arXiv subjects

Tohru Eguchi

Publications and source records attributed to Tohru Eguchi.

At least 19 recordsLinked to original sources

arXiv2018

Duality in N=4 Liouville Theory and Moonshine Phenomena

We consider the ${\cal N}=4$ Liouville theory by varying the linear dilaton coupling constant $\cal{Q}$. It is known that at two different values of coupling constant ${\cal Q}=\sqrt{2\over N},-(N-1)\sqrt{2\over N}$ system exhibits two different small ${\cal N}=4$ superconformal symmetries with central charge $c=6$ and $c=6(N-1)$, respectively. In the context of string theory these two theories are considered to describe Coulumb and Higgs branch of the theory and expected to be dual to each other. We study the Mathieu and umbral moonshine phenomena in these two theories and discuss their dual description. We mainly consider the case of $A_N$ type modular invariants.

hep-th↗

arXiv2016

Professor Nambu, String Theory and Moonshine Phenomenon

I first recall the last occasion of meeting the late Professor Yoichiro Nambu in a hospital in Osaka. I then present a brief introduction to the moonshine phenomenon in string theory which is under recent investigations.

hep-th↗

arXiv2014

Compact Formulas for the Completed Mock Modular Forms

In this paper we present a new compact expression of the elliptic genus of SL(2)/U(1)-supercoset theory by making use of the `spectral flow method' of the path-integral evaluation. This new expression is written in a form like a Poincare series with a non-holomorphic Gaussian damping factor, and manifestly shows the modular and spectral flow properties of a real analytic Jacobi form. As a related problem, we present similar compact formulas for the modular completions of various mock modular forms which appear in the representation theory of N=2,4 superconformal algebras. We further discuss the generalization to the cases of arbitrary spin-structures, that is, the world-sheet fermions with twisted boundary conditions parameterized by a continuous parameter. This parameter is naturally identified with the `u-variable' in the Appell-Lerch sum.

hep-th↗

arXiv2013

Enriques moonshine

We propose a new moonshine phenomenon associated with the elliptic genus of the Enriques surface (1/2 of the elliptic genus of K3) with the symmetry group given by the Mathieu group M12.

hep-th↗

arXiv2012

N=2 Moonshine

We construct a model of moonshine phenomenon based on the use of N=2 superconformal algebra. We consider an extremal Jacobi form of weight 0 and index 2, and expand it in terms of N=2 massless and massive representations. We find the multiplicities of massive representations are decomposed into a sum of dimensions of irreducible representations of the group L2(11).

hep-th↗

arXiv2012

Twisted elliptic genus for K3 and Borcherds product

We further discuss the relation between the elliptic genus of K3 surface and the Mathieu group M24. We find that some of the twisted elliptic genera for K3 surface, defined for conjugacy classes of the Mathieu group M24, can be represented in a very simple manner in terms of the eta-product of the corresponding conjugacy classes. It is shown that our formula is a consequence of the identity between the Borcherds product and additive lift of some Siegel modular forms.

hep-th↗

arXiv2011

Non-holomorphic Modular Forms and SL(2,R)/U(1) Superconformal Field Theory

We study the torus partition function of the SL(2,R)/U(1) SUSY gauged WZW model coupled to N=2 U(1) current. Starting from the path-integral formulation of the theory, we introduce an infra-red regularization which preserves good modular properties and discuss the decomposition of the partition function in terms of the N=2 characters of discrete (BPS) and continuous (non-BPS) representations. Contrary to our naive expectation, we find a non-holomorphic dependence (dependence on \barτ) in the expansion coefficients of continuous representations. This non-holomorphicity appears in such a way that the anomalous modular behaviors of the discrete (BPS) characters are compensated by the transformation law of the non-holomorphic coefficients of the continuous (non-BPS) characters. Discrete characters together with the non-holomorphic continuous characters combine into real analytic Jacobi forms and these combinations exactly agree with the "modular completion" of discrete characters known in the theory of Mock theta functions \cite{Zwegers}. We consider this to be a general phenomenon: we expect to encounter "holomorphic anomaly" (\barτ-dependence) in string partition function on non-compact target manifolds. The anomaly occurs due to the incompatibility of holomorphy and modular invariance of the theory. Appearance of non-holomorphicity in SL(2,R)/U(1) elliptic genus has recently been observed by Troost \cite{Troost}.

hep-th↗

arXiv2010

Note on Twisted Elliptic Genus of K3 Surface

We discuss the possibility of Mathieu group M24 acting as symmetry group on the K3 elliptic genus as proposed recently by Ooguri, Tachikawa and one of the present authors. One way of testing this proposal is to derive the twisted elliptic genera for all conjugacy classes of M24 so that we can determine the unique decomposition of expansion coefficients of K3 elliptic genus into irreducible representations of M24. In this paper we obtain all the hitherto unknown twisted elliptic genera and find a strong evidence of Mathieu moonshine.

hep-th↗

arXiv2010

Notes on the K3 Surface and the Mathieu group M_24

We point out that the elliptic genus of the K3 surface has a natural decomposition in terms of dimensions of irreducible representations of the largest Mathieu group M_24. The reason is yet a mystery.

hep-th↗

arXiv2010

Seiberg-Witten theory, matrix model and AGT relation

We discuss the Penner-type matrix model which has been proposed to explain the AGT relation between the 2-dimensional Liouville theory and 4-dimensional N=2 superconformal gauge theories. In our previous communication we have obtained the spectral curve of the matrix model and showed that it agrees with that derived from M-theory. We have also discussed the decoupling limit of massive flavors and proposed new matrix models which describe Seiberg-Witten theory with flavors N_f=2,3. In this article we explicitly evaluate the free energy of these matrix models and show that they in fact reproduce the amplitudes of Seiberg-Witten theory.

hep-th↗

arXiv2010

N=2 Superconformal Algebra and the Entropy of Calabi-Yau Manifolds

We use the representation theory of N=2 superconformal algebra to study the elliptic genera of Calabi-Yau (CY) D-folds. We compute the entropy of CY manifolds from the growth rate of multiplicities of the massive (non-BPS) representations in the decomposition of their elliptic genera. We find that the entropy of CY manifolds of complex dimension D behaves differently depending on whether D is even or odd. When D is odd, CY entropy coincides with the entropy of the corresponding hyperKahler (D-3)-folds due to a structural theorem on Jacobi forms. In particular, we find that the Calabi-Yau 3-fold has a vanishing entropy. At D>3, using our previous results on hyperKahler manifolds, we find $S_{CY_D} \sim 2π\sqrt{{(D-3)^2\over 2(D-1)}n}$. When D is even, we find the behavior of CY entropy behaving as $S_{CY_D}\sim 2 π\sqrt{{D-1\over 2}n}$. These agree with Cardy's formula at large D.

hep-th↗

arXiv2010

Penner Type Matrix Model and Seiberg-Witten Theory

We discuss the Penner type matrix model recently proposed by Dijkgraaf and Vafa for a possible explanation of the relation between four-dimensional gauge theory and Liouville theory by making use of the connection of the matrix model to two-dimensional CFT. We first consider the relation of gauge couplings defined in UV and IR regimes of N_f = 4, N = 2 supersymmetric gauge theory being related as $q_{\rm UV}={\vartheta_2(q_{\rm IR})^4/\vartheta_3(q_{\rm IR})^4}$. We then use this relation to discuss the action of modular transformation on the matrix model and determine its spectral curve. We also discuss the decoupling of massive flavors from the N_f = 4 matrix model and derive matrix models describing asymptotically free N = 2 gauge theories. We find that the Penner type matrix theory reproduces correctly the standard results of N = 2 supersymmetric gauge theories.

hep-th↗

arXiv2010

N=4 Superconformal Algebra and the Entropy of HyperKahler Manifolds

We study the elliptic genera of hyperKahler manifolds using the representation theory of N=4 superconformal algebra. We consider the decomposition of the elliptic genera in terms of N=4 irreducible characters, and derive the rate of increase of the multiplicities of half-BPS representations making use of Rademacher expansion. Exponential increase of the multiplicity suggests that we can associate the notion of an entropy to the geometry of hyperKahler manifolds. In the case of symmetric products of K3 surfaces our entropy agrees with the black hole entropy of D5-D1 system.

hep-th↗

arXiv2009

Superconformal Algebras and Mock Theta Functions 2. Rademacher Expansion for K3 Surface

The elliptic genera of the K3 surfaces, both compact and non-compact cases, are studied by using the theory of mock theta functions. We decompose the elliptic genus in terms of the N=4 superconformal characters at level-1, and present an exact formula for the coefficients of the massive (non-BPS) representations using the Poincare-Maass series.

math-ph↗

arXiv2009

Superconformal Algebras and Mock Theta Functions

It is known that characters of BPS representations of extended superconformal algebras do not have good modular properties due to extra singular vectors coming from the BPS condition. In order to improve their modular properties we apply the method of Zwegers which has recently been developed to analyze modular properties of mock theta functions. We consider the case of N=4 superconformal algebra at general levels and obtain the decomposition of characters of BPS representations into a sum of simple Jacobi forms and an infinite series of non-BPS representations. We apply our method to study elliptic genera of hyper-Kahler manifolds in higher dimensions. In particular we determine the elliptic genera in the case of complex 4 dimensions of the Hilbert scheme of points on K3 surfaces K^{[2]} and complex tori A^{[[3]]}.

math-ph↗

arXiv2008

Modular Forms and Elliptic Genera for ALE Spaces

When we describe string propagation on non-compact or singular Calabi-Yau manifolds by CFT, continuous as well as discrete representations appear in the theory. These representations mix in an intricate way under the modular transformations. In this article, we propose a method of combining discrete and continuous representations so that the resulting combinations have a simpler modular behavior and can be used as conformal blocks of the theory. We compute elliptic genera of ALE spaces and obtain results which agree with those suggested from the decompactification of K3 surface. Consistency of our approach is assured by some remarkable identity of theta functions. We include in the appendix some new materials on the representation theory of ${\cal N}=4$ superconformal algebra.

hep-th↗

arXiv2007

Rigid Limit in N=2 Supergravity and Weak-Gravity Conjecture

We analyze the coupled N=2 supergravity and Yang-Mills system using holomorphy, near the rigid limit where the former decouples from the latter. We find that there appears generically a new mass scale around g M_{pl} where g is the gauge coupling constant and M_{pl} is the Planck scale. This is in accord with the weak-gravity conjecture proposed recently. We also study the scale dependence of the gauge theory prepotential from its embedding into supergravity.

hep-th↗

arXiv2007

Liouville Field, Modular Forms and Elliptic Genera

When we describe non-compact or singular Calabi-Yau manifolds by CFT, continuous as well as discrete representations appear in the theory. These representations mix in an intricate way under the modular transformations. In this article, we propose a method of combining discrete and continuous representations so that the resulting combinations have a simpler modular behavior and can be used as conformal blocks of the theory. We compute elliptic genera of ALE spaces and obtain results which agree with those suggested from the decompactification of K3 surface. Consistency of our approach is assured by some remarkable identity of theta functions whose proof, by D. Zagier, is included in an appendix.

hep-th↗