K-Theory of Hilbert Schemes as a Formal Quantum Field Theory

Zhou, Jian

doi:10.48550/arxiv.1803.06080

Cited by 5 publications

(6 citation statements)

References 48 publications

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“…Explicit calculations in [A,Section 6], [EGL,Section 5], [K,Section 8] , [Sc,Section 5], [Z,Section 7] answer Question 5 in the affirmative for several values of the parameters ℓ, k i and rank α i . We will investigate the general case in future work.…”

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confidence: 99%

The virtual K-theory of Quot schemes of surfaces

Arbesfeld,

Johnson,

Lim

et al. 2020

Preprint

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We study virtual invariants of Quot schemes parametrizing quotients of dimension at most 1 of the trivial sheaf of rank N on nonsingular projective surfaces. We conjecture that the generating series of virtual K-theoretic invariants are given by rational functions. We prove rationality for several geometries including punctual quotients for all surfaces and dimension 1 quotients for surfaces X with pg > 0. We also show that the generating series of virtual cobordism classes can be irrational.Given a K-theory class on X of rank r, we associate natural series of virtual Segre and Verlinde numbers. We show that the Segre and Verlinde series match in the following cases:(i) Quot schemes of dimension 0 quotients, (ii) Hilbert schemes of points and curves over surfaces with pg > 0, (iii) Quot schemes of minimal elliptic surfaces for quotients supported on fiber classes. Moreover, for punctual quotients of the trivial sheaf of rank N , we prove a new symmetry of the Segre/Verlinde series exchanging r and N . The Segre/Verlinde statements have analogues for punctual Quot schemes over curves.

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confidence: 99%

The virtual K-theory of Quot schemes of surfaces

Arbesfeld,

Johnson,

Lim

et al. 2020

Preprint

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“…Formal quantum field theory. Let us recall the notion of a formal quantum field theory introduced in[28]. It consists of observable algebra and correlation functions.…”

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confidence: 99%

Hermitian One-Matrix Model and KP Hierarchy

Zhou

2018

Preprint

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“…of the BKP hierarchy, one can consider the formal quantum field theory associated to τ A (see [44] for an introduction of the notion of formal quantum field theory).…”

Section: Relation To the Free Energymentioning

confidence: 99%

BKP Hierarchy, Affine Coordinates, and a Formula for Connected Bosonic $N$-Point Functions

Wang,

Yang

2022

Preprint

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We derive a formula for the connected n-point functions of a taufunction of the BKP hierarchy in terms of its affine coordinates. This is a BKPanalogue of a formula for KP tau-functions proved by Zhou in [43]. Moreover, we prove a simple relation between the KP-affine coordinates of a tau-function τ (t) of the KdV hierarchy and the BKP-affine coordinates of τ (t/2). As applications, we present a new algorithm to compute the free energies of the Witten-Kontsevich tau-function and the Brézin-Gross-Witten tau-function.

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K-Theory of Hilbert Schemes as a Formal Quantum Field Theory

Cited by 5 publications

References 48 publications

The virtual K-theory of Quot schemes of surfaces

The virtual K-theory of Quot schemes of surfaces

Hermitian One-Matrix Model and KP Hierarchy

BKP Hierarchy, Affine Coordinates, and a Formula for Connected Bosonic $N$-Point Functions

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