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Donaldson-Witten theory, surface operators and mock modular forms
High Energy Physics - Theory · Submitted 16 Oct 2018
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim We show that for specific K\"ahler surfaces of Kodaira dimension $-\infty$ the integral localizes at the cusp at infinity of the Coulomb branch of the theory.
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The abstract, sentence by sentence
We revisit the $u$-plane integral of the topologically twisted $\mathcal{N}=2$ super Yang-Mills theory, the Donaldson-Witten theory, on a closed four-manifold $X$ with embedded surfaces that support supersymmetric surface operators. This integral mathematically corresponds to the generating function of the ramified Donaldson invariants of $X$. By including a $\overline{\mathcal{Q}}$-exact deformation to the $u$-plane integral we are able to re-express its integrand in terms of a total derivative with respect to an indefinite theta function, a special kind of mock modular form. We show that for specific K\"ahler surfaces of Kodaira dimension $-\infty$ the integral localizes at the cusp at infinity of the Coulomb branch of the theory.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We revisit the $u$-plane integral of the topologically twisted $\mathcal{N}=2$ super Yang-Mills theory, the Donaldson-Witten theory, on a closed four-manifold $X$ with embedded surfaces that support supersymmetric surface operators. | contribution 0.54 | 0.52 | 0.05 | asserted 0.79 |
| s1 | This integral mathematically corresponds to the generating function of the ramified Donaldson invariants of $X$. | background 0.78 | 0.17 | 0.07 | asserted 1.00 |
| s2 | By including a $\overline{\mathcal{Q}}$-exact deformation to the $u$-plane integral we are able to re-express its integrand in terms of a total derivative with respect to an indefinite theta function, a special kind of mock modular form. | method 0.67 | 0.94 | 0.37 | asserted 0.99 |
| s3 | We show that for specific K\"ahler surfaces of Kodaira dimension $-\infty$ the integral localizes at the cusp at infinity of the Coulomb branch of the theory. | result 1.00 | 0.93 | 0.25 | asserted 1.00 |
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The paper
Original abstract
We revisit the $u$-plane integral of the topologically twisted $\mathcal{N}=2$ super Yang-Mills theory, the Donaldson-Witten theory, on a closed four-manifold $X$ with embedded surfaces that support supersymmetric surface operators. This integral mathematically corresponds to the generating function of the ramified Donaldson invariants of $X$. By including a $\overline{\mathcal{Q}}$-exact deformation to the $u$-plane integral we are able to re-express its integrand in terms of a total derivative with respect to an indefinite theta function, a special kind of mock modular form. We show that for specific K\"ahler surfaces of Kodaira dimension $-\infty$ the integral localizes at the cusp at infinity of the Coulomb branch of the theory.
Human review
How the literature uses it
4 arXiv papers cite this in 5 sentences
Papers citing it · 3 sentences without a confident label
Builds on it 1 paper
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The result can be readily extended to the case with surface defects [ref] or theories with matter hypermultiplets [ref] and even class $𝒮$ theories [ref] , although we leave this for future work.
The $u$-plane integral, mock modularity and enumerative geometry · Conclusions · 0.91
Cites it as background 1 paper
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This operator has appeared previously in the context of the CohFT interpretation of Witten-like indices [ref] , and more recently for the evaluation of Coulomb branch integrals using indefinite theta functions in [ref] .
Renormalization and BRST symmetry in Donaldson-Witten theory · Introduction · 0.89
What it cites · 61 sentences without a confident label
Builds on 4 papers
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Using the same argumentation as in [ref] , but for the ramified theory, the difference of the Coulomb branch between two neighboring chambers is given by a term $ΔΦ̃_^J_1J_0 =Φ̃_^J_1-Φ̃_^J_0$ which reads ΔΦ̃_^J_1J_0 = ∫_/Γ^0(4) dτ∧dτ̅ f̃ (Ψ̃_^J_1-Ψ̃_^J_0 ), with the contribution from the cusp at $i∞$ giving the following result ΔΦ̃_^J_1J_0(p,) = 4 [ ν̃(τ) e^2pu + ^2G(u) + ^2H(u)_^J_1J_0(τ, ;α) ]_q^0.
Donaldson-Witten theory and indefinite theta functions · Wall-crossing formula · 0.93
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In this paper we continue towards this paradigm by further investigating the ramified $u$-plane integral inspired by the works of Tan [ref] .
Integration Over The u-Plane In Donaldson Theory With Surface Operators · Introduction · 0.88
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In this paper we continue towards this paradigm by further investigating the ramified $u$-plane integral inspired by the works of Tan [ref] .
Supersymmetric Surface Operators, Four-Manifold Theory and Invariants in Various Dimensions · Introduction · 0.88
Uses the method or tool of 1 paper
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To this end we will add a $$-exact operator $𝖨̃_+()$ to the partition function that is defined accrording to the lines of [ref] as [equation] Since the physical operators of the (ramified or not) Donaldson-Witten theory belong to the $G$-equivariant cohomology of $X$ we are definitely allowed to do so as long as any insertion is $$-exact.
Donaldson-Witten theory and indefinite theta functions · The Grassmann and photon path integral · 0.86
Compares with 5 papers
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This comes in very close analogy to what was found in [ref] where the usual Donaldson invariants were obtained (for specific Kähler surfaces) by a very similar simple formula as well and at the limit of vanishing volume for the embedded surface our result reduces to the one of the usual Donaldson-Witten theory.
Donaldson-Witten theory and indefinite theta functions · Discussion and conclusion · 0.99
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We already recognize two contact terms, unlike in [ref] where there is only one, with the second one here being precisely due to the presence of the surface defect.
Integration over the u-plane in Donaldson theory · The Grassmann and photon path integral · 0.99
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We already recognize two contact terms, unlike in [ref] where there is only one, with the second one here being precisely due to the presence of the surface defect.
Issues in Topological Gauge Theory · The Grassmann and photon path integral · 0.99
Cites as background 34 papers
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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].
Integration Over The u-Plane In Donaldson Theory With Surface Operators · The ramified $u$-plane integral as a total derivative · 0.88
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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 and indefinite theta functions · The ramified $u$-plane integral as a total derivative · 0.88
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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].
Integration over the u-plane in Donaldson theory · The ramified $u$-plane integral as a total derivative · 0.88
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