Seiberg-Witten Theory: I

December 07, 2021

This series of posts will be a documentation of my attempts to understand Seiberg-Witten theory. This particular post will focus on the linear algebra foundations: Clifford algebras and the spinor representation. I am using Marcolli’s Seiberg-Witten Gauge Theory, Nicolaescu’s Notes on Seiberg-Witten Theory and Salamon’s Spin Geometry and Seiberg-Witten Invariants as the main sources to learn. Apologies for errors, and comments are welcome.

The real Clifford algebra on \(n\) generators and \(+_n\) signature \(\mathrm{Cliff}_{n, 0}\) is the algebra generated by \(n\) symbols \(e_1, \cdots, e_n\) satisfying the anticommutation relations \(\{e_i, e_j\} = e_i e_j + e_j e_i = -2 \delta_{ij}\). This is closely related to the exterior algebra on \(n\) generators \(\Lambda_n\) generated by \(n\) symbols \(e_1, \cdots, e_n\) satisfying the anticommutation relations \(\{e_i, e_j\} = e_i \wedge e_j + e_j \wedge e_i = 0\), except in the former we have the identities \(e_i^2 = -1\) for all \(1 \leq i \leq n\); indeed they are isomorphic as graded vector spaces with basis consisting of monomials on \(e_1, \cdots, e_n\) with degree of each \(e_i\) either \(0\) or \(1\). In particular, \(\dim \mathrm{Cliff}_{n, 0} = \dim \Lambda_n = 2^n\)

One might think of \(\mathrm{Cliff}_{n, 0}\) as trying to mimic the quaternions \(\mathrm{Cliff}_{2, 0} \cong \Bbb H\) (identify \(e_1 \mapsto i, e_2 \mapsto j, e_1 e_2 \mapsto k\)) in higher dimensions. Recall that \(\Bbb H\) carries a natural norm \(\|a + bi + cj + dk\| = a^2 + b^2 + c^2 + d^2\), and therefore one could look at the unit-norm quaternions, which would be the locus of points \((a, b, c, d)\) satisfying \(a^2 + b^2 + c^2 + d^2 = 1\) the underlying space of such being \(S^3 \subset \Bbb H \cong \Bbb R^4\). But the quaternionic multiplication now restricts to \(S^3\), making it a Lie group, just like the complex multiplication on \(\Bbb C\) makes the unit circle \(S^1 \subset \Bbb C \cong \Bbb R^2\) a Lie group. There is a natural representation \(\Bbb H \to M_2(\Bbb C)\) of the quaternion algebra in the space of \(2 \times 2\) matrices given by the so called Pauli matrices,

\[\displaystyle 1 \mapsto \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}, \, i \mapsto \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}, \, j \mapsto \begin{pmatrix} i & 0 \\ 0 & -i \end{pmatrix}, \, k \mapsto \begin{pmatrix} 0 & i \\ -i & 0 \end{pmatrix}\]

The image of the unit sphere in \(\Bbb H \cong \Bbb R^4\) under this representation consists of exactly the \(2 \times 2\) unitary matrices (that is, \(A^\dagger A = I\)) with determinant \(1\), that is, the special unitary group \(SU(2)\). Thus, \(S^3\) with the Lie group structure given by quaternionic multiplication is isomorphic to the familiar Lie group \(SU(2)\).

We shall construct similar representations \(\mathrm{Cliff}_{2m, 0} \to M_{2^m}(\Bbb C)\). First we introduce the complexified Clifford algebra \(\mathrm{Cliff}_{n, 0} \otimes \Bbb C\) which is defined mutatis mutandis except the coefficients associated to the standard basis of monomials can be taken to be complex numbers now. In this setup, let \(e_1, \cdots, e_m, f_1, \cdots, f_m\) denote the standard generators of \(\mathrm{Cliff}_{2m, 0}\). Define \(v_k = (e_k + i f_k)/\sqrt{2}\) for all \(1 \leq k \leq m\). Then the subspace \(V\) spanned by \(v_1, \cdots, v_m\) has the property that the Clifford multiplication restricts to the exterior algebra on \(V\), i.e., \(V\) is totally isotropic with respect to the complexified dot product \(z \cdot w = \sum_i z_i w_i\). Therefore, we can decompose \(\Bbb C^{2m} = V \oplus V^*\) by considering the map \(\Bbb C^n \to V^*\) sending \(z\) to the linear functional \(z \cdot (-)\) on \(V\), which has kernel precisely \(V\). There’s a natural way to make the exterior algebra \(\Lambda_m\) generated by \(v_1, \cdots, v_m\) a module over \(\mathrm{Cliff}_{2m, 0}\) by defining the action of elements of \(\Bbb C^{2m} = V \oplus V^*\) as follows:

\[\begin{gather*} v \cdot (w_1 \wedge \cdots \wedge w_k) = v \wedge w_1 \wedge \cdots \wedge w_k \\ \phi \cdot (w_1 \wedge \cdots \wedge w_k) = \sum_{j = 1}^k (-1)^j \phi(w_j) w_1 \wedge \cdots \wedge \widehat{w_j} \wedge \cdots \wedge w_k \end{gather*}\]

For any \(v \in V, \phi \in V^*\). The complex spinor representation is simply this representation \(\varphi : \mathrm{Cliff}_{2m, 0}\otimes \Bbb C \to \mathrm{End}(\Lambda_m) \cong M_{2^m}(\Bbb C)\). We can make this a bit more explicit by choosing a basis for everything: the copy of \(V^*\) sitting inside \(\Bbb C^{2m}\) is spanned by the conjugates \(\overline{v}_k = (e_k - i f_k)/\sqrt{2}\), \(1 \leq k \leq m\). Indeed one can check the relations \(v_i \cdot v_j = \overline{v}_i \cdot \overline{v}_j = 0\) and \(\overline{v}_i \cdot v_j = v_i \cdot \overline{v}_j = \delta_{ij}\) for all \(1 \leq i, j \leq m\). The representation restricts to a representation of \(\mathrm{Cliff}_{2m, 0} \subset \mathrm{Cliff}_{2m, 0} \otimes \Bbb C\) by \(\varphi(e_k) = v_k \wedge \cdot - \iota_{\overline{v}_k}(\cdot)\) and \(\varphi(f_k) = i(v_k \wedge \cdot + \iota_{\overline{v}_k}(\cdot))\), where \(\wedge\) denotes the exterior product and \(\iota\) denotes the contraction operator.

At this point it is a good idea to switch to coordinate-free language to consolidate various points. Let \((V, \langle \cdot, \cdot \rangle)\) be an even-dimensional real inner product space and \(J\) be a complex structure on \(V\), that is, \(J \in \mathrm{End}(V)\) such that \(J^2 = -I\) which is moreover compatible with the inner product in the sense that \(J\) is an isometry, i.e., \(\langle Jv, Jw\rangle = \langle v, w\rangle\). Then the complexification of \(V\) splits as \(V \otimes \Bbb C = V^{1, 0} \oplus V^{0, 1}\) where \(V^{1, 0}, V^{0, 1}\) are the eigenspaces of \(J\) with eigenvalue \(\pm i\) respectively. The real Clifford algebra \(\mathrm{Cliff}(V) = \bigoplus_{n \geq 0} V^{\otimes n}/(v \otimes w + w \otimes v + 2\langle v, w\rangle)\) is a quotient of the full tensor algebra by two-sided ideal generated by the appropriate commutation relations. The complex Clifford algebra \(\mathrm{Cliff}(V) \otimes \Bbb C\) has a representation on the exterior algebra \(\Lambda^\bullet V^{1, 0}\) by \(v \mapsto v \wedge \cdot\) if \(v \in V^{1, 0}\) and \(\overline{v} \mapsto \iota_{\overline{v}}\) if \(\overline{v} \in V^{0, 1}\), where we treat \(\overline{v}\) as the covector \(\langle \overline{v}, \cdot \rangle_{\Bbb C}\) acting on \(V^{0, 1}\) where \(\langle \cdot, \cdot \rangle_{\Bbb C}\) is the complexified inner product. The representation map \(\mathrm{Cliff}(V) \otimes \Bbb C \to \mathrm{End}(\Lambda^{\bullet} V^{1, 0})\) is in fact an isomorphism: first note \(\mathrm{Cliff}(V_1 \oplus V_2) \cong \mathrm{Cliff}(V_1) \widehat{\otimes} \mathrm{Cliff}(V_2)\) and \(\mathrm{End}(\Lambda^{\bullet} (V_1 \oplus V_2)) \cong \mathrm{End}(\Lambda^{\bullet} V_1) \widehat{\otimes} \mathrm{End}(\Lambda^\bullet V_2)\). By decomposing \(V^{1, 0}\) into \(1\)-dimensional subspaces we obtain that the representation map is obtained from direct sum of the representation maps of the form \(\mathrm{Cliff}(L) \otimes \Bbb C \to \mathrm{End}(\Lambda^{\bullet} L^{1, 0})\) where \(L\) is \(1\)-dimensional, which are clearly isomorphisms.

This shows \(\mathrm{Cliff}(V) \otimes \Bbb C \cong \mathrm{End}(\Lambda^{\bullet} V^{1, 0})\) which is a matrix ring, and therefore \(\Lambda^{\bullet} V^{1, 0}\) must be a simple module over it, i.e., it is an irreducible representation of \(\mathrm{Cliff}(V) \otimes \Bbb C\). One can restrict this representation to the real Clifford algebra, and it follows from the coordinate computations above that it is explicitly given by \(\mathrm{Cliff}(V) \to \mathrm{End}(\Lambda^{\bullet} V^{1, 0})\), \(v \mapsto (v - Jv) \wedge \cdot - \iota_{(v - Jv)^*}\) where \(w^* = \langle w, \cdot \rangle\) is the covector associated to \(w\). From now onwards we denote \(S = \Lambda^{\bullet} V^{1, 0}\) to be this complex spinor representation of \(\mathrm{Cliff}_{2m, 0}\) of complex dimension \(2^m\).

From the discussions before, \(\mathrm{Cliff}(V) \cong \Lambda^\bullet(V)\) as graded vector spaces for any real inner product space \(V\), even though the algebra structure is very much different. Under this identification we can segregate the subspaces of \(\mathrm{Cliff}(V)\) generated by even-length words on a chosen orthobasis \(e_1, \cdots, e_n\) and the one generated by odd-length words: \(\mathrm{Cliff}(V) \cong \mathrm{Cliff}^+(V) \oplus \mathrm{Cliff}^-(V)\). The Clifford algebra as well as the exterior algebra is a superalgebra in the sense that it admits such a decomposition graded by \(\Bbb Z/2 = \{+, -\}\) in a way that the multiplication of the algebra preserves these “islands” modulo \(2\), i.e., \(++ = +\), \(+- = -+ = -\), \(-- = +\). Finally, (multiples of) the highest degree term \(\varepsilon := e_1 \cdots e_n\) in \(\mathrm{Cliff}(V)\) is called the volume form in analogy with that of the exterior algebra. In case \(n = 2m\), observe that \(\varepsilon^2 = (-1)^m\), therefore the matrix of \(\varepsilon\) acting on \(S\) has two distinct eigenvalues, \(\pm i^m\) (the isomorphism of central simple algebras \(\mathrm{Cliff}(V) \otimes \Bbb C \to \mathrm{End}(S)\) implies \(\varepsilon\) cannot act by a scalar as it is not a scalar). Therefore, there is a decomposition \(S = S^+ \oplus S^-\) into the eigenspaces of \(\varepsilon\). Observe that for any \(1 \leq i \leq 2m\), \(\varepsilon e_i = -e_i \varepsilon\), so for all \(v \in V\), \(\varepsilon v = - v \varepsilon\). As a corollary, any \(v \in V \subset \mathrm{Cliff}(V)\) acts on \(S\) by switching \(S^{\pm}\).

At this point, we stop and make a general definition. Given a real inner product space \((V, \langle \cdot, \cdot\rangle)\) of dimension \(2m\), a spinc structure on \(V\) is choice of a representation \(\rho : V \to \mathrm{End}(W)\) in a complex \(2^m\)-dimensional Hermitian inner product space \(W\) such that \(\rho(v) + \rho(v)^\dagger = 0\) and \(\rho(v) \rho(v)^\dagger = \|v\|^2 \mathrm{id}_W\). Equivalently, it is a representation \(\phi : \mathrm{Cliff}(V) \to \mathrm{End}(W)\) of \(\star\)-algebras, which restricts to \(\rho\) on \(V\). To see the equivalence, one can define \(\phi(v_1 \cdots v_n) := \rho(v_1) \cdots \rho(v_n)\); then it is easy to check \(\phi(vw + wv) = -\langle v, w\rangle\mathrm{id}_W\) from the properties of \(\phi\). The spinor representation \(\mathrm{Cliff}(V) \to \mathrm{End}(S)\) defines a canonical spinc structure on any real inner product space. It is moreover easily checked that for any spinc structure, the representation map \(\phi\) is an isomorphism; indeed, \(\mathrm{Cliff}(V)\) is a (central) simple algebra, hence the map is injective, and by dimension reasons (\(\dim W = 2^m\)) it must also be surjective. Finally, it follows from the Skolem-Noether theorem that any two spinc structures on a real inner product space are isomorphic in the sense that the corresponding representations are conjugate.

Eventually we shall discuss the notion of a spinor bundle \(S\) on a Riemannian manifold \((M, g)\) of dimension \(2m\), which shall be a module over the Clifford algebra bundle \(\mathrm{Cliff}(TM)\) of rank \(2^m\). The main subtlety is that even though this looks like the above construction parametrized over a manifold \(M\), a spinor bundle need not always exist; existence of such a thing is equivalent to the more familiar notion of existence of a spinc structure on \(M\). This will be discussed in detail in a future post. Up next, some discussions regarding the \(\mathrm{Spin}(n)\) and \(\mathrm{Spin}^{\Bbb C}(n)\) groups.