Correlation Functions In Conformal Field Theory Research Articles

Differential equations for Carrollian amplitudes

Differential equations are powerful tools in the study of correlation functions in conformal field theories (CFTs). Carrollian amplitudes behave as correlation functions of Carrollian CFT that holographically describes asymptotically flat spacetime. We derive linear differential equations satisfied by Carrollian MHV gluon and graviton amplitudes. We obtain non-distributional solutions for both the gluon and graviton cases. We perform various consistency checks for these differential equations, including compatibility with conformal Carrollian symmetries.

Conformal field theory approach to parton fractional quantum Hall trial wave functions

We show that all lowest Landau-level projected and unprojected chiral parton type fractional quantum Hall ground and edge-state trial wave functions, which take the form of products of integer quantum Hall wave functions, can be expressed as conformal field theory (CFT) correlation functions, where we can associate a chiral algebra to each parton state such that the CFT defined by the algebra is the “smallest” such CFT that can generate the corresponding ground and edge-state trial wave functions (assuming that the corresponding chiral algebra does indeed define a physically “sensible” CFT). A field-theoretic generalization of Laughlin's plasma analogy, known as generalized screening, is formulated for these states. If this holds, along with an additional assumption, we argue that the inner products of edge-state trial wave functions, for parton states where the “densest” trial wave function is unique, can be expressed as matrix elements of an exponentiated local action operator of the CFT, generalizing the result of Dubail [], which implies the equality between edge-state and entanglement level counting to state counting in the corresponding CFT. We numerically test this result in the case of the unprojected ν=2/5 composite fermion state and the bosonic ν=1ϕ22 parton state. We discuss how Read's arguments [] still apply, implying that conformal blocks of the CFT defined by the corresponding chiral algebra are valid quasihole trial wave functions, with the adiabatic braiding statistics given by the monodromy of these functions, assuming the existence of a quasiparticle trapping Hamiltonian. Generalizations of these constructions are discussed, with particular attention given to simple current constructions. It is shown that all chiral composite fermion wave functions can be expressed as CFT correlation functions without explicit symmetrization or antisymmetrization and that the ground, edge, and certain quasihole trial wave functions of the ϕnm parton states can be expressed as the conformal blocks of the U(1)⊗SU(n)m WZW models. Finally, we discuss the relation of the ϕ2k series with the Read-Rezayi series, where several examples of quasihole braiding statistics calculations are given. Published by the American Physical Society 2024

Genus two correlation functions in CFTs with $$T\bar T$$ deformation

Since the definition of $$T\bar T$$ deformation in the curved Riemann surface is obstructive in the literature, we propose a way to do the deformation in the genus two Riemann surfaces by sewing prescription. We construct the correlation functions of conformal field theories (CFTs) on genus two Riemann surfaces with the $$T\bar T$$ deformation in terms of the perturbative CFT approach. Thanks to sewing construction to form higher genus Riemann surfaces from lower genus ones and conformal symmetry, we systematically obtain the first-order $$T\bar T$$ correction to the genus two correlation functions in the $$T\bar T$$ deformed CFTs, e.g., partition function and one/higher-point correlation functions.

Multiple-SLE κ connectivity weights for rectangles, hexagons, and octagons

In previous work, two of the authors determined, completely and rigorously, a solution space for a homogeneous system of 2N + 3 linear partial differential equations (PDEs) in 2N variables that arises in conformal field theory (CFT) and multiple Schramm–Löwner evolution (SLE κ ). The system comprises 2N null-state equations and three conformal ward identities that govern CFT correlation functions of 2N one-leg boundary operators or SLE κ partition functions. M Bauer et al conjectured a formula, expressed in terms of ‘pure SLE κ partition functions,’ for the probability that the growing curves of a multiple-SLE κ process join in a particular connectivity. In a previous article, we rigorously define certain elements of , which we call ‘connectivity weights,’ argue that they are in fact pure SLE κ partition functions, and show how to find explicit formulas for them in terms of Coulomb gas contour integrals. Our formal definition of the connectivity weights immediately leads to a method for finding explicit expressions for them. However, this method gives very complicated formulas where simpler versions may be available, and it is not applicable for certain values of κ ∈ (0, 8) corresponding to well-known critical lattice models in statistical mechanics. In this article, we determine expressions for all connectivity weights in for N ∈ {1, 2, 3, 4} (those with N ∈ {3, 4} are new) and for so-called ‘rainbow connectivity weights’ in for all . We verify these formulas by explicitly showing that they satisfy the formal definition of a connectivity weight. In appendix B, we investigate logarithmic singularities of some of these expressions, appearing for certain values of κ predicted by logarithmic CFT.

Note on higher-point correlation functions of the $$T\bar T$$ or $$J\bar T$$ deformed CFTs

We investigate generic n-point correlation functions of conformal field theories (CFTs), with $$T\bar T$$ and $$J\bar T$$ deformations, in terms of the perturbative CFT approach. We systematically obtain the first order correction to the generic correlation functions of CFTs with $$T\bar T$$ or $$J\bar T$$ deformation. We compute the out of time ordered correlation function (OTOC) in the Ising model with $$T\bar T$$ or $$J\bar T$$ deformation, which confirms that these deformations do not change the integrable property up to the first order level.

A Solution Space for a System of Null-State Partial Differential Equations: Part 2

This article is the second of four that completely characterize a solution space $\mathcal{S}_N$ for a homogeneous system of $2N+3$ linear partial differential equations (PDEs) in 2N variables that arises in conformal field theory (CFT) and multiple Schramm-Lowner evolution (SLE). The system comprises $2N$ null-state equations and three conformal Ward identities which govern CFT correlation functions of $2N$ one-leg boundary operators. In the first article (part I), we use methods of analysis and linear algebra to prove that $\dim\mathcal{S}_N\leq C_N$, with $C_N$ the $N$th Catalan number. The analysis of that article is complete except for the proof of a lemma that it invokes. This article provides the proof. The lemma states that if every interval among $(x_2,x_3),$ $(x_3,x_4), ... ,(x_{2N-1},x_{2N})$ is a two-leg interval of $F\in\mathcal{S}_N$, then $F$ vanishes. Proving this lemma by contradiction, we show that the existence of such a nonzero function implies the existence of a non-vanishing CFT two-point function involving primary operators with different conformal weights, an impossibility. This proof (which is rigorous in spite of our occasional reference to CFT) involves two different types of estimates, those that give the asymptotic behavior of F as the length of one interval vanishes, and those that give this behavior as the lengths of two intervals vanish simultaneously. We derive these estimates by using Green functions to rewrite certain null-state PDEs as integral equations, combining other null-state PDEs to obtain Schauder interior estimates, and then repeatedly integrating the integral equations with these estimates until we obtain optimal bounds. In the case where two adjacent interval lengths vanish, we use a Green function that contains the Jacobi heat kernel as its essential ingredient.

A Solution Space for a System of Null-State Partial Differential Equations: Part 3

This article is the third of four that completely and rigorously characterize a solution space \({\mathcal{S}_N}\) for a homogeneous system of 2N + 3 linear partial differential equations (PDEs) in 2N variables that arises in conformal field theory (CFT) and multiple Schramm-Löwner evolution (SLE κ ). The system comprises 2N null-state equations and three conformal Ward identities that govern CFT correlation functions of 2N one-leg boundary operators. In the first two articles (Flores and Kleban, in Commun Math Phys, arXiv:1212.2301, 2012; Commun Math Phys, arXiv:1404.0035, 2014), we use methods of analysis and linear algebra to prove that dim \({\mathcal{S}_N\leq C_N}\), with C N the Nth Catalan number.Extending these results, we prove in this article that dim \({\mathcal{S}_N=C_N}\) and \({\mathcal{S}_N}\) entirely consists of (real-valued) solutions constructed with the CFT Coulomb gas (contour integral) formalism. In order to prove this claim, we show that a certain set of C N such solutions is linearly independent. Because the formulas for these solutions are complicated, we prove linear independence indirectly. We use the linear injective map of Lemma 15 in Flores and Kleban (Commun Math Phys, arXiv:1212.2301, 2012) to send each solution of the mentioned set to a vector in \({\mathbb{R}^{C_N}}\), whose components we find as inner products of elements in a Temperley–Lieb algebra. We gather these vectors together as columns of a symmetric \({C_N\times C_N}\) matrix, with the form of a meander matrix. If the determinant of this matrix does not vanish, then the set of C N Coulomb gas solutions is linearly independent. And if this determinant does vanish, then we construct an alternative set of C N Coulomb gas solutions and follow a similar procedure to show that this set is linearly independent. The latter situation is closely related to CFT minimal models. We emphasize that, although the system of PDEs arises in CFT in away that is typically non-rigorous, our treatment of this system here and in Flores and Kleban (Commun Math Phys, arXiv:1212.2301, 2012; Commun Math Phys, arXiv:1404.0035, 2014; Commun Math Phys, arXiv:1405.2747, 2014) is completely rigorous.

A Solution Space for a System of Null-State Partial Differential Equations: Part 4

This article is the last of four that completely characterize a solution space $\mathcal{S}_N$ for a homogeneous system of $2N+3$ linear partial differential equations (PDEs) in $2N$ variables that arises in conformal field theory (CFT) and multiple Schramm-Loewner evolution (SLE). The system comprises $2N$ null-state equations and three conformal Ward identities that govern CFT correlation functions of $2N$ one-leg boundary operators. In the first two articles, we use methods of analysis and linear algebra to prove that $\dim\mathcal{S}_N\leq C_N$, with $C_N$ the $N$th Catalan number. Building on these results in the third article, we prove that $\dim\mathcal{S}_N=C_N$ and $\mathcal{S}_N$ is spanned by (real-valued) solutions constructed with the Coulomb gas (contour integral) formalism of CFT. In this article, we use these results to prove some facts concerning the solution space $\mathcal{S}_N$. First, we show that each of its elements equals a sum of at most two distinct Frobenius series in powers of the difference between two adjacent points (unless $8/\kappa$ is odd, in which case a logarithmic term may appear). This establishes an important element in the operator product expansion (OPE) for one-leg boundary operators, assumed in CFT. We also identify particular elements of $\mathcal{S}_N$, which we call connectivity weights, and exploit their special properties to conjecture a formula for the probability that the curves of a multiple-SLE$_\kappa$ process join in a particular connectivity. Finally, we propose a reason for why the "exceptional speeds" (certain $\kappa$ values that appeared in the analysis of the Coulomb gas solutions in the third article) and the minimal models of CFT are connected.

Schramm-Loewner evolution martingales in coset conformal field theory

Schramm-Loewner evolution (SLE) and conformal field theory (CFT) are popular and widely used instruments to study critical behavior of two-dimensional models, but they use different objects. While SLE has natural connection with lattice models and is suitable for strict proofs, it lacks computational and predictive power of conformal field theory. To provide a way for the concurrent use of SLE and CFT, CFT correlation functions, which are martingales with respect to SLE, are considered. A relation between parameters of Schramm-Loewner evolution on coset space and algebraic data of coset conformal field theory is revealed. The consistency of this approach with the behavior of parafermionic and minimal models is tested. Coset models are connected with off-critical massive field theories and implications of SLE are discussed.

Calculus on manifolds of conformal maps and CFT

In conformal field theory (CFT) on simply connected domains of the Riemann sphere, the natural conformal symmetries under self-maps are extended, in a certain way, to local symmetries under general conformal maps, and this is at the basis of the powerful techniques of CFT. Conformal maps of simply connected domains naturally have the structure of an infinite-dimensional groupoid, which generalizes the finite-dimensional group of self-maps. We put a topological structure on the space of conformal maps on simply connected domains, which makes it into a topological groupoid. Further, we (almost) extend this to a local manifold structure based on the infinite-dimensional Fréchet topological vector space of holomorphic functions on a given domain A. From this, we develop the notion of conformal A-differentiability at the identity. Our main conclusion is that quadratic differentials characterizing cotangent elements on the local manifold enjoy properties similar to those of the holomorphic stress–energy tensor of CFT; these properties underpin the local symmetries of CFT. Applying the general formalism to CFT correlation functions, we show that the stress–energy tensor is exactly such a quadratic differential. This is at the basis of constructing the stress–energy tensor in conformal loop ensembles. It also clarifies the relation between Cardy's boundary conditions for CFT on simply connected domains, and the expression of the stress–energy tensor in terms of metric variations.

Quantum Hall quasielectron operators in conformal field theory

In the conformal field theory (CFT) approach to the quantum Hall effect, the multi-electron wave functions are expressed as correlation functions in certain rational CFTs. While this approach has led to a well-understood description of the fractionally charged quasihole excitations, the quasielectrons have turned out to be much harder to handle. In particular, forming quasielectron states requires non-local operators, in sharp contrast to quasiholes that can be created by local chiral vertex operators. In both cases, the operators are strongly constrained by general requirements of symmetry, braiding and fusion. Here we construct a quasielectron operator satisfying these demands and show that it reproduces known good quasiparticle wave functions, as well as predicts new ones. In particular we propose explicit wave functions for quasielectron excitations of the Moore-Read Pfaffian state. Further, this operator allows us to explicitly express the composite fermion wave functions in the positive Jain series in hierarchical form, thus settling a longtime controversy. We also critically discuss the status of the fractional statistics of quasiparticles in the Abelian hierarchical quantum Hall states, and argue that our construction of localized quasielectron states sheds new light on their statistics. At the technical level we introduce a generalized normal ordering, that allows us to "fuse" an electron operator with the inverse of an hole operator, and also an alternative approach to the background charge needed to neutralize CFT correlators. As a result we get a fully holomorphic CFT representation of a large set of quantum Hall wave functions.

On Conformal Field Theory of SLE(κ,ρ)

SLE(κ ρ), a generalization of chordal Schramm-Löwner evolution (SLE), is discussed from the point of view of statistical mechanics and conformal field theory (CFT). Certain ratios of CFT correlation functions are shown to be martingales. The interpretation is that SLE(κ ρ) describes an interface in a statistical mechanics model whose boundary conditions are created in the Coulomb gas formalism by vertex operators with charges α j = $$\alpha_j = \frac{\rho_j}{2 \sqrt{\kappa}}$$ . The total charge vanishes and therefore the partition function has a simple product form. We also suggest a generalization of SLE(κ ρ)