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Taizan Watari

Publications and source records attributed to Taizan Watari.

At least 19 recordsLinked to original sources
arXiv2026

Notes on Gepner Construction

In this note, we describe the Gepner construction purely as orbifold (rather than a beta-prescription or orbifold-like process). This is done by distinguishing orbifolding by a chiral symmetry from that by a left-right diagonal symmetry, and also by exploiting an extra degree of freedom available in orbifolds of Type 0 SCFTs. We also describe by using the Arf theory how the extra freedom in Type 0 SCFT orbifolds in general (not just in the Gepner construction) should be implemented in higher genus amplitudes. This note also determines when a combinatorial data set of a Gepner construction has a mirror Gepner construction, and also provides two thinking frameworks to determine the SCFT vertex operators of the Gepner model SCFTs with phase ambiguity and branch cuts under control.

hep-th↗
arXiv2024

Notes on Characterizations of 2d Rational SCFTs: Algebraicity, Mirror Symmetry and Complex Multiplication

These notes combine results from two papers by the present authors viz., Part I (arXiv:2205.10299) and Part II (arXiv:2212.13028) into one streamlined version for better readability, along with a review on theory of complex multiplication for non-singular complex projective varieties and complex tori that is aimed at string theorists. We think that it is worth posting this edition as a separate entry in arXiv for those reasons, although this edition contains no essential progress beyond Part I and Part II. S. Gukov and C. Vafa proposed a characterization of rational N=(1,1) superconformal field theories (SCFTs) on 1+1 dimensions with Ricci-flat Kahler target spaces in terms of the Hodge structure of the target space, extending an earlier observation by G. Moore. We refined this idea and obtained a conjectural statement on necessary and sufficient conditions for such SCFTs to be rational, which we indeed prove to be true in the case the target space is T^4. In the refined statement, the algebraicity of the geometric data of the target space turns out to be essential, and the Strominger--Yau--Zaslow fibration in the mirror correspondence also plays a vital role.

hep-th↗
arXiv2023

A note on varieties of weak CM-type

CM-type projective varieties X of complex dimension n are characterized by their CM-type rational Hodge structures on the cohomology groups. One may impose such a condition in a weakest form when the canonical bundle of X is trivial; the rational Hodge structure on the level-n subspace of $H^n(X;Q)$ is required to be of CM-type. This brief note addresses the question whether this weak condition implies that the Hodge structure on the entire $H^\ast(X;Q)$ is of CM-type. We study in particular abelian varieties when the dimension of the level-n subspace is two or four, and K3 $\times T^2$. It turns out that the answer is affirmative. Moreover, such an abelian variety is always isogenous to a product of CM-type elliptic curves or abelian surfaces. This extends a result of Shioda and Mitani in 1974.

math.AG↗
arXiv2022

Towards Hodge Theoretic Characterizations of 2d Rational SCFTs: II

A characterization of rational superconformal field theories (SCFTs) on 1+1 dimensions with Ricci-flat Kahler targets was proposed by S. Gukov and C. Vafa in terms of the Hodge structure of the target space. The article [arXiv:2205.10299] refined this idea and extracted a set of necessary conditions for a $T^4$-target N=(1,1) SCFT to be rational; only a partial effort was made, however, to study whether it also constitutes a sufficient condition. It turns out that the set of conditions in [arXiv:2205.10299] is not sufficient, and that it becomes a set of necessary and sufficient conditions by adding one more condition in the case of $T^4$. The Strominger--Yau--Zaslow fibration in the mirror correspondence plays an essential role there. At the end, we also propose a refined version of Gukov--Vafa's idea for general Ricci-flat Kahler target spaces.

hep-th↗
arXiv2022

Direct computation of period polynomials and classification of K3-fibred Calabi--Yau threefolds

One can assign to four-dimensional N=2 supersymmetric Heterotic string vacua a set of classification invariants including a lattice $\Lambda_S$ and vector-valued modular forms. Some of the classification invariants are constrained by the condition that the Coulomb branch monodromy matrices should be integer-valued. We computed numerically the period polynomials of meromorphic cusp forms for some rank-1 $\Lambda_S$; we then computed the monodromy matrices and extracted general patterns of the constraints on the invariants. The constraints we got imply that a large fraction of the Heterotic string vacua we studied satisfy the necessary conditions for a non-linear sigma model interpretation in the dual Type IIA description. Our computation can also be used to identify diffeomorphism classes of real six-dimensional manifolds that cannot be realized by K3-fibred Calabi--Yau threefolds.

hep-th↗
arXiv2022

Towards Hodge Theoretic Characterizations of 2d Rational SCFTs

The study of rational conformal field theories in the moduli space is of particular interest since these theories correspond to points in moduli space where the algebraic and arithmetic structure are usually richer, while also being points where non--trivial physics occurs (such as in the study of attractor black holes and BPS states at rational points). This has led to various attempts to characterize and classify such rational points. In this paper, a conjectured characterization by Gukov--Vafa of rational conformal field theories whose target space is a Ricci flat K\"ahler manifold is analyzed carefully for the case of toroidal compactifications. We refine the conjectured statement as well as making an effort to verify it, using $T^4$ compactification as a test case. Seven common properties in terms of Hodge theory (including complex multiplication) have been identified for $T^4$-target rational conformal field theories. By imposing three properties out of the seven, however, there remain $\mathcal N = (1,1)$ SCFTs that are not rational. Open questions, implications and future lines of work are discussed.

hep-th↗
arXiv2021

Direct Computation of Monodromy Matrices and Classification of 4d N=2 Heterotic--IIA Dual Vacua

We compute the monodromy matrices on the special geometry of 4d N=2 Heterotic--IIA dual vacua in some simple cases by numerical evaluation of the period integrals, without assuming geometric background. The integrality of the monodromy matrices constrains some classification invariants of the string vacua. We also mention some mathematical open problems on period polynomials for modular form with poles.

hep-th↗
arXiv2020

W=0 Complex Structure Moduli Stabilization on CM-type K3 x K3 Orbifolds:---Arithmetic, Geometry and Particle Physics---

It is an important question in string compactification whether complex structure moduli stabilization inevitably ends up with a vacuum expectation value of the superpotential < W > of the order of the Planck scale cubed. Any thoughts on volume stabilization and inflation in string theory, as well as on phenomenology of supersymmetric Standard Models, will be affected by the answer to this question. In this work, we follow an idea for making < W > $\simeq$ 0 where the internal manifold has a vacuum complex structure with arithmetic characterization, and address Calabi--Yau fourfold compactification of F-theory. The moduli space of K3 x K3 orbifolds contain infinitely many such vacua. Arithmetic conditions for a < W > =0 flux are worked out, and then all the K3 moduli have supersymmetric mass. Possible gauge groups, matter representations and discrete symmetries are studied for the case of Z_2-orbifolds.

hep-th↗
arXiv2020

Witten Anomaly in 4d Heterotic Compactificaitons with N = 2 Supersymmetry

We showed that there is no SU(2) Witten anomaly in a large class of 4d N = 2 supersymmetric Heterotic string compactifications. The consistency conditions we consider are the modularity of the new supersymmetric index, the integrality of BPS indices, and the discrete Peccei-Quinn shift symmetries. We also found an example where these conditions are not sufficient to show that the theory is anomaly free. This suggests that there are more conditions in the worldsheet SCFT essential for consistent string compactifications.

hep-th↗
arXiv2019

Modular Parametrization as Polyakov Path Integral: Cases with CM Elliptic Curves as Target Spaces

For an elliptic curve E over an abelian extension k/K with CM by K of Shimura type, the L-functions of its [k:K] Galois representations are Mellin transforms of Hecke theta functions; a modular parametrization (surjective map) from a modular curve to E pulls back the 1-forms on E to give the Hecke theta functions. This article refines the study of our earlier work and shows that certain class of chiral correlation functions in Type II string theory with [E]_C (E as real analytic manifold) as a target space yield the same Hecke theta functions as objects on the modular curve. The Kahler parameter of the target space [E]_C in string theory plays the role of the index (partially ordered) set in defining the projective/direct limit of modular curves.

hep-th↗
arXiv2019

Modular Forms as Classification Invariants of 4D N=2 Heterotic--IIA Dual Vacua

We focus on 4D $\mathcal{N}=2$ string vacua described both by perturbative Heterotic theory and by Type IIA theory; a Calabi--Yau three-fold $X_{\rm IIA}$ in the Type IIA language is further assumed to have a regular K3-fibration. It is well-known that one can assign a modular form $Φ$ to such a vacuum by counting perturbative BPS states in Heterotic theory or collecting Noether--Lefschetz numbers associated with the K3-fibration of $X_{\mathrm{IIA}}$. In this article, we expand the observations and ideas (using gauge threshold correction) in the literature and formulate a modular form $Ψ$ with full generality for the class of vacua above, which can be used along with $Φ$ for the purpose of classification of those vacua. Topological invariants of $X_{\mathrm{IIA}}$ can be extracted from $Φ$ and $Ψ$, and even a pair of diffeomorphic Calabi--Yau's with different Kähler cones may be distinguished by introducing the notion of "the set of $Ψ$'s for Higgs cascades/for curve classes". We illustrated these ideas by simple examples.

hep-th↗
arXiv2018

Vector-Like Pairs and Brill--Noether Theory

How likely is it that there are particles in a vector-like pair of representations in low-energy spectrum, when neither symmetry nor anomaly consideration motivates their presence? We address this question in the context of supersymmetric and geometric phase compactification of F-theory and Heterotic dual. Quantisation of the number of generations (or net chiralities in more general term) is also discussed along the way. Self-dual nature of the fourth cohomology of Calabi--Yau fourfolds is essential for the latter issue, while we employ Brill--Noether theory to set upper bounds on the number $\ell$ of vector-like pairs of chiral multiplets in the SU(5) 5+5bar representations. For typical topological choices of geometry for F-theory compactification for SU(5) unification, the range of $0 \leq \ell \lesssim 4$ for perturbative unification is not in immediate conflict with what is already understood about F-theory compactification at this moment.

hep-th↗
arXiv2018

String-theory Realization of Modular Forms for Elliptic Curves with Complex Multiplication

It is known that the L-function of an elliptic curve defined over Q is given by the Mellin transform of a modular form of weight 2. Does that modular form have anything to do with string theory? In this article, we address a question along this line for elliptic curves that have complex multiplication defined over number fields. So long as we use diagonal rational N=(2,2) superconformal field theories for the string-theory realizations of the elliptic curves, the weight-2 modular form turns out to be the Boltzmann-weighted (q^{L_0-c/24}-weighted) sum of U(1) charges with F e^{ πi F} insertion computed in the Ramond sector.

hep-th↗
arXiv2017

Pure Natural Inflation

We point out that a simple inflationary model in which the axionic inflaton couples to a pure Yang-Mills theory may give the scalar spectral index (n_s) and tensor-to-scalar ratio (r) in complete agreement with the current observational data.

hep-ph↗
arXiv2017

Revisiting arithmetic solutions to the $W=0$ condition

The gravitino mass is expected not to be much smaller than the Planck scale for a large fraction of vacua in flux compactifications. There is no continuous parameter to tune even by hand, and it seems that the gravitino mass can be small only as a result of accidental cancellation among period integrals weighted by integer-valued flux quanta. DeWolfe et.al. (2005) proposed to pay close attention to vacua where the Hodge decomposition is possible within a number field, so that the precise cancellation takes place as a result of algebra. We focus on a subclass of those vacua---those with complex multiplications---and explore more on the idea in this article. It turns out, in Type IIB compactifications, that those vacua admit non-trivial supersymmetric flux configurations if and only if the reflex field of the Weil intermediate Jacobian is isomorphic to the quadratic imaginary field generated by the axidilaton vacuum expectation value. We also found that flux statistics is highly enriched on such vacua, as F-term conditions become linearly dependent.

hep-th↗
arXiv2016

Heterotic-Type IIA Duality and Degenerations of K3 Surfaces

We study the duality between four-dimensional N=2 compactifications of heterotic and type IIA string theories. Via adiabatic fibration of the duality in six dimensions, type IIA string theory compactified on a K3-fibred Calabi-Yau threefold has a potential heterotic dual compactification. This adiabatic picture fails whenever the K3 fibre degenerates into multiple components over points in the base of the fibration. Guided by monodromy, we identify such degenerate K3 fibres as solitons generalizing the NS5-brane in heterotic string theory. The theory of degenerations of K3 surfaces can then be used to find which solitons can be present on the heterotic side. Similar to small instanton transitions, these solitons escort singular transitions between different Calabi-Yau threefolds. Starting from well-known examples of heterotic--type IIA duality, such transitions can take us to type IIA compactifications with unknown heterotic duals.

hep-th↗
arXiv2015

Statistics of Flux Vacua for Particle Physics

Supersymmetric flux compactification of F-theory in the geometric phase yields numerous vacua, and provides an ensemble of low-energy effective theories with different symmetry, matter multiplicity and Lagrangian parameters. Theoretical tools have already been developed so that we can study how the statistics of flux vacua depend on the choice of symmetry and some of Lagrangian parameters. In this article, we estimate the fraction of i) vacua that have a U(1) symmetry for spontaneous R-parity violation, and ii) those that realise ideas which achieve hierarchical eigenvalues of the Yukawa matrices. We also learn a lesson that the number of flux vacua is reduced very much when the unbroken $U(1)_Y$ symmetry is obtained from a non-trivial Mordell--Weil group, while it is not when $U(1)_Y$ is in SU(5) unification. It also turns out that vacua with an approximate U(1) symmetry forms a locus of accumulation points of the flux vacua distribution.

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