Corrigendum and addendum to “Moduli spaces of framed symplectic and orthogonal bundles on P2 and the K-theoretic Nekrasov partition functions” [J. Geom. Phys. 106 (2016) 284–304]

We consider an elementary, and largely unexplored, combinatorial problem in low-dimensional topology: for a compact surface [Formula: see text], with a finite set of points [Formula: see text] fixed on its boundary, how many configurations of disjoint arcs are there on [Formula: see text] whose boundary is [Formula: see text]? We find that this enumerative problem, counting curves on surfaces, has a rich structure. We show that such curve counts obey an effective recursion, in the general spirit of topological recursion, and exhibit quasi-polynomial behavior. This “elementary curve-counting” is in fact related to a more advanced notion of “curve-counting” from algebraic geometry or symplectic geometry. The asymptotics of this enumerative problem are closely related to the asymptotics of volumes of moduli spaces of curves, and the quasi-polynomials governing the enumerative problem encode intersection numbers on moduli spaces. Among several other results, we show that generating functions and differential forms for these curve counts exhibit structure that is reminiscent of the mathematical physics of free energies, partition functions and quantum curves.

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The Partition Function in the Four-Dimensional Schwarz-Type Topological Half-Flat Two-Form Gravity

Modern Physics Letters A ◽

10.1142/s021773239700039x ◽

1997 ◽

Vol 12 (06) ◽

pp. 381-392 ◽

Cited By ~ 1

Author(s):

Mitsuko Abe

Keyword(s):

Partition Function ◽

Gravity Model ◽

Moduli Spaces ◽

Partition Functions ◽

K3 Surface ◽

Analytic Manifold ◽

Rham Complex ◽

De Rham ◽

Complex Analytic Manifold ◽

Equivalent Class

We derive the partition functions of the Schwarz-type four-dimensional topological half-flat two-form gravity model on K3-surface or T4 up to on-shell one-loop corrections. In this model the bosonic moduli spaces describe an equivalent class of a trio of the Einstein–Kähler forms (the hyper-Kähler forms). The integrand of the partition function is represented by the product of some [Formula: see text]-torsions. [Formula: see text]-torsion is the extension of R-torsion for the de Rham complex to that for the [Formula: see text]-complex of a complex analytic manifold.

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Instantons and vortices on noncommutative toric varieties

Reviews in Mathematical Physics ◽

10.1142/s0129055x14300088 ◽

2014 ◽

Vol 26 (09) ◽

pp. 1430008 ◽

Cited By ~ 2

Author(s):

Lucio S. Cirio ◽

Giovanni Landi ◽

Richard J. Szabo

Keyword(s):

Moduli Spaces ◽

Toric Varieties ◽

Partition Functions ◽

Toric Geometry ◽

Monoidal Categories ◽

Klein Quadric ◽

Noncommutative Algebraic Geometry ◽

Fundamental Matter ◽

Braided Monoidal Categories ◽

Noncommutative Gauge Theories

We elaborate on the quantization of toric varieties by combining techniques from toric geometry, isospectral deformations and noncommutative geometry in braided monoidal categories, and the construction of instantons thereon by combining methods from noncommutative algebraic geometry and a quantized twistor theory. We classify the real structures on a toric noncommutative deformation of the Klein quadric and use this to derive a new noncommutative four-sphere which is the unique deformation compatible with the noncommutative twistor correspondence. We extend the computation of equivariant instanton partition functions to noncommutative gauge theories with both adjoint and fundamental matter fields, finding agreement with the classical results in all instances. We construct moduli spaces of noncommutative vortices from the moduli of invariant instantons, and derive corresponding equivariant partition functions which also agree with those of the classical limit.

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Decomposition of BPS moduli spaces and asymptotics of supersymmetric partition functions

Journal of High Energy Physics ◽

10.1007/jhep01(2022)062 ◽

2022 ◽

Vol 2022 (1) ◽

Author(s):

Arash Arabi Ardehali ◽

Junho Hong

Keyword(s):

Partition Function ◽

Moduli Space ◽

Moduli Spaces ◽

Gauge Theories ◽

Asymptotic Expression ◽

Effective Field ◽

Asymptotic Result ◽

Partition Functions ◽

Abelian Gauge ◽

Scale Hierarchy

Abstract We present a prototype for Wilsonian analysis of asymptotics of supersymmetric partition functions of non-abelian gauge theories. Localization allows expressing such partition functions as an integral over a BPS moduli space. When the limit of interest introduces a scale hierarchy in the problem, asymptotics of the partition function is obtained in the Wilsonian approach by i) decomposing (in some suitable scheme) the BPS moduli space into various patches according to the set of light fields (lighter than the scheme dependent cut-off Λ) they support, ii) localizing the partition function of the effective field theory on each patch (with cut-offs set by the scheme), and iii) summing up the contributions of all patches to obtain the final asymptotic result (which is scheme-independent and accurate as Λ → ∞). Our prototype concerns the Cardy-like asymptotics of the 4d superconformal index, which has been of interest recently for its application to black hole microstate counting in AdS5/CFT4. As a byproduct of our analysis we obtain the most general asymptotic expression for the index of gauge theories in the Cardy-like limit, encompassing and extending all previous results.

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Partition Functions of $$\mathcal{N}=(2,2)$$ Supersymmetric Sigma Models and Special Geometry on the Moduli Spaces of Calabi-Yau Manifolds

Theoretical and Mathematical Physics ◽

10.1134/s0040577919110060 ◽

2019 ◽

Vol 201 (2) ◽

pp. 1606-1613 ◽

Cited By ~ 4

Author(s):

A. A. Belavin ◽

B. A. Eremin

Keyword(s):

Sigma Models ◽

Moduli Spaces ◽

Partition Functions ◽

Special Geometry ◽

Calabi Yau Manifolds

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More on topological vertex formalism for 5-brane webs with O5-plane

Journal of High Energy Physics ◽

10.1007/jhep04(2021)292 ◽

2021 ◽

Vol 2021 (4) ◽

Author(s):

Hirotaka Hayashi ◽

Rui-Dong Zhu

Keyword(s):

Gauge Theory ◽

Partition Function ◽

Vertex Operator ◽

Vertex Function ◽

Gauge Theories ◽

Partition Functions ◽

Topological Vertex ◽

Concrete Form ◽

Nekrasov Partition Functions ◽

Chern Simons

Abstract We propose a concrete form of a vertex function, which we call O-vertex, for the intersection between an O5-plane and a 5-brane in the topological vertex formalism, as an extension of the work of [1]. Using the O-vertex it is possible to compute the Nekrasov partition functions of 5d theories realized on any 5-brane web diagrams with O5-planes. We apply our proposal to 5-brane webs with an O5-plane and compute the partition functions of pure SO(N) gauge theories and the pure G2 gauge theory. The obtained results agree with the results known in the literature. We also compute the partition function of the pure SU(3) gauge theory with the Chern-Simons level 9. At the end we rewrite the O-vertex in a form of a vertex operator.

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