Holomorphic Observables Research Articles

Discrete Symplectic Fermions on Double Dimers and Their Virasoro Representation

AbstractA discrete version of the conformal field theory of symplectic fermions is introduced and discussed. Specifically, discrete symplectic fermions are realised as holomorphic observables in the double-dimer model. Using techniques of discrete complex analysis, the space of local fields of discrete symplectic fermions on the square lattice is proven to carry a representation of the Virasoro algebra with central charge $$-2$$ - 2 .

Complex Langevin: Correctness criteria, boundary terms, and spectrum

The Complex Langevin (CL) method to simulate “complex probabilities” ideally produces expectation values for the observables that converge to a limit equal to the expectation values obtained with the original complex “probability” measure. The situation may be spoiled in two ways: failure to converge and convergence to the wrong limit. It was found long ago that “wrong convergence” is caused by boundary terms; nonconvergence may arise from bad spectral properties of the various evolution operators related to the CL process. Here, we propose a class of criteria that allow one to rule out boundary terms and at the same time bad spectrum. Ruling out boundary terms in the equilibrium distribution arising from a CL simulation implies that the so-called convergence conditions are fulfilled. This in turn has been shown to guarantee that the expectation values of holomorphic observables are given by complex linear combinations of exp(−S) over various integration cycles. If the spectrum is pathological, however, the CL simulation in general does not reproduce the integral over the desired real cycle. Published by the American Physical Society 2024

The distribution of Gaussian multiplicative chaos on the unit interval

We consider a subcritical Gaussian multiplicative chaos (GMC) measure defined on the unit interval $[0,1]$ and prove an exact formula for the fractional moments of the total mass of this measure. Our formula includes the case where log-singularities (also called insertion points) are added in $0$ and $1$, the most general case predicted by the Selberg integral. The idea to perform this computation is to introduce certain auxiliary functions resembling holomorphic observables of conformal field theory that will be solutions of hypergeometric equations. Solving these equations then provides nontrivial relations that completely determine the moments we wish to compute. We also include a detailed discussion of the so-called reflection coefficients appearing in tail expansions of GMC measures and in Liouville theory. Our theorem provides an exact value for one of these coefficients. Lastly, we mention some additional applications to small deviations for GMC measures, to the behavior of the maximum of the log-correlated field on the interval and to random hermitian matrices.

Holomorphic Spinor Observables in the Critical Ising Model

We introduce a new version of discrete holomorphic observables for the critical planar Ising model. These observables are holomorphic spinors defined on double covers of the original multiply connected domain. We compute their scaling limits, and show their relation to the ratios of spin correlations, thus providing a rigorous proof to a number of formulae for those ratios predicted by CFT arguments.

Conformal invariance in random cluster models. I. Holmorphic fermions in the Ising model

We construct discrete holomorphic observables in the Ising model at criticality and show that they have conformally covariant scaling limits (as mesh of the lattice tends to zero). In the sequel those observables are used to construct conformally invariant scaling limits of interfaces. Though Ising model is often cited as a classical example of conformal invariance, it seems that ours is the first paper where it is actually established.

Discrete Holomorphicity at Two-Dimensional Critical Points

After a brief review of the historical role of analyticity in the study of critical phenomena, an account is given of recent discoveries of discretely holomorphic observables in critical two-dimensional lattice models. These are objects whose correlation functions satisfy a discrete version of the Cauchy-Riemann relations. Their existence appears to have a deep relation with the integrability of the model, and they are presumably the lattice versions of the truly holomorphic observables appearing in the conformal field theory (CFT) describing the continuum limit. This hypothesis sheds light on the connection between CFT and integrability, and, if verified, can also be used to prove that the scaling limit of certain discrete curves in these models is described by Schramm-Loewner evolution (SLE).

Holomorphic parafermions in the Potts model and stochastic Loewner evolution

We analyse parafermionic operators in theQ-state Potts model from three different perspectives. First, we explicitly constructlattice holomorphic observables in the Fortuin–Kasteleyn representation,and point out some special simplifying features of the particular caseQ = 2 (Ising model). In particular, away from criticality, we find a lattice generalizationof the massive Majorana fermion equation. We also compare the parafermionicscaling dimensions with known results from conformal field theory and Coulombgas methods in the continuum. Finally, we show that expectation values ofthese parafermions correspond to local observables of the stochastic Loewnerevolution process which is conjectured to describe the scaling limit of theQ-state Potts model.

An exact elliptic superpotential for [formula omitted] deformations of finite [formula omitted] gauge theories

We study relevant deformations of the N=2 superconformal theory on the world-volume of N D3-branes at an A k−1 singularity. In particular, we determine the vacuum structure of the mass-deformed theory with N=1 supersymmetry and show how the different vacua are permuted by an extended duality symmetry. We then obtain exact, modular covariant formulae (for all k, N and arbitrary gauge couplings) for the holomorphic observables in the massive vacua in two different ways: by lifting to M-theory, and by compactification to three dimensions and subsequent use of mirror symmetry. In the latter case, we find an exact superpotential for the model which coincides with a certain combination of the quadratic Hamiltonians of the spin generalization of the elliptic Calogero–Moser integrable system.

Softly-broken Script N = 4 supersymmetry in the large-N limit

We calculate the exact values of the holomorphic observables of {\cal N}=4 supersymmetric SU(N) Yang-Mills theory deformed by mass terms which preserve {\cal N}=1 SUSY. These include the chiral condensates in each massive vacuum of the theory as well as the central charge which determines the tension of BPS saturated domain walls interpolating between these vacua. Several unexpected features emerge in the large-N limit, including anomalous modular properties under an SL(2,Z) duality group which acts on a complexification of the 't Hooft coupling \lambda=g^{2}N/4\pi. We discuss our results in the context of the AdS/CFT correspondence.