arXiv subjects

Zhi-Cong Ong

Publications and source records attributed to Zhi-Cong Ong.

4 recordsLinked to original sources

arXiv2025

Topological 5d $\mathcal {N} = 2$ Gauge Theory: Novel Floer Homologies, their Dualities, and an $A_\infty$-category of Three-Manifolds

We show how one can define novel gauge-theoretic Floer homologies of four, three, and two-manifolds from the physics of a certain topologically-twisted 5d ${\cal N}=2$ gauge theory via its supersymmetric quantum mechanics interpretation. They are associated with Vafa-Witten, Hitchin, and $G_{\mathbb{C}}$-BF configurations on the four, three, and two-manifolds, respectively. We also show how one can define novel symplectic Floer homologies of Hitchin spaces, which in turn will allow us to derive novel Atiyah-Floer correspondences that relate our gauge-theoretic Floer homologies to symplectic intersection Floer homologies of Higgs bundles. Furthermore, topological invariance and 5d "S-duality" suggest a web of relations and a Langlands duality amongst these novel Floer homologies and their loop/toroidal group generalizations. Last but not least, via a 2d gauged Landau-Ginzburg model interpretation of the 5d theory, we derive, from the soliton string theory that it defines and the 5d partition function, a Fukaya-Seidel type $A_\infty$-category of Hitchin configurations on three-manifolds -- thereby categorifying the aforementioned Floer homology of three-manifolds -- and its novel Atiyah-Floer type correspondence. Our work therefore furnishes purely physical proofs and generalizations of the mathematical conjectures by Haydys [1], Abouzaid-Manolescu [2], and Bousseau [3], and more.

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arXiv2024

Topological-Holomorphic ${\mathcal N} =4$ Gauge Theory: From Langlands Duality of Holomorphic Invariants to Mirror Symmetry of Quasi-topological Strings

We perform a topological-holomorphic twist of $\mathcal{N}=4$ supersymmetric gauge theory on a four-manifold of the form $M_4=Σ_1 \times Σ_2$ with Riemann surfaces $Σ_{1,2}$, and unravel the mathematical implications of its physics. In particular, we consider different linear combinations of the resulting scalar supercharges under $S$-duality, where this will allow us to derive novel topological and holomorphic invariants of $M_4$ and their Langlands duals. As the twisted theory can be topological along $Σ_1$ whence we can dimensionally reduce it to 2d, via the effective sigma-model on $Σ_2$, we can also relate these 4d invariants and their Langlands duals to the mirror symmetry of Higgs bundles and that of quasi-topological strings described by the sheaf of chiral differential operators. As an offshoot, we would be able to obtain a fundamental understanding from 4d gauge theory, why chiral differential operators are purely perturbative objects.

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arXiv2023

Vafa-Witten Theory: Invariants, Floer Homologies, Higgs Bundles, a Geometric Langlands Correspondence, and Categorification

We revisit Vafa-Witten theory in the more general setting whereby the underlying moduli space is not that of instantons, but of the full Vafa-Witten equations. We physically derive (i) a novel Vafa-Witten four-manifold invariant associated with this moduli space, (ii) their relation to Gromov-Witten invariants, (iii) a novel Vafa-Witten Floer homology assigned to three-manifold boundaries, (iv) a novel Vafa-Witten Atiyah-Floer correspondence, (v) a proof and generalization of a conjecture by Abouzaid-Manolescu in [1] about the hypercohomology of a perverse sheaf of vanishing cycles, (vi) a Langlands duality of these invariants, Floer homologies and hypercohomology, and (vii) a quantum geometric Langlands correspondence with purely imaginary parameter that specializes to the classical correspondence in the zero-coupling limit, where Higgs bundles feature in (ii), (iv), (vi) and (vii). We also explain how these invariants and homologies will be categorified in the process, and discuss their higher categorification. We therefore relate differential and enumerative geometry, topology and geometric representation theory in mathematics, via a maximally-supersymmetric topological quantum field theory with electric-magnetic duality in physics.

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arXiv2022

The $u$-plane integral, mock modularity and enumerative geometry

We revisit the low-energy effective $U(1)$ action of topologically twisted $\mathcal N=2$ SYM theory with gauge group of rank one on a generic oriented smooth 4-manifold $X$ with nontrivial fundamental group. After including a specific new set of $\mathcal Q$-exact operators to the known action, we express the integrand of the path integral of the low-energy $U(1)$ theory as an anti-holomorphic derivative. This allows us to use the theory of mock modular forms and indefinite theta functions for the explicit evaluation of correlation functions of the theory, including but not restricted to those that physically reproduce Donaldson invariants, thus facilitating the computations compared to previously used methods. As an explicit check of our results, we compute the path integral for the product ruled surfaces $X=Σ_g \times \mathbb{CP}^1$ for the reduction on either factor and compare the results with existing literature. In the case of reduction on the Riemann surface $Σ_g$, via an equivalent topological A-model on $\mathbb{CP}^1$, we will be able to express the generating function of genus zero Gromov-Witten invariants of the moduli space of flat rank one connections over $Σ_g$ in terms of an indefinite theta function, whence we would be able to make concrete numerical predictions of these enumerative invariants in terms of modular data, thereby allowing us to derive results in enumerative geometry from number theory.

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