Modular Covariance Research Articles

Warped CFT duals of the Plebański-Demiański family of solutions

In this paper, we analyze the symmetry properties of the complete family of type D spacetimes generalized form the Plebański-Demiański solution in four dimensions holographically in terms of a warped CFT. The generalized Plebański-Demiański solutions are black hole-like spacetimes characterized by seven physical parameters. Most of the black holes in four dimensions are included within this family. Generically consider a solution with horizon in this family, we figure out the possible warped conformal symmetry attached to the horizon. The horizon can be either extremal or non-extremal. In the extremal case, the near horizon region can be mapped to an infinite spacetime with geometry given by a warped and twist product of AdS2 and S2. The new boundary conditions for AdS2 as well as their higher dimensional uplifts are applied here to manifest the asymptotic symmetry as the warped conformal symmetry. In the non-extremal case, the global warped conformal symmetry is singled out by analyzing the scalar wave equation with constant frequency. The local warped conformal symmetries are represented by the charge algebra associated to the vector fields which preserve the scalar wave equation as well as its frequency. In defining the variation of the covariant charges, a proper counterterm is introduced for consistency conditions which is supposed to be suitable for all the solutions within the family. As a consistency check, the horizon entropy is reproduced by the entropy formula of the warped CFT by using its modular covariance and the central terms derived in the bulk spacetimes.

Comparative morphology in the context of facial reduction: Modularity in primate, dog, and bat crania.

Biological variation in the mammalian skull is the product of a series of factors including changes in gene expression, developmental timing, and environmental pressures. When considering the diversity of extant mammalian crania, it is important to understand these mechanisms that contribute to cranial growth and in turn, how differences in cranial morphology have been attained. Various researchers, including Dr. Sue Herring, have proposed a variety of mechanisms to explain the process of cranial growth. This work has set the foundation on which modern analysis of craniofacial morphology happens today. This study focused on the analysis of modularity in three mammalian taxa, all of which exhibit facial reduction. Specifically, we examined facial reduction as a morphological phenomenon through the use of two-module and six-module modularity hypotheses. We recorded three-dimensional coordinate data for 55 cranial landmarks that allowed us to analyze differences in cranial shape in these three taxa (primates n = 88, bats n = 64, dogs n = 81). When assessing modularity within the two-module modularity hypothesis specifically, dogs exhibited the lowest levels of modularity, while bats and primates both showed a slightly more modular covariance structure. We further assessed modularity in the same sample using the Goswami six-module model, where again dogs exhibited a low degree of modularity, with bats and primates being more moderate. We then broke the sample into subsets by analyzing each morphotype separately. We hypothesized that the modularity would be more pronounced in the brachycephalic morphotype. Surprisingly, we found that in brachycephalic dogs, normocephalic dogs, brachycephalic primates, and normocephalic primates, there was a moderate degree of modularity. Brachycephalic bats had a low degree of modularity, while normocephalic bats were the most modular group observed in this study. Based on these results, it is evident that facial reduction is a complex and multifaceted phenomenon with unique morphological changes observed in each of the three taxa studied.

Exploring duality symmetries, multicriticality and RG flows at c = 2

In this work, we study the realization of non-invertible duality symmetries along the toroidal branch of the c = 2 conformal manifold. A systematic procedure to construct symmetry defects is implemented to show that all Rational Conformal Field Theories along this branch enjoy duality symmetries. Furthermore, we delve into an in-depth analysis of two representative cases of multicritical theories, where the toroidal branch meets various orbifold branches. For these particular examples, the categorical data and the defect Hilbert spaces associated with the duality symmetries are obtained by resorting to modular covariance. Finally, we study the interplay between these novel symmetries and the various exactly marginal and relevant deformations, including some representative examples of Renormalization Group flows where the infrared is constrained by the non-invertible symmetries and their anomalies.

BMS modular covariance and structure constants

Two-dimensional (2d) field theories invariant under the Bondi-Metzner-Sachs algebra, or 2d BMSFTs in short, are putative holographic duals of Einstein gravity in 3d asymptotically flat spacetimes. When defined on a torus, these field theories come equipped with a modified modular structure. We use the modular covariance of the BMS torus two-point function to develop formulae for different three-point structure constants of the field theory. These structure constants indicate that BMSFTs follow the eigenstate thermalization hypothesis, albeit with some interesting changes to usual 2d CFTs. The singularity structures of the structure constants contain information on perturbations of cosmological horizons in 3d asymptotically flat spacetimes, which we show can also be obtained as a limit of BTZ quasinormal modes.

The Task-Dependent Modular Covariance Networks Unveiled by Multiple-Way Fusion-Based Analysis.

Cognitive processes induced by the specific task are underpinned by intrinsic anatomical structures with functional neural activation patterns. However, current covariance network analysis still pays much attention to brain morphologies or baseline activity due to the lack of an effective method for capturing the structural-functional covarying during tasks. Here, a multimodal covariance network (MCN) construction method was proposed to identify inter-regional covariations of the structural skeleton and functional activities by simultaneous magnetic resonance imaging and electroencephalogram (EEG). Results from two independent cohorts confirmed that MCNs could capture cognition-specific hierarchical modules in joint comprehensive multimodal features well, especially when time-resolved EEG was further integrated. The quantitative evaluation further demonstrates significantly larger modularity of MCN integrating fine-grained features from EEG. The application to the discovery cohort identified prominent modular covarying across the default mode and salience networks at rest, while the visual oddball task was accomplished by synchronous structural-functional cooperation within networks associated with attention control and working memory updating. Strikingly, the results of an external validation cohort showed a different covariant pattern corresponding to decision-specific cognitive modules. Overall, the results suggested that multimodal covariance analysis provides a reliable definition of multistate neural cognitive networks, further discloses modular-specific structural and functional co-variation.

Modular covariants of cyclic groups of order p

Let G be a cyclic group of order p and let V,W be kG-modules. We study the modules of covariants k[V,W]G=(S(V⁎)⊗W)G. Recall that G has exactly p inequivalent indecomposable kG-modules, denoted Vn (n=1,…,p) and Vn has dimension n. For any n, we show that k[V2,Vn]G is a free k[V2]G-module (recovering a result of Broer and Chuai [1]) and we give an explicit set of covariants generating k[V2,Vn]G freely over k[V2]G. For any n, we show that k[V3,Vn]G is a Cohen-Macaulay k[V3]G-module (again recovering a result of Broer and Chuai) and we give an explicit set of covariants which generate k[V3,Vn]G freely over a homogeneous system of parameters for k[V3]G. We also use our results to compute a minimal generating set for the transfer ideal of k[V3]G over a homogeneous system of parameters.

Notes on massless scalar field partition functions, modular invariance and Eisenstein series

The partition function of a massless scalar field on a Euclidean spacetime manifold ℝd−1 × \U0001d54b2 and with momentum operator in the compact spatial dimension coupled through a purely imaginary chemical potential is computed. It is modular covariant and admits a simple expression in terms of a real analytic SL(2, ℤ) Eisenstein series with s = (d + 1)/2. Different techniques for computing the partition function illustrate complementary aspects of the Eisenstein series: the functional approach gives its series representation, the operator approach yields its Fourier series, while the proper time/heat kernel/world-line approach shows that it is the Mellin transform of a Riemann theta function. High/low temperature duality is generalized to the case of a non-vanishing chemical potential. By clarifying the dependence of the partition function on the geometry of the torus, we discuss how modular covariance is a consequence of full SL(2, ℤ) invariance. When the spacetime manifold is ℝp × \U0001d54bq+1, the partition function is given in terms of a SL(q + 1, ℤ) Eisenstein series again with s = (d + 1)/2. In this case, we obtain the high/low temperature duality through a suitably adapted dual parametrization of the lattice defining the torus. On \U0001d54bd+1, the computation is more subtle. An additional divergence leads to an harmonic anomaly.

Bardeen-Cooper-Schrieffer pairing of composite fermions

Topological pairing of composite fermions has led to remarkable ideas, such as excitations obeying non-Abelian braid statistics and topological quantum computation. We construct a $p$-wave paired Bardeen-Cooper-Schrieffer (BCS) wave function for composite fermions in the torus geometry, which is a convenient geometry for formulating momentum space pairing as well as for revealing the underlying composite-fermion Fermi sea. Following the standard BCS approach, we minimize the Coulomb interaction energy at half filling in the lowest and the second Landau levels, which correspond to filling factors $\nu=1/2$ and $\nu=5/2$ in GaAs quantum wells, by optimizing two variational parameters that are analogous to the gap and the Debye cut-off energy of the BCS theory. Our results show no evidence for pairing at $\nu=1/2$ but a clear evidence for pairing at $\nu=5/2$. To a good approximation, the highest overlap between the exact Coulomb ground state at $\nu=5/2$ and the BCS state is obtained for parameters that minimize the energy of the latter, thereby providing support for the physics of composite-fermion pairing as the mechanism for the $5/2$ fractional quantum Hall effect. We discuss the issue of modular covariance of the composite-fermion BCS wave function, and calculate its Hall viscosity and pair correlation function. By similar methods, we look for but do not find an instability to $s$-wave pairing for a spin-singlet composite-fermion Fermi sea at half-filled lowest Landau level in a system where the Zeeman splitting has been set to zero.

Three dimensional pure gravity and generalized Hecke operators

In this paper, we study mathematical functions of relevance to pure gravity in AdS3. Modular covariance places stringent constraints on the space of such functions; modular invariance places even stronger constraints on how they may be combined into physically viable candidate partition functions. We explicitly detail the list of holomorphic and anti-holomorphic functions that serve as candidates for chiral and anti-chiral partition functions and note that modular covariance is only consistent with such functions when the left (resp. right) central charge is an integer multiple of 8, c ∈ 8ℕ. We then find related constraints on the symmetry group of the corresponding topological, Chern-Simons, theory in the bulk of AdS. The symmetry group of the theory can be one of two choices: either SO(2; 1) × SO(2; 1) or its three-fold diagonal cover. We introduce the generalized Hecke operators which map the modular covariant functions to the modular covariant functions. With these mathematical results, we obtain conjectural partition functions for extremal CFT2s, and the corresponding microcanonical entropies, when the chiral central charges are multiples of eight. Finally, we compute subleading corrections to the Beckenstein-Hawking entropy in the bulk gravitational theory with these conjectural partition functions.

Universal dynamics of heavy operators in CFT2

We obtain an asymptotic formula for the average value of the operator product expansion coefficients of any unitary, compact two dimensional CFT with c > 1. This formula is valid when one or more of the operators has large dimension or — in the presence of a twist gap — has large spin. Our formula is universal in the sense that it depends only on the central charge and not on any other details of the theory. This result unifies all previous asymptotic formulas for CFT2 structure constants, including those derived from crossing symmetry of four point functions, modular covariance of torus correlation functions, and higher genus modular invariance. We determine this formula at finite central charge by deriving crossing kernels for higher genus crossing equations, which give analytic control over the structure constants even in the absence of exact knowledge of the conformal blocks. The higher genus modular kernels are obtained by sewing together the elementary kernels for four-point crossing and modular transforms of torus one-point functions. Our asymptotic formula is related to the DOZZ formula for the structure constants of Liouville theory, and makes precise the sense in which Liouville theory governs the universal dynamics of heavy operators in any CFT. The large central charge limit provides a link with 3D gravity, where the averaging over heavy states corresponds to a coarse-graining over black hole microstates in holographic theories. Our formula also provides an improved understanding of the Eigenstate Thermalization Hypothesis (ETH) in CFT2, and suggests that ETH can be generalized to other kinematic regimes in two dimensional CFTs.

Degree bounds for modular covariants

Abstract Let V , W {V,W} be representations of a cyclic group G of prime order p over a field 𝕜 {\Bbbk} of characteristic p. The module of covariants 𝕜 ⁢ [ V , W ] G {\Bbbk[V,W]^{G}} is the set of G-equivariant polynomial maps V → W {V\rightarrow W} , and is a module over 𝕜 ⁢ [ V ] G {\Bbbk[V]^{G}} . We give a formula for the Noether bound β ⁢ ( 𝕜 ⁢ [ V , W ] G , 𝕜 ⁢ [ V ] G ) {\beta(\Bbbk[V,W]^{G},\Bbbk[V]^{G})} , i.e. the minimal degree d such that 𝕜 ⁢ [ V , W ] G {\Bbbk[V,W]^{G}} is generated over 𝕜 ⁢ [ V ] G {\Bbbk[V]^{G}} by elements of degree at most d.

Structure constants from modularity in warped CFT

We derive a universal formula for the asymptotic growth of the mean value of three-point coefficient for Warped Conformal Field Theories (WCFTs), and provide a holographic calculation in BTZ black holes. WCFTs are two dimensional quantum field theories featuring a chiral Virasoro and U(1) Kac-Moody algebra, and are conjectured to be holographically dual to quantum gravity on asymptotically AdS3 spacetime with Compère- Song-Strominger boundary conditions. The WCFT calculation amounts to the calculation of one-point functions on torus, whose high temperature limit can be approximated by using modular covariance of WCFT, similar to the derivation of Cardy formula. The bulk process is given by a tadpole diagram, with a massive spinning particle propagates from the infinity to the horizon, and splits into particle and antiparticle which annihilate after going around the horizon of BTZ black holes. The agreement between the bulk and WCFT calculations indicates that the black hole geometries in asymptotically AdS3 spacetimes can emerge upon coarse-graining over microstates in WCFTs, similar to the results of Kraus and Maloney in the context of AdS/CFT [1].

Scale and Möbius Covariance in Two-Dimensional Haag–Kastler Net

Given a two-dimensional Haag–Kastler net which is Poincaré-dilation covariant with additional properties, we prove that it can be extended to a Möbius covariant net. Additional properties are either a certain condition on modular covariance, or a variant of strong additivity. The proof relies neither on the existence of stress-energy tensor nor any assumption on scaling dimensions. We exhibit some examples of Poincaré-dilation covariant net which cannot be extended to a Möbius covariant net, and discuss the obstructions.

Modular covariance properties of composite fermions on the torus

In this work we show that the composite fermion construction for the torus geometry is modular covariant. We show that this is the case both before and after projection, and that modular covariance properties are preserved under both exact projection and under JK projection which was recently introduced by Pu, Wu, and Jain (PRB 96, 195302 (2017)). It is crucial for the modular properties to hold that the CF state is a proper state, i.e. that there are no holes in the occupied $\Lambda$-levels.

Modular covariance and uniqueness of Joverline{T} deformed CFTs

We study families of two dimensional quantum field theories, labeled by a dimensionful parameter μ, that contain a holomorphic conserved U(1) current J(z). We assume that these theories can be consistently defined on a torus, so their partition sum, with a chemical potential for the charge that couples to J, is modular covariant. We further require that in these theories, the energy of a state at finite μ is a function only of μ, and of the energy, momentum and charge of the corresponding state at μ = 0, where the theory becomes conformal. We show that under these conditions, the torus partition sum of the theory at μ = 0 uniquely determines the partition sum (and thus the spectrum) of the perturbed theory, to all orders in μ, to be that of a mu Joverline{T} deformed conformal field theory (CFT). We derive a flow equation for the Joverline{T} deformed partition sum, and use it to study non-perturbative effects. We find non-perturbative ambiguities for any non-zero value of μ, and comment on their possible relations to holography.

A note on the complex SYK model and warped CFTs

We discuss the connections between the complex SYK model at the conformal limit and warped conformal field theories. Both theories have an SL(2, ℝ) × U(1) global symmetry. We present comparisons on symmetries, correlation functions, the effective action and the entropy formula. We also use modular covariance to reinterpret results in the complex SYK model.

Entanglement, replicas, and Thetas

We compute the single-interval Rényi entropy (replica partition function) for free fermions in 1+1d at finite temperature and finite spatial size by two methods: (i) using the higher-genus partition function on the replica Riemann surface, and (ii) using twist operators on the torus. We compare the two answers for a restricted set of spin structures, leading to a non-trivial proposed equivalence between higher-genus Siegel Θ-functions and Jacobi θ-functions. We exhibit this proposal and provide substantial evidence for it. The resulting expressions can be elegantly written in terms of Jacobi forms. Thereafter we argue that the correct Rényi entropy for modular-invariant free-fermion theories, such as the Ising model and the Dirac CFT, is given by the higher-genus computation summed over all spin structures. The result satisfies the physical checks of modular covariance, the thermal entropy relation, and Bose-Fermi equivalence.

An elliptic triptych

We clarify three aspects of non-compact elliptic genera. Firstly, we give a path integral derivation of the elliptic genus of the cigar conformal field theory from its non-linear sigma-model description. The result is a manifestly modular sum over a lattice. Secondly, we discuss supersymmetric quantum mechanics with a continuous spectrum. We regulate the theory and analyze the dependence on the temperature of the trace weighted by the fermion number. The dependence is dictated by the regulator. From a detailed analysis of the dependence on the infrared boundary conditions, we argue that in noncompact elliptic genera right-moving supersymmetry combined with modular covariance is anomalous. Thirdly, we further clarify the relation between the flat space elliptic genus and the infinite level limit of the cigar elliptic genus.

A cardy formula for three-point coefficients or how the black hole got its spots

Modular covariance of torus one-point functions constrains the three point function coefficients of a two dimensional CFT. This leads to an asymptotic formula for the average value of light-heavy-heavy three point coefficients, generalizing Cardy’s formula for the high energy density of states. The derivation uses certain asymptotic properties of one-point conformal blocks on the torus. Our asymptotic formula matches a dual AdS3 computation of one point functions in a black hole background. This is evidence that the BTZ black hole geometry emerges upon course-graining over a suitable family of heavy microstates.

Universal volume of groups and anomaly of Vogel’s symmetry

We show that integral representation of universal volume function of compact simple Lie groups gives rise to six analytic functions on $CP^2$, which transform as two triplets under group of permutations of Vogel's projective parameters. This substitutes expected invariance under permutations of universal parameters by more complicated covariance. We provide an analytical continuation of these functions and particularly calculate their change under permutations of parameters. This last relation is universal generalization, for an arbitrary simple Lie group and an arbitrary point in Vogel's plane, of the Kinkelin's reflection relation on Barnes' $G(1+N)$ function. Kinkelin's relation gives asymmetry of the $G(1+N)$ function (which is essentially the volume function for $SU(N)$ groups) under $N\leftrightarrow -N$ transformation (which is equivalent of the permutation of parameters, for $SU(N)$ groups), and coincides with universal relation on permutations at the $SU(N)$ line on Vogel's plane. These results are also applicable to universal partition function of Chern-Simons theory on three-dimensional sphere. This effect is analogous to modular covariance, instead of invariance, of partition functions of appropriate gauge theories under modular transformation of couplings.