AGT and Exact-Correspondence Dictionaries: Status and Limits

The AGT correspondence identifies normalized four-dimensional $N=2$ Omega-background partition functions with two-dimensional conformal blocks. In its basic $A_1$ example, the instanton sum of $SU(2)$ theory on a four-punctured sphere matches a Virasoro four-point block. The equality requires a precise map of Coulomb, mass, coupling, and Omega parameters, plus removal of Abelian and normalization factors; it is not an equivalence of every observable in the parent four- and two-dimensional theories.

Required background. Use Omega-background instanton counting and the construction of protected operator algebras.

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

The SU(2), Nf=4 dictionary

Consider the conformal $SU(2)$ gauge theory associated with a four-punctured sphere. Let $q$ be the cross-ratio in the chosen weak-coupling frame, $a$ the Coulomb parameter, $m_i$ the four puncture masses, and $\epsilon_1,\epsilon_2$ the Omega parameters. Define

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

The Liouville internal and external momenta are

$$\alpha=\frac Q2+\frac{a}{\sqrt{\epsilon_1\epsilon_2}}, \qquad \alpha_i=\frac Q2+\frac{m_i}{\sqrt{\epsilon_1\epsilon_2}},$$

with conformal weights

$$\Delta(\alpha)=\alpha(Q-\alpha), \qquad \Delta_i=\alpha_i(Q-\alpha_i).$$

This convention takes $m_i$ to be the class-S puncture masses. The four fundamental-hypermultiplet masses in a chosen Lagrangian frame are linear combinations of them; $SO(8)$ triality changes that linear map between duality frames. On a real Liouville contour, $a$ is often continued to an imaginary variable, so factors of $i$ seen in other conventions are contour choices rather than contradictory dictionaries.

The remaining map is

Four-dimensional quantity	Two-dimensional quantity
UV gauge coupling $q=e^{2\pi i\tau_{\mathrm{UV}}}$	Four-puncture cross-ratio in the selected channel
Coulomb parameter $a$	Internal momentum $\alpha$
Puncture masses $m_i$	External momenta $\alpha_i$
$\epsilon_1/\epsilon_2$	$b^2$
$\epsilon_1\epsilon_2$	Overall conversion from masses to momenta
Instanton number	Virasoro descendant level

The equality is between the instanton series and the conformal block after a declared $U(1)$ factor is removed:

$$Z_{\mathrm{inst}}^{U(2)}(a,m_i,q;\epsilon_1,\epsilon_2) =Z_{U(1)}(m_i,q;\epsilon_1,\epsilon_2) \mathcal F_{\Delta} \!\left[\begin{matrix}\Delta_3&\Delta_2\\ \Delta_4&\Delta_1\end{matrix};q\right].$$

The exponent in the common form $Z_{U(1)}=(1-q)^\nu$ depends on the puncture-mass and free-boson normalization. Quoting the block without $\nu$ and those conventions is not a reproducible equality Alday, Gaiotto, and Tachikawa 2010, §§2–3.

A first-level coefficient check

Normalize the Virasoro block to begin with one. Its level-one term is

$$\mathcal F_\Delta(q) =1+q\, \frac{(\Delta+\Delta_1-\Delta_2) \bigl(\Delta+\Delta_3-\Delta_4\bigr)}{2\Delta} +O(q^2)$$

Equivalently, define

$$A=\Delta+\Delta_1-\Delta_2, \qquad B=\Delta+\Delta_3-\Delta_4.$$

Then the coefficient is $AB/(2\Delta)$. Expanding the one-instanton fixed points and the declared $U(1)$ factor must reproduce this value after the parameter map. This single comparison detects many mass-shift and normalization errors.

From blocks to full sphere correlators

On the round four-sphere, $\epsilon_1=\epsilon_2=1/r$ and hence $b=1$. Pestun’s Coulomb integral has the form

$$Z_{S^4} =\int da\, \left|Z_{\mathrm{cl}}Z_{\mathrm{1-loop}}Z_{\mathrm{inst}}\right|^2.$$

The corresponding Liouville four-point function is

$$\int d\alpha\, C(\alpha_1,\alpha_2,\alpha) C(Q-\alpha,\alpha_3,\alpha_4) \left|\mathcal F_\alpha(q)\right|^2.$$

The one-loop factors match the Liouville three-point structure constants after normalization, while the classical gauge factor matches the leading conformal-block power. The integration contour and reflection identification $\alpha\sim Q-\alpha$ must be aligned.

Defects and extended dictionaries

A half-BPS surface defect can map to insertion of a degenerate Virasoro primary. The null-vector equation then becomes a differential equation for the defect partition function. Loop operators act as difference operators on $a$ and correspond to Verlinde-type operators on conformal blocks. These refinements test more than the vacuum function because they compare operator actions.

Higher-rank class-S theories lead to Toda conformal blocks, while irregular punctures lead to asymptotically free or Argyres–Douglas limits. Each extension needs its own puncture, contour, $U(1)$, and normalization analysis; the $A_1$ dictionary cannot simply be copied.

Status and limits

The instanton/block identity has extensive coefficient checks and mathematical proofs in broad $A_1$ settings Alba et al. 2011. The full physical picture is supported by class S, localization, crossing/S-duality, and defect extensions. Still, the correspondence directly controls protected functions and operator actions. It does not imply that Liouville theory has the complete spectrum or locality structure of the four-dimensional gauge theory.

Check every application for:

• physical versus equivariant mass shifts by $(\epsilon_1+\epsilon_2)/2$;
• $U(2)$ versus $SU(2)$ and the decoupled $U(1)$ factor;
• normalization of conformal blocks and three-point functions;
• puncture masses versus Lagrangian hypermultiplet masses;
• the UV coupling coordinate and analytic-continuation path;
• global gauge form and allowed line/defect sectors;
• the precise protected quantity being equated.

Exercises

Compute the Liouville central charge at the self-dual Omega point $\epsilon_1=\epsilon_2$.

Solution

Then $b=1$, so $Q=2$ and $c_{\mathrm L}=1+6Q^2=25$.

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.