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The Dirac quantisation condition for fluxes on four-manifolds. (English) Zbl 0964.57027

The paper gives physically motivated interpretations of some facts about connections, curvature and characteristic classes on complex line and spinor bundles over a closed Riemannian 4-manifold \(M\). The first part deals with a line bundle \(L\) carrying a \(U_1\)-connection with curvature 2-form \(F\). Physicists consider \(F\) as an electromagnetic field acting on a charged bosonic particle which is represented by a section of \(L\). The corresponding de Rham cohomology class \([F]\) is essentially the Chern class \(c_1(L)\), up to a constant \(2\pi\hbar q\) where \(q\) denotes the electric charge of the particle. In particular, \[ {q\over 2\pi\hbar} \int_\Sigma F = c_1(L)[\Sigma] \in \mathbb Z \tag{1} \] for any integral 2-cycle \(\Sigma\) representing a homology class \([\Sigma] \in H_2(M,\mathbb Z)\). Thus \(\int_\Sigma F\) is quantized (taking values in the discrete set \(2\pi\hbar q\mathbb Z\)) for topological reasons, and it is called the flux of the electromagnetic field through the “surface” \(\Sigma\). Locally, any \(U_1\)-connection on \(L\) takes the form \(\partial + iA\) for some real valued 1-form \(A\) (connection form or potential) with \(F = dA\). If \(\Sigma\) is no longer closed (a cycle) but has some boundary \(\gamma = \partial\Sigma\), one wishes to see a sort of Stokes theorem \(\int_\Sigma F = \int_\gamma A\). But \(A\) is not globally defined, so a priori \(\int_\gamma A\) has no meaning. In the present paper a substitute is defined by patching together the local expressions of \(A\) using the transition functions. However, this expression is determined uniquely only up to an integral multiple of \({2\pi\hbar\over q}\), so the above “Stokes theorem” makes sense only in the exponential form \[ \exp \Biggl({iq\over\hbar} \int_\Sigma F \Biggr)= \exp \Biggl({iq\over\hbar}\int_\gamma A \Biggr). \tag{2} \] Geometrically, the expression \({q\over\hbar}\int_\gamma A\) is the holonomy of the connection with respect to \(\gamma\), i.e. the rotation angle of the parallel transport around \(\gamma\) within the line bundle \(L\); the exponentiation underlines that this angle is determined only up to multiples of \(2\pi\). If \(M\) has a \(Spin\) structure, i.e. a \(Spin\)-principal bundle \(P_{Spin}\) which is a twofold covering of the oriented orthonormal frame bundle \(P_{SO}\), then there is an associated vector bundle \(S = P_{Spin} \times_{Spin} W\) (with \(Spin: = Spin(4)\)) where \(W\) is a complex vector space on which the Clifford algebra \(Cl(\mathbb R^4)\) acts by linear transformations. Its sections are called neutral spinors by physicists. Such a structure exists only if \(w_2(M) = 0\) (where \(w_2\) is the 2nd Stiefel-Whitney class) which fails for many 4-manifolds, e.g. for \(\mathbb C P^2\). But what always exists on a closed oriented 4-manifold is a \(Spin_c\)-structure where \(Spin_c = (Spin \times U_1)/\pm I\) (where \(U_1 = S^1 \subset \mathbb C\) denotes the unit circle). This is a \(Spin_c\)-principal bundle \(P_{Spin_c}\) which is a twofold covering of \(P_{SO} \times Q'\) for some \(U_1\)-principle bundle (circle bundle) \(Q'\). In fact, the topological condition for the existence of a \(Spin_c\)-structure is that \(w_2(M) \in H^2(M,\mathbb Z/2\mathbb Z)\) lifts to an integral cohomology class \(w\); by a theorem of Hirzebruch and H. Hopf this is always true in dimension 4. Thus a spinor bundle \(S\) might not exist but there is always a complex spinor bundle \(S^c = P_{Spin_c} \times_{Spin_c} W\). Sections of \(S^c\) are called electrically charged spinors. If even a \(Spin\)-structure \(P_{Spin}\) exists on \(M\), the corresponding \(Spin_c\)-structures are obtained as \(P_{Spin_c} = (P_{Spin} \times Q)/\pm I\) for another circle bundle \(Q\) whose 2-fold covering is the above \(Q'\) (i.e. we have \(L' = L\otimes L = L^2\) for the line bundles \(L\) and \(L'\) corresponding to \(Q\) and \(Q'\)). Thus a connection on \(P_{Spin_c}\) is given by the Levi-Civita connection on \(P_{Spin}\) (which is a twofold covering of \(P_{SO}\)) together with a connection on \(Q\) (or \(L\)) whose connection and curvature forms are again called \(A\) and \(F\), and the above theory applies to \(\int_\Sigma F\). On \(Q'\) (or \(L'\)), a connection \(\partial + iA'\) is obtained by putting \(A' = 2A\). In the general case when only a \(Spin_c\)-structure exists on \(M\), the authors still use the same forms \(A\) and \(F\) (interpreted as electromagnetic field acting on charged spinors) although \(Q\) does not exist anymore. This is justified since \(Q'\) is still there: we may start with a connection \(\partial + iA'\) on \(Q'\) and put \(A = {1\over 2}A'\) (locally) and \(F = {1\over 2}F'\) (globally). Consequently, we now have for any integral 2-cycle \(\Sigma\): \[ {q\over 2\pi\hbar}\int_\Sigma F = {1\over 2}c_1(Q')[\Sigma] \in {1\over 2} \mathbb Z \] Reduction modulo 2 gives \(\tilde c_1(Q') = w_2(Q') = w_2(M)\) (consider the exact sequence (D.4), p. 391 in Lawson-Michelson, and note that the map \(w_2 + \tilde c_1\) vanishes on \(P_{SO} \times Q\) since this is the image of \(P_{Spin_c}\) under \(\xi\)). Moreover, \(w_2(M)[\Sigma]\) is the self intersection number \(I(\Sigma,\Sigma)\) modulo 2. To show this last statement the authors analyse the intersection form modulo 2. Here is an alternative argument if \(\Sigma \subset M\) is a smooth surface: then \(I(\Sigma,\Sigma)\) is the number of zeros for a section of the normal bundle \(N\Sigma\) (counted with multiplicities), hence modulo 2 it is \(w_2(N\Sigma)\), but on the other hand \(w_2(M)[\Sigma] = w_2(T\Sigma + N\Sigma) = w_2(N\Sigma)\) since \(w_2(T\Sigma) = w_2(\Sigma)= 0\) (an oriented surface carries a spin structure). The conclusion is \[ {q\over 2\pi\hbar}\int_\Sigma F + {1\over 2}w[\Sigma] \in \mathbb Z. \tag{3} \] Moreover, there is another “exponential Stokes formula” combining (2) and (3) if \(\Sigma\) has a boundary.

MSC:

57R20 Characteristic classes and numbers in differential topology
81R25 Spinor and twistor methods applied to problems in quantum theory
53C65 Integral geometry
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