Flat Connections for Characters in Irrational Conformal Field Theory

Following the paradigm on the sphere, we begin the study of irrational conformal field theory (ICFT) on the torus. In particular, we find that the affine-Virasoro characters of ICFT satisfy heat-like differential equations with flat connections. As a first example, we solve the system for the general $g/h$ coset construction, obtaining an integral representation for the general coset characters. In a second application, we solve for the high-level characters of the general ICFT on simple $g$, noting a simplification for the subspace of theories which possess a non-trivial symmetry group. Finally, we give a geometric formulation of the system in which the flat connections are generalized Laplacians on the centrally-extended loop group.Comment: harvmac (answer b to question) 40 pages. LBL-35718, UCB-PTH-94/1

Singular vectors by Fusions in affine su(2)

Explicit expressions for the singular vectors in the highest weight representations of $A_1^{(1)}$ are obtained using the fusion formalism of conformal field theory.Comment: 7 page

Fusion and singular vectors in A1{(1)} highest weight cyclic modules

We show how the interplay between the fusion formalism of conformal field theory and the Knizhnik--Zamolodchikov equation leads to explicit formulae for the singular vectors in the highest weight representations of A1{(1)}.Comment: 42 page

Physical States in G/G Models and 2d Gravity

An analysis of the BRST cohomology of the G/G topological models is performed for the case of $A_1^{(1)}$. Invoking a special free field parametrization of the various currents, the cohomology on the corresponding Fock space is extracted. We employ the singular vector structure and fusion rules to translate the latter into the cohomology on the space of irreducible representations. Using the physical states we calculate the characters and partition function, and verify the index interpretation. We twist the energy-momentum tensor to establish an intriguing correspondence between the ${SL(2)\over SL(2)}$ model with level $k={p\over q}-2$ and $(p,q)$ models coupled to gravity.Comment: 42 page

GG-Strands

A $G$-strand is a map $g(t,{s}):\,\mathbb{R}\times\mathbb{R}\to G$ for a Lie group $G$ that follows from Hamilton's principle for a certain class of $G$-invariant Lagrangians. The SO(3)-strand is the $G$-strand version of the rigid body equation and it may be regarded physically as a continuous spin chain. Here, $SO(3)_K$-strand dynamics for ellipsoidal rotations is derived as an Euler-Poincar\'e system for a certain class of variations and recast as a Lie-Poisson system for coadjoint flow with the same Hamiltonian structure as for a perfect complex fluid. For a special Hamiltonian, the $SO(3)_K$-strand is mapped into a completely integrable generalization of the classical chiral model for the SO(3)-strand. Analogous results are obtained for the $Sp(2)$-strand. The $Sp(2)$-strand is the $G$-strand version of the $Sp(2)$ Bloch-Iserles ordinary differential equation, whose solutions exhibit dynamical sorting. Numerical solutions show nonlinear interactions of coherent wave-like solutions in both cases. ${\rm Diff}(\mathbb{R})$-strand equations on the diffeomorphism group $G={\rm Diff}(\mathbb{R})$ are also introduced and shown to admit solutions with singular support (e.g., peakons).Comment: 35 pages, 5 figures, 3rd version. To appear in J Nonlin Sc

Антиутопия как диагноз будущему

Материалы XIV Междунар. науч. конф. студентов, магистрантов, аспирантов и молодых ученых, Гомель, 13–14 мая 2021 г

On the geometry of classically integrable two-dimensional non-linear sigma models

A master equation expressing the classical integrability of two-dimensional non-linear sigma models is found. The geometrical properties of this equation are outlined. In particular, a closer connection between integrability and T-duality transformations is emphasised. Finally, a whole new class of integrable non-linear sigma models is found and all their corresponding Lax pairs depend on a spectral parameter.Comment: 16 pages. Major changes (almost a new paper). To be published in Nuclear Physics B (2010

Improving the predictive potential of diffusion MRI in schizophrenia using normative models-Towards subject-level classification.

Diffusion MRI studies consistently report group differences in white matter between individuals diagnosed with schizophrenia and healthy controls. Nevertheless, the abnormalities found at the group-level are often not observed at the individual level. Among the different approaches aiming to study white matter abnormalities at the subject level, normative modeling analysis takes a step towards subject-level predictions by identifying affected brain locations in individual subjects based on extreme deviations from a normative range. Here, we leveraged a large harmonized diffusion MRI dataset from 512 healthy controls and 601 individuals diagnosed with schizophrenia, to study whether normative modeling can improve subject-level predictions from a binary classifier. To this aim, individual deviations from a normative model of standard (fractional anisotropy) and advanced (free-water) dMRI measures, were calculated by means of age and sex-adjusted z-scores relative to control data, in 18 white matter regions. Even though larger effect sizes are found when testing for group differences in z-scores than are found with raw values (p < .001), predictions based on summary z-score measures achieved low predictive power (AUC < 0.63). Instead, we find that combining information from the different white matter tracts, while using multiple imaging measures simultaneously, improves prediction performance (the best predictor achieved AUC = 0.726). Our findings suggest that extreme deviations from a normative model are not optimal features for prediction. However, including the complete distribution of deviations across multiple imaging measures improves prediction, and could aid in subject-level classification