10-08 Notes

Generators for the cohomology of \(B\mathcal{G}\)

Following the previous lecture the goal is to identify generators for the cohomology of \(B\mathcal{G} = \text{Map} [\Sigma , BU_n]\). Consider a principal bundle \(SU_n \rightarrow P \rightarrow \Sigma\) which is the bundle of frames \((v_1,..,v_n)\) of a vector bundle \(E\) with volume \(1\). The volume element is the same as a trivialization of the determinant line bundle \(\Delta=\text{det} E\). Let \(\delta\) be a unitary connection on \(\Delta\) and consider \[\mathcal{A}^{\delta}=\{A | A \text{ a unitary connection on E s.t.} \text{det}(A) = \delta \}.\] \(\mathcal{A}^{\delta}\) is an affine space modeled by \(\Omega^1(su(E))\) where \(su(E)\) is the traceless skew hermitian endomorphisms of \(E\).

The subgroup \(\mathcal{G}^\delta\) of the gauge transformations of \(E\) preserving \(\mathcal{A}^\delta\) is the space of sections of the bundle of unitary endomorphisms of \(E\) with determinant \(1\). Now let \(\mathcal{B}^\delta = \mathcal{A}^\delta/\mathcal{G}^\delta\) and consider the fibration \((\mathcal{A}^\delta \times E)/\mathcal{G}^\delta \stackrel{\pi}{\rightarrow} \mathcal{B}^\delta \times \Sigma\). The fiber above a point is given by \(\pi^{-1}([A]\times \sigma)=E|_{\sigma}/Stab(A)\). In trying to make this a vector bundle the stabilizer of \(A\) needs to be gotten rid of. As the center \(Z(SU_n) \subset \mathcal{G}^{\delta}\) is clearly in the center a first attempt is to get rid of this part. To this end, consider instead the fibration \[(\mathcal{A}^{\delta}\times \text{End}^0(E))/\mathcal{G}^{\delta} \stackrel{\pi}{\rightarrow} \mathcal{B}^{\delta} \times \Sigma,\] where \(End^0(E)\) denotes the bundle of traceless endomorphisms of \(E\). Now as \(Z(SU_n)\) acts trivially on \(\text{End}^0(E)\) that part of the stabilizer has been taken care of. However there is still an issue in the case where \(\text{Stab}(A) \neq Z(SU_n)\). Hence it can only be said that \(\mathcal{E}|_{\mathcal{B}_*^{\delta} \times \Sigma}\) is a vector bundle, where \(\mathcal{E}= (\mathcal{A}^{\delta}\times \text{End}^0(E))/\mathcal{G}^{\delta}\) and \(\mathcal{B}_*^{\delta}=\{[A]|\text{Stab}(A)=Z(SU(n))\}\) is the space of irreducible connections.

Now the Chern classes \(c_i(\mathcal{E}) \in H^{2i}(\mathcal{B}_*^{\delta} \times \Sigma)\) can be decomposed using Kunneth formula and used to define \(\mu_i : H_*(\Sigma) \rightarrow H^*(\mathcal{B}_*^{\delta})\) via \[\mu_i(a) = c_i(\mathcal{E})/a \in H^{2i-\text{deg}(a)}.\] Here \(c_i(\mathcal{E})/a\) denotes the evaluation of the \(H^{\text{deg}a}(\Sigma) \otimes H^{2i-\text{deg}a}(\mathcal{B}_*^{\delta})\) component of \(c_i(\mathcal{E})\) on \(a\). In the case that there is another vector bundle \(\mathcal{V}\) such that \(\text{End}^0(\mathcal{V}) = \mathcal{E}\), the Chern classes \(c_i(\mathcal{E})\) can be written as polynomials in \(c_i(\mathcal{V})\). This formula can be inverted in general, regardless of whether \(\mathcal{V}\) exists to define \(c_i(\mathcal{V})\). The convention is to use \(c_i(\mathcal{V})\) rather than \(c_i(\mathcal{E})\) to define \(\mu_i\). Since the original bundle \(E\) was arbitrary and \(\mathcal{B}=\mathcal{A}/\mathcal{G}\) is a classifying space for \(\mathcal{G}\) the \(\mu_i\)’s give the required generators for \(H^*(B\mathcal{G})\).

Stabilizers in the rank 2 case

Consider a rank \(2\) bundle with a unitary connection \(A\) on it. It was shown in lecture 4 that \(\text{Stab}_{p_0}(A)=\text{Centralizer of}\text{ hol}_{p_0}(A)\). Hence \(\text{Stab}(A) = \{\pm 1\}, U_1 \text{ or } SU_2\). Assuming \(\langle c_1(E), [\Sigma] \rangle = 1\) and \(\text{vol}\Sigma = 2\pi\) these three cases can be dealt with as follows –

\(\mathbf{Case 1}\): \(\text{Stab}(A) = SU_2\). In this case the holonomy would be \(\{\pm 1\}\) and since it changes continuously it must be locally constant and the connection is flat (\(F_A = 0\)). Thus it cannot be true that \(\langle c_1(E), [\Sigma] \rangle = 1\) and this case cannot happen.

\(\mathbf{Case 2}\): \(\text{Stab}(A) = U_1\). In this case the bundle splits \(E = L_1 \oplus L_2\) and \(A\) preserves the splitting. The curvature operator splits as \(*F_A = \begin{bmatrix} i\lambda_1 & 0 \\ 0 & i\lambda_2 \end{bmatrix}\), where \(\lambda_i = -c_1(L_i)\) are not necessarily distinct. As \(\lambda_1+\lambda_2=-c_1(E)=-1\), if \(\lambda_1=-h\) the energy is given by \(\mathcal{E}(A)=2h^2-2h+1\).

\(\mathbf{Case 3}\): \(\text{Stab}(A) = \pm 1\). In this case \(E\) would have full holonomy \(SU_2\) and hence it cannot split. As the eigenspaces of \(*F_A\) give a holomorphic splitting of \(E\) it must be the case that there is just one eigenspace \(*F_A = \begin{bmatrix} i\lambda & 0 \\ 0 & i\lambda \end{bmatrix}\). Here \(\lambda = -\frac{1}{2}c_1(E) = -\frac{1}{2}\) and hence the energy is \(\mathcal{E}(A)=\frac{1}{2}\).

Hessian of YMF

Since the goal is to understand the topology of \(N^0(n,k)=\{A \in A^{\delta}| \text{rank}(E)=n, c_1(E)=k\}/\mathcal{G}\) using the Yang Mills functional, the Morse index and nullity of the critical points must first be understood. The computation to this end gives \(F_{A+ta} = F_A + td_A +\frac{t^2}{2}[a,a]\), where \(A\) is a fixed unitary connection, \(a \in \Omega^1(su(E))\) and \([a,a](x,y) \doteq [a(x),a(y)]\) by definition. The energy is \[\mathcal{E}(A+ta) = \mathcal(A) + 2t \langle F_A , d_Aa\rangle + t^2(\langle d_Aa, d_Aa \rangle + \langle F_A,[a,a]\rangle) + O(t^3),\] and the Hessian can be computed as \[\begin{aligned} \text{Hessian}_A(a) &= \langle d_Aa,d_Aa \rangle + \langle F_A , [a,a]\rangle \\ &= \langle d_Aa,d_Aa \rangle + \int (*F_A , [a,a]) \\ &= \langle d_Aa,d_Aa \rangle + \int ([a,*F_A] , a) \\ &= \langle d_Aa,d_Aa \rangle + \int (**[*F_A,a] , a) \\ &= \langle L_Aa , a \rangle, \end{aligned}\] where \(L_A a = d_A^*d_Aa + *[*F_A,a]\). The Hessian can be viewed as an operator on the tangent space \(\Omega^1(\Sigma,su(E))\). It will be shown in the next lecture that the tangent space to the gauge orbit \(T_A(\mathcal{G}.A)\) is in the nullspace of \(L_A\). However \(L_A\) can be replaced by \(H_A = L_A + d_Ad_A^*\) to get an operator with the same number of negative eigenvalues and the nullity of \(L_A\) restricted to a slice (\(\text{ker} L_A|_{T_A(\mathcal{G}.A)^{\perp}}\)). The operator \(H_A\) acts on \(\Omega^1(\Sigma,su(E))\). In the case where \(E=L_1\oplus L_2\) is a sum of line bundles \(su(E) = i \mathbb{R} \oplus L_1^* \otimes L_2 = i\mathbb{R} \oplus V\), and there is a decomposition \(\Omega^1(su(E)) = \Omega^1(\Sigma) \oplus \Omega^1(V) = \Omega^1(\Sigma) \oplus \Omega^{0,1}(V) \oplus \Omega^{1,0}(V)\). In the next lecture the nullity and index of \(H_A\) will be studied along the lines of the Kodaira-Nakano calculation.