About Monopoles

Recently, monopoles was mentioned in a youtube episode of Hossenfelder’s show. For, me, it had always been more natural that monopoles do not exist. Let me explain. Electromagnetism is defined if one has a geometry with an exterior derivative d. This is very general and works for delta sets. The theory postulates the existence of a 1-form A that satisfies d*A=0 which in physics is called Coulomb gauge. Now because of the property d d=0, the electromagnetic field F=dA satisfies dF=0 and d* F= d* dA = (d*d + d d*) A = L A = j, where j is again a 1-form. The j encodes charge and current. The two equations dF=0, d*F=j are what we call the Maxwell equations. They become an awkward list of 4 equations if the geometry is 4 dimensional space time and one separates space and time and the 2-form F which has in 4 dimensional has 6 components is split into an electric part E and magnetic part B, which encodes 6=3+3. The operator L=d*d+dd* is the 1-form Laplacian. In the continuum it is called d’Alembert operator. If is of course just the Hodge Laplacian for the geometry, where D =d+d* is the Dirac operator. In vacuum, when j=0, the Maxwell equations are just L A = 0, which means that A is a harmonic 1-form and this can be seen as the wave equation similarly as DA=0 can be seen as the Dirac equation.

Having seen this classical theory, one can wonder how a monopole situation dF=i can make sense. It tells that d is no more an exterior derivative or then, that there is no vector potential A any more as if F=dA, then dF=ddA=0 by the property of exterior derivative. So, for me, it always has been (since my student times actually) been a bit foolish to even contemplate about magnetic monopoles. It messes with a serious mathematical construct, namely calculus on manifolds, which crucially depends on exterior calculus. And who wants to get rid of things like Stokes theorem or differential forms? I want to argue in this post that there is actually a frame work, where one can think about monopoles as naturally existing. Monopole theory is just electromagnetism in the dual geometry. But as we see (and have seen historically since Poincare), duality needs some new frame work. The dual of a q-manifold is only after some clever definitions again a q-manifold. As Heegard had pointed out, the first approaches of Poincare in this respect had been flawed and we see the difficulty already in small dimensions. The dual of the smallest 3-sphere, the 8-cell is the tesseract, which is NOT a simplicial complex any more. We artificially have to impose the 2 dimensional faces as 4-gons and the 3 dimensional facets as cubes. Now, for the 8-cell tesseract pair this was easy to imagine but if we are given a 25-dimensional manifold in the form of a simplicial complex, how do we see the 17-dimensional subcells of the dual manifold? Well, it is possible and easy once one sees it but not obvious at all.

Having worked more recently on “duality” in finite settings, it occurred to me however, that magnetic monopoles still could make sense in a natural geometric settings. It is just calculus on the dual manifold but that is a bit more strange. Famously, the main founder of topology, Henry Poincare struggled for a while with duality. If you have a triangulation of a q-dimensional manifold, then this can be naturally described as a finite abstract simplicial complex for which every unit sphere S(x) = d U(x) is a (q-1)-sphere. The U(x) is the star of a simplex x, the set of simplices which contain x and dU(x) is the boundary, the complement of U(x) in the closure (the smallest closed set containing U(x)). Geometry of manifolds becomes very simple and natural in the context of simplicial complexes, a geometry with only one axiom: “a finite set of non-empty sets closed under the operation of taking finite non-empty subsets”, a geometry which also comes with a canonical topology, the Alexandrov topology. To get the manifold notion, one only needed to clarify what a sphere is and a q-sphere is a q-manifold with the property that removing a star from it produces a contractible complex. A complex G is contractible if there is a star U(x) in it such that both its unit sphere S(x) and G-U(x) are both contractible. All these definitions are recursive and very convenient to prove theorems. There is hardly any match for any competing discrete theory. Of course, we never, ever would touch the continuum when working in a combinatorial setting.

Any way, here is how one can see a monopole theory in a finite geometry. First of all, it follows from the definition of a q-manifold that for every k-simplex x in G, the intersection of all unit spheres of its vertices in x is a (q-k-1)-sphere. We think about this sphere as bounding a virtual (q-k)-dimensional cell. This is analog to what in the continuum are known as CW complexes. In the discrete what is nice is that we just have perfect duality. The number of the number of $k$-simplices in G is the same than the number of (q-k) cells in the dual. In the case when the manifold is orientable we have also a direct correspondence of Betti numbers because we just have switched d with d*. The poset structure of simplices in G is dual to the poset structure of virtual simplices in the dual. The duality of cohomology is known as Poincare Duality (what Poincare struggled to formulate properly and which still to this day is a bit awkward in topology books and only seems natural in de Rham cohomology settings, where one has a Hodge dual operator, provided that the manifold is orientable. Apropos, we every summer are following the traces of de Rham.

So, what are monopole mathematics then. It is just electromagnetism in the dual space! The Maxwell equations dF=0, d*F=j become now d*F=0, dF=i, where i is the monopole current. We see however that when we look at things like that, we just have a parallel electromagnetic theory. But it only works if Poincare duality holds and the manifold is orientable. By the way, the original electromagnetic theory assumed that the manifold is simply connected implying the first cohomology vanishing and allowing to conclude from dF=0 that F=dA (as closed 2-forms are exact 2-forms).

So, what would be the mathematicians explanation for not seeing monopoles? ‘

It is because it is just the usual electromagnetic theory looked at in the Poincare mirror!

Because it is just electromagnetism seen in the Poincare duality mirror!