Knizhnik Zamolodchikov Research Articles

GHOSTS AND CONGRUENCES FOR -APPROXIMATIONS OF HYPERGEOMETRIC PERIODS

AbstractWe prove general Dwork-type congruences for constant terms attached to tuples of Laurent polynomials. We apply this result to establishing arithmetic and p-adic analytic properties of functions originating from polynomial solutions modulo $p^s$ of hypergeometric and Knizhnik–Zamolodchikov (KZ) equations, solutions which come as coefficients of master polynomials and whose coefficients are integers. As an application, we show that the simplest example of a p-adic KZ connection has an invariant line subbundle while its complex analog has no nontrivial subbundles due to the irreducibility of its monodromy representation.

Reduction cohomology of Riemann surfaces

We study the algebraic conditions leading to the chain property of complexes for vertex operator algebra n-point functions (with their convergence assumed) with differential being defined through reduction formulas. The notion of the reduction cohomology of Riemann surfaces is introduced. Algebraic, geometrical, and cohomological meanings of reduction formulas are clarified. A counterpart of the Bott–Segal theorem for Riemann surfaces in terms of the reductions cohomology is proven. It is shown that the reduction cohomology is given by the cohomology of n-point connections over the vertex operator algebra bundle defined on a genus g Riemann surface [Formula: see text]. The reduction cohomology for a vertex operator algebra with formal parameters identified with local coordinates around marked points on [Formula: see text] is found in terms of the space of analytical continuations of solutions to Knizhnik–Zamolodchikov equations. For the reduction cohomology, the Euler–Poincaré formula is derived. Examples for various genera and vertex operator cluster algebras are provided.

Anyonic defect branes and conformal blocks in twisted equivariant differential (TED) K-theory

We demonstrate that twisted equivariant differential K-theory of transverse complex curves accommodates exotic charges of the form expected of codimension[Formula: see text]2 defect branes, such as of [Formula: see text]-branes in IIB/F-theory on [Formula: see text]-type orbifold singularities, but also of their dual 3-brane defects of class-S theories on M5-branes. These branes have been argued, within F-theory and the AGT correspondence, to carry special [Formula: see text]-monodromy charges not seen for other branes, at least partially reflected in conformal blocks of the [Formula: see text]-WZW model over their transverse punctured complex curve. Indeed, it has been argued that all “exotic” branes of string theory are defect branes carrying such U-duality monodromy charges — but none of these had previously been identified in the expected brane charge quantization law given by K-theory. Here we observe that it is the subtle (and previously somewhat neglected) twisting of equivariant K-theory by flat complex line bundles appearing inside orbi-singularities (“inner local systems”) that makes the secondary Chern character on a punctured plane inside an [Formula: see text]-type singularity evaluate to the twisted holomorphic de Rham cohomology which Feigin, Schechtman and Varchenko showed realizes [Formula: see text]-conformal blocks, here in degree 1 — in fact it gives the direct sum of these over all admissible fractional levels [Formula: see text]. The remaining higher-degree [Formula: see text]-conformal blocks appear similarly if we assume our previously discussed “Hypothesis H” about brane charge quantization in M-theory. Since conformal blocks — and hence these twisted equivariant secondary Chern characters — solve the Knizhnik–Zamolodchikov equation and thus constitute representations of the braid group of motions of defect branes inside their transverse space, this provides a concrete first-principles realization of anyon statistics of — and hence of topological quantum computation on — defect branes in string/M-theory.

Non-Lorentzian Kač-Moody algebras

We investigate two dimensional (2d) quantum field theories which exhibit Non-Lorentzian Kač-Moody (NLKM) algebras as their underlying symmetry. Our investigations encompass both 2d Galilean (speed of light c → ∞) and Carrollian (c → 0) CFTs with additional number of infinite non-Abelian currents, stemming from an isomorphism between the two algebras. We alternate between an intrinsic and a limiting analysis. Our NLKM algebra is constructed first through a contraction and then derived from an intrinsically Carrollian perspective. We then go on to use the symmetries to derive a Non-Lorentzian (NL) Sugawara construction and ultimately write down the NL equivalent of the Knizhnik Zamolodchikov equations. All of these are also derived from contractions, thus providing a robust cross-check of our analyses.

Isomonodromic Deformations: Confluence, Reduction and Quantisation

In this paper we study the isomonodromic deformations of systems of differential equations with poles of any order on the Riemann sphere as Hamiltonian flows on the product of co-adjoint orbits of the truncated current algebra, also called generalised Takiff algebra. Our motivation is to produce confluent versions of the celebrated Knizhnik–Zamolodchikov equations and explain how their quasiclassical solution can be expressed via the isomonodromic tau -function. In order to achieve this, we study the confluence cascade of r+ 1 simple poles to give rise to a singularity of arbitrary Poincaré rank r as a Poisson morphism and explicitly compute the isomonodromic Hamiltonians. In loving memory of Igor KricheverA great man and outstanding mathematician

Comparison of quantizations of symmetric spaces: cyclotomic Knizhnik–Zamolodchikov equations and Letzter–Kolb coideals

Abstract We establish an equivalence between two approaches to quantization of irreducible symmetric spaces of compact type within the framework of quasi-coactions, one based on the Enriquez–Etingof cyclotomic Knizhnik–Zamolodchikov (KZ) equations and the other on the Letzter–Kolb coideals. This equivalence can be upgraded to that of ribbon braided quasi-coactions, and then the associated reflection operators (K-matrices) become a tangible invariant of the quantization. As an application we obtain a Kohno–Drinfeld type theorem on type $\mathrm {B}$ braid group representations defined by the monodromy of KZ-equations and by the Balagović–Kolb universal K-matrices. The cases of Hermitian and non-Hermitian symmetric spaces are significantly different. In particular, in the latter case a quasi-coaction is essentially unique, while in the former we show that there is a one-parameter family of mutually nonequivalent quasi-coactions.

Stokes Phenomenon and Reflection Equations

In this paper, we study the Stokes phenomenon of the cyclotomic Knizhnik–Zamolodchikov equation associated to the symmetric pair \((\mathfrak {gl}_n,\mathfrak {so}_n)\), and prove that its two types of Stokes matrices satisfy the Yang–Baxter and reflection equations respectively. We briefly discuss its isomonodromy deformation, and its relations with cyclotomic associators, twists and quantum symmetric pairs.

Ising-like and Fibonacci anyons from KZ-equations

In this work we present solutions to Knizhnik-Zamolodchikov (KZ) equations corresponding to conformal block wavefunctions of non-Abelian Ising-like and Fibonacci Anyons. We solve these equations around regular singular points in configuration space in terms of hypergeometric functions and derive explicit monodromy representations of the braid group action. This confirms the correct non-Abelian statistics of the solutions. One novelty of our approach is that we explicitly keep track of spin basis states and identify conformal blocks uniquely with such states at relevant points in moduli space.

Knizhnik–Zamolodchikov functor for degenerate double affine Hecke algebras: algebraic theory

In this article, we define an algebraic version of the Knizhnik–Zamolodchikov (KZ) functor for the degenerate double affine Hecke algebras (a.k.a. trigonometric Cherednik algebras). We compare it with the KZ monodromy functor constructed by Varagnolo–Vasserot. We prove the double centraliser property for our functor and give a characterisation of its kernel. We establish these results for a family of algebras, called quiver double Hecke algebras, which includes the degenerate double affine Hecke algebras as special cases.

Quantum nonequilibrium dynamics from Knizhnik-Zamolodchikov equations

We consider a set of non-stationary quantum models. We show that their dynamics can be studied using links to Knizhnik-Zamolodchikov (KZ) equations for correlation functions in conformal field theories. We specifically consider the boundary Wess-Zumino-Novikov-Witten model, where equations for correlators of primary fields are defined by an extension of KZ equations and explore the links to dynamical systems. As an example of the workability of the proposed method, we provide an exact solution to a dynamical system that is a specific multi-level generalization of the two-level Landau-Zenner system known in the literature as the Demkov-Osherov model. The method can be used to study the nonequilibrium dynamics in various multi-level systems from the solution of the corresponding KZ equations.

Surface defects in gauge theory and KZ equation

We study the regular surface defect in the \(\Omega \)-deformed four-dimensional supersymmetric gauge theory with gauge group SU(N) with 2N hypermultiplets in fundamental representation. We prove its vacuum expectation value obeys the Knizhnik–Zamolodchikov equation for the 4-point conformal block of the \(\widehat{{\mathfrak {sl}}}_{N}\)-current algebra, originally introduced in the context of two-dimensional conformal field theory. The level and the vertex operators are determined by the parameters of the \(\Omega \)-background and the masses of the hypermultiplets; the cross-ratio of the 4 points is determined by the complexified gauge coupling. We clarify that in a somewhat subtle way the branching rule is parametrized by the Coulomb moduli. This is an example of the BPS/CFT relation.

N-point spherical functions and asymptotic boundary KZB equations

Let G be a split real connected Lie group with finite center. In the first part of the paper we define and study formal elementary spherical functions. They are formal power series analogues of elementary spherical functions on G in which the role of the quasi-simple admissible G-representations is replaced by Verma modules. For generic highest weight we express the formal elementary spherical functions in terms of Harish-Chandra series and integrate them to spherical functions on the regular part of G. We show that they produce eigenstates for spin versions of quantum hyperbolic Calogero–Moser systems. In the second part of the paper we define and study special subclasses of global and formal elementary spherical functions, which we call global and formal N-point spherical functions. Formal N-point spherical functions arise as limits of correlation functions for boundary Wess–Zumino–Witten conformal field theory on the cylinder when the position variables tend to infinity. We construct global N-point spherical functions in terms of compositions of equivariant differential intertwiners associated with principal series representations, and express them in terms of Eisenstein integrals. We show that the eigenstates of the quantum spin Calogero–Moser system associated to N-point spherical functions are also common eigenfunctions of a commuting family of first-order differential operators, which we call asymptotic boundary Knizhnik–Zamolodchikov–Bernard operators. These operators are explicitly given in terms of folded classical dynamical r-matrices and associated dynamical k-matrices.

Folded and contracted solutions of coupled classical dynamical Yang–Baxter and reflection equations

In this paper we give a concrete recipe how to construct triples of algebra-valued meromorphic functions on a complex vector space a satisfying three coupled classical dynamical Yang–Baxter equations and an associated classical dynamical reflection equation. Such triples provide the local factors of a consistent system of first order differential operators on a, generalising asymptotic boundary Knizhnik–Zamolodchikov–Bernard (KZB) equations.The recipe involves folding and contracting a-invariant and θ-twisted symmetric classical dynamical r-matrices along an involutive automorphism θ. In case of the universal enveloping algebra of a simple Lie algebra g we determine the subclass of Schiffmann’s classical dynamical r-matrices which are a-invariant and θ-twisted.The paper starts with a section highlighting the connections between asymptotic (boundary) KZB equations, representation theory of semisimple Lie groups, and integrable quantum field theories.

Tensor Structure on the Kazhdan–Lusztig Category for Affine 𝔤𝔩(1|1)

AbstractWe show that the Kazhdan–Lusztig category $KL_k$ of level-$k$ finite-length modules with highest-weight composition factors for the affine Lie superalgebra $\widehat{\mathfrak{gl}(1|1)}$ has vertex algebraic braided tensor supercategory structure and that its full subcategory $\mathcal{O}_k^{fin}$ of objects with semisimple Cartan subalgebra actions is a tensor subcategory. We show that every simple $\widehat{\mathfrak{gl}(1|1)}$-module in $KL_k$ has a projective cover in ${\mathcal{O}}_k^{fin}$, and we determine all fusion rules involving simple and projective objects in ${\mathcal{O}}_k^{fin}$. Then using Knizhnik–Zamolodchikov equations, we prove that $KL_k$ and $\mathcal{O}_k^{fin}$ are rigid. As an application of the tensor supercategory structure on $\mathcal{O}_k^{fin}$, we study certain module categories for the affine Lie superalgebra $\widehat{\mathfrak{sl}(2|1)}$ at levels $1$ and $-\frac{1}{2}$. In particular, we obtain a tensor category of $\widehat{\mathfrak{sl}(2|1)}$-modules at level $-\frac{1}{2}$ that includes relaxed highest-weight modules and their images under spectral flow.

The open XXZ chain at Δ = −1/2 and the boundary quantum Knizhnik–Zamolodchikov equations

The open XXZ spin chain with the anisotropy parameter and diagonal boundary magnetic fields that depend on a parameter x is studied. For real x > 0, the exact finite-size ground-state eigenvalue of the spin-chain Hamiltonian is explicitly computed. In a suitable normalisation, the ground-state components are characterised as polynomials in x with integer coefficients. Linear sum rules and special components of this eigenvector are explicitly computed in terms of determinant formulas. These results follow from the construction of a contour-integral solution to the boundary quantum Knizhnik–Zamolodchikov equations associated with the R-matrix and diagonal K-matrices of the six-vertex model. A relation between this solution and a weighted enumeration of totally-symmetric alternating sign matrices is conjectured.

Elliptic stable envelopes

We construct stable envelopes in equivariant elliptic cohomology of Nakajima quiver varieties. In particular, this gives an elliptic generalization of the results of Maulik and Okounkov [Astérisque 408 (2019), ix+209]. We apply them to the computation of the monodromy of q q -difference equations arising in the enumerative K-theory of rational curves in Nakajima varieties, including the quantum Knizhnik–Zamolodchikov equations.

Quantization of Integrable Systems with Spectral Parameter on a Riemann Surface

Given an integrable system defined by a Lax representation with spectral parameter on a Riemann surface, we construct a unitary projective representation of the corresponding Lie algebra of Hamiltonian vector fields by means of operators of covariant derivatives with respect to the Knizhnik–Zamolodchikov connection. It is a Dirac-type prequantization of the integrable system from a physical point of view. Simultaneously, it establishes a correspondence between integrable systems in question and conformal field theories. In the present paper, we focus on systems whose spectral curves possess a holomorphic involution. Examples are presented by Hitchin systems of the types Bn, $${{C}_{n}}$$ , $${{D}_{n}}$$ , and also of the type An on hyperelliptic curves.

Q-SCHUR ALGEBRAS CORRESPONDING TO HECKE ALGEBRAS OF TYPE B

In this paper the authors investigate the q-Schur algebras of type B that were constructed earlier using coideal subalgebras for the quantum group of type Α. The authors present a coordinate algebra type construction that allows us to realize these q-Schur algebras as the duals of the dth graded components of certain graded coalgebras. Under suitable conditions an isomorphism theorem is proved that demonstrates that the representation theory reduces to the q-Schur algebra of type Α. This enables the authors to address the questions of cellularity, quasi-hereditariness and representation type of these algebras. Later it is shown that these algebras realize the 1-faithful quasi hereditary covers of the Hecke algebras of type Β. As a further consequence, the authors demonstrate that these algebras are Morita equivalent to the category 𝒪 for rational Cherednik algebras for the Weyl group of type Β. In particular, we have introduced a Schur-type functor that identifies the type Β Knizhnik–Zamolodchikov functor.

A note on the Drinfeld associator for genus-zero superstring amplitudes in twisted de Rham theory

The string corrections of tree-level open-string amplitudes can be described by Selberg integrals satisfying a Knizhnik–Zamolodchikov (KZ) equation. This allows for a recursion of the α′-expansion of tree-level string corrections in the number of external states using the Drinfeld associator. While the feasibility of this recursion is well-known, we provide a mathematical description in terms of twisted de Rham theory and intersection numbers of twisted forms. In particular, this leads to purely combinatorial expressions for the matrix representation of the Lie algebra generators appearing in the KZ equation in terms of directed graphs. This, in turn, admits efficient algorithms for symbolic and numerical computations using adjacency matrices of directed graphs and is a crucial step towards analogous recursions and algorithms at higher genera.

Correlation functions of integrable O(n) spin chains

We study the correlation functions of the integrable O(n) spin chain in the thermodynamic limit. We addressed the problem of solving functional equations of the quantum Knizhnik Zamolodchikov type for density matrix related to the O(n) spin chain. We give the explicit solution for two-sites density matrix elements for the O(n) which are then evaluated for the n=3,4,…,8 cases at zero temperature.