Affine sl(N) conformal blocks from N=2 SU(N) gauge theories

Can Kozcaz; Filippo Passerini; Niclas Wyllard; Sara Pasquetti

REVIEW

Affine sl(N) conformal blocks from N=2 SU(N) gauge theories

Can Kozcaz, Sara Pasquetti, Filippo Passerini, Niclas Wyllard

High Energy Physics - Theory · Submitted 8 Aug 2010 · 53 references

Pith's verdict: Honest, useful sl(N) extension of Alday–Tachikawa; the checks are real but stop short of the terms that would actually test the guessed K† insertion.

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arXiv ID1008.1412 (version 2)

Pith IDFT43HF4L, Pith's permanent record of this paper and its review

CitationCan Kozcaz, Filippo Passerini, Niclas Wyllard, Sara Pasquetti. "Affine sl(N) conformal blocks from N=2 SU(N) gauge theories." arXiv:1008.1412 (2010). https://pith.science/paper/FT43HF4L

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Editor's note

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This is the sl(N) version of Alday–Tachikawa: instanton partition functions of N=2 SU(N) quivers with a full surface operator should equal affine sl(N) conformal blocks with a particular insertion. The SU(2) case was [1]; this paper does the general rank, gives an explicit dictionary (4.30), (4.33), and adds a non-conformal section that recasts the Gaiotto state as a Whittaker-vector condition (5.20), (5.28). That last part is the piece I'd actually point people to — it's clean, it matches what Braverman had already proven on the math side, and it's the kind of statement that generalizes without much fuss. The paper is also candid: it says plainly that the relation is still a conjecture, and it flags the overlap with [22] rather than hiding it. That counts for something in a program where a lot of the literature is loose about what has been checked and what has been guessed. The soft spot is K†, Eq. (4.16). The authors call it 'a guess based on the conjecture for sl(2) [1] together with sl(N) covariance' and say they will only check parts of it. That is exactly right, and it means the evidence is thinner than the length of the paper suggests. The leading hypergeometric terms mostly test the denominators, and the denominators are what fixed the dictionary — so the leading-order agreement is partly circular. The genuinely independent evidence is the numerator matching, the four-point x1^n x2 terms in (A.13)–(A.16), and the factorised five-point terms. What's missing is the discriminating case: §4.2 explicitly sets aside the i=l and i=l+1 cross terms because the generators don't commute or act nontrivially on x_l. Those are precisely where the J^{il} pieces and the zero-mode factorisation (4.17) of K† would earn their keep. Until someone computes them, K† is a motivated ansatz, not a derived object. There's also a small tension with §6's remark that 'the conformal blocks are completely determined by the symmetry algebra alone' — if so, the insertion shouldn't be free. Minor, but worth a sentence. None of this is fatal; it's the normal shape of a conjecture paper in the AGT program, and the authors are upfront about it. Send it out. The referee should push on the §4.2 cross terms and on whether K† can be fixed by a symmetry principle rather than by matching.

Referee report

Major objections (3)

[§4.1–4.2, Eq. (4.16)] The operator K† is introduced as 'a guess based on the conjecture for sl(2) [1] together with sl(N) covariance,' and the authors state that 'in our explicit examples below we will only check parts of these expressions.' The most discriminating test would be the five-point cross terms with i=l and i=l+1, which §4.2 explicitly sets aside because the relevant generators do not commute or act non-trivially on x_l. Those are precisely the terms where the unverified J^{il} and zero-mode pieces of K† enter. As it stands, the evidence for the full K† consists of leading terms (which use little of it), the four-point x_1^n x_2 terms (A.13)–(A.16), and the factorised five-point terms. I would ask the authors to state this scope limitation prominently in §4.1 rather than only in the closing sentence, and, if feasible, to compute at least one of the skipped cross-term cases and compare to (2.19). Without that, the claim that the guessed K† is the correct insertion rests on indirect evidence.
[§4.1, Eqs. (4.30), (4.33)] The dictionary between affine data and gauge data is fixed by matching the denominators of the leading hypergeometric terms in (2.15). Consequently the leading-order agreement is partly by construction, and the genuinely independent evidence is the numerator structure and the subleading terms of Appendix A.2. This is a legitimate way to fix a dictionary, but the paper should say so explicitly so that a reader can weigh the evidence correctly. In particular, the statement in §4.1 that the maps (4.29), (4.30), (4.33) 'lead to complete agreement' should be qualified: the denominators are matched by choice, and the non-trivial content is the numerator and higher-order matching.
[§6 vs §4] Section 6 states that 'the conformal blocks are completely determined by the symmetry algebra alone,' yet the construction of §4 requires an ad hoc insertion K† whose form is not fixed by the algebra. These two statements sit uneasily together. The authors should clarify the logical status of K†: is it part of the definition of the object being compared to the instanton series, or is it a modification of the algebra action? If the former, the sentence in §6 should be softened; if the latter, the sense in which the block is 'determined by the algebra' needs restatement.

Minor objections (6)

[Eq. (2.12)] In the i=N line the summation is written as 's∑_{n=1} e^{ǫ1 s}', which appears to be a typographical transposition of 'n∑_{s=1} e^{ǫ1 s}' as in the i≤N−1 line. Please check.
[§4.1] The sentence 'These expression are a guess based on the conjecture for sl(2) [1] together with sl(N) covariance' should read 'These expressions are a guess…'.
[§2, §6, Appendix A.2] Several typographical errors should be corrected: 'classiﬁciation' in §2, 'reducton' in §6, and 'deﬁnied' in the caption of (A.3).
[§3.4] The claim that the instanton partition function is insensitive to the distinction between 2d and 4d defects for SU(2) but sensitive to it already at the quiver level is stated without a proof of the negative. The SU(2) Nf=4 agreement is checked, but the quiver statement rests on the observation that 'it is not straightforward to find a relation.' This should be softened to a conjecture or supported by an explicit obstruction.
[§5] The non-conformal checks in §5.1–5.2 are leading-order in each variable; the paper should state this scope explicitly, as it does for the conformal case, so that the reader does not overestimate the extent of the verification.
[§2.2] The remark that 'it is not completely obvious that the problems solved by the mathematicians is really equivalent to the physics problem' is well taken and could usefully be repeated where the localization result is first used to define the surface-operator partition function.

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The paper

The paper claims

Conformal blocks of the affine $sl(N)$ current algebra, with a specific operator inserted, are proposed to equal the instanton partition functions of $N=2$ $SU(N)$ quiver gauge theories in the presence of a full surface operator.

The reading

This paper proposes that a two-dimensional conformal block built from the affine $sl(N)$ current algebra, with a particular exponential operator inserted at one point, equals the four-dimensional instanton partition function of an $N=2$ $SU(N)$ quiver gauge theory in the presence of a full surface operator, up to a prefactor. The proposal lifts a rank-one statement made shortly before for $SU(2)$ to gauge groups of any rank, and supplies the dictionary that translates between the two sides. The authors do not prove the equality; they verify it term by term against explicit instanton expressions for the four-point block on the sphere, the five-point block of a two-node quiver, and the one-point block on the torus. They also extend the construction to asymptotically free theories, where the surface-operator partition function is argued to arise from a Whittaker-like state in the affine module. If the correspondence holds, a class of four-dimensional defect partition functions becomes computable from current-algebra data alone, with no separate gauge-theory integral to perform.

What carries the argument

Two ingredients carry the argument. First, the character formula for instanton counting in the presence of a full surface operator, taken from the mathematics literature, which replaces the usual sum over $N$-vectors of Young tableaux by a periodic structure $lambda_i equiv lambda_{i+N}$, $a_i equiv a_{i+N}$ with instanton numbers $k_i = sum_{j geq 1} lambda^{i-j+1}_j$; this is what produces the $N$ separate expansion variables $y_i$ and the characteristic floor functions $lfloor cdot/N rfloor$ that single out $y_N$. Second, the decomposition of the affine conformal block over a complete set of descendants with the inverse Gram (Shapovalov) matrix $X^{-1}$, which turns the block into a sum of hypergeometric functions whose parameters are read off from the differential-operator realisation of the $sl(N)$ generators on $N-1$ isospin variables. The operator $K^dagger$, an exponential of raising and Cartan modes with coefficients $1/(2n-1)$, is the device that aligns the argument of each hypergeometric function with the instanton series; its zero-mode part factorises as $e^{J^1_+} e^{-J^{2,1}_0} cdots e^{-J^{N-1,N-2}_0}$.

Extended reading notes

Core claim

The central claim is that the instanton partition function of a conformal $N=2$ $SU(N)$ quiver gauge theory with a full surface operator is, up to a prefactor, an affine $sl(N)$ conformal block. Concretely, the authors conjecture that the four-point block $sum_{n,A;n',A'} langle j_1| V_{chi_2}(1,1) |n,A;j rangle X^{-1}_{n,A;n',A'}(j) langle n',A';j| V_{chi_3}(x,z) |j_4 rangle$ equals the $N_f = 2N$ instanton partition function with a full surface operator, where the external insertions $V_chi$ carry the special weights $chi_i = kappa_i Lambda_{N-1}$ (or $kappa_i Lambda_1$) and are dressed by an operator $K$ or $K^dagger$. The dictionary identifies the $N-1$ surface-operator couplings with the isospin coordinates through $y_1 = x_1$, $y_{i+1} = x_{i+1}/x_i$, $y_N = z/x_{N-1}$, the internal affine weight with the Coulomb moduli through $j_i = -1/2 + (a_i - a_{i+1})/(2 epsilon_1)$, and the level with the deformation parameters through $k = -N - epsilon_2/epsilon_1$. The matching is exhibited explicitly for the hypergeometric pieces of the four-point block, for the four hypergeometric families of the five-point block, and for the torus one-point block, and it is extended to pure $SU(N)$ via a state annihilated by $J^1_+$, $J^{N-1,-1}$ and the intermediate raising operators.

Load-bearing premise

The dressing operator $K^dagger$ written down in equation (4.16) is a guess fixed only by $sl(N)$ covariance and by analogy with the $SU(2)$ case, and the reported checks exercise only those pieces of it that act on the states examined.

Editorial extensions

If this is right

Any conformal $N=2$ $SU(N)$ quiver gauge theory with a full surface operator acquires a two-dimensional computational route: expand the affine block instead of evaluating localisation integrals over the defect moduli space.
The $N-1$ continuous parameters of the surface operator are identified with ratios of isospin variables, so the defect's coupling space is mapped onto the coordinate space of the current-algebra insertion.
Asymptotically free $SU(N)$ theories with a full surface operator obtain a Whittaker-vector construction analogous to the Gaiotto state, with the state fixed by $J^1_+|vec{x},z;jrangle = sqrt{x_1}|vec{x},z;jrangle$ and $J^{N-1,-1}|vec{x},z;jrangle = sqrt{z/x_{N-1}}|vec{x},z;jrangle$.
The subsector with $k_N = 0$ depends only on $epsilon_1$ and not $epsilon_2$, matching the known statement that this piece is an eigenfunction of the quantum trigonometric Calogero-Sutherland model and explaining the absence of $epsilon_2$ as a restriction to zero modes of the affine current.
The comparison of two-dimensional-defect and four-dimensional-defect realisations becomes a sharp question at higher rank, since the two agree for $SU(2)$ but appear to differ already for a two-node $SU(2)$ quiver.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The $K^dagger$ insertion reads like a stand-in for spectral flow. The paper itself observes that replacing one primary by its spectrally flowed version could substitute for $K$, which suggests the operator is bookkeeping for a choice of affine coordinate patch rather than additional physical input; if so, the correspondence may be stated more invariantly without it.
If the equality holds beyond the checked orders, the factorised structure of the affine block predicts recursive patterns in the surface-operator instanton series that localisation could test independently, without ever computing a conformal block.
The failure of the two-dimensional/three-dimensional defect coincidence at quiver level for $SU(2)$ may be an artefact of $SU(2)$ having a single Levi type; a rank-two check with a simple surface operator breaking $SU(3)$ to $SU(2) times U(1)$ would settle whether the coincidence is generic or accidental.
Because the dictionary fixes $j_i$ from differences $a_i - a_{i+1}$, the affine block is sensitive only to the $SU(N)$ Cartan of the Coulomb branch, so the overall $U(1)$ decoupled from the $N_f = 2N$ matter content should appear only as a prefactor, which is a check the paper does not spell out.

Original abstract

Recently Alday and Tachikawa proposed a relation between conformal blocks in a two-dimensional theory with affine sl(2) symmetry and instanton partition functions in four-dimensional conformal N=2 SU(2) quiver gauge theories in the presence of a certain surface operator. In this paper we extend this proposal to a relation between conformal blocks in theories with affine sl(N) symmetry and instanton partition functions in conformal N=2 SU(N) quiver gauge theories in the presence of a surface operator. We also discuss the extension to non-conformal N=2 SU(N) theories.

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What the abstract itself states, as probabilities: claims first 0.08 · states limitation 0.11 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.03 · compares with prior results 0.82 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v2. Results and the main claim are underlined; show every role.

Recently Alday and Tachikawa proposed a relation between conformal blocks in a two-dimensional theory with affine sl(2) symmetry and instanton partition functions in four-dimensional conformal N=2 SU(2) quiver gauge theories in the presence of a certain surface operator. In this paper we extend this proposal to a relation between conformal blocks in theories with affine sl(N) symmetry and instanton partition functions in conformal N=2 SU(N) quiver gauge theories in the presence of a surface operator. We also discuss the extension to non-conformal N=2 SU(N) theories.

background contribution method result interpretation main claim

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#	sentence	role	the authors' own work	states the main conclusion	strength
s0	Recently Alday and Tachikawa proposed a relation between conformal blocks in a two-dimensional theory with affine sl(2) symmetry and instanton partition functions in four-dimensional conformal N=2 SU(2) quiver gauge theories in the presence of a certain surface operator.	background 1.00	0.03	0.05	asserted 0.99
s1	In this paper we extend this proposal to a relation between conformal blocks in theories with affine sl(N) symmetry and instanton partition functions in conformal N=2 SU(N) quiver gauge theories in the presence of a surface operator.	contribution 1.00	0.90	0.86	asserted 0.99
s2	We also discuss the extension to non-conformal N=2 SU(N) theories.	contribution 0.97	0.82	0.07	asserted 0.71

What kind of paper the classifier reports this is: theoretical result 0.96 · evidence from proof or derivation 0.57.

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R. Dijkgraaf and C. Vafa, “Toda theories, matrix models, topo logical strings, and N = 2 Gauge Systems,” arXiv:0909.2453 [hep-th]

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L. Feher, L. O’Raifeartaigh, P. Ruelle, I. Tsutsui, and A. Wipf, “ On Hamiltonian reductions of the Wess-Zumino-Novikov-Witten theories,” Phys. Rept. 222 (1992) 1–64; J. de Boer and T. Tjin, “The relation between quantum W algebras and Lie algebras,” Commun. Math. Phys. 160 (1994) 317–332, arXiv:hep-th/9302006

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A. M. Polyakov, “Gauge transformations and diﬀeomorphisms,” Int. J. Mod. Phys. A5 (1990) 833; M. Bershadsky, “Conformal ﬁeld theories via Hamiltonian reduction ,” Commun. Math. Phys. 139 (1991) 71–82

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A. A. Belavin, A. M. Polyakov, and A. B. Zamolodchikov, “Inﬁnite conformal symmetry in two-dimensional quantum ﬁeld theory,” Nucl. Phys. B241 (1984) 333–380. 40

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Forward citations

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How the literature uses it

52 arXiv papers cite this in 112 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 84 sentences without a confident label

Builds on it 2 papers

In this section we extend the analysis in [ref] to general $N$.

Instanton partition functions in N=2 SU(N) gauge theories with a general surface operator, and their W-algebra duals · The case of $N=1+(N-1)$ surface operators · 1.00
This relation is extended to the theory with the generalized $W$ algebra [ref] [ref] [ref] , which is obtained from the quantum Drinfeld-Sokolov reduction applied to an affine Lie algebra $(n)$ [ref] .

Matrix model from N = 2 orbifold partition function · String theory perspective · 0.95

Uses its method or tool 1 paper

These can be defined for the quasi-superconformal algebras in a way completely analogous to the construction in [ref] (see also [ref] and section 5 in [ref] ).

Instanton partition functions in N=2 SU(N) gauge theories with a general surface operator, and their W-algebra duals · The case of $N=1+…+1+2$ surface operators · 0.82

Compares with it 4 papers

We note that to make our results compatible with those in [ref] the following redefinition of the parameters is required [equation] and [equation]

Conformal blocks of Chiral fields in N=2 SUSY CFT and Affine Laumon Spaces · Affine $sl(2)$ blocks and instanton partition functions · 0.98
For instance, the specialization of the primary fields $V_χ$, $χ=κΛ_1, κΛ_N-1$ in [ref] for the simple punctures agrees with the choice of the spectral type $(N-1,1)$.

A quantum isomonodromy equation and its application to N=2 SU(N) gauge theories · Summary and discussions · 0.81
For example in the case corresponding to $N=1+…+1$ with $k_N=0$ the character obtained in [ref] reduces to the character in [ref] (see e.g. [ref] for a discussion).

Instanton partition functions in N=2 SU(N) gauge theories with a general surface operator, and their W-algebra duals · Preliminaries and statement of the main conjecture · 0.94

Cites it as support 2 papers

We are now going to propose the following conjecture: There exists a choice of twist elements $_e(τ_e,x_e)$ such that we have [equation] assuming that [equation] Evidence for this conjecture is provided by the computations performed in [ref] in the cases $C=C_1,1$ and $C=C_0,4$.

Surface Operators and Separation of Variables · Relation to conformal field theory · 0.99
This expectation is based on the computations in [ref] for $N=2$ and in [ref] for $N=3$.

Instanton partition functions in N=2 SU(N) gauge theories with a general surface operator, and their W-algebra duals · The case of $N=1+(N-1)$ surface operators · 0.96

Cites it as background 18 papers

The AGT correspondence [ref] relates $= 2$ supersymmetric gauge theories in four dimensions and conformal field theories with W-algebra symmetry in two dimensions [ref] .

Compactification on the \Omega-background and the AGT correspondence · Introduction · 0.99
On the conformal blocks in Toda theory which correspond to the instanton part of partition function in gauge theory, the discussion and calculation have been developed [ref] .

Notes on 3-point functions of A_{N-1} Toda theory and AGT-W relation for SU(N) quiver · Introduction · 0.96
There are other surface operators which, according to [ref] , changes the 2D theories from Liouville or Toda theories to those with affine Lie algebra symmetry.

AGT on the S-duality Wall · Introduction · 0.87

What it cites · 77 sentences without a confident label

Builds on 1 paper

In this paper we extend the proposal in [ref] to a relation between conformal blocks in theories with affine $(N)$ symmetry and instanton partition functions in conformal $=2$ $(N)$ quiver gauge theories in the presence of a surface operator arising from a $4d$ defect.

Affine SL(2) conformal blocks from 4d gauge theories · Introduction · 1.00

Uses the method or tool of 1 paper

However, since the $$ operator is crucial when matching the higher-point conformal blocks to instanton partition functions in quiver gauge theories, we will insert a $$ operator, following the prescription in [ref] (note that the expression for $$ written in [ref] is equal to $(1,1)$ in our notation).

Affine SL(2) conformal blocks from 4d gauge theories · Four-point conformal block on the sphere · 0.98

Compares with 4 papers

Whereas instanton partition functions built from the character ([ref]) are intimately connected with the affine $(N)$ algebra the results in [ref] are based on the ordinary $(N)$ algebra.

Laumon Spaces and the Calogero-Sutherland Integrable System · $(N)$ instanton counting in the presence of a full surface operator · 0.99
An important thing to note is that the $y_N$-independent terms only depend on $_1$ and not on $_2$, which is similar to the setting in [ref] (see also [ref] ).

Quantization of Integrable Systems and Four Dimensional Gauge Theories · $(N)$ instanton counting in the presence of a full surface operator · 0.92
An important thing to note is that the $y_N$-independent terms only depend on $_1$ and not on $_2$, which is similar to the setting in [ref] (see also [ref] ).

Nekrasov Functions and Exact Bohr-Sommerfeld Integrals · $(N)$ instanton counting in the presence of a full surface operator · 0.92

Cites as background 23 papers

In [ref] it was conjectured that a simple surface operator in a (mass-deformed) conformal $(2)$ theory has a dual description in the Liouville theory in terms of the insertion of a certain degenerate field.

Loop and surface operators in N=2 gauge theory and Liouville modular geometry · $(N)$ instanton counting in the presence of a simple surface operator · 0.86
In [ref] the following terminology was used: a full surface operator corresponds to the breaking of $(N)$ to $(1)^N-1$ and depends on $N-1$ continuous parameters (this is the maximal number of parameters possible), whereas a simple surface operator corresponds to the breaking of $(N)$ to $(N-1)×(1)$ and depends on one parameter.

Affine SL(2) conformal blocks from 4d gauge theories · Surface operators and instanton counting · 0.96
Such surface operators were first considered in [ref] and have been further studied in [ref] .

Deformed Prepotential, Quantum Integrable System and Liouville Field Theory · Introduction · 0.82

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

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