Quantization via mirror symmetry
This is a write-up of the author's \(8\)th Takagi lectures on brane quantization and mirror symmetry delivered at Kyoto University in 2010. The style is descriptive and expository with focus on ideas and concepts rather than technicalities, and computations are mostly sketched. The brane quantization of a symplectic manifold \((M,\omega)\) is a new approach developed by the author and \textit{E. Witten} in [Adv. Theor. Math. Phys. 13, No. 5, 1445--1518 (2009; Zbl 1247.81378)]. It requires a choice of complexification \(Y\) of \(M\) with a non-degenerate holomorphic \(2\)-form \(\Omega\) that restricts to \(\omega\) on \(M\), and a unitary line bundle over \(Y\) that restricts to the prequantum line bundle of geometric quantization, with a connection of curvature \(\operatorname{Re}\Omega\). The Hilbert space of states is then given by the space of morphisms in the Fukaya category of \(Y\) (open strings) \(\mathcal{H}=\Hom(\mathcal{B}_{cc},\mathcal{B}')\), where \(\mathcal{B}'\) is a Lagrangian object (\(A\)-brane) supported on \(M\), and \(\mathcal{B}_{cc}\) is a canonical coisotropic object filling the entire \(Y\). Two classes of examples are considered in which the brane quantization can be carried out explicitly. The first are coadjoint orbits of a real Lie group \(G\) with the Kirillov-Kostant-Souriau form, the setting of the geometric quantization and the orbit method of the representation theory. A natural complexification is provided by switching to the complexified group \(G_{\mathbb{C}}\) (e.g. \(\mathrm{SL}(2,\mathbb{C})\) for \(\mathrm{SU}(2)\) and \(\mathrm{SL}(2,\mathbb{R})\)). Brane quantization offers new insights into representation theory explaining, for instance, why some unitary representations do not correspond to orbits (e.g. the complementary series representations of \(\mathrm{SL}(2,\mathbb{R})\)), and vice versa (e.g. the minimal orbits of \(\mathrm{SO}(p,q)\) with \(p,q\geq4\)). It also leads to new conditions for \((M,\omega)\) to be quantizable, e.g. it suffices that \(Y\) admits a Calabi-Yau metric for which \(\Omega\) is a Kähler form. The second class of examples are moduli spaces of flat \(G\) connections on Riemann surfaces, this is the setting of the Chern-Simons gauge theory. The two classes are unified if one allows punctured surfaces since coadjoint orbits can be identified with the moduli of flat connections over the punctured disk. The focus of the lectures is on the mirror picture, where the Fukaya category is replaced by the bounded derived category of coherent sheaves on the mirror \(\widetilde{Y}\) of \(Y\), and the Hilbert space can be identified with \(\mathrm{Ext}_{\widetilde{Y}}^*(\widetilde{\mathcal{B}}_{cc}, \widetilde{\mathcal{B}}')\). In the simplest case the mirror \(B\)-brane \(\widetilde{\mathcal{B}}'\) is the skyscraper sheaf of a point in \(\widetilde{Y}\), and \(\widetilde{\mathcal{B}}_{cc}\) can be described in terms of Ströminger-Yau-Zaslov (SYZ) fibrations of \(Y\) and \(\widetilde{Y}\). When restriction of \(\operatorname{Re}\Omega\) to a generic fiber \(F\) of the SYZ fibration of \(Y\) is non-trivial, \(\widetilde{\mathcal{B}}_{cc}\) is supported on all of \(\widetilde{Y}\) and has rank \(\frac1{n!}\int_F(\operatorname{Re}\Omega)^{\wedge n}\). Since in general mirror symmetry is mostly conjectural, explicit computations can be carried out only when \(M\) and then \(Y\) have additional structure. In the lectures \(Y\) is usually hyper-Kähler with \(\operatorname{Re}\Omega\) and \(-\operatorname{Im}\Omega\) being the two symplectic forms related by its complex structure (this happens e.g. if \(M\) is Kähler). Then \(\mathcal{B}_{cc}\) and \(\mathcal{B}'\) can be identified with half-BPS boundary conditions in the \(\mathcal{N}=(4,4)\) sigma-model of \(Y\) that often preserve supersymmetry in two different \(A\)-models and one \(B\)-model of \(Y\). That makes them what is called \((B,A,A)\) branes. If SYZ fibers are also \((B,A,A)\) branes then the mirror map transforms \(\mathcal{B}_{cc}\) and \(\mathcal{B}'\) into \(B\)-branes for all complex structures on \(\widetilde{Y}\), called \((B,B,B)\) branes, and the mirrors are explicitly computable via the Fourier-Moukai transform. Since \(B\)-model computations can also be done explicitly this allows one to complete the brane quantization in such examples. Two toy examples are quantized along these lines, \(2\)-tori and \(K3\) surfaces, before the main example of the lectures is taken up, the Chern-Simons gauge theory with a compact Lie group \(G\) on a closed Riemann surface \(\Sigma\). The symplectic manifold \(M\) is the moduli of flat \(G\)-connections on \(\Sigma\). The complexification \(Y\) is obtained by switching to the complexified group \(G_{\mathbb{C}}\), and can be identified with the moduli of Higgs \(G\)-bundles on \(\Sigma\), the Hitchin moduli space \(\mathcal{M}_H(G,\Sigma)\). It is hyper-Kähler, and its mirror is \(\mathcal{M}_H(^LG,\Sigma)\), where \(^LG\) is the Langlands dual of \(G\). Both of them fiber over the same vector space under Hitchin maps providing the SYZ fibrations with \((B,A,A)\) fibers. Since \(G\) is compact, \(\mathcal{B}_{cc}\) is defined (and the moduli are quantizable) only for a discrete set of symplectic structures indexed by an integer \(k\) called level. The quantum Hilbert space \(\mathcal{H}\) is then finite-dimensional with the dimension given by the celebrated Verlinde formula. Brane quantization recovers it in a highly non-trivial way from mirror calculations as a polynomial with rational coefficients. Moreover, the coefficients of large powers of \(k\) get interpreted geometrically in terms of branes on \(\mathcal{M}_H(G,\Sigma)\), and of small powers in terms of branes on its mirror \(\mathcal{M}_H(^LG,\Sigma)\).
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