{"arxiv_id":"1006.3313","incoming":{"papers":6,"sentences":9,"groups":[{"kind":"extends","label":"Builds on it","papers":1,"sentences":1,"quotes":[{"arxiv_id":"1810.07057","title":"Donaldson-Witten theory, surface operators and mock modular forms","url":"/paper/1810.07057","section":"Introduction","sentence":"In this paper we continue towards this paradigm by further investigating the ramified $u$-plane integral inspired by the works of Tan [ref] .","intent_p":0.88,"contests_p":null}]},{"kind":"background","label":"Cites it as background","papers":6,"sentences":7,"quotes":[{"arxiv_id":"1008.0574","title":"Localization with a Surface Operator, Irregular Conformal Blocks and Open Topological String","url":"/paper/1008.0574","section":"Introduction","sentence":"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).","intent_p":0.98,"contests_p":null},{"arxiv_id":"1302.7297","title":"Defects in Cohomological Gauge Theory and Donaldson-Thomas Invariants","url":"/paper/1302.7297","section":"Surface operators in four dimensional $=2$ Yang–Mills","sentence":"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] .","intent_p":0.81,"contests_p":null},{"arxiv_id":"1810.07057","title":"Donaldson-Witten theory, surface operators and mock modular forms","url":"/paper/1810.07057","section":"Review of surface operators in four dimensions","sentence":"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] .","intent_p":0.91,"contests_p":null},{"arxiv_id":"1307.2578","title":"Surface Defects and Resolvents","url":"/paper/1307.2578","section":"Introduction","sentence":"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] .","intent_p":1.0,"contests_p":null},{"arxiv_id":"1105.0357","title":"Instanton counting with a surface operator and the chain-saw quiver","url":"/paper/1105.0357","section":"Defining data of a surface operator","sentence":"In this section we review the basic properties of the surface operator following [ref] [See also [ref] for the surface operators in $𝒩=2$ theories.].","intent_p":0.96,"contests_p":null}]}],"unlabelled_sentences":1},"outgoing":{"papers":18,"sentences":80,"groups":[{"kind":"extends","label":"Builds on","papers":4,"sentences":5,"quotes":[{"arxiv_id":"0912.4261","title":"Integration Over The u-Plane In Donaldson Theory With Surface Operators","url":"/paper/0912.4261","section":"Introduction","sentence":"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.","intent_p":0.93,"contests_p":null},{"arxiv_id":"math/0005268","title":"Higher Type Adjunction Inequalities in Seiberg-Witten Theory","url":"/paper/math/0005268","section":"Introduction","sentence":"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).","intent_p":0.88,"contests_p":null},{"arxiv_id":"math/9811087","title":"The symplectic Thom conjecture","url":"/paper/math/9811087","section":"Introduction","sentence":"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).","intent_p":0.88,"contests_p":null},{"arxiv_id":"0806.1053","title":"Knot homology groups from instantons","url":"/paper/0806.1053","section":"Introduction","sentence":"In $$7, the final section, by further applying our physical insights and results obtained hitherto, we will first provide – via the topological gauge theory with nontrivially-embedded surface operators on a general four-manifold with boundaries – a physical derivation of certain key properties of knot homology groups from “ramified” instantons defined and proved by Kronheimer and Mrowka in [ref] .","intent_p":0.83,"contests_p":null}]},{"kind":"comparison","label":"Compares with","papers":5,"sentences":7,"quotes":[{"arxiv_id":"math/9811087","title":"The symplectic Thom conjecture","url":"/paper/math/9811087","section":"Introduction","sentence":"This observation matches exactly the claim in Theorem 1.3 of [ref] for four-manifolds with $b_2^+ =1$.","intent_p":0.92,"contests_p":null},{"arxiv_id":"math/0005268","title":"Higher Type Adjunction Inequalities in Seiberg-Witten Theory","url":"/paper/math/0005268","section":"Introduction","sentence":"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.","intent_p":0.84,"contests_p":null},{"arxiv_id":"dg-ga/9606005","title":"Lectures on Gromov invariants for symplectic 4-manifolds","url":"/paper/dg-ga/9606005","section":"Introduction","sentence":"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] .","intent_p":0.8,"contests_p":null},{"arxiv_id":"0804.1371","title":"Seiberg-Witten Floer homology and symplectic forms on S^1 X M^3","url":"/paper/0804.1371","section":"Introduction","sentence":"Note that this easy-to-reach but nevertheless important conclusion about $HM_∗(M, π(ŝ))$ has also been derived via a distinct and highly-involved mathematical approach in the Main Theorem of [ref] , where “$e$” and “$s_e$” therein correspond to $π(c_1(E))$ and $π(ŝ)$ herein.","intent_p":0.89,"contests_p":null},{"arxiv_id":"0806.1053","title":"Knot homology groups from instantons","url":"/paper/0806.1053","section":"Introduction","sentence":"This result has also been proved via a distinct mathematical approach by Kronheimer and Mrowka as Proposition 3.27 in [ref] .","intent_p":0.86,"contests_p":null}]},{"kind":"supports_this_paper","label":"Cites as support","papers":1,"sentences":1,"quotes":[{"arxiv_id":"0804.1371","title":"Seiberg-Witten Floer homology and symplectic forms on S^1 X M^3","url":"/paper/0804.1371","section":"Introduction","sentence":"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.","intent_p":0.96,"contests_p":null}]}],"unlabelled_sentences":67},"claims":[],"thresholds":{"intent_p":0.8,"supports_p":0.95,"contests_p":0.7},"method":"Citing sentences come from each paper's LaTeX source; every \\cite marks the sentence and the works it cites. Pith's classifier typed each sentence. A label is shown at intent probability 0.8 or more (0.95 for support), a contest at 0.7 or more. Measured accuracy against two blind model families is at https://pith.science/accuracy. The quoted sentences are the citing authors' own.","held_up":null,"_links":{"html":"https://pith.science/paper/1006.3313"}}