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CFT exercises for the needs of AGT
High Energy Physics - Theory · Submitted 14 Aug 2009
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim We collect simplest expressions of this type for N=3 and for the two lowest descendant levels, together with the detailed derivations, which can be now computerized and used in more general studies of conformal blocks and AGT relations at higher levels.
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The abstract, sentence by sentence
An explicit check of the AGT relation between the W_N-symmetry controlled conformal blocks and U(N) Nekrasov functions requires knowledge of the Shapovalov matrix and various triple correlators for W-algebra descendants. We collect simplest expressions of this type for N=3 and for the two lowest descendant levels, together with the detailed derivations, which can be now computerized and used in more general studies of conformal blocks and AGT relations at higher levels.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | An explicit check of the AGT relation between the W_N-symmetry controlled conformal blocks and U(N) Nekrasov functions requires knowledge of the Shapovalov matrix and various triple correlators for W-algebra descendants. | background 0.90 | 0.09 | 0.14 | asserted 1.00 |
| s1 | We collect simplest expressions of this type for N=3 and for the two lowest descendant levels, together with the detailed derivations, which can be now computerized and used in more general studies of conformal blocks and AGT relations at higher levels. | contribution 0.98 | 0.94 | 0.46 | asserted 0.96 |
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The paper
Original abstract
An explicit check of the AGT relation between the W_N-symmetry controlled conformal blocks and U(N) Nekrasov functions requires knowledge of the Shapovalov matrix and various triple correlators for W-algebra descendants. We collect simplest expressions of this type for N=3 and for the two lowest descendant levels, together with the detailed derivations, which can be now computerized and used in more general studies of conformal blocks and AGT relations at higher levels.
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References
How the literature uses it
36 arXiv papers cite this in 68 sentences
Papers citing it · 37 sentences without a confident label
Builds on it 1 paper
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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.
On non-conformal limit of the AGT relations · Introduction · 0.81
Uses its method or tool 6 papers
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The key is the Mironov-Mironov-Morozov-Morozov recursion relations [ref] for a descendant $V_a$: [equation]
On AGT Conjecture for Pure Super Yang-Mills and W-algebra · recursion relations and asymptotic behavior · 0.99
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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.
2D CFT blocks for the 4D class $\mathcal{S}_k$ theories · Comparisons of the curves with the blocks · 0.86
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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] .
On AGT relation in the case of U(3) · Conformal block at level two · 0.93
Uses its data or values 2 papers
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The level-$1$ Shapovalov matrix is [ref] [ref] [equation] where the label is $𝒴=(Y_L,Y_W)$.
On AGT Conjecture for Pure Super Yang-Mills and W-algebra · Check · 0.80
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Therefore, we return to a more straightforward approach and just take the true value of the $3$-point function from the conformal field theory analysis, see [ref] and s.[ref] below.
On AGT relation in the case of U(3) · General case, all $α⃗≠0$, $α⃗_1$ and $α⃗_3$ special · 0.87
Compares with it 1 paper
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$[ selecton; paper $N$ central charge dimension rule: 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; ; ]$
On AGT relation in the case of U(3) · Appendix: Normalization conventions · 0.93
Cites it as background 16 papers
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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')
On AGT relation in the case of U(3) · $4$-point conformal block for the $W_3$ algebra · 0.94
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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] .
Finalizing the proof of AGT relations with the help of the generalized Jack polynomials · Introduction · 0.99
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In particular, the AGT relations [ref] provide a connection to $2d$ conformal theories [ref] and, perhaps, further to the generic stringy AdS/CFT correspondence.
On elementary proof of AGT relations from six dimensions · Introduction · 0.98
What it cites · 20 sentences without a confident label
Compares with 2 papers
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In alternative normalization, accepted also in [ref] , with $α$ and $Q$ multiplied by $√(2)$ while $W$ and $w$ simultaneously divided by $√(8)$, one would get instead of ([ref]) D= ϰ(Δ-1/5) + 1/5, ϰ= (1-15Q^2/4)^-1, c=2(1-12Q^2) In this case the $ϰ$-dependent factor is also included into the definition of the $W^(3)$ operator: $W→√(κ)W$, so that the eigenvalues of the two operators are $Δ= α^2+β̃^2-Q^2$ and $w = √(ϰ)α(α^2-3β̃^2)$.
The Power of Nekrasov Functions · Switching on $c≠2$ · 0.80
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In particular, with our present normalization a factor of $√(2)$ would appear in the AGT relation, $Q = ϵ√(2)$, hence the other choice made in the AGT-oriented papers [ref] and [ref] .
On Combinatorial Expansions of Conformal Blocks · Free field with $c≠1$ · 0.88
Cites as background 2 papers
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As we mentioned in the Introduction, this subject has a direct relation to the AGT conjecture [ref] , but we leave it beyond the scope of the present discussion.
Liouville Correlation Functions from Four-dimensional Gauge Theories · $Γ̅$ vertices, structure constants and Shapovalov matrix · 0.99
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An important story about relation between holomorphic and bilinear sums is provided by the theory of NSR and heterotic superstrings and GSO projection [ref] . ] Elements of Verma module are now labeled by generalized Young diagrams $Y$, which depend on particular choice of the algebra.
NSR Superstring Measures Revisited · Extended conformal algebra · 0.96
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