Magnificent Four with Colors

Nekrasov, Nikita; Piazzalunga, Nicolò

doi:10.1007/s00220-019-03426-3

Cited by 64 publications

(130 citation statements)

References 81 publications

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“…, b d ) (where we use the coordinate of the corner closest to the origin). 10 We show that the contribution of b to the constant term of V + π is even. This contribution comes from the following terms:…”

Section: 2mentioning

confidence: 71%

“…This result could be viewed as an indication that dimensions 3 and 4 are special. Perhaps it is also related to Nekrasov's comment in his paper "Magnificent four" [10]:…”

Section: Introductionmentioning

confidence: 89%

“…where M (q) still denotes the MacMahon function. For X = C 4 , this conjecture is a special case of a more general K-theoretic conjecture of Nekrasov [10]. However, our conjecture makes sense on (and is motivated by) an analogous conjecture on projective Calabi-Yau fourfolds.…”

Section: Introductionmentioning

confidence: 90%

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Counting zero-dimensional subschemes in higher dimensions

Cao

Kool

2019

Journal of Geometry and Physics

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Section: 2mentioning

confidence: 71%

“…This result could be viewed as an indication that dimensions 3 and 4 are special. Perhaps it is also related to Nekrasov's comment in his paper "Magnificent four" [10]:…”

Section: Introductionmentioning

confidence: 89%

See 1 more Smart Citation

Counting zero-dimensional subschemes in higher dimensions

Cao

Kool

2019

Journal of Geometry and Physics

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“…There is a related work due to Nekrasov [19], where he proposes a conjectural formula for a very general equivariant K-theoretical partition function on Hilbert schemes of points on C 4 . Specializations of his partition function seem related to our Conjecture 1.8.…”

Section: Introductionmentioning

confidence: 99%

Zero-dimensional Donaldson–Thomas invariants of Calabi–Yau 4-folds

Cao

Kool

2018

Advances in Mathematics

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We study Hilbert schemes of points on a smooth projective Calabi-Yau 4-fold X. We define DT 4 invariants by integrating the Euler class of a tautological vector bundle L [n] against the virtual class. We conjecture a formula for their generating series, which we prove in certain cases when L corresponds to a smooth divisor on X. A parallel equivariant conjecture for toric Calabi-Yau 4-folds is proposed. This conjecture is proved for smooth toric divisors and verified for more general toric divisors in many examples.Combining the equivariant conjecture with a vertex calculation, we find explicit positive rational weights, which can be assigned to solid partitions. The weighted generating function of solid partitions is given by exp(M (q) − 1), where M (q) denotes the MacMahon function.The virtual class (1.1) depends on the choice of orientation o(L). On each connected component of Hilb n (X), there are two choices of orientations, which affects the corresponding contribution to the class (1.1) by a sign. We review facts about the DT 4 virtual class in Section 2.1.In order to define the invariants, we require insertions. Let L be a line bundle on X and denote by L [n] the tautological (rank n) vector bundle over Hilb n (X) with fibre H 0 (L| Z ) over Z ∈ Hilb n (X). Then it makes sense to define the following: Definition 1.1. Let X be a smooth projective Calabi-Yau 4-fold and let L be a line bundle on X. Let L be the determinant line bundle of Hilb n (X) with quadratic form Q induced from Serre duality. Suppose L is given an orientation o(L). We define DT 4 (X, L, n ; o(L)) := [Hilb n (X)] vir o(L) e(L [n] ) ∈ Z, if n 1,where e(−) denotes the Euler class. We also set DT 4 (X, L, 0 ; o(L)) := 1.We make the following conjecture for the generating series of these invariants: 1 2 YALONG CAO AND MARTIJN KOOL Conjecture 1.2 (Conjecture 2.2). Let X be a smooth projective Calabi-Yau 4-fold and L be a line bundle on X. There exist choices of orientation such that ∞ n=0 DT 4 (X, L, n ; o(L)) q n = M (−q) X c1(L)·c3(X) , where

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“…This construction can be used to define K-theoretic and refined Donaldson-Thomas invariants for toric threefolds [32], as well as certain generalizations [5,33,34]. When applied to our moduli spaces, it provides a mathematical description of the protected spin character, by identifying them with refined Donaldson-Thomas invariants of framed quiver representations.…”

Section: Localizationmentioning

confidence: 99%

Quantum line defects and refined BPS spectra

Cirafici

2019

Lett Math Phys

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In this note we study refined BPS invariants associated with certain quantum line defects in quantum field theories of class S. Such defects can be specified via geometric engineering in the UV by assigning a path on a certain curve. In the IR they are described by framed BPS quivers. We study the associated BPS spectral problem, including the spin content. The relevant BPS invariants arise from the K-theoretic enumerative geometry of the moduli spaces of quiver representations, adapting a construction by Nekrasov and Okounkov. In particular refined framed BPS states are described via Euler characteristics of certain complexes of sheaves.

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Magnificent Four with Colors

Cited by 64 publications

References 81 publications

Counting zero-dimensional subschemes in higher dimensions

Counting zero-dimensional subschemes in higher dimensions

Zero-dimensional Donaldson–Thomas invariants of Calabi–Yau 4-folds

Quantum line defects and refined BPS spectra

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