# M2 Project 2025-2026 ## Algebraic Geometry and Mathematical Physics **Instructor**: Taro Kimura (office: A332) The deep interplay between algebraic geometry and mathematical physics has led to profound insights in both fields over the decades. In particular, string theory, gauge theory, and mirror symmetry have revealed rich geometric structures of underlying physical theories, motivating the study of moduli spaces of bundles, sheaves, curves, and surfaces on algebraic varieties. Moreover, various techniques from enumerative geometry, derived categories, and geometric representation theory now play central roles in formulating and understanding a number of interesting physical phenomena. This is a (incomplete) list of potential topics related to Algebraic Geometry and Mathematical Physics for M2 project of Math4Phys. One can select one of them (all these are related to each other though), depending on their own interest and background, or if you have a concrete idea related to these topics, it could be also possible. ### Moduli spaces and gauge theory The notion of moduli spaces plays a central role in the context of algebraic geometry and mathematical physics. One of the important results in this direction is the correspondence between the stable vector bundle and the Hermitian Yang--Mills connection (Donaldson--Uhlenbeck--Yau; Kobayashi--Hitchin). Another important example is the so-called instanton ((anti-)self-dual Yang--Mills connection). For the case on $\mathbb{S}^4$, there exists an interesting description of the moduli space based on the Atiyah--Drinfeld--Hitchin--Manin (ADHM) construction, which has also an interesting link to twistor theory. ##### References - Donaldson, Kronheimer, *The Geometry of Four-Manifolds*, https://global.oup.com/academic/product/the-geometry-of-four-manifolds-9780198502692 - Freed, Uhlenbeck, *Instantons and Four-Manifolds*, https://doi.org/10.1007/978-1-4613-9703-8 - Kobayashi, *Differential Geometry of Complex Vector Bundles*, https://www.mathsoc.jp/assets/pdf/publications/pubmsj/Vol15.pdf - Nakajima, *Lectures on Hilbert Schemes of Points on Surfaces*, https://doi.org/10.1090/ulect/018 ### Enumerative geometry and enumerative invariants The purpose of the enumerative geometry is to count up some objects, for example, rational curves in algebraic varieties, having a number of applications in particular in string theory and gauge theory. These interactions between algebraic geometry and mathematical physics motivate various enumerative invariants, e.g., Gromov--Witten, Gopakumar--Vafa, Donaldson--Thomas, Pandharipande--Thomas, and their equivariant, K-theoretic, motivic, and categorical versions. ##### References - Cox, Katz, *Mirror Symmetry and Algebraic Geometry*, https://doi.org/10.1090/surv/068 - Kock, Vainsencher, *An Invitation to Quantum Cohomology*, https://doi.org/10.1007/978-0-8176-4495-6 ### Quiver variety and geometric representation theory Quiver variety is a cental object in the context of geometric representation theory, which is an interdisciplinary arena of algebraic geometry and representation theory. There are a number of applications in mathematical physics, e.g., integrable system, geometric Langlands correspondence. ##### References - Brion, *Representations of quivers*, https://hal.science/cel-00440026/ - Kirillov, *Quiver Representations and Quiver Varieties*, https://doi.org/10.1090/gsm/174 - Reineke, *Moduli of representations of quivers*, https://arxiv.org/abs/0802.2147 - Etingof, Liu, *Hitchin systems and their quantization*, https://arxiv.org/abs/2409.09505 - Okounkov, *Lectures on K-theoretic computations in enumerative geometry*, https://arxiv.org/abs/1512.07363