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Supersymmetric Surface Operators, Four-Manifold Theory and Invariants in Various Dimensions
High Energy Physics - Theory · Submitted 16 Jun 2010
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim Elegant physical proofs of various seminal theorems in four-manifold theory obtained by Ozsvath-Szabo [2,3] and Taubes [4], will be furnished.
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We continue our program initiated in [arXiv:0912.4261] to consider supersymmetric surface operators in a topologically-twisted N=2 pure SU(2) gauge theory, and apply them to the study of four-manifolds and related invariants. Elegant physical proofs of various seminal theorems in four-manifold theory obtained by Ozsvath-Szabo [2,3] and Taubes [4], will be furnished. In particular, we will show that Taubes' groundbreaking and difficult result -- that the ordinary Seiberg-Witten invariants are in fact the Gromov invariants which count pseudo-holomorphic curves embedded in a symplectic four-manifold X -- nonetheless lends itself to a simple and concrete physical derivation in the presence of "ordinary" surface operators. As an offshoot, we will be led to several interesting and mathematically novel identities among the Gromov and "ramified" Seiberg-Witten invariants of X, which in certain cases, also involve the instanton and monopole Floer homologies of its three-submanifold. Via these identities, and a physical formulation of the "ramified" Donaldson invariants of four-manifolds with boundaries, we will uncover completely new and economical ways of deriving and understanding various important mathematical results concerning (i) knot homology groups from "ramified" instantons by Kronheimer-Mrowka [5]; and (ii) monopole Floer homology and Seiberg-Witten theory on symplectic four-manifolds by Kutluhan-Taubes [4,6]. Supersymmetry, as well as other physical concepts such as R-invariance, electric-magnetic duality, spontaneous gauge symmetry-breaking and localization onto supersymmetric configurations in topologically-twisted quantum field theories, play a pivotal role in our story.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We continue our program initiated in [arXiv:0912.4261] to consider supersymmetric surface operators in a topologically-twisted N=2 pure SU(2) gauge theory, and apply them to the study of four-manifolds and related invariants. | background 0.60 | 0.36 | 0.07 | asserted 0.85 |
| s1 | Elegant physical proofs of various seminal theorems in four-manifold theory obtained by Ozsvath-Szabo [2,3] and Taubes [4], will be furnished. | contribution 0.57 | 0.59 | 0.22 | asserted 0.99 |
| s2 | In particular, we will show that Taubes' groundbreaking and difficult result -- that the ordinary Seiberg-Witten invariants are in fact the Gromov invariants which count pseudo-holomorphic curves embedded in a symplectic four-manifold X -- nonetheless lends itself to a simple and concrete physical derivation in the presence of "ordinary" surface operators. | result 0.66 | 0.50 | 0.40 | asserted 0.99 |
| s3 | As an offshoot, we will be led to several interesting and mathematically novel identities among the Gromov and "ramified" Seiberg-Witten invariants of X, which in certain cases, also involve the instanton and monopole Floer homologies of its three-submanifold. | result 0.90 | 0.74 | 0.12 | asserted 0.89 |
| s4 | Via these identities, and a physical formulation of the "ramified" Donaldson invariants of four-manifolds with boundaries, we will uncover completely new and economical ways of deriving and understanding various important mathematical results concerning (i) knot homology groups from "ramified" instantons by Kronheimer-Mrowka [5]; and (ii) monopole Floer homology and Seiberg-Witten theory on symplectic four-manifolds by Kutluhan-Taubes [4,6]. | result 0.50 | 0.80 | 0.26 | asserted 1.00 |
| s5 | Supersymmetry, as well as other physical concepts such as R-invariance, electric-magnetic duality, spontaneous gauge symmetry-breaking and localization onto supersymmetric configurations in topologically-twisted quantum field theories, play a pivotal role in our story. | method 0.80 | 0.21 | 0.05 | asserted 1.00 |
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Original abstract
We continue our program initiated in [arXiv:0912.4261] to consider supersymmetric surface operators in a topologically-twisted N=2 pure SU(2) gauge theory, and apply them to the study of four-manifolds and related invariants. Elegant physical proofs of various seminal theorems in four-manifold theory obtained by Ozsvath-Szabo [2,3] and Taubes [4], will be furnished. In particular, we will show that Taubes' groundbreaking and difficult result -- that the ordinary Seiberg-Witten invariants are in fact the Gromov invariants which count pseudo-holomorphic curves embedded in a symplectic four-manifold X -- nonetheless lends itself to a simple and concrete physical derivation in the presence of "ordinary" surface operators. As an offshoot, we will be led to several interesting and mathematically novel identities among the Gromov and "ramified" Seiberg-Witten invariants of X, which in certain cases, also involve the instanton and monopole Floer homologies of its three-submanifold. Via these identities, and a physical formulation of the "ramified" Donaldson invariants of four-manifolds with boundaries, we will uncover completely new and economical ways of deriving and understanding various important mathematical results concerning (i) knot homology groups from "ramified" instantons by Kronheimer-Mrowka [5]; and (ii) monopole Floer homology and Seiberg-Witten theory on symplectic four-manifolds by Kutluhan-Taubes [4,6]. Supersymmetry, as well as other physical concepts such as R-invariance, electric-magnetic duality, spontaneous gauge symmetry-breaking and localization onto supersymmetric configurations in topologically-twisted quantum field theories, play a pivotal role in our story.
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References
How the literature uses it
6 arXiv papers cite this in 9 sentences
Papers citing it · 1 sentence without a confident label
Builds on it 1 paper
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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] .
Donaldson-Witten theory, surface operators and mock modular forms · Introduction · 0.88
Cites it as background 6 papers
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Recall that one of the ways to define the surface operator is to prescribe a singular behavior of the gauge field [ref] (see also [ref] for the surface operators in $𝒩=2$ theories and [ref] for more mathematical formulation).
Localization with a Surface Operator, Irregular Conformal Blocks and Open Topological String · Introduction · 0.98
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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
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As explained in detail in [ref] in principle $α∈𝕋 = 𝗍/Λ_cochar.$ where $Λ_cochar. = Hom(U(1),𝕋)$ is the co-character lattice [ref] .
Donaldson-Witten theory, surface operators and mock modular forms · Review of surface operators in four dimensions · 0.91
What it cites · 67 sentences without a confident label
Builds on 4 papers
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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.
Integration Over The u-Plane In Donaldson Theory With Surface Operators · Introduction · 0.93
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In fact, our physical proof asserts that ([ref]) should also hold for $Σ$ with $Σ∩Σ= 0$, and not just for $Σ$ with $Σ∩Σ< 0$ (as stipulated in Corollary 1.7 of [ref] ); this physical assertion is indeed consistent with Theorem 1.1 of [ref] for $X$ of SW simple-type (for which ([ref]) is also valid).
Higher Type Adjunction Inequalities in Seiberg-Witten Theory · Introduction · 0.88
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In fact, our physical proof asserts that ([ref]) should also hold for $Σ$ with $Σ∩Σ= 0$, and not just for $Σ$ with $Σ∩Σ< 0$ (as stipulated in Corollary 1.7 of [ref] ); this physical assertion is indeed consistent with Theorem 1.1 of [ref] for $X$ of SW simple-type (for which ([ref]) is also valid).
The symplectic Thom conjecture · Introduction · 0.88
Compares with 5 papers
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This observation matches exactly the claim in Theorem 1.3 of [ref] for four-manifolds with $b_2^+ =1$.
The symplectic Thom conjecture · Introduction · 0.92
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Consequently, after noting that $s' = s - δ_Σ$ because $ϵ=-1$, we find that ([ref]) and ([ref]), with ([ref]) and ([ref]), is nothing but Theorem 1.6 of [ref] ; here, $a_d$, $p$ and $(/2 + p)$ can be consistently identified as $U$, $d$ and $d'$ in Theorem 1.6 of [ref] , respectively.
Higher Type Adjunction Inequalities in Seiberg-Witten Theory · Introduction · 0.84
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In fact, ([ref]) is consistent with the relation $Gr_0(A) = ±1$ proved in Proposition 3.18 of [ref] , while ([ref]) – in light of ([ref]) – is consistent with the relation $|Gr(K) |= 1$ proved in Theorem 3.10 of [ref] .
Lectures on Gromov invariants for symplectic 4-manifolds · Introduction · 0.80
Cites as support 1 paper
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Here, $w = w_2(E_M)$ is the second Stiefel-Whitney class of the $SO(3)$ gauge bundle $E_M$ over $M$, and according to the Main Theorem of [ref] and $$40.1 of [ref] , the first Chern class of the projection $π(ŝ)$ to $M$ of the $$-structure $ŝ$ on $X$ is necessarily non-torsion.
Seiberg-Witten Floer homology and symplectic forms on S^1 X M^3 · Introduction · 0.96
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