fields on the sphere is interesting geometrically because we may visualize someinteresting field line patterns via stereographic projection. However, mathematically it is notso interesting because the complex potentials are the same as those of the familiar fields on theplane. In this section, we explain a beautiful result: the relation of electrostatics to a class of7 ( z ) = z ; pole at infinity. f ( z ) = z ; pole at infinity. f ( z ) = ( z − z + 1) / ( z + 1) ;pole at z = i . Figure 2: Various fields on the sphere: a dipole, quadrupole, and some rational function. Fieldlines are in black, equipotentials are gray. The hue corresponds to the phase of f ( z ).functions one typically encounters when studying the period of a pendulum, for example. Inparticular, we embark on the study of electric fields on the torus (that is, the surface of a donut)which necessarily leads to elliptic functions.The basic analytic difficulty that we face is to obtain a useful parameterization of the torus.The torus can be characterized as the surface of revolution obtained from taking a round circle S in, say, the XZ -plane and rotating it about the Z -axis. Suppose the circle’s center lies onthe XY -plane at a distance a from the origin along the X -axis, and that the circle has radius b , where 0 < b < a . Then the equation of the torus is (cid:0) a − (cid:112) X + Y (cid:1) + Z = b . If we were to attempt to use vector calculus to study fields on this surface, it would be an-alytically excruciating. In accordance with our theme, the strategy will be to study it as a2-dimensional problem and rephrase it in terms of a holomorphic function.The first step is to search for a conformal map. We give here a geometric construction ofthe map known as the Clifford embedding. It should be clear that the torus, as a set, is theCartesian product S × S . The embedding of the torus into 3-dimensional space breaks thesymmetry between the two S ’s: because of their placement, effectively, we only rotate one ofthem about the Z -axis; the other is unaffected by the rotation. This asymmetry is reflected inthe complicated nature of the algebraic equation for the torus in three dimensions.The key idea of the Clifford embedding is to find a representation of the torus that preservesthe symmetry between the S ’s. The standard way to represent S as a round circle of radius R is S = { ( R cos φ, R sin φ ) ∈ R | φ ∈ [0 , π ) } . This naturally lives in a plane, R . For later convenience, we will set R = 1 / √
2. So, using theabove representation, the natural way to represent the Cartesian product is S × S = (cid:26)(cid:18) √ φ, √ φ, √ θ, √ θ (cid:19) ∈ R | ( θ, φ ) ∈ [0 , π ) × [0 , π ) (cid:27) . This torus sits in 4-dimensional space R , the set of all 4-tuples of real numbers ( x , x , x , x ). Itis clear that since we have ( x , x , x , x ) = (cos φ, sin φ, cos θ, sin θ ) / √
2, then from trigonometricidentities x + x + x + x = 1 . Note that the angular coordinates ( θ, φ ) of this section are completely distinct from the spherical coordinatesof the same name in the previous section. S × S is contained within the 3-sphere S in four dimensions: S = { ( x , x , x , x ) ∈ R | x + x + x + x = 1 } . That is, S × S ⊂ S . This embedding of the torus into S is known as the Clifford embedding,and the torus as the Clifford torus.We now must relate this abstract torus in S with the familiar one in R . The key, once again,is stereographic projection. We have seen that we may stereographically project an S to a plane.This projection can be extended to any sphere, in any number of dimensions. In particular, the3-dimensional sphere S , although harder to visualize, can be projected stereographically fromits north or south pole onto its 3-dimensional equatorial ‘plane’.Let’s construct this projection explicitly. Since visualization and Euclidean geometry cannotaid us anymore, we will use a different strategy. First notice that, given that the round S hasbeen placed symmetrically inside R , the equational ‘plane’ is all points of R with with x = 0.This describes an R . We start with the north pole N = (0 , , ,
1) and consider some pointP = ( x , x , x , x ) ∈ S . We consider a line which passes through N and P, and intersects thehyperplane x = 0 at some point P (cid:48) = ( X, Y, Z,
X, Y , and Z in terms of x , x , x , and x as follows. The line NP is given in parametric form as the set of pointsNP = { ( tx , tx , tx , t ( x − ∈ R | t ∈ ( −∞ , ∞ ) } . (13)The line L intersects the desired hyperplane when its fourth coordinate vanishes. This happensat 1 + t ( x −
1) = 0, or t = 1 / (1 − x ). Inserting this value of t for the other coordinates, wemay solve for ( X, Y, Z, X = x − x , (14) Y = x − x , (15) Z = x − x . (16)This is the stereographic projection of S minus the north pole down to R . We are almostdone: if we consider the Clifford torus in S , ( x , x , x , x ) = (cos φ, sin φ, cos θ, sin θ ) / √ R : X = cos φ √ − sin θ ,Y = sin φ √ − sin θ ,Z = cos θ √ − sin θ . This is a somewhat strange parameterization of the set, but some experimentation with theequations reveals that ( √ − (cid:112) X + Y ) + Z = 1 . This identifies this set as the familiar torus in R .The 3-dimensional version of stereographic projection is also a conformal map. We omitthe proof here since the details are unimportant for our purposes. In four dimensions, theinfinitesimal line element is ds = dx + dx + dx + dx .
9f we substitute in ( x , x , x , x ) in terms of θ and φ on the Clifford torus, we obtain theinfinitesimal distance on the Clifford torus: ds = 12 ( dθ + dφ ) . (17)Remarkably, this is, up to a constant, simply the usual distance on the plane with coordinates( θ, φ ). Since the angular variables only vary between 0 and 2 π , we have found that the the torus,given by its complicated algebraic equation, conformally maps (via stereographic projection)onto the interior of a square!This is quite an enchanting mathematical story, but we must return to the physics. Atthis point, the strategy is familiar. Since the electrostatic equation ∇ ϕ = 0 is conformallyinvariant, we can take electrostatics on the torus and map it to the ( θ, φ )-plane. Introducingthe complex coordinate z = ( φ + iθ ) / π on the plane, the equation becomes ∂ z ∂ z ϕ = 0, whichcan be solved in terms of a complex potential f ( z ) by ϕ ( z, z ) = Re f ( z ).The novel feature here is that θ and φ are angular variables, so θ and θ + 2 π are consideredto be equivalent points, and similarly for φ . In terms of z , this means that z , z + 1, and z + i must be considered equivalent. Thus, for f to be a well defined function on the torus, it mustsatisfy f ( z ) = f ( z + 1) = f ( z + i ) . (18)In other words, it must be doubly periodic . A meromorphic, doubly periodic function is calledan elliptic function . Clearly, such a function is fixed by its behavior in the “fundamental square”( θ, φ ) ∈ [0 , π ) × [0 , π ).Actually, the double periodicity may be relaxed slightly: if f ( z + 1) and f ( z + i ) only differfrom f ( z ) by constants, this is still permissible, since it is only the field that is physicallyobservable, not the potential. We refer to such functions as quasi-periodic.At this point, we may borrow some results from the general theory of elliptic functions to aidin our search for electric fields. We will use the so-called Weierstrass elliptic functions, becausetheir relationship to the physics is more transparent in this case. The basic building block forall elliptic functions is the “Weierstrass ℘ -function” ℘ ( z ) = 1 z + (cid:88) ∗ ( m,n ) ∈ Z × Z (cid:104) z + m + in ) − m + in ) (cid:105) , where the star in the summation implies that the value n = m = 0 must be omitted. Thedouble sum may be shown to converge. It is doubly periodic by construction — a periodic shiftsimply permutes the terms. ℘ ( z ) has a double pole at zero and its periodic equivalents, but isregular everywhere else. It also has has two zeroes in the “fundamental square”, and, of course,their periodic equivalents.General results in elliptic function theory ensure that any elliptic function is completelyspecified by its poles and zeroes. This, in turn, implies that they are all rational expressionsin ℘ and ℘ (cid:48) . The idea is that one may use the poles and zeroes of ℘ and ℘ (cid:48) to engineer thedivergence or vanishing of the appropriate order at the appropriate points.Let us finally turn to the computation of some fields. We start with a point charge. Thepoint charge field is characterized by a logarithmic singularity. Since ℘ diverges as z − , a pointcharge q placed at z = 0 generates the complex potential f ( z ) = − q π(cid:15) ln ℘ ( z ) . (19)Since we are on a compact surface, we should not expect the point charge to come alone. Inthis case, the other sources come from the zeroes of ℘ ( z ). In particular, this complex potentialdescribes a point charge + q at z = 0, and two other point charges, each of charge − q/ ℘ ( z ). 10hile it is not possible to have an isolated point charge on the torus, it is in fact possibleto have an isolated dipole. If the dipole has dipole moment p , the complex potential is f ( z ) = p (cid:90) z ℘ ( z (cid:48) ) dz (cid:48) := − pζ W ( z ) , (20)the antiderivative of ℘ , often called the Weierstrass zeta function ζ W ( z ). This antiderivative isin fact quasi-periodic, but as we have explained this does not impact the physical field. Theway to understand this result is that ℘ ( z ) has a double pole at z = 0, but is regular everywhereelse, so its integral will have only a simple pole at z = 0 (the hallmark of a dipole), and beregular everywhere else. This regularity means that the dipole is in fact isolated. f ( z ) of theform ℘ (cid:48) /℘ corresponds to a non-isolated dipole.These fields have been constructed on the abstract Clifford torus, using conformal invariance.Using stereographic projection, one can project them back to the physical torus in R , whichgives a nice way to visualize the elliptic functions, as we see in Figure 3. Two dipoles. f ( z ) = ℘ ( z ). Symmetric arrangement of fourquadrupoles. Figure 3: Various fields on the torus–two dipoles, a quadrupole, and four quadrupoles distributedsymmetrically. Field lines are in black, equipotentials are gray. The hue corresponds to thephase of f ( z ). We conclude this article by directing the interested reader to the relevant mathematical litera-ture, and pointing out some natural extensions of our constructions in these contexts.The main theme, of course, is the mathematics of Riemann surfaces. For a rather accessibleintroduction that covers an impressive amount of ground, see [1]. Classic references on the theoryof theta functions on Riemann surfaces (which is really what underlies elliptic function theoryand its generalizations) are [3], [7]. There is also the textbook [2], and more mathematicallyinclined readers may wish to investigate [5]. Finally, for the relevant material on electrostatics,see the standard texts [4], [6].For readers that plan on consulting references such as [5], we will explain how our construc-tions fit into the broader theory. A major theme for us was the fact that a meromorphic functionon the surface of interest is essentially characterized by its poles and zeroes — physically, thesecorresponded to multipole-type sources for electric fields. One of the central questions in thetheory of Riemann surfaces is to what extent this holds in general — that is, given zeroes andpoles on a general Riemann surface, are there any functions with this behavior, and if so, howmany linearly independent ones? The answer is provided by the so-called Riemann-Roch the-orem and is the natural starting point for the theory of divisors and line bundles in algebraicgeometry.A related question is, after we have determined that functions exist with the prescribedzeroes and poles, can we construct them explicitly? It turns out that this is the case if one11ntroduces the so-called theta functions. These are naturally defined on an auxiliary space calledthe Jacobian variety, and the theory of the Abel-Jacobi map explains under what conditions onecan “pull-back” the theta functions to the Riemann surface itself to give concrete descriptionsof meromorphic functions. In fact, ℘ ( z ) may be written in terms of theta functions.We now remark on the natural extensions of our work. Perhaps the most obvious one is anextension of these constructions to Riemann surfaces of higher genus. The inherent difficulty indealing with such surfaces is the lack of explicit parametric descriptions — one typically mustinvoke the full machinery of algebraic geometry and regard the surfaces as projective varietiesto make any progress. Nonetheless, it would be interesting to attempt to describe the geometryof higher genus Riemann surface in an approach similar to ours: finding a convenient conformalmap which allows for visualization in R , giving a concrete representation of the geometricstructures on the Riemann surface in terms of electric fields.Another possibility is to study the effect of varying the complex structure. Let us brieflyexplain what this means. Typically, given a real surface, there are actually several inequiv-alent ways to combine its coordinates into a local complex coordinate z . Each way of doingthis is called a complex structure, and typically complex structures come in continuous families(so-called moduli spaces). In this language, our Clifford embedding only covered one point inthe moduli space, and by introducing additional parameters, it can be generalized to describeinequivalent complex structures. The fields, depending on holomorphic quantities, would cor-respondingly distort as the complex structure is varied, and this would be an interesting effect.The extension of this to higher genus would be highly nontrivial. Acknowledgements
S.T. thanks C.E. for the opportunity to work on this project, useful discussions, and encour-agement.
References [1] B. Eynard,
Lecture Notes on Compact Riemann Surfaces (2018) https://arxiv.org/abs/1805.06405 .[2] H.M. Farkas and I. Kra,
Riemann Surfaces . Graduate Texts in Mathematics. Springer, 2ndedition, 1992.[3] J. D. Fay,
Theta Functions on Riemann Surfaces , Lecture Notes in Mathematics, SpringerBerlin Heidelberg, Vol. , 1973.[4] David J. Griffiths, Introduction to Electrodynamics. Upper Saddle River, N.J: Prentice Hall,1999.[5] P. Griffiths and J. Harris,
Principles of Algebraic Geometry . John Wiley and Sons Inc., NewYork, 1994.[6] J. D. Jackson, 1925-2016. Classical Electrodynamics. New York: Wiley, 1999.[7] D. Mumford,