{"arxiv_id":"hep-th/0005247","incoming":{"papers":142,"sentences":279,"groups":[{"kind":"extends","label":"Builds on it","papers":2,"sentences":2,"quotes":[{"arxiv_id":"1612.09191","title":"Open Gauged Sigma Models, Equivariant Branes, and Equivariant Homological Mirror Symmetry","url":"/paper/1612.09191","section":"Quantum Corrections","sentence":"We shall now investigate quantum effects for the alternative formulation of equivariant B-branes given in Section 4.3,[We shall not study the quantum effects for the first formulation, since the main quantum correction is the running of the FI parameters, and the FI parameters do not enter the boundary conditions in that formalism.] since we shall use this formulation for the proof of mirror symmetry in the following section.[The following is a generalization of the analysis given in Section 6 of [ref] to the case of multiple $U(1)$ gauge groups. ]","intent_p":0.98,"contests_p":null},{"arxiv_id":"1808.02040","title":"A 3d Gauge Theory/Quantum K-Theory Correspondence","url":"/paper/1808.02040","section":"Defect entropies and vortex counting","sentence":"[ref] from sheaves on $X$ to sheaves on the loop space $LX$.","intent_p":0.86,"contests_p":null}]},{"kind":"method_or_tool","label":"Uses its method or tool","papers":10,"sentences":10,"quotes":[{"arxiv_id":"2012.04664","title":"Nonabelian mirrors and Gromov-Witten invariants","url":"/paper/2012.04664","section":"Picard-Fuchs equations for periods","sentence":"To find Picard-Fuchs equations for periods of mirrors, in this section, we study the open string theory with A-boundary in the Landau-Ginzburg model [ref] , and these boundary configurations are called A-branes or Lagrangian submanifolds in the context of LG models denoted by $γ_i$.","intent_p":0.9,"contests_p":null},{"arxiv_id":"1301.4437","title":"The Physical Mirror Equivalence for the Local P^2","url":"/paper/1301.4437","section":"The Double Fibration","sentence":"Following [ref] , we can rewrite the elliptic fibration in an almost Weierstrass form as follows.","intent_p":0.99,"contests_p":null},{"arxiv_id":"2503.16599","title":"Symmetries and M-theory-like Vacua in Four Dimensions","url":"/paper/2503.16599","section":"The x46 LG orientifold model","sentence":"The integral of such a one-form over a wedge is then given by the contour integral [ref] [equation] with $k=0,1,2$.","intent_p":0.86,"contests_p":null},{"arxiv_id":"1612.06874","title":"$E_6$ Yukawa couplings in F-theory as D-brane instanton effects","url":"/paper/1612.06874","section":"Mirror description","sentence":"The previous results can also be justified, to some extent, from the point of view of the IIA mirror description of the conifold using the techniques in [ref] .","intent_p":0.95,"contests_p":null},{"arxiv_id":"1706.05741","title":"Topological defects in string theory orbifolds with target spaces $\\mathbb{C}/\\mathbb{Z}_N$ and $S^1/\\mathbb{Z}_2$","url":"/paper/1706.05741","section":"A-type supersymmetry","sentence":"We obtain the A-type classification following the work of [ref] .","intent_p":0.87,"contests_p":null}]},{"kind":"data_used","label":"Uses its data or values","papers":1,"sentences":1,"quotes":[{"arxiv_id":"2007.02353","title":"Topological defects and SUSY RG flow","url":"/paper/2007.02353","section":"Branes and flows","sentence":"For instance, in [ref] the Witten index was computed for all such boundary conditions and it was found that it is $±1$ for two neighboring smallest wedges $(b,x_c, b+1)$ and $(b+1, x_c, b+2)$.","intent_p":0.82,"contests_p":null}]},{"kind":"comparison","label":"Compares with it","papers":8,"sentences":8,"quotes":[{"arxiv_id":"1704.01730","title":"Quasimap SYZ for toric Calabi-Yau manifolds","url":"/paper/1704.01730","section":"Wall-crossing for quasidisk invariants and the SYZ mirror","sentence":"Notice that the quasimap SYZ mirror family [ref] is entirely written in terms of symplectic-geometric information such as complexified Kähler parameters and quasidisk invariants of $X$, and it coincides with the mirror [ref] predicted by physical arguments [ref] .","intent_p":0.97,"contests_p":null},{"arxiv_id":"1412.7231","title":"SYZ mirror symmetry for toric varieties","url":"/paper/1412.7231","section":"The SYZ construction","sentence":"Notice that the SYZ mirror family [ref] is entirely written in terms of complexified Kähler parameters and disk counting invariants of $X$, and it agrees with the prediction [ref] using physical arguments [ref] .","intent_p":0.94,"contests_p":null},{"arxiv_id":"1603.08382","title":"Boundaries, Mirror Symmetry, and Symplectic Duality in 3d $\\mathcal{N}=4$ Gauge Theory","url":"/paper/1603.08382","section":"Two Dimensions","sentence":"In the presence of mass and FI parameters, the boundary conditions in 3d $=4$ theories share many properties with boundary conditions in A-twisted Landau-Ginzburg models [ref] , which generate Fukaya-Seidel categories [ref] .","intent_p":0.92,"contests_p":null},{"arxiv_id":"1608.02955","title":"Supersymmetric localization in two dimensions","url":"/paper/1608.02955","section":"Localization on the hemisphere","sentence":"For theories that flow to NLSMs on Kähler manifolds, the large-volume limit reproduces the known geometric expression [ref] , however the localization formula contains all quantum corrections (to be compared with [ref] ).","intent_p":0.91,"contests_p":null},{"arxiv_id":"1612.09191","title":"Open Gauged Sigma Models, Equivariant Branes, and Equivariant Homological Mirror Symmetry","url":"/paper/1612.09191","section":"The Gauged Nonlinear Sigma Model with Boundaries","sentence":"Before ending this section, we note that the action $S+S_r$ ([ref] + [ref]) and supersymmetry variations [ref], [ref] and [ref] reduce to the familiar formulae given in [ref] for GLSMs and NLSMs with boundary, in the appropriate limits.","intent_p":0.89,"contests_p":null}]},{"kind":"supports_this_paper","label":"Cites it as support","papers":3,"sentences":3,"quotes":[{"arxiv_id":"0911.1316","title":"Surface Operators in N=2 4d Gauge Theories","url":"/paper/0911.1316","section":"Hitchin systems from surface operators","sentence":"As proven in [ref] and reviewed in the appendix [ref] the correlation function of the 2d theory on a half-cylinder of radius $R$, with such a boundary at one end and a choice of Ramond vacuum at the other end, is a flat section of the spectral connection $∇$ with the same $R, ζ$!","intent_p":0.97,"contests_p":null},{"arxiv_id":"1103.5832","title":"Classification of complete N=2 supersymmetric theories in 4 dimensions","url":"/paper/1103.5832","section":"Lagrangian $A$–branes as ideal triangulations","sentence":"[ref] it was shown that the Lagrangian $A$–branes ${γ_i}$ span the relative homology group $H_1(,)$ (where $⊂$ is the region near the punctures where $Re(e^iθ W)≫1$), so all the axioms for an ideal triangulation are satisfied.","intent_p":0.96,"contests_p":null},{"arxiv_id":"2012.09769","title":"Exponential BPS graphs and D-brane counting on toric Calabi-Yau threefolds: Part II","url":"/paper/2012.09769","section":"A view from the Fukaya category","sentence":"Note that this definition of morphisms is in agreement with the fact, found in [ref] that the open string index between A-branes in a LG model satisfying $(W(p_k))< (W(p_k'))$ always vanishes.","intent_p":0.98,"contests_p":null}]},{"kind":"background","label":"Cites it as background","papers":65,"sentences":78,"quotes":[{"arxiv_id":"1001.0447","title":"Open String Invariants and Mirror Curve of the Resolved Conifold","url":"/paper/1001.0447","section":"Introduction","sentence":"For open string, Aganagic and Vafa [ref] conjectured a relationship between genus $0$ open string invariants and the equation for the mirror curve given by the Hori-Iqbal-Vafa construction [ref] .","intent_p":0.87,"contests_p":null},{"arxiv_id":"1811.12385","title":"B-brane transport in anomalous (2,2) models and localization","url":"/paper/1811.12385","section":"Introduction","sentence":"B-branes on anomalous models have been studied in the context of mirror symmetry of massive theories [ref] and the interpretation in terms of flows in the moduli space in [ref] for Fano and general type hypersurfaces in $^N$.[In mathematics similar cases have been analyzed in [ref] , but mostly limited to Fano varieties or general type hypersurfaces in $^N$.] We want to unify [ref] with the categorical approach of [ref] .","intent_p":0.88,"contests_p":null},{"arxiv_id":"1704.01730","title":"Quasimap SYZ for toric Calabi-Yau manifolds","url":"/paper/1704.01730","section":"Toric Calabi-Yau manifolds and their physical mirrors","sentence":"The mirror of a toric Calabi-Yau manifold $X$ is predicted to be a family of affine hypersurfaces in $^2 ×(^×)^n-1$ [ref] explicitly written as [equation] where the coefficients $Č_i ∈$ are constants (without loss of generality, we will set $Č_m = 1$) subject to the constraints [equation] here $t = (t_1, …, t_r)$ are coordinates on the mirror complex moduli $ℳ̌_B := 𝕂^∨⊗_ ^×≅(^×)^r$. $X̌_t$ is Calabi-Yau since [equation] is a nowhere vanishing holomorphic volume form on $X̌_t$.","intent_p":0.89,"contests_p":null},{"arxiv_id":"1609.01723","title":"Brane Brick Models in the Mirror","url":"/paper/1609.01723","section":"Mirror Approach to Brane Tilings and Brane Brick Models","sentence":"Its mirror geometry [ref] is another $n$-fold $𝒲$ given by a double fibration over the complex $W$ plane [ W = P(x_1,…, x_n-1); W = uv ]","intent_p":0.83,"contests_p":null},{"arxiv_id":"2010.11837","title":"Categorical Wall-Crossing in Landau-Ginzburg Models","url":"/paper/2010.11837","section":"Wall-Crossing of BPS Indices","sentence":"Such branes have been analyzed in great detail in references, [ref] .","intent_p":0.89,"contests_p":null}]}],"unlabelled_sentences":177},"outgoing":{"papers":0,"sentences":0,"groups":[],"unlabelled_sentences":0},"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":{"arxiv_id":"hep-th/0005247","kind":"model","paper_version":2,"prompt_version":"held-up-v3.5","generated_at":"2026-10-27T15:33:40+00:00","loaded_at":"2026-10-28T01:02:26.173890+00:00","evidence":"substantial","account":[],"groups":[{"kind":"extends","label":"Builds on it","papers":2,"entries":[{"p":0.98,"quote":null,"citing":[{"self":false,"year":2016,"title":"Open Gauged Sigma Models, Equivariant Branes, and Equivariant Homological Mirror Symmetry","arxiv_id":"1612.09191"}],"verified":true,"paraphrase":"Generalized the analysis given in Section 6 of this paper to the case of multiple U(1) gauge groups."},{"p":0.99,"quote":null,"citing":[{"self":false,"year":2022,"title":"Electric-Magnetic Duality in a Class of $G_2$-Compactifications of M-theory","arxiv_id":"2210.08628"}],"verified":true,"paraphrase":"Generalized this paper's result to expect the physics to be described by n D3-branes probing an I*_3 singularity."}],"papers_all":3,"self_papers":0,"self_entries":[]},{"kind":"method_or_tool","label":"Uses its method or tool","papers":2,"entries":[{"p":0.9,"quote":null,"citing":[{"self":false,"year":2020,"title":"Nonabelian mirrors and Gromov-Witten invariants","arxiv_id":"2012.04664"}],"verified":true,"paraphrase":"Studied the open string theory with A-boundary in the Landau-Ginzburg model of this paper to find Picard-Fuchs equations for periods of mirrors."},{"p":0.99,"quote":null,"citing":[{"self":false,"year":2013,"title":"The Physical Mirror Equivalence for the Local P^2","arxiv_id":"1301.4437"}],"verified":true,"paraphrase":"Followed this paper to rewrite the elliptic fibration in an almost Weierstrass form."}],"papers_all":10,"self_papers":0,"self_entries":[]}],"counts":{"by_kind":{"extends":3,"data_used":1,"comparison":8,"method_or_tool":10,"supports_this_paper":3},"dropped":{"relied_on":0,"unknown_ids":0,"account_lines":0,"contest_vetoed":0,"below_threshold":0,"contest_drawback":0,"contest_without_basis":0,"contest_not_one_sentence":0,"contest_on_group_citation":0},"papers_read":22,"self_papers":14,"citing_papers":142,"sentences_read":26,"citing_sentences":278,"unambiguous_read":26,"typed_independent":21,"independent_papers":128,"by_kind_independent":{"extends":3,"data_used":1,"comparison":7,"method_or_tool":9,"supports_this_paper":2}},"input_sha256":"cac1514fe5bc32dd10fc4c43c33761a5416e9eff666f344f7606bf370f1cb486","method":"For a paper with five or more citing arXiv papers, Pith selected up to 40 citing sentences its citation classifier typed, read those whose citation markers point at this paper alone, and wrote one paraphrase for each later paper in each group, each resting on one sentence. A later paper is listed under a label only when the classifier gives one of its sentences that label at intent probability 0.8 or more (0.95 for support), Pith's reading of the sentence finds that the citing paper itself did it, and a second check of the paraphrase against the sentence passes. The section lists uses only: building on the paper, using its method, tool, data or values, comparing with it and citing it as support. It does not list criticism or disagreement, so their absence says nothing about how the paper has been received. Code counted the papers, wrote the summary from the listed entries, set the evidence level and linked each entry to its citing paper. Quotes are copied from the citing sentence and shown only when the citing paper's licence is CC BY, CC BY-SA or CC0; otherwise the paraphrase is shown. Papers that share an author with this one are listed apart. A paper with fewer citing papers gets a line counted by code, which names uses only: a later paper whose only label is a contest is counted as not described.","line":null,"summary":"4 later papers by other authors are listed below: 2 built on it and 2 used its method or tool."},"_links":{"html":"https://pith.science/paper/hep-th/0005247"}}