Integration Over The u-Plane In Donaldson Theory With Surface Operators

Meng-Chwan Tan

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Integration Over The u-Plane In Donaldson Theory With Surface Operators

Meng-Chwan Tan

High Energy Physics - Theory · Submitted 21 Dec 2009

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim Elegant physical proofs -- rooted in R-anomaly cancellations and modular invariance over the u-plane -- of various seminal results in four-dimensional geometric topology obtained by Kronheimer and Mrowka [1,2] -- such as a universal formula relating the "ramified" and ordinary Donaldson invariants, and a generalization of the celebrated Thom conjecture -- will be furnished.

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What the abstract itself states, as probabilities: claims first 0.38 · states limitation 0.08 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.07 · compares with prior results 0.87 · mentions code or data availability 0.02.

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We generalize the analysis by Moore and Witten in [arXiv:hep-th/9709193], and consider integration over the u-plane in Donaldson theory with surface operators on a smooth four-manifold X. Several novel aspects will be developed in the process; like a physical interpretation of the "ramified" Donaldson and Seiberg-Witten invariants, and the concept of curved surface operators which are necessarily topological at the outset. Elegant physical proofs -- rooted in R-anomaly cancellations and modular invariance over the u-plane -- of various seminal results in four-dimensional geometric topology obtained by Kronheimer and Mrowka [1,2] -- such as a universal formula relating the "ramified" and ordinary Donaldson invariants, and a generalization of the celebrated Thom conjecture -- will be furnished. Wall-crossing and blow-up formulas of these "ramified" invariants which have not been computed in the mathematical literature before, as well as a generalization and a Seiberg-Witten analog of the universal formula as implied by an electric-magnetic duality of trivially-embedded surface operators in X, will also be presented, among other things.

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#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We generalize the analysis by Moore and Witten in [arXiv:hep-th/9709193], and consider integration over the u-plane in Donaldson theory with surface operators on a smooth four-manifold X.	contribution 0.92	0.72	0.06	asserted 0.92
s1	Several novel aspects will be developed in the process; like a physical interpretation of the "ramified" Donaldson and Seiberg-Witten invariants, and the concept of curved surface operators which are necessarily topological at the outset.	contribution 0.97	0.74	0.08	asserted 0.76
s2	Elegant physical proofs -- rooted in R-anomaly cancellations and modular invariance over the u-plane -- of various seminal results in four-dimensional geometric topology obtained by Kronheimer and Mrowka [1,2] -- such as a universal formula relating the "ramified" and ordinary Donaldson invariants, and a generalization of the celebrated Thom conjecture -- will be furnished.	result 0.68	0.86	0.37	asserted 1.00
s3	Wall-crossing and blow-up formulas of these "ramified" invariants which have not been computed in the mathematical literature before, as well as a generalization and a Seiberg-Witten analog of the universal formula as implied by an electric-magnetic duality of trivially-embedded surface operators in X, will also be presented, among other things.	contribution 0.51	0.86	0.19	asserted 0.99

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Original abstract

We generalize the analysis by Moore and Witten in [arXiv:hep-th/9709193], and consider integration over the u-plane in Donaldson theory with surface operators on a smooth four-manifold X. Several novel aspects will be developed in the process; like a physical interpretation of the "ramified" Donaldson and Seiberg-Witten invariants, and the concept of curved surface operators which are necessarily topological at the outset. Elegant physical proofs -- rooted in R-anomaly cancellations and modular invariance over the u-plane -- of various seminal results in four-dimensional geometric topology obtained by Kronheimer and Mrowka [1,2] -- such as a universal formula relating the "ramified" and ordinary Donaldson invariants, and a generalization of the celebrated Thom conjecture -- will be furnished. Wall-crossing and blow-up formulas of these "ramified" invariants which have not been computed in the mathematical literature before, as well as a generalization and a Seiberg-Witten analog of the universal formula as implied by an electric-magnetic duality of trivially-embedded surface operators in X, will also be presented, among other things.

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References

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Cited by 3 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Surface Defects and Resolvents
hep-th 2013-07 accept novelty 7.0 of 10

The effective twisted superpotential of a large class of BPS surface defects is obtained by replacing traces of the bulk scalar in the 2d one-loop result with the exact 4d resolvent, reproducing and extending brane-co...
Instanton counting with a surface operator and the chain-saw quiver
hep-th 2011-05 conditional novelty 6.0 of 10

The instanton partition function with a general surface operator is computed from chain-saw quiver fixed points and shown to match the norm of a Whittaker vector in the corresponding W-algebra Verma module for partiti...
Affine SL(2) conformal blocks from 4d gauge theories
hep-th 2010-05 conditional novelty 6.0 of 10

The Nekrasov partition function of N=2 SU(2) quiver theories with a full surface operator matches an affine SL(2) conformal block dressed by a fixed operator K, and in the limit ǫ2 to 0 gives eigenfunctions of the qua...

How the literature uses it

10 arXiv papers cite this in 47 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 32 sentences without a confident label

Builds on it 2 papers

In this paper, we continue the program initiated in [ref] ; we consider arbitrarily-embedded surface operators in a topologically-twisted $=2$ pure $SU(2)$ gauge theory, and apply them to the study of four-manifolds and invariants in two, three and four dimensions.

Supersymmetric Surface Operators, Four-Manifold Theory and Invariants in Various Dimensions · Introduction · 0.93
In this paper we continue towards this paradigm by further investigating the ramified $u$-plane integral inspired by the works of Tan [ref] .

Donaldson-Witten theory, surface operators and mock modular forms · Introduction · 0.88

Cites it as background 7 papers

Let us begin by discussing the domain of integration $/Γ^0(4)$ which is the union of six images of the fundamental domain of $SL(2,)$ [ref] This is depicted in Figure [ref].

Donaldson-Witten theory, surface operators and mock modular forms · The ramified $u$-plane integral as a total derivative · 0.88
In the case of $SU(2)$ the problem has been set up by Kronheimer and Mrowka in [ref] and discussed from the field theory point of view in [ref] .

Defects in Cohomological Gauge Theory and Donaldson-Thomas Invariants · Surface operators in four dimensional $=2$ Yang–Mills · 0.81
BPS surface defects (a.k.a. surface operators) in $=2$ four-dimensional gauge theories are an interesting theoretical subject which ties together many subjects in mathematical physics [ref] .

Surface Defects and Resolvents · Introduction · 1.00

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Note that the blowup factors ([ref]) and ([ref]) coincide with those for the generating function of the ordinary Donaldson invariants computed in [ref] .

The blowup formula for Donaldson invariants · Introduction · 0.98
Note that the blowup factors ([ref]) and ([ref]) coincide with those for the generating function of the ordinary Donaldson invariants computed in [ref] .

Jacobi forms and the structure of Donaldson invariants for 4-manifolds with b_+=1 · Introduction · 0.98

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