The 2-Leg Vertex in K-theoretic DT Theory

Kononov, Ya.; Okounkov, Andreĭ; Osinenko, A.

doi:10.1007/s00220-021-03936-z

Cited by 10 publications

(14 citation statements)

References 12 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…[29]). This last generating series agrees with the expression of the unrefined limit of the topological vertex of Iqbal-Kozçaz-Vafa [25] and can be seen as a specialization both of the motivic invariants of Morrison-Mozgovoy-Nagao-Szendrői [35] and of the K-theoretic invariants of Kononov-Okounkov-Osinenko [26].…”

Section: 4supporting

confidence: 81%

See 1 more Smart Citation

Double nested Hilbert schemes and the local stable pairs theory of curves

Monavari¹

2021

Preprint

View full text Add to dashboard Cite

We propose a variation of the classical Hilbert scheme of points -the double nested Hilbert scheme of points -which parametrizes flags of zero-dimensional subschemes whose nesting is dictated by a Young diagram. Over a smooth quasi-projective curve, we compute the generating series of topological Euler characteristic of these spaces, by exploiting the combinatorics of reversed plane partitions. Moreover, we realize this moduli space as the zero locus of a section of a vector bundle over a smooth ambient space, which therefore admits a virtual fundamental class. We apply this construction to the stable pair theory of a local curve, that is the total space of the direct sum of two line bundles over a curve. We show that the invariants localize to virtual intersection numbers on double nested Hilbert scheme of points on the curve, and that the localized contributions to the invariants are controlled by three universal series for every Young diagram, which can be explicitly determined after the anti-diagonal restriction of the equivariant parameters. Under the anti-diagonal restriction, the invariants are matched with the Gromov-Witten invariants of local curves of Bryan-Pandharipande, as predicted by the MNOP correspondence. Finally, we discuss K-theoretic refinements à la Nekrasov-Okounkov.

show abstract

Section: 4supporting

confidence: 81%

“…K-theoretic refinement. K-theoretic refinement of Donaldson-Thomas theory and stable pair theory attracted much attention recently, both in Mathematics and String Theory: see for example [58,1,3,17] for Calabi-Yau threefolds, [37,39,11] for Calabi-Yau fourfolds and [38,44,26,45] for local curves.…”

mentioning

confidence: 99%

Double nested Hilbert schemes and the local stable pairs theory of curves

Monavari¹

2021

Preprint

View full text Add to dashboard Cite

show abstract

“…As shown in [53], the full K-theoretic PT vertex glues to give the same result. Indeed, the resulting PT partition function is independent of the additional parameters corresponding to the torus action on the conifold.…”

Section: Hilbert Series Of the Hilbert Schemementioning

confidence: 59%

“…The equivariant Euler characteristics (3.22) are then identified with the bare 1-leg vertex and following the notation of e.g. [53] we have:…”

Section: Adhm Vorticesmentioning

confidence: 99%

Blocks and Vortices in the 3d ADHM Quiver Gauge Theory

Crew,

Dorey,

Zhang

2020

Preprint

View full text Add to dashboard Cite

We study the hemisphere partition function of a three-dimensional N = 4 supersymmetric U (N ) gauge theory with one adjoint and one fundamental hypermultipletthe ADHM quiver theory. In particular, we propose a distinguished set of UV boundary conditions which yield Verma modules of the quantised chiral rings of the Higgs and Coulomb branches. In line with a recent proposal by two of the authors in collaboration with M. Bullimore, we show explicitly that the hemisphere partition functions recover the characters of these modules in two limits, and realise blocks gluing exactly to the partition functions of the theory on closed three-manifolds. We study the geometry of the vortex moduli space and investigate the interpretation of the vortex partition functions as equivariant indices of quasimaps to the Hilbert scheme of points in C 2 . Along the way we find interesting relations between our hemisphere blocks and related quantities in topological string theory and equivariant quantum K-theory.

show abstract

“…3.1.8. K-theoretic refinement K-theoretic refinement of Donaldson-Thomas theory and stable pair theory attracted much attention recently, both in Mathematics and String Theory: see for example [178,1,3,76] for Calabi-Yau threefolds, [140,143,36] for Calabi-Yau fourfolds and [142,149,109,150] for local curves. A scheme X with a perfect obstruction theory is endowed not only with a virtual fundamental class, but also with a virtual structure sheaf O vir X ∈ K 0 (X).…”

Section: The Local Gw Theory Of Curvesmentioning

confidence: 99%

Equivariant Enumerative Geometry and Donaldson-Thomas Theory

Monavari¹

View full text Add to dashboard Cite

Chapter 1. Introduction 1.1 A bit of history 1.2 Donaldson-Thomas theory 1.3 Pandharipande-Thomas theory of local curves 1.4 Higher rank K-theoretic Donaldson-Thomas theory of points 1.5 Donaldson-Thomas theory of Calabi-Yau 4-folds 1.6 Nested Quot schemes 1.7 Organization of the thesis 1.8 Cross-reference 1.9 Conventions I Donaldson-Thomas theory in dimension three Chapter 2. Localization in Equivariant Enumerative Geometry 2.1 Une promenade equivariante 2.2 Une promenade virtuelle Chapter 3. Double nested Hilbert schemes and the local PT theory of curves 3.1 Introduction 3.2 Double nested Hilbert schemes 3.3 Moduli space of stable pairs 3.4 K-theory class of the perfect obstruction theory 3.5 Universality 3.6 The leading term 3.7 Toric computations 3.8 Summing the theory up 3.9 K-theoretic refinement 3.10 Appendix: The combinatorial identities Chapter 4. Higher rank K-theoretic Donaldson-Thomas theory of points 4.1 Introduction 4.2 The local Quot scheme: critical and equivariant structure 4.3 Invariants attached to the local Quot scheme 4.4 Higher rank vertex on the local Quot scheme 4.5 The higher rank K-theoretic DT partition function 8 CONTENTS 4.6 The higher rank cohomological DT partition function 4.7 Elliptic Donaldson-Thomas invariants 4.8 Higher rank DT invariants of compact toric 3-folds II Donaldson-Thomas theory in dimension four Chapter 5. Donaldson-Thomas theory of Calabi-Yau 4-folds 5.1 Taking square roots 5.2 Oh-Thomas virtual structures Chapter 6. K-theoretic DT/PT for toric Calabi-Yau 4-folds 6.1 Introduction 6.2 K-theoretic vertex formalism 6.3 Limits of K-theoretic conjecture 6.4 Hilbert schemes of points 6.5 Local resolved conifold 6.6 Appendix: Combinatorial sign rule Chapter 7. Stable pair invariants of local Calabi-Yau 4-folds 7.1 Introduction 7.2 Moduli spaces 7.3 Invariants 7.4 Appendix: Tables III Nested Quot schemes Chapter 8. On the motive of the nested Quot scheme of points on a curve 8.1 Introduction 8.2 Nested Quot schemes of points 8.3 Tangent space and smoothness in the case of curves 8.4 Bia lynicki-Birula decomposition 8.5 The motive of the nested Quot scheme on a curve 8.6 Appendix: Motivic exponentials Chapter 9. A note on the smoothness of the nested Quot scheme of points 9.1 Introduction 9.2 Properties of the moduli space 9.3 Proof of the theorem Samenvatting Acknowledgements Curriculum vitae Bibliography

show abstract

The 2-Leg Vertex in K-theoretic DT Theory

Cited by 10 publications

References 12 publications

Double nested Hilbert schemes and the local stable pairs theory of curves

Double nested Hilbert schemes and the local stable pairs theory of curves

Blocks and Vortices in the 3d ADHM Quiver Gauge Theory

Equivariant Enumerative Geometry and Donaldson-Thomas Theory

Contact Info

Product

Resources

About