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Instanton moduli spaces and \(\mathcal W\)-algebras. (English) Zbl 1435.14001

Astérisque 385. Paris: Société Mathématique de France (SMF) (ISBN 978-2-85629-848-0/pbk). vii, 128 p. (2016).
The paper under review describes the intersection cohomology of the moduli spaces of framed Uhlenbeck spaces together with the Poincaré pairing on them in terms of representation theory of vertex operator algebras.
Let \(G\) be an almost simple simply-connected complex algebraic group with a simply laced Lie algebra \(\mathfrak{g}\). The latter means that the corresponding Dynkin diagram has only simple edges. Let \(\mathfrak h\) be a Cartan subalgebra of \(\mathfrak g\).
Let \(\mathbb P^2\) be the projective plane and let \(l_{\infty}\) be a fixed line, called the line at infinity. Let \(\mathrm{Bun}^d_G\) be the moduli space of \(G\)-bundles on \(\mathbb P^2\) trivially framed along \(l_\infty\) with instanton number \(d\). Its dimension equals \(2dh^{\vee}\), where \(h^{\vee}\) is the dual Coxeter number of \(G\).
Let \(\mathcal U^d_G\) be the Uhlenbeck partial compactification of \(\mathrm{Bun}^d_G\). There is a natural action of the group \(G\times GL(2)\) on \(\mathcal U^d_G\), where \(G\) acts by changing the trivialization at \(l_{\infty}\) and \(GL(2)\) acts on the projective plane preserving the line \(l_{\infty}\). Restricting to the diagonal subgroup of \(GL(2)\) one obtains the corresponding action of the group \(\mathbb G=G\times\mathbb C^*\times \mathbb C^*\).
Let \(\mathrm{IH^*_{\mathbb G}}(\mathcal U^d_G)\) be the \(\mathbb G\)-equivariant intersection cohomology, which is a module over the algebra \(\mathbf A_G=H^*_{\mathbb G}(\mathrm pt)\). Let \(\mathbf F_G\) be the field of fractions of \(\mathbf A_G\). \(\mathbf A_G\) can be identified with the algebra of polynomial functions on \(\mathfrak{h}\times \mathbb C^2\). The elements of \(\mathfrak h\times\mathbb C^2\) are denoted by \((\boldsymbol a, \varepsilon_1, \varepsilon_2)\). There is a natural symmetric Poincaré pairing \[ \mathrm{IH^*_{\mathbb G}}(\mathcal U^d_G)\underset{\mathbf A_G}{\otimes}\mathrm{IH^*_{\mathbb G}}(\mathcal U^d_G)\to \mathbf F_G. \] For every non negative integer \(d\) there is a canonical unit cohomology class \(|1^d\rangle\in \mathrm{IH^*_{\mathbb G}}(\mathcal U^d_G)\).
For every \(k\in \mathbb C\) (called level) there is a vertex operator algebra \(\mathcal W_k(\mathfrak g)\), which can be canonically embedded into the Heisenberg algebra \(\mathfrak{Heis}_{k+h^\vee}(\mathfrak h)\) of level \(k+h^\vee\). The algebra \(\mathcal W_k(\mathfrak g)\) is known to be generated by certain elements \(W^{(\kappa)}\), \(\kappa=1,\dots l\), \(l=\mathrm{rank}(\mathfrak g)\) (cf. [E. Frenkel and D. Ben-Zvi, Vertex algebras and algebraic curves. 2nd revised and expanded ed. Providence, RI: American Mathematical Society (AMS) (2004; Zbl 1106.17035)]).
For every \(\lambda \in \mathfrak h^*\) there is a Verma module \(M(\lambda)\) over \(\mathcal W_k(\mathfrak g)\) equipped with a natural Kac-Shapovalov bilinear form (cf. [T. Arakawa, Invent. Math. 169, No. 2, 219–320 (2007; Zbl 1172.17019)]).
For \(\boldsymbol a\in \mathfrak h\), define \[ M^d_{\mathbf F_G}(\boldsymbol a)=\mathrm{IH}^*_{\mathbb G}(\mathcal U^d_G)\underset{\mathbf A_G}{\otimes} \mathbf F_G,\quad M_{\mathbf{F}_G}(\boldsymbol a)=\bigoplus_{d=0}^\infty M^d_{\mathbf F_G}(\boldsymbol a). \] Equivalently \(M^d_{\mathbf F_G}(\boldsymbol a)=\mathrm{IH}^*_{\mathbb G, c}(\mathcal U^d_G)\underset{\mathbf A_G}{\otimes} \mathbf F_G\), where \(c\) stays for cohomology with compact support.
Consider the invariant bilinear form on \(\mathfrak g\) defined by the condition \((\alpha,\alpha)=2\) for every root of \(\mathfrak g\) and use it to identify \(\mathfrak h\) with \(\mathfrak h^*\). Let \(k=-h^{\vee} -\frac{\varepsilon_2}{\varepsilon_1}\).
The main result of the paper is Theorem 1.4.1. Its main points are the following.
1)
There is a \(\mathcal W_k(\mathfrak g)\)-module structure on \(M_{\mathbf{F}_G}(\boldsymbol a)\) such that this \(\mathcal W_k(\mathfrak g)\)-module is isomorphic to the Verma module \(M(\lambda)\) over \(\mathcal W_k(\mathfrak g)\) with weight \(\lambda=\frac{\boldsymbol a}{\varepsilon_1}-\rho\), where \(\rho\) is the half-sum of the positive roots of \(\mathfrak g\).
2)
The Kac-Shapovalov form on \(M(\lambda)\) corresponds to a twisted Poincaré pairing on \(M_{\mathbf{F}_G}(\boldsymbol a)\).
3)
The grading by \(d\) corresponds to the grading by eigenvalues of the Virasoro generator \(L_0\).
4)
For \(d\ge 1\), \(n>0\) \[ W_n^{(\kappa)}|1^d\rangle = \begin{cases} \pm\varepsilon_1^{-1}\varepsilon_2^{-h^{\vee}+1}|1^{d-1}\rangle & \text{if }\kappa=l\text{ and }n=1,\\ 0 &\text{otherwise}. \end{cases} \]
The authors construct the representation of \(\mathcal W_k(\mathfrak g)\) on \(\bigoplus_{d} \mathrm{IH}^*_{\mathbb G}(\mathcal U^d_G)\underset{\mathbf A_G}{\otimes} \mathbf F_G\) by constructing first an action of the Heisenberg algebra \(\mathfrak{Heis}_{k+h^\vee}(\mathfrak h)\) and using the embedding \(\mathcal W_k(\mathfrak g)\subset \mathfrak{Heis}_{k+h^\vee}(\mathfrak h)\). The action of the Heisenberg algebra is obtained by reducing the question with the help of the the hyperbolic restriction to a situation where the authors apply the results from [D. Maulik and A. Okounkov, Quantum groups and quantum cohomology. Paris: Société Mathématique de France (SMF) (2019; Zbl 1422.14002)] for \(G=SL(2)\). In order to check the relations the authors provide two possibilities. The first one is reducing the question to the case of \(G=SL(3)\), which is possible due to the assumption of simply lacedness of \(G\), and applying again the results of [D. Maulik and A. Okounkov, Quantum groups and quantum cohomology. Paris: Société Mathématique de France (SMF) (2019; Zbl 1422.14002)] for \(SL(3)\). The second way to check the relations is using \(R\)-matrices.
The paper under review consists of 8 chapters and two appendices. Section 1 is an introduction. Here the authors introduce the main objects of study, formulate the main result, Theorem 1.4.1 mentioned above, sketch its proof, give an extensive overview of previous works concerning both the results and mathematical techniques involved in the proof, and comment on open questions and further work. In a short Chapter 2 some generalities about Uhlenbeck spaces are collected. Chapter 3 deals with hyperbolic restriction in general. In Chapter 4 the hyperbolic restriction on Uhlenbeck spaces is considered. Chapter 5 connects the discussion in Chapter 4 to constructions from [D. Maulik and A. Okounkov, Quantum groups and quantum cohomology. Paris: Société Mathématique de France (SMF) (2019; Zbl 1422.14002)] for \(G\) being of type \(A\). The representation of \(\mathcal W_k(\mathfrak g)\) on \(\bigoplus_{d} \mathrm{IH}^*_{\mathbb G}(\mathcal U^d_G)\underset{\mathbf A_G}{\otimes} \mathbf F_G\) is constructed in Chapter 6. Geometric \(R\)-matrices are discussed in Chapter 7, which gives an alternative proof (Proposition 7.7.2) of the Heisenberg commutation relations from Chapter 6 (Proposition 6.3.8). An integral (non-localized) version of the main result over the ring \(\mathbf A_G\) is presented in Chapter 8, which in particular is needed to prove the fourth statement of Theorem 1.4.1. Appendix A deals with the exactness of hyperbolic restriction and is used in Chapter 4. One of the main results of the paper, Theorem 4.6.1, is proved in this appendix. Appendix B introduces the integral form \(\mathcal W_{\mathbf A}(\mathfrak g)\) of the \(\mathcal W\)-algebra such that tensoring with \(\mathbf F=\mathbb C(\varepsilon_1, \varepsilon_2)\) gives \(\mathcal W_k(\mathfrak g)\) with \(k=-h^\vee - \frac{\varepsilon_2}{\varepsilon_1}\). This appendix is used in Chapter 6, where representations of Heisenberg and Virasoro algebras on non-localized equivariant cohomology groups are constructed, and in Chapter 8.
The paper is written in a very condensed language. As a simultaneous reading the reviewer recommends the lecture notes [H. Nakajima, IAS/Park City Mathematics Series 24, 381–436 (2017; Zbl 1403.14036)], where some further insights and explanations on the geometric part of the paper are provided.

MSC:

14-02 Research exposition (monographs, survey articles) pertaining to algebraic geometry
17-02 Research exposition (monographs, survey articles) pertaining to nonassociative rings and algebras
14D21 Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory)
14D20 Algebraic moduli problems, moduli of vector bundles
14F43 Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies)
17B69 Vertex operators; vertex operator algebras and related structures
17B68 Virasoro and related algebras
14J60 Vector bundles on surfaces and higher-dimensional varieties, and their moduli
14C05 Parametrization (Chow and Hilbert schemes)
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