[Paper Review] Asymptotically free N=2 theories and irregular conformal blocks

Davide Gaiotto|ArXiv.org|Aug 3, 2009

Black Holes and Theoretical PhysicsPhysics and Astronomy10 references143 citations

TL;DR

This paper establishes a correspondence between instanton partition functions in asymptotically free $σ=2$ gauge theories and conformal blocks in 2D CFT by introducing irregular conformal blocks associated with higher-order poles in the quadratic differential $φ_2(z)$. The authors define exotic Virasoro states that reproduce the gauge theory partition functions order by order, with exact match for $SU(2)$ theories with $N_f = 0,1,2,3$ and a consistent spurious factor in $N_f=2$ cases, extending the AGT correspondence to asymptotically free theories via irregular punctures.

ABSTRACT

A surprising connection between N=2 gauge theory instanton partition functions and conformal blocks has been recently proposed. We illustrate through simple examples the generalization to asymptotically free N=2 gauge theories

Motivation & Objective

• To extend the AGT correspondence from superconformal to asymptotically free $σ=2$ gauge theories by identifying their instanton partition functions with conformal blocks.
• To define new types of Virasoro states—irregular states—associated with higher-order poles ($z^{-3}, z^{-4}$) in the quadratic differential $φ_2(z)$, which correspond to irregular punctures in the 6D $(2,0)$ theory.
• To show that these irregular conformal blocks reproduce the instanton partition functions of $SU(2)$ gauge theories with $N_f = 0,1,2,3$ flavors, including the correct $q$-dependence and spurious factors.
• To provide a systematic method for constructing such states recursively via Virasoro algebra constraints, ensuring consistency at each level.

Proposed method

• Define exotic states $|\Delta, \Lambda^2\rangle$ in the Virasoro Verma module that are eigenvectors of $L_1$ and annihilated by $L_2$, with $L_n|\cdots\rangle = 0$ for $n > 2$, to model $z^{-3}$ poles.
• Construct higher-order irregular states for $z^{-4}$ poles by requiring $L_n|\cdots\rangle = 0$ for $n > 3$, and solve recursively for descendants at each level using the Virasoro algebra.
• Use the inner product $\langle \Delta, \Lambda^2 | \Delta, \Lambda^2 \rangle = \sum \Lambda^{4n} |v_n|^2$ to compute conformal blocks that match the instanton partition function order by order.
• Identify the instanton counting parameter $q$ with $\Lambda^4$ for $N_f=0$, $\Lambda^2$ for $N_f=2$ in the second realization, and $-2\Lambda$ for $N_f=3$, matching known results.
• Handle spurious factors via asymptotic limits of $N_f=4$ results, showing they vanish for $N_f < 2$ but remain finite for $N_f=2$ due to mass scaling.
• Use the six-dimensional $(2,0)$ SCFT compactification with codimension-two defects to realize the gauge theories, where irregular punctures arise from merging regular punctures in the UV limit.

Experimental results

Research questions

• RQ1 Can the AGT correspondence be extended to asymptotically free $σ=2$ gauge theories using irregular conformal blocks?
• RQ2 How can Virasoro states be defined to reproduce the singular behavior of the quadratic differential $\phi_2(z)$ with poles of order 3 and 4?
• RQ3 What is the role of spurious factors in the conformal block when matching to instanton partition functions in $N_f=2$ theories?
• RQ4 Do the recursive equations for irregular states have a unique formal solution at all levels, as suggested by explicit computation up to level 8?
• RQ5 Can the boundary conditions for $\sqrt{\phi_2(z)}$ at irregular punctures be consistently formulated in terms of fixed coefficients in the Laurent expansion?

Key findings

• The inner product $\langle \Delta, \Lambda^2 | \Delta, \Lambda^2 \rangle$ computed from the irregular state $|\Delta, \Lambda^2\rangle$ matches the instanton partition function for $SU(2)$ with $N_f=0$, with $q = \Lambda^4$, order by order up to level 8.
• For $N_f=1$, the conformal block with a $z^{-3}$ and a $z^{-4}$ pole reproduces the correct instanton partition function, with $q = \Lambda^4$, after identifying the state via $L_1|\psi\rangle = \Lambda^2|\psi\rangle$ and $L_2|\psi\rangle = 0$.
• In the $N_f=2$ case with two $z^{-4}$ poles, the conformal block matches the instanton partition function with $q = 4\Lambda^2$, up to a spurious factor conjectured to be $\exp(2\Lambda^2)$.
• In the second realization of $N_f=2$, the three-point function with one irregular state and two regular states reproduces the instanton partition function exactly, with $q = \Lambda^2$ and no spurious factor.
• For $N_f=3$, the three-point function with one $z^{-4}$ pole and two regular punctures reproduces the instanton partition function with $q = -2\Lambda$, up to a spurious factor $\exp(1/b + b - 2m_1)$, which matches the asymptotic limit of the $N_f=4$ factor.
• The authors conjecture that irregular states exist as formal power series satisfying $L_n|\psi\rangle = 0$ for $n > N$, with $L_1, \dots, L_{N-1}$ acting as multiplication by constants, and that such states are unique at each level.