AGT and Exact-Correspondence Dictionaries: Status and Limits

In its sharpest form, the AGT correspondence is a coefficient-by-coefficient identity between a normalized two-dimensional conformal block and a four-dimensional equivariant instanton series after a precise parameter map and an Abelian factor are fixed. This page freezes the original $A_1$ four-puncture convention, derives the level-one coefficient from both sides, and tests it with exact rational arithmetic. The result is a reproducible protected-function dictionary—not an equivalence of every observable in the two parent theories.

Required background. Use Omega-background instanton counting for the Young-diagram sum and protected operator algebras for the meaning of a protected sector.

Helpful background. Class S and non-Lagrangian interfaces explain why a punctured surface organizes duality frames.

The four-puncture dictionary

Begin with physical, dimension-one parameters and divide every Coulomb, mass, and Omega parameter by the same scale $\sqrt{\epsilon_1\epsilon_2}$. After dropping hats, the original convention is

$$\epsilon_1=b, \qquad \epsilon_2=b^{-1}, \qquad Q=b+b^{-1}, \qquad c_{\mathrm L}=1+6Q^2.$$

For complex Omega parameters, the branch of the common square root is part of the convention. The exact benchmark below avoids that ambiguity by setting $\epsilon_1=2$ and $\epsilon_2=1/2$ explicitly.

The physical class-S theory is $SU(2)$ gauge theory with four flavors, but the localization formula used in the original comparison is a $U(2)$ fixed-point sum with

$$(a_1,a_2)=(a,-a).$$

Its oriented matter content is two antifundamentals followed by two fundamentals. In the notation of Alday, Gaiotto, and Tachikawa 2010, §3.2, eqs. (12)–(15),

$$\begin{aligned} \mu_1&=m_0+\widetilde m_0, &\mu_2&=m_0-\widetilde m_0,\\ \mu_3&=m_1+\widetilde m_1, &\mu_4&=m_1-\widetilde m_1. \end{aligned}$$

This orientation matters in the $U(2)$ regulator: replacing a fundamental of mass $m$ by an antifundamental uses $m\mapsto Q-m$. The pseudoreality of the $SU(2)$ doublet does not license dropping that shift inside the auxiliary $U(2)$ formula.

To remove a common source of index errors, label the external weights by their positions in

$$\left\langle V_{\alpha_0}(\infty) V_{m_0}(1) V_{m_1}(q) V_{\alpha_1}(0) \right\rangle .$$

The complete momentum and weight map is

Channel or position	Liouville momentum	Conformal weight
internal	$\alpha=Q/2+a$	$\Delta=Q^2/4-a^2$
$\infty$	$\alpha_0=Q/2+\widetilde m_0$	$\Delta_\infty=Q^2/4-\widetilde m_0^2$
$1$	$m_0$	$\Delta_{z=1}=m_0(Q-m_0)$
$q$	$m_1$	$\Delta_q=m_1(Q-m_1)$
$0$	$\alpha_1=Q/2+\widetilde m_1$	$\Delta_0=Q^2/4-\widetilde m_1^2$

In particular, $m_0$ and $m_1$ are already Liouville momenta in this convention; adding another $Q/2$ would be a double shift. The weak-coupling coordinate is

$$q=e^{2\pi i\tau_{\mathrm{UV}}}$$

in the specified ultraviolet renormalization scheme. It is not the exponentiated infrared coupling of the Seiberg–Witten curve.

Normalize the Virasoro block by $\mathcal F(0)=1$. The frozen identity is

$$Z_{\mathrm{inst}}^{U(2),N_f=4} =(1-q)^\nu\, \mathcal F_\Delta \bigl(\Delta_\infty,\Delta_{z=1},\Delta_q,\Delta_0;q\bigr), \qquad \nu=2m_0(Q-m_1).$$

Thus the Virasoro block is the $SU(2)$ part of the auxiliary $U(2)$ instanton sum after the Coulomb-independent factor $(1-q)^\nu$ is divided out. The exponent, matter orientation, and position labels are part of the equality, not optional annotations.

Why the level-one factors have this order

At level one, the internal Verma module has one descendant, $L_{-1}|\Delta\rangle$, whose Gram matrix is

$$\langle\Delta|L_1L_{-1}|\Delta\rangle=2\Delta.$$

The global conformal Ward identity gives the normalized three-point factors on the two sides of the sewing channel,

$$R_{\infty,1}=\Delta+\Delta_{z=1}-\Delta_\infty, \qquad R_{q,0}=\Delta+\Delta_q-\Delta_0.$$

Inserting the inverse Gram matrix therefore yields

$$\mathcal F(q)=1+qF_1+O(q^2), \qquad F_1= \frac{ \bigl(\Delta+\Delta_{z=1}-\Delta_\infty\bigr) \bigl(\Delta+\Delta_q-\Delta_0\bigr) }{2\Delta}.$$

This is the level-one specialization of the sewing formula in Alday, Gaiotto, and Tachikawa 2010, Appendix A.1, eq. (56). A $2\times2$ array of external weights is easy to reverse because conventions differ across the literature; explicit positions make the answer unambiguous.

The two one-box fixed points

Write

$$Z_{\mathrm{inst}}^{U(2),N_f=4}=1+qz_1+O(q^2).$$

At instanton number one, the single box lies in either $Y_1$ or $Y_2$. The vector, fundamental, and antifundamental factors of Alday, Gaiotto, and Tachikawa 2010, Appendix B.1, eqs. (75)–(84) give

$$z_1= \sum_{i=1}^{2} \frac{ (a_i+\mu_1)(a_i+\mu_2) (a_i+Q-\mu_3)(a_i+Q-\mu_4) }{ \epsilon_1\epsilon_2 \displaystyle\prod_{j\ne i}(a_j-a_i)(a_i-a_j+Q) }.$$

For $\epsilon_1\epsilon_2=1$, define

$$N(x)=(x+\mu_1)(x+\mu_2)(x+Q-\mu_3)(x+Q-\mu_4).$$

Then the two fixed points reduce to

$$z_1= -\frac{N(a)}{2a(2a+Q)} +\frac{N(-a)}{2a(Q-2a)}.$$

The Abelian factor expands as

$$(1-q)^\nu=1-\nu q+O(q^2).$$

Consequently the exact acceptance identity is

$$z_1=F_1-\nu, \qquad\text{or equivalently}\qquad z_1+\nu=F_1.$$

The plus sign in the last expression is forced by the binomial expansion. Forgetting the $U(1)$ factor produces a discrepancy of exactly $-\nu$ at this order.

An exact rational benchmark

Choose the nonpolar rational point

$$\epsilon_1=2, \quad \epsilon_2=\frac12, \quad Q=\frac52, \quad a=\frac13, \quad m_0=\frac23, \quad \widetilde m_0=\frac15, \quad m_1=\frac34, \quad \widetilde m_1=\frac17.$$

The mass and weight maps give

Quantity	Exact value
$(\mu_1,\mu_2,\mu_3,\mu_4)$	$(13/15,\ 7/15,\ 25/28,\ 17/28)$
$\Delta$	$209/144$
$(\Delta_\infty,\Delta_{z=1},\Delta_q,\Delta_0)$	$(609/400,\ 11/9,\ 21/16,\ 1209/784)$
one box in $Y_1$	$-91443/46550$
one box in $Y_2$	$14017/121275$
$z_1$	$-1217173/658350$
$\nu$	$7/3$
$z_1+\nu$ and $F_1$	$318977/658350$

Both sides are exact fractions; no fitted tolerance is involved. The machine-readable benchmark and its flat CSV projection retain the inputs, both fixed points, the positional weights, source locators, and the claim ceiling. A deterministic fixed-point generator and an independently implemented Ward-identity verifier also reject wrong matter orientation, a missing or sign-flipped Abelian exponent, reversed external weights, altered instanton grading, coincident-eigenvalue inputs in the separated fixed-point chart, and a genuine $\Delta=0$ pole input.

This point is deliberately outside the physical Liouville contour. It is an exact test of a meromorphic $q^1$ coefficient, not a numerical evaluation of a physical Liouville correlator.

Formal series, analytic functions, and poles

The most conservative statement is equality of formal series in the weak-coupling coordinate $q$, with coefficients meromorphic in $a$, the masses, and the Omega parameters. At level one, the summed result has the Virasoro pole $\Delta=0$. The apparent $a=0$ poles of the two separate fixed points cancel in their sum. For the benchmark masses,

$$\lim_{a\to0}z_1=-\frac{98947}{55125},$$

which equals $F_1-\nu$ evaluated at $a=0$. Thus $a=0$ is a removable singularity of this separated fixed-point chart, whereas $a=Q/2$ gives $\Delta=0$ and a genuine level-one pole at the same masses. A checker should test the summed rational function before inferring a physical singularity from one chart contribution.

Higher levels introduce the corresponding Kac-determinant and localization pole loci. The dictionary is first stated for generic parameters and then extended meromorphically where the continuation is defined.

As a formal power series, $(1-q)^\nu$ has a unique binomial expansion. As an analytic function, it means $\exp[\nu\Log(1-q)]$ and requires a branch of the logarithm. Continuation around $q=1$ can have monodromy when $\nu\notin\mathbb Z$. A one-coefficient benchmark establishes neither the all-level identity nor a global analytic continuation.

From a block to the four-sphere integral

At the round-sphere point $b=1$, write $a_{\mathrm E}\in\mathfrak t_{\mathbb R}$ for the real localization coordinate and $a_\Omega$ for the Nekrasov Coulomb variable denoted by $a$ above. Pestun’s contour relates them by $a_\Omega=i a_{\mathrm E}$, or equivalently $a_\Omega=iP$ on the Liouville contour. The Cartan integral may then be written schematically as

$$Z_{S^4}\propto \frac1{|W|} \int_{\mathfrak t_{\mathbb R}} da_{\mathrm E}\, \Delta_{\mathrm{VdM}}(a_{\mathrm E})^2 e^{-S_{\mathrm{cl}}(a_{\mathrm E})} Z_{\mathrm{1-loop}}^{S^4}(a_{\mathrm E},m) \left|Z_{\mathrm{inst}}(a_\Omega=i a_{\mathrm E},m,q)\right|^2.$$

Integrating over the full Cartan and dividing by the Weyl-group order is equivalent to integrating over one Weyl chamber. For $SU(2)$, $\Delta_{\mathrm{VdM}}^2=(2a_{\mathrm E})^2$ in this coordinate. A convention may absorb this factor into the vector determinant, but it must occur exactly once. Likewise, $Z_{\mathrm{1-loop}}^{S^4}$ is already the full-sphere determinant; one takes a further modulus square only after defining a holomorphic hemisphere factor. These details are explicit in Pestun 2012, eqs. (1.3)–(1.4) and §§4.5–5.

On the Liouville side, reflection identifies $\alpha$ with $Q-\alpha$. One may integrate over $\alpha=Q/2+iP$ with $P\geq0$, or over the full $P\in\mathbb R$ line with the compensating factor $1/2$—not both. The DOZZ structure constants supply the three-point factors, the gauge one-loop determinant matches them only after external normalizations are fixed, and the classical factor matches the leading block power. The precise external factors and the conversion to $a^2da\,|Z_{\mathrm{full}}|^2$ appear in Alday, Gaiotto, and Tachikawa 2010, §4.1, eqs. (28)–(34).

Extensions need new dictionaries

A surface defect may lead to a degenerate or affine conformal-block problem, while a loop operator acts by a difference operator that is represented by a Liouville loop operator. These are additional observables with their own parameters and global data, not automatic consequences of the vacuum block Alday et al. 2010, §§1.2 and 2.2; Drukker et al. 2010, §§4.2–4.3.

Likewise, higher-rank quivers motivate Toda and $W_N$ blocks, and asymptotically free limits motivate irregular conformal blocks. The original papers present these as further proposals and worked families, not as substitutions that preserve every $A_1$ convention unchanged Wyllard 2009; Gaiotto 2009.

Status and claim ceiling

The phrase “AGT is proved” is too coarse. The status depends on the object and formulation:

Statement	Evidence level	Safe conclusion
The rational $q^1$ fixture on this page	exact independent calculation	The frozen level-one identity and its normalization pass at the stated point.
Original $A_1$ four-point proposal	conjecture with extensive coefficient and localization checks	The specified protected functions agree in the tested dictionary; this wording alone is not an all-observable theorem.
Young-diagram expansion in $\mathrm{Vir}\otimes\mathcal H$	all-level algebraic construction for the stated generic sphere linear-quiver setting	The special orthogonal basis reproduces the factorized bifundamental matrix elements and inverse vector norms.
Four-sphere/Liouville gluing	localization plus the Liouville three-point and block decomposition, with convention-dependent external factors	The specified integrated partition function matches the specified correlator after contour, measure, and normalization choices are aligned.
Defect, irregular, or higher-rank variants	formulation-specific proposals, derivations, and checks	Each variant needs a new dictionary and its own status statement.

The original paper calls the correspondence a conjecture. Alba et al. prove existence and uniqueness of a special orthogonal basis and obtain the dressed Virasoro-block combinatorial expansion in a precise $\mathrm{Vir}\otimes\mathcal H$ setting; that result does not by itself prove the six-dimensional origin, every contour statement, or all extensions Alba et al. 2011, §§1–3.

The literature-status summary above was checked through 4 September 2026. In every application, retain the exact protected object, global gauge form, line or defect sector, mass origin, $U(1)$ factor, block normalization, coupling coordinate, contour, analytic-continuation path, and strongest claim actually supported.

The chapter-wide protected-export comparison records the quantity, discarded information, ambiguity, uncertainty, evidence status, and version boundary for this case.

Common pitfalls

Calling the raw $U(2)$ sum the $SU(2)$ answer. The auxiliary fixed-point formula contains a Coulomb-independent Abelian contribution. Divide by the declared $(1-q)^\nu$ before identifying the Virasoro block.

Shifting a mass twice. Physical and equivariant hypermultiplet masses can differ by $Q/2$, while fundamental and antifundamental orientations are related by $m\mapsto Q-m$. Translate between conventions once and record which symbols were shifted.

Using an infrared coupling for $q$. The four-puncture cross-ratio equals the exponentiated ultraviolet coupling in the Nekrasov scheme. Its relation to an infrared modular parameter is nontrivial.

Promoting one coefficient to a global theorem. An exact $q^1$ match is a strong normalization test, but it proves neither the higher coefficients nor convergence and continuation of the summed functions.

Forgetting global data for defects. A local Lie-algebra dictionary does not determine the genuine line lattice, global gauge form, discrete theta angle, or allowed defect sectors.

Exercises

Derive $F_1$ from the level-one Gram matrix and the two three-point Ward identities.

Solution

The only level-one state is $L_{-1}|\Delta\rangle$, with norm $2\Delta$. The normalized left and right three-point matrix elements are $\Delta+\Delta_{z=1}-\Delta_\infty$ and $\Delta+\Delta_q-\Delta_0$. Sewing inserts the inverse Gram matrix, so

$$F_1= \frac{ (\Delta+\Delta_{z=1}-\Delta_\infty) (\Delta+\Delta_q-\Delta_0) }{2\Delta}.$$

Use the rational fixture to verify the two Ward factors and the final block coefficient.

Solution

Substitution gives

$$\Delta+\Delta_{z=1}-\Delta_\infty=\frac{259}{225}, \qquad \Delta+\Delta_q-\Delta_0=\frac{8621}{7056}.$$

Therefore

$$F_1= \frac{(259/225)(8621/7056)}{2(209/144)} =\frac{318977}{658350}.$$

Suppose the Abelian factor is omitted when comparing the one-instanton sum with the block. Predict the discrepancy before evaluating either side.

Solution

Because $(1-q)^\nu=1-\nu q+O(q^2)$ and $Z_{\mathrm{inst}}=(1-q)^\nu\mathcal F$, one has $z_1=F_1-\nu$. The raw instanton coefficient minus the block coefficient is therefore $-\nu$. At the benchmark point it is $-7/3$.

Explain why integrating over both $P\in\mathbb R$ and its Liouville reflection image can double count the internal channel.

Solution

On the physical contour, $\alpha=Q/2+iP$ and the reflection $\alpha\sim Q-\alpha$ sends $P\mapsto-P$. Thus the two halves of the full real $P$ line describe the same reflected primary. One may integrate over $P\geq0$ or use the full line with a factor $1/2$; using the full line without that factor counts each generic channel twice.

References

• Alba, V. A., V. A. Fateev, A. V. Litvinov, and G. M. Tarnopolsky. “On Combinatorial Expansion of the Conformal Blocks Arising from AGT Conjecture.” Letters in Mathematical Physics 98 (2011): 33–64. DOI; Open PDF.
• Alday, L. F., D. Gaiotto, and Y. Tachikawa. “Liouville Correlation Functions from Four-Dimensional Gauge Theories.” Letters in Mathematical Physics 91 (2010): 167–197. DOI; Open PDF.
• Alday, L. F., D. Gaiotto, S. Gukov, Y. Tachikawa, and H. Verlinde. “Loop and Surface Operators in $N=2$ Gauge Theory and Liouville Modular Geometry.” Journal of High Energy Physics 2010, no. 1 (2010): 113. DOI; Open PDF.
• Drukker, N., J. Gomis, T. Okuda, and J. Teschner. “Gauge Theory Loop Operators and Liouville Theory.” Journal of High Energy Physics 2010, no. 2 (2010): 057. DOI; Open PDF.
• Gaiotto, D. “Asymptotically Free $N=2$ Theories and Irregular Conformal Blocks.” Journal of Physics: Conference Series 462 (2013): 012014. DOI; Open PDF.
• Pestun, V. “Localization of Gauge Theory on a Four-Sphere and Supersymmetric Wilson Loops.” Communications in Mathematical Physics 313 (2012): 71–129. DOI; Open PDF.
• Wyllard, N. “$A_{N-1}$ Conformal Toda Field Theory Correlation Functions from Conformal $N=2$ $SU(N)$ Quiver Gauge Theories.” Journal of High Energy Physics 2009, no. 11 (2009): 002. DOI; Open PDF.