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.Comment: 13 page

ZZ-Branes of N=2 Super-Liouville Theory

We study conformal boundary conditions and corresponding one-point functions of the N=2 super-Liouville theory using both conformal and modular bootstrap methods. We have found both continuous (`FZZT-branes') and discrete (`ZZ-branes') boundary conditions. In particular, we identify two different types of the discrete ZZ-brane solutions, which are associated with degenerate fields of the N=2 super-Liouville theory.Comment: 26 page

Free Field Representations and Screening Operators for the N=4N=4 Doubly Extended Superconformal Algebras

We present explicit free field representations for the $N=4$ doubly extended superconformal algebra, $\tilde{\cal{A}}_{\gamma}$. This algebra generalizes and contains all previous $N=4$ superconformal algebras. We have found $\tilde{\cal{A}}_{\gamma}$ to be obtained by hamiltonian reduction of the Lie superalgebra $D(2|1;\alpha)$. In addition, screening operators are explicitly given and the associated singular vectors identified. We use this to present a natural conjecture for the Kac determinant generalizing a previous conjecture by Kent and Riggs for the singly extended case. The results support and illuminate several aspects of the characters of this algebra previously obtained by Taormina and one of us.Comment: 15 pages, Late

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 \bar{\tau}) 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" (\bar{\tau}-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}.Comment: 39+1 pages, no figure; v2 a reference added, some points are clarified, typos corrected, version to appear in JHE

Extended SL(2,R)/U(1) characters, or modular properties of a simple non-rational conformal field theory

We define extended SL(2,R)/U(1) characters which include a sum over winding sectors. By embedding these characters into similarly extended characters of N=2 algebras, we show that they have nice modular transformation properties. We calculate the modular matrices of this simple but non-trivial non-rational conformal field theory explicitly . As a result, we show that discrete SL(2,R) representations mix with continuous SL(2,R) representations under modular transformations in the coset conformal field theory. We comment upon the significance of our results for a general theory of non-rational conformal field theories.Comment: JHEP style, 25 pages, 2 figures, v2: minor corrections, reference added, version to appear in JHE

Crosscap states in N=2 Liouville theory

We construct crosscap states in the N = 2 Liouville theory from the modular bootstrap method. We verify our results by comparing it with the calculation from the minisuperspace approximation and by checking the consistency with the conformal bootstrap equation. Various overlaps with other known branes are studied. We further discuss the topological nature of the discrete terms in the crosscap wavefunction and their connection with the Landau-Ginzburg approach in a nontrivial dilaton background. We find that it can be mapped to the Landau-Ginzburg theory with a negative power superpotential by a simple change of variables, extending the known duality to the open string sector. Possible applications to the two-dimensional noncritical string theories and supersymmetric orientifolds in the higher dimension are also discussed.Comment: 35pages, v2:references added, typos corrected, v3:consistency check with the conformal bootstrap is added, v4:typos corrected published version in NP

Prospective association of soft drink consumption with depressive symptoms.

OBJECTIVE: Consumption of soft drinks has become a serious public health issue worldwide. However, prospective evidence is limited regarding the relationship between soft drink consumption and depression, especially in Asia. The aim of this study was to investigate the prospective association between soft drink consumption and the development of depressive symptoms. METHODS: We evaluated an occupational cohort of 935 adults in Japan (2012-2016), who were free from depressive symptoms at baseline and attended a 3-y follow-up assessment. Soft drink consumption was assessed using a self-administered diet history questionnaire. Depressive symptoms were assessed using the Center for Epidemiologic Studies Depression scale. Odds ratios (ORs) and 95% confidence intervals (CIs) were estimated from multivariate logistic regression analysis controlling for sociodemographic, lifestyle, dietary, and occupational covariates. RESULTS: Over the 3-y study period, 16.9% (158 cases) of the study participants reported depressive symptoms. Higher soft drink consumption was associated with higher odds of depressive symptoms. The multivariable-adjusted OR was 1.91 (95% CI, 1.11-3.29; Ptrend = 0.015) when comparing soft drink consumption of ≥4 cups/wk with consumption of <1 cup/wk. CONCLUSION: The present results suggested that greater consumption of soft drinks would increase the likelihood of exhibiting depressive symptoms.This study was supported by JSPS KAKENHI Grant Numbers 25293146, 25702006, Practical Research Project for Life-Style related Diseases including Cardiovascular Diseases and Diabetes Mellitus (15ek0210021h0002) from the Japan Agency for Medical Research and Development, and the Industrial Health Foundation. F.I. was funded by the Medical Research Council Epidemiology Unit, United Kingdom (MC_UU_12015/5)

Common Peak Approach Using Mass Spectrometry Data Sets for Predicting the Effects of Anticancer Drugs on Breast Cancer

We propose a method for biomarker discovery from mass spectrometry data, improving the common peak approach developed by Fushiki et al. (BMC Bioinformatics, 7:358, 2006). The common peak method is a simple way to select the sensible peaks that are shared with many subjects among all detected peaks by combining a standard spectrum alignment and kernel density estimates. The key idea of our proposed method is to apply the common peak approach to each class label separately. Hence, the proposed method gains more informative peaks for predicting class labels, while minor peaks associated with specific subjects are deleted correctly. We used a SELDI-TOF MS data set from laser microdissected cancer tissues for predicting the treatment effects of neoadjuvant therapy using an anticancer drug on breast cancer patients. The AdaBoost algorithm is adopted for pattern recognition, based on the set of candidate peaks selected by the proposed method. The analysis gives good performance in the sense of test errors for classifying the class labels for a given feature vector of selected peak values

SL(2,R)/U(1) Supercoset and Elliptic Genera of Non-compact Calabi-Yau Manifolds

We first discuss the relationship between the SL(2;R)/U(1) supercoset and N=2 Liouville theory and make a precise correspondence between their representations. We shall show that the discrete unitary representations of SL(2;R)/U(1) theory correspond exactly to those massless representations of N=2 Liouville theory which are closed under modular transformations and studied in our previous work hep-th/0311141. It is known that toroidal partition functions of SL(2;R)/U(1) theory (2D Black Hole) contain two parts, continuous and discrete representations. The contribution of continuous representations is proportional to the space-time volume and is divergent in the infinite-volume limit while the part of discrete representations is volume-independent. In order to see clearly the contribution of discrete representations we consider elliptic genus which projects out the contributions of continuous representations: making use of the SL(2;R)/U(1), we compute elliptic genera for various non-compact space-times such as the conifold, ALE spaces, Calabi-Yau 3-folds with A_n singularities etc. We find that these elliptic genera in general have a complex modular property and are not Jacobi forms as opposed to the cases of compact Calabi-Yau manifolds.Comment: 39 pages, no figure; v2 references added, minor corrections; v3 typos corrected, to appear in JHEP; v4 typos corrected in eqs. (3.22) and (3.44