A new shell formula unifies and delivers explicit closed-form expressions plus recursions for instanton partition functions in 5d SYM and multiple gauge origami configurations using arbitrary-dimensional Young diagrams.
Magnificent Four
5 Pith papers cite this work. Polarity classification is still indexing.
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abstract
We present a statistical mechanical model whose random variables are solid partitions, i.e. Young diagrams built by stacking up four dimensional hypercubes. Equivalently, it can be viewed as the model of random tessellations of ${\bf R}^{3}$ by squashed cubes of four fixed orientations. The model computes the refined index of a system of D0-branes in the presence of $D8$-$\overline{D8}$ system, with a $B$-field strong enough to support the bound states. Mathematically, it is the equivariant K-theoretic version of integration over the Hilbert scheme of points on ${\bf C}^{4}$ and its higher rank analogues, albeit the definition is real, not complex analytic. The model is a mother of all random partition models, including the equivariant Donaldson-Thomas theory and the four dimensional instanton counting. Finally, a version of our model with infinite solid partitions with four fixed plane partition asymptotics is the vertex contribution to the equivariant count of instantons on toric Calabi-Yau fourfolds. The conjectured partition function of the model is presented. We have checked it up to six instantons (which is one step beyond the checks of the celebrated P.~MacMahon's failed conjectures of the early XX century). A specialization of the formula is our earlier (2004) conjecture on the equivariant K-theoretic Donaldson-Thomas theory, proven by A.~Okounkov in 2015.
verdicts
UNVERDICTED 5representative citing papers
Proposes and proves for 5D an expression for charge functions of odd-dimensional partitions whose poles mark addable and removable boxes.
Derives a residue formula for elliptic genera in 2d (0,1) gauge theories that recovers the Jeffrey-Kirwan prescription for (0,2) theories and applies it to the Gukov-Pei-Putrov model to study its phase structure.
Proves the Cao-Maulik-Toda conjecture equating GV invariants of fiber classes on a CY4 fibered over a curve with those of the smooth fiber under an orientation compatibility assumption.
Proposes comet-shaped quiver gauge theories for surface defects with nested instantons in 4D gauge theories on T^2 × T*C_{g,k} and gives conjectural explicit formulae for the virtual equivariant elliptic genus of bundles over nested Hilbert schemes of points on the affine plane.
citing papers explorer
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Shell formulas for instantons and gauge origami
A new shell formula unifies and delivers explicit closed-form expressions plus recursions for instanton partition functions in 5d SYM and multiple gauge origami configurations using arbitrary-dimensional Young diagrams.
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Charge functions for odd dimensional partitions
Proposes and proves for 5D an expression for charge functions of odd-dimensional partitions whose poles mark addable and removable boxes.
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Elliptic Genera of 2d $\mathcal{N}=(0,1)$ Gauge Theories
Derives a residue formula for elliptic genera in 2d (0,1) gauge theories that recovers the Jeffrey-Kirwan prescription for (0,2) theories and applies it to the Gukov-Pei-Putrov model to study its phase structure.
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Gopakumar-Vafa invariants of fiber classes on Calabi-Yau 4-folds fibered over curves
Proves the Cao-Maulik-Toda conjecture equating GV invariants of fiber classes on a CY4 fibered over a curve with those of the smooth fiber under an orientation compatibility assumption.
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Defects, nested instantons and comet shaped quivers
Proposes comet-shaped quiver gauge theories for surface defects with nested instantons in 4D gauge theories on T^2 × T*C_{g,k} and gives conjectural explicit formulae for the virtual equivariant elliptic genus of bundles over nested Hilbert schemes of points on the affine plane.