Holomorphic Part Research Articles

Holographic interpretation of 1-point toroidal block in the semiclassical limit

We propose the holographic interpretation of the 1-point conformal block on a torus in the semiclassical regime. To this end we consider the linearized version of the block and find its coefficients by means of the perturbation procedure around natural seed configuration corresponding to the zero-point block. From the AdS/CFT perspective the linearized block is given by the geodesic length of the tadpole graph embedded into the thermal AdS plus the holomorphic part of the thermal AdS action.

Blobbed topological recursion: properties and applications

AbstractWe study the set of solutions (ωg,n)g⩾0,n⩾1of abstract loop equations. We prove that ωg,nis determined by its purely holomorphic part: this results in a decomposition that we call “blobbed topological recursion”. This is a generalisation of the theory of the topological recursion, in which the initial data (ω0,1, ω0,2) is enriched by non-zero symmetric holomorphic forms innvariables (φg,n)2g−2+n>0. In particular, we establish for any solution of abstract loop equations: (1) a graphical representation of ωg,nin terms of φg,n; (2) a graphical representation of ωg,nin terms of intersection numbers on the moduli space of curves; (3) variational formulas under infinitesimal transformation of φg,n; (4) a definition for the free energies ωg,0=Fgrespecting the variational formulas. We discuss in detail the application to the multi-trace matrix model and enumeration of stuffed maps.

Modular Forms and Weierstrass Mock Modular Forms

Alfes, Griffin, Ono, and Rolen have shown that the harmonic Maass forms arising from Weierstrass ζ-functions associated to modular elliptic curves “encode” the vanishing and nonvanishing for central values and derivatives of twisted Hasse-Weil L-functions for elliptic curves. Previously, Martin and Ono proved that there are exactly five weight 2 newforms with complex multiplication that are eta-quotients. In this paper, we construct a canonical harmonic Maass form for these five curves with complex multiplication. The holomorphic part of this harmonic Maass form arises from the Weierstrass ζ-function and is referred to as the Weierstrass mock modular form. We prove that the Weierstrass mock modular form for these five curves is itself an eta-quotient or a twist of one. Using this construction, we also obtain p-adic formulas for the corresponding weight 2 newform using Atkin’s U-operator.

Overpartition rank differences modulo 7 by Maass forms

Using that the overpartition rank function is the holomorphic part of a harmonic Maass form, we deduce formulas for the rank differences modulo 7. To do so we make improvements on the current state of the overpartition rank function in terms of harmonic Maass forms by giving simple formulas for the transformations under SL2(Z) as well as formulas for orders at cusps.

Evolution of Smooth Shapes and Integrable Systems

We consider a homotopic evolution in the space of smooth shapes starting from the unit circle. Based on the Lowner–Kufarev equation, we give a Hamiltonian formulation of this evolution and provide conservation laws. The symmetries of the evolution are given by the Virasoro algebra. The ‘positive’ Virasoro generators span the holomorphic part of the complexified vector bundle over the space of conformal embeddings of the unit disk into the complex plane and smooth on the boundary. In the covariant formulation, they are conserved along the Hamiltonian flow. The ‘negative’ Virasoro generators can be recovered by an iterative method making use of the canonical Poisson structure. We study an embedding of the Lowner–Kufarev trajectories into the Segal–Wilson Grassmannian, construct the \(\tau \)-function, and the Baker–Akhiezer function which are related to a class of solutions to the KP hierarchy.

Pairings of harmonic Maass–Jacobi forms involving special values of partial L-functions

We prove that, for a given Jacobi integral F, there is a harmonic Maass–Jacobi form such that its holomorphic part is F, and that the converse is also true. As an application, we construct a pairing between two Jacobi integrals that is defined by special values of partial L-functions of skew-holomorphic Jacobi cusp forms. We obtain connections between this pairing and the Petersson inner product for skew-holomorphic Jacobi cusp forms. This result can be considered as an analogue of the Haberland formula of elliptic modular forms for Jacobi forms.

Polynomial and rational inequalities on analytic Jordan arcs and domains

In this paper we prove an asymptotically sharp Bernstein-type inequality for polynomials on analytic Jordan arcs. Also a general statement on mapping of a domain bounded by finitely many Jordan curves onto a complement to a system of the same number of arcs with rational function is presented here. This fact, as well as, Borwein–Erdélyi inequality for derivative of rational functions on the unit circle, Gonchar–Grigorjan estimate of the norm of holomorphic part of meromorphic functions and Totik's construction of fast decreasing polynomials play key roles in the proof of the main result.

N = 2 theories from cluster algebras

We propose a new description of 3d $\mathcal{N}=2$ theories which do not admit conventional Lagrangians. Given a quiver $Q$ and a mutation sequence $m$ on it, we define a 3d $\mathcal{N}=2$ theory $\mathcal{T}[(Q,m)]$ in such a way that the $S^3_b$ partition function of the theory coincides with the cluster partition function defined from the pair $(Q, m)$. Our formalism includes the case where 3d $\mathcal{N}=2$ theories arise from the compactification of the 6d $(2,0)$ $A_{N-1}$ theory on a large class of 3-manifolds $M$, including complements of arbitrary links in $S^3$. In this case the quiver is defined from a 2d ideal triangulation, the mutation sequence represents an element of the mapping class group, and the 3-manifold is equipped with a canonical ideal triangulation. Our partition function then coincides with that of the holomorphic part of the $SL(N)$ Chern-Simons partition function on $M$.

Spinor representation for loop quantum gravity

We perform a quantization of the loop gravity phase space purely in terms of spinorial variables, which have recently been shown to provide a direct link between spin network states and simplicial geometries. The natural Hilbert space to represent these spinors is the Bargmann space of holomorphic square-integrable functions over complex numbers. We show the unitary equivalence between the resulting generalized Bargmann space and the standard loop quantum gravity Hilbert space by explicitly constructing the unitary map. The latter maps SU(2)-holonomies, when written as a function of spinors, to their holomorphic part. We analyze the properties of this map in detail. We show that the subspace of gauge invariant states can be characterized particularly easy in this representation of loop gravity. Furthermore, this map provides a tool to efficiently calculate physical quantities since integrals over the group are exchanged for straightforward integrals over the complex plane.

Sharp estimates for functions with a pole and logarithmic singularity

We consider functions with a pole and a logarithmic singularity. We obtain sharp estimates for the Schwarzian and the Taylor coefficients of the holomorphic part of such functions. We also describe geometric properties of conformal mappings of the exterior of the unit disc with a cut that connects some boundary point with the point at infinity.

A Maass Lifting of $\Theta^3$ and Class Numbers of Real and imaginary Quadratic Fields

We give an explicit construct of a harmonic weak Maass form $F_{\Theta}$ that is a lift of $\Theta^3$, where $\Theta$ is the classical Jacobi theta function. Just as the Fourier coefficients of $\Theta^3$ are related to class numbers of imaginary quadratic fields, the Fourier coefficients of the holomorphic part of $F_{\Theta}$ are associated to class numbers of real quadratic fields.

The non-compact elliptic genus: mock or modular

We analyze various perspectives on the elliptic genus of non-compact supersymmetric coset conformal field theories with central charge larger than three. We calculate the holomorphic part of the elliptic genus via a free field description of the model, and show that it agrees with algebraic expectations. The holomorphic part of the elliptic genus is directly related to an Appell-Lerch sum and behaves anomalously under modular transformation properties. We analyze the origin of the anomaly by calculating the elliptic genus through a path integral in a coset conformal field theory. The path integral codes both the holomorphic part of the elliptic genus, and a non-holomorphic remainder that finds its origin in the continuous spectrum of the non-compact model. The remainder term can be shown to agree with a function that mathematicians introduced to parameterize the difference between mock theta functions and Jacobi forms. The holomorphic part of the elliptic genus thus has a path integral completion which renders it non-holomorphic and modular.

P -adic coupling of mock modular forms and shadows

A "mock modular form" is the holomorphic part of a harmonic Maass form f. The nonholomorphic part of f is a period integral of its "shadow," a cusp form g. A direct method for relating the coefficients of shadows and mock modular forms is not known. We solve these problems when the shadow is an integer weight newform. Our solution is p-adic, and it relies on our definition of an algebraic "regularized mock modular form." As an application, we consider the modular solution to the cubic "arithmetic-geometric mean."

Modularity and the distinct rank function

If R(ω,q) denotes Dyson’s partition rank generating function, due to work of Bringmann and Ono, it is known that for roots of unity ω≠1, R(ω,q) is the “holomorphic part” of a harmonic weak Maass form. Dating back to Ramanujan, it is also known that \(\widehat{R}(\omega,q):=R(\omega,q^{-1})\) is given by Eichler integrals and modular forms. In analogy to these results, more recently Monks and Ono have shown that modular forms arise in a natural way from G(ω,q), the generating function for ranks of partitions into distinct parts. Moreover, Monks and Ono pose the following problem: determine whether the function \(\widehat{G}(\omega,q):=G(\omega,q^{-1})\) appears naturally in the theory of modular forms. Here we answer this question of Monks and Ono, and show that \(\widehat{G}(\omega,q)\) , when combined with \(\widehat{G}(\omega^{-1},q)\) and a twisted third-order mock theta of Ramanujan, form a weight 1 modular form. We provide a more general result on the modularity of certain expressions involving basic hypergeometric series and then show that our result on \(\widehat {G}(\omega,q)\) may be deduced from this as a special case.

Dirac operator zero-modes on a torus

We study Dirac operator zero-modes on a torus for gauge background with uniform field strengths. Under the basic translations of the torus coordinates the wave functions are subject to twisted periodic conditions. In suitable torus coordinates the zero-mode wave functions can be related to holomorphic functions of the complex torus coordinates. Half of the twisted boundary conditions for the holomorphic part of the zero-mode wave function can be made periodic or anti-periodic. The remaining half is until coordinate dependent but diagonal. We completely solve the twisted boundary conditions and construct the zero-mode wave functions. The chirality and the degeneracy of the zero-modes are uniquely determined by the gauge background and are consistent with the index theorem.

On supersymmetry breaking and the Dijkgraaf-Vafa conjecture

We investigate the Dijkgraaf-Vafa proposal when supersymmetry is broken. We consider U(N) SYM with chiral adjoint matter where the coupling constants in the tree-level superpotential are promoted to chiral spurions. The holomorphic part of the low-energy glueball superpotential can still be analyzed. We compute the holomorphic supersymmetry breaking contributions using methods of the geometry underlying the N=1 effective gauge theory viewed as a Whitham system. We also study the change in the effective glueball superpotential using perturbative supergraph techniques in the presence of spurions.

The f(q) mock theta function conjecture and partition ranks

In 1944, Freeman Dyson initiated the study of ranks of integer partitions. Here we solve the classical problem of obtaining formulas for Ne(n) (resp. No(n)), the number of partitions of n with even (resp. odd) rank. Thanks to Rademacher’s celebrated formula for the partition function, this problem is equivalent to that of obtaining a formula for the coefficients of the mock theta function f(q), a problem with its own long history dating to Ramanujan’s last letter to Hardy. Little was known about this problem until Dragonette in 1952 obtained asymptotic results. In 1966, G.E. Andrews refined Dragonette’s results, and conjectured an exact formula for the coefficients of f(q). By constructing a weak Maass-Poincare series whose “holomorphic part” is q-1f(q24), we prove the Andrews-Dragonette conjecture, and as a consequence obtain the desired formulas for Ne(n) and No(n).

Analysis of a Crack in a Finite Thermopiezoelectric Plate under Heat Flux

The mechanical and electric fields in a finite thermopiezoelectric plate containing an isolated crack are formulated by applications of the Stroh’s formulism and conformal transformation. The general form of the solution is constructed consisting of a holomorphic part in terms of Laurent series of each mapping planes, and a nonholomorphic part in integral form due to the crack. The approximate solution is obtained by least square method for a rectangular plate in which supplementary functions are introduced concerning its four corners for the purpose of accelerating the convergency of the Laurent series. The coefficients of the Laurent series of the solution, both for thermal field and electro-mechanical field, are exhibited for a crack problem, and the accuracy of the approximation is explored subsequently. The stress and electric displacement (SED) intensity factors are given for varying the plate size and the crack site. For specified crack length, considerable enhancement of SED intensity factors may be attained as the plate size increases owing to the mechanical and electric fields formed under uniform heat flow.

Complex matrix model and fermion phase space for bubbling AdS geometries

We study a relation between droplet configurations in the bubbling AdS geometries and a complex matrix model that describes the dynamics of a class of chiral primary operators in dual N=4 super Yang Mills (SYM). We show rigorously that a singlet holomorphic sector of the complex matrix model is equivalent to a holomorphic part of two-dimensional free fermions, and establish an exact correspondence between the singlet holomorphic sector of the complex matrix model and one-dimensional free fermions. Based on this correspondence, we find a relation of the singlet holomorphic operators of the complex matrix model to the Wigner phase space distribution. By using this relation and the AdS/CFT duality, we give a further evidence that the droplets in the bubbling AdS geometries are identified with those in the phase space of the one-dimensional fermions. We also show that the above correspondence actually maps the operators of N=4 SYM corresponding to the (dual) giant gravitons to the droplet configurations proposed in the literature.

Solutions holomorphes locale et globale pour un opérateur différentiel linéaire à plusieurs variables Fuchsiennes

We consider linear partial differential equations with several Fuchsian variables in the sense of M.S. Baouendi and C. Goulaouic [1]. For a holomorphic Fuchsian operator with holomorphic Fuchsian principal part, we prove existence and uniqueness of a holomorphic local solution. Our theorem generalizes the results of ([3, 1, 11]), precises the one of [4] and reduces the proof of their theorems to the proof of the fixed-point theorem. For a holomorphic Fuchsian operator with constant Fuchsian principal part, we establish the existence and uniqueness of a holomorphic global solution. Our aim is to simplify its proof. The methods of proof are based on the application of the fixed-point theorem in some Banach spaces defined by majorant functions that are suitable to this kind of equations.