{"arxiv_id":"0908.2064","incoming":{"papers":36,"sentences":68,"groups":[{"kind":"extends","label":"Builds on it","papers":1,"sentences":1,"quotes":[{"arxiv_id":"0909.2052","title":"On non-conformal limit of the AGT relations","url":"/paper/0909.2052","section":"Introduction","sentence":"A similar analysis is possible for conformal blocks with more external states and for some $W$-algebra blocks $N_c>2$, in the last case the results of [ref] should be used.","intent_p":0.81,"contests_p":null}]},{"kind":"method_or_tool","label":"Uses its method or tool","papers":6,"sentences":8,"quotes":[{"arxiv_id":"0912.4789","title":"On AGT Conjecture for Pure Super Yang-Mills and W-algebra","url":"/paper/0912.4789","section":"recursion relations and asymptotic behavior","sentence":"The key is the Mironov-Mironov-Morozov-Morozov recursion relations [ref] for a descendant $V_a$: [equation]","intent_p":0.99,"contests_p":null},{"arxiv_id":"1703.00736","title":"2D CFT blocks for the 4D class $\\mathcal{S}_k$ theories","url":"/paper/1703.00736","section":"Comparisons of the curves with the blocks","sentence":"We refer to appendix [ref] for the computations of the $_2$ and $_3$ blocks relevant for the comparison with the curve coefficients and to [ref] for an overview of the techniques needed for these computations.","intent_p":0.86,"contests_p":null},{"arxiv_id":"0908.2569","title":"On AGT relation in the case of U(3)","url":"/paper/0908.2569","section":"Conformal block at level two","sentence":"The conformal block $B^(2)$ is obtained from the general expression ([ref]) by substitution of the inverted $5×5$ Shapovalov matrix ([ref]) and five pairs triple vertices from [ref] .","intent_p":0.93,"contests_p":null},{"arxiv_id":"1007.0601","title":"Analysis of correlation functions in Toda theory and AGT-W relation for SU(3) quiver","url":"/paper/1007.0601","section":"2-dim CFT: W-algebra, Toda theory, and correlation functions","sentence":"In order to calculate general $n$-point functions, we often use the following decomposition [ref] : _n(z_n) ,⋯,_3(z_3) ,_2(z_2) ,_1(z_1) =∑_,Y_1,Y_2 (z_2-z_1)^Δ_V-Δ__2-Δ__1 _n(z_n) ,⋯,_3(z_3) ,_-Y_1(z_1)V_(z_1) ×(S^-1_)_Y_1,Y_2 V_2Qρ⃗- |(_-Y_2)^† _2(1)|_1 , where the index $$ labels the primary fields and $Y_1,Y_2$ labels the descendants of the primary fields, such as _-Y(z) V_(z):= L_-n_1(z) ⋯L_-n_l(z) W_-n'_1(z) ⋯W_-n'_w(z) V_(z) .","intent_p":0.82,"contests_p":null},{"arxiv_id":"1203.1427","title":"Generalized Whittaker states for instanton counting with fundamental hypermultiplets","url":"/paper/1203.1427","section":"Decoupling limit of $W_3$ conformal block","sentence":"The key ingredient in our computation is the recursion relations for the three point conformal blocks which were developed by Russian group [ref] : [equation] where $Y⃗,α⃗$ is the label for the descendants.","intent_p":1.0,"contests_p":null}]},{"kind":"data_used","label":"Uses its data or 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exponential 1/λ ε/Q = κ_1/|ρ| κ_1Q⃗ κ_2; ∑_i α⃗_i = primary ; ; [ref] 2 1+6Q^2 α(Q-α) Q :e^2αϕ: 2 √(2) Q -1/2; ; ; [ref] N (N-1)(1+ α⃗(2Q⃗ -α⃗)/2 2Q⃗ :e^α⃗ϕ⃗: 1 1/|ρ|= √(12/N(N^2-1)) Q⃗ -1; + N(N+1)ε^2); ; [ref] 2 1+6Q^2/ϵ_1ϵ_2 α(Q-α)/ϵ_1ϵ_2 Q :e^αϕ: 1 1/2√(2) 1/2Q 2/ϵ_1ϵ_2; ; ; [ref] 2 1-12Q^2 α(α-2Q)/2 2Q :e^αϕ: 1 |ρ_SU(2)|^-1=√(2) Q 1; ; ; [ref] 3 2-12Q^2 α^2+β(β-2Q)/2 (0,2Q) :e^αϕ_1+βϕ_2: 1 |ρ_SU(3)|^-1=1/√(2) (0,Q) 1; ; ; present 3 2-24Q^2 α^2+β̃^2-Q^2 (0,2Q) :e^√(2)(αϕ_1+βϕ_2): √(2) 1 (0,√(2)Q) 1; paper β̃= β-Q; ; ; [ref] and; present N (N-1)( 1 + α⃗(α⃗-2Q⃗)/-ϵ_1ϵ_2 2Q⃗ :e^√(2)α⃗ϕ⃗: √(2) √(2)/|ρ⃗ | √(2)Q⃗ 1/-ϵ_1ϵ_2; paper +N(N+1)ε^2/ϵ_1ϵ_2) =α̃^2-Q^2/-ϵ_1ϵ_2; ; ]$","intent_p":0.93,"contests_p":null}]},{"kind":"background","label":"Cites it as background","papers":16,"sentences":19,"quotes":[{"arxiv_id":"0908.2569","title":"On AGT relation in the case of U(3)","url":"/paper/0908.2569","section":"$4$-point conformal block for the $W_3$ algebra","sentence":"As reviewed in [ref] and (in far more detail) in [ref] , the conformal block has the structure B^α⃗_α⃗_1,α⃗_2,α⃗_3,α⃗_4(x) =∑_|Y=|Y'| x^|Y| B^α⃗_α⃗_1,α⃗_2,α⃗_3,α⃗_4 (Y,Y') =∑_|Y|=|Y'| x^|Y| Γ̅_α⃗_1,α⃗_2;α⃗(Y) Q_α⃗^-1(Y,Y') Γ_α⃗,α⃗_3,α⃗_4(Y')","intent_p":0.94,"contests_p":null},{"arxiv_id":"1307.2576","title":"Finalizing the proof of AGT relations with the help of the generalized Jack polynomials","url":"/paper/1307.2576","section":"Introduction","sentence":"AGT relations [ref] - [ref] identify two a priori different classes of theories: $2d$ conformal theories [ref] - [ref] and instanton calculus in multi-dimensional Yang-Mills [ref] .","intent_p":0.99,"contests_p":null},{"arxiv_id":"1512.06701","title":"On elementary proof of AGT relations from six dimensions","url":"/paper/1512.06701","section":"Introduction","sentence":"In particular, the AGT relations [ref] provide a connection to $2d$ conformal theories [ref] and, perhaps, further to the generic stringy AdS/CFT correspondence.","intent_p":0.98,"contests_p":null},{"arxiv_id":"1003.5752","title":"Matrix model version of AGT conjecture and generalized Selberg integrals","url":"/paper/1003.5752","section":"Introduction","sentence":"The coefficients $C$ can be straightforwardly found by standard CFT methods, see [ref] for a detailed review.","intent_p":0.9,"contests_p":null},{"arxiv_id":"1603.00304","title":"Spectral duality in elliptic systems, six-dimensional gauge theories and topological strings","url":"/paper/1603.00304","section":"A primer: 5d $SU(2)$ theory and $q$-Virasoro 4-point conformal block","sentence":"The 4-point conformal block [ref] provided by the Dotsenko-Fateev (conformal) matrix model [ref] and AGT related [ref] to the $N_f=4$ $SU(2)$ SUSY gauge theories [ref] , is associated with the square brane diagram [ref] .","intent_p":0.87,"contests_p":null}]}],"unlabelled_sentences":37},"outgoing":{"papers":7,"sentences":25,"groups":[{"kind":"comparison","label":"Compares 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algebra.","intent_p":0.96,"contests_p":null}]}],"unlabelled_sentences":20},"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. 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