Superintegrability of dd-dimensional Conformal Blocks

We observe that conformal blocks of scalar 4-point functions in a $d$-dimensional conformal field theory can mapped to eigenfunctions of a 2-particle hyperbolic Calogero-Sutherland Hamiltonian. The latter describes two coupled P\"oschl-Teller particles. Their interaction, whose strength depends smoothly on the dimension $d$, is known to be superintegrable. Our observation enables us to exploit the rich mathematical literature on Calogero-Sutherland models in deriving various results for conformal field theory. These include an explicit construction of conformal blocks in terms of Heckman-Opdam hypergeometric functions and a remarkable duality that relates the blocks of theories in different dimensions.Comment: 5 page

Chiral Primaries in Strange Metals

It was suggested recently that the study of 1-dimensional QCD with fermions in the adjoint representation could lead to an interesting toy model for strange metals and their holographic formulation. In the high density regime, the infrared physics of this theory is described by a constrained free fermion theory with an emergent {\cal N}=(2,2) superconformal symmetry. In order to narrow the choice of potential holographic duals, we initiate a systematic search for chiral primaries in this model. We argue that the bosonic part of the superconformal algebra can be extended to a coset chiral algebra of the form {\cal W}_N =\SO(2N^2-2)_1/\SU(N)_{2N}. In terms of this algebra the spectrum of the low energy theory decomposes into a finite number of sectors which are parametrized by special necklaces. We compute the corresponding characters and partition functions and determine the set of chiral primaries for N \leq 5.Comment: 42 pages, 5 figure

Chiral Ring of Strange Metals: The Multicolor Limit

The low energy limit of a dense 2D adjoint QCD is described by a family of ${\cal N}=(2,2)$ supersymmetric coset conformal field theories. In previous work we constructed chiral primaries for a small number $N < 6$ of colors. Our aim in the present note is to determine the chiral ring in the multicolor limit where $N$ is sent to infinity. We shall find that chiral primaries are labeled by partitions and identify the ring they generate as the ring of Schur polynomials. Our findings impose strong constraints on the possible dual description through string theory in an $AdS_3$ compactification.Comment: 25 pages, 4 figure

Towards a full solution of the large N double-scaled SYK model

We compute the exact, all energy scale, 4-point function of the large $N$ double-scaled SYK model, by using only combinatorial tools and relating the correlation functions to sums over chord diagrams. We apply the result to obtain corrections to the maximal Lyapunov exponent at low temperatures. We present the rules for the non-perturbative diagrammatic description of correlation functions of the entire model. The latter indicate that the model can be solved by a reduction of a quantum deformation of SL$(2)$, that generalizes the Schwarzian to the complete range of energies.Comment: 52+28 pages, 14 figures; v2: references revised, typos corrected, changed normalization of SL(2)_q 6j symbo

Toda 3-Point Functions From Topological Strings II

In arXiv:1409.6313 we proposed a formula for the 3-point structure constants of Toda field theory, derived using topological strings and the AGT-W correspondence from the partition functions of the non-Lagrangian $T_N$ theories on $S^4$. In this article, we show how the semi-degeneration of one of the three primary fields on the Toda side corresponds to a particular Higgsing of the $T_N$ theories and derive the well-known formula by Fateev and Litvinov.Comment: 43 pages, 14 figures,v2: published in JHE

Quantum groups, non-commutative AdS2AdS_2, and chords in the double-scaled SYK model

We study the double-scaling limit of SYK (DS-SYK) model and elucidate the underlying quantum group symmetry. The DS-SYK model is characterized by a parameter $q$, and in the $q\rightarrow 1$ and low-energy limit it goes over to the familiar Schwarzian theory. We relate the chord and transfer-matrix picture to the motion of a ``boundary particle" on the Euclidean Poincar{\'e} disk, which underlies the single-sided Schwarzian model. $AdS_2$ carries an action of $\mathfrak{s}\mathfrak{l}(2,{\mathbb R}) \simeq \mathfrak{s}\mathfrak{u}(1,1)$, and we argue that the symmetry of the full DS-SYK model is a certain $q$-deformation of the latter, namely $\mathcal{U}_{\sqrt q}(\mathfrak{s}\mathfrak{u}(1,1))$. We do this by obtaining the effective Hamiltonian of the DS-SYK as a (reduction of) particle moving on a lattice deformation of $AdS_2$, which has this $\mathcal{U}_{\sqrt q}(\mathfrak{s}\mathfrak{u}(1,1))$ algebra as its symmetry. We also exhibit the connection to non-commutative geometry of $q$-homogeneous spaces, by obtaining the effective Hamiltonian of the DS-SYK as a (reduction of) particle moving on a non-commutative deformation of $AdS_3$. There are families of possibly distinct $q$-deformed $AdS_2$ spaces, and we point out which are relevant for the DS-SYK model.Comment: 70 pages, 6 figure

Calogero-Sutherland Approach to Defect Blocks

Extended objects such as line or surface operators, interfaces or boundaries play an important role in conformal field theory. Here we propose a systematic approach to the relevant conformal blocks which are argued to coincide with the wave functions of an integrable multi-particle Calogero-Sutherland problem. This generalizes a recent observation in 1602.01858 and makes extensive mathematical results from the modern theory of multi-variable hypergeometric functions available for studies of conformal defects. Applications range from several new relations with scalar four-point blocks to a Euclidean inversion formula for defect correlators.Comment: v2: changes for clarit

Conformal Field Theory and Functions of Hypergeometric Type

AbstractConformal field theory provides a universal description of various phenomena in natural sciences. Its development, swift and successful, belongs to the major highlights of theoretical physics of the late XX century. In contrast, advances of the theory of hypergeometric functions always assumed a slower pace throughout the centuries of its existence. Functional identities studied by this mathematical discipline are fascinating both in their complexity and beauty. This thesis investigates the interrelation of two subjects through a direct analysis of three CFT problems: two-point functions of the 2d strange metal CFT, three-point functions of primaries of the non-rational Toda CFT and kinematical parts of Mellin amplitudes for scalar four-point functions in general dimensions. We flash out various generalizations of hypergeometric functions as a natural mathematical language for two of these problems. Several new methods inspired by extensions of classical results on hypergeometric functions, are presented.ZusammenfassungDie konforme Feldtheorie (CFT) bietet eine universelle Beschreibung verschiedener Phänomene in den Naturwissenschaften. Ihre schnelle und erfolgreiche Entwicklung gehört zu den wichtigsten Höhepunkten der theoretischen Physik des späten 20. Jahrhunderts. Demgegenüber ging der Fortschritt der hypergeometrischen Funktionen durch die Jahrhunderte langsamer vonstatten. Funktionale Identitäten, die von dieser mathematsichen Disziplin untersucht werden, sind faszinierend sowohl in ihrer Komplexität, als auch ihrer Schönheit. Diese Arbeit untersucht das Zusammenspiel dieser beiden Themen anhand der direkten Analyse dreier CFT-Problemen: Zweipunktfunktionen der zweidimensionalen ’strange metal CFT’, Dreipunktfunktionen von primären Feldern der nichtrationalen Toda CFT und kinematischen Teilen von Mellin-Amplituden für skalare Vierpunktfunktionen in beliebigen Dimensionen. Wir heben verschiedene Verallgemeinerungen der hypergeometrischen Funktionen als eine natürliche mathematische Sprache für zwei dieser Probleme hervor. Einige neue Methoden, die durch klassische Resultate über hypergeometrische Funktionen inspiriert wurden, werden vorgestellt