Modular Invariance Research Articles

3-Preprojective Algebras of Type D

AbstractWe present a family of selfinjective algebras of type D, which arise from the 3-preprojective algebras of type A by taking a $$\mathbb {Z}_3$$ Z 3 -quotient. We show that a subset of these are themselves 3-preprojective algebras, and that the associated 2-representation-finite algebras are fractional Calabi-Yau. In addition, we show our work is connected to modular invariants for SU(3).

Multiboundary wormholes and OPE statistics

We derive higher moments in the statistical distribution of OPE coefficients in holographic 2D CFTs, and show that such moments correspond to multiboundary Euclidean wormholes in pure 3D gravity. The nth cyclic non-Gaussian contraction of heavy-heavy-light OPE coefficients follows from crossing symmetry of the thermal n-point function. We derive universal expressions for the cubic and quartic moments and demonstrate that their scaling with the microcanonical entropy agrees with a generalization of the Eigenstate Thermalization Hypothesis. Motivated by this result, we conjecture that the full statistical ensemble of OPE data is fixed by three premises: typicality, crossing symmetry and modular invariance. Together, these properties give predictions for non-factorizing observables, such as the generalized spectral form factor. Using the Virasoro TQFT, we match these connected averages to new on-shell wormhole topologies with multiple boundary components. Lastly, we study and clarify examples where the statistics of heavy operators are not universal and depend on the light operator spectrum. We give a gravitational interpretation to these corrections in terms of Wilson loops winding around non-trivial cycles in the bulk.

Solving the strong CP problem without axions

We formulate general conditions under which the strong CP problem is solved by spontaneous CP violation. Quark-mass matrix elements are polynomials in the CP-breaking order parameters, engineered such that their determinant is a real constant. This scheme permits only a limited number of textures. These conditions can be realized in supersymmetric theories with CP as an anomaly-free local flavor symmetry, suggesting a unified solution to the strong CP problem and the flavor puzzle. Our solution can be implemented using either modular invariance or a local U(1) symmetry. We present modular-invariant realizations where matter fields are assigned small modular weights ±2 (±1), utilising higher levels N = 2 (N = 3). Heavy quarks are in general not required, but their presence allows for models where colored particles fill non-singlet representations of the flavor group.

Orbifolded elliptic genera of non-compact models

We revisit the flavored elliptic genus of the N=2 superconformal cigar model and generalize the analysis of the path integral result to the case of real central charge. It gives rise to a non-holomorphic modular covariant function generalizing completed mock modular forms. We also compute the genus for angular orbifolds of the cigar and Liouville theory and decompose it in terms of discrete and continuous contributions. The orbifolded elliptic genus at fractional level is a completed mock modular form with a shadow related to U(1) modular invariants at rational radius squared. We take the limit of the orbifolded genera towards a weighted ground state index and carefully interpret the contributions. We stress that the orbifold cigar and Liouville theories have a maximal and a minimal radius, respectively.

Non-holomorphic modular flavor symmetry

The formalism of non-holomorphic modular flavor symmetry is developed, and the Yukawa couplings are level N polyharmonic Maaß forms satisfying the Laplacian condition. We find that the integer (even) weight polyharmonic Maaß forms of level N can be decomposed into multiplets of the finite modular group ΓN′\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {\\Gamma}_N^{\\prime } $$\\end{document} (ΓN). The original modular invariance approach is extended by the presence of negative weight polyharmonic Maaß forms. The non-holomorphic modular flavor symmetry can be consistently combined with the generalized CP symmetry. We present three example models for lepton sector based on the Γ3 ≅ A4 modular symmetry, the charged lepton masses and the neutrino oscillation data can be accommodated very well, and the predictions for the leptonic CP violation phases and the effective Majorana neutrino mass are studied.

Thermodynamic formulation of vacuum energy density in flat spacetime and potential implications for the cosmological constant

We propose a thermodynamical definition of the vacuum energy density ρvac, defined as 〈vac|Tμν|vac〉 = − ρvacgμν, in quantum field theory in flat Minkowski space in D spacetime dimensions, which can be computed in the limit of high temperature, namely in the limit β = 1/T → 0. It takes the form ρvac = const ∙ mD where m is a fundamental mass scale and “const” is a computable constant which can be positive or negative depending on interaction couplings. Due to modular invariance ρvac can also be computed in a different non-thermodynamic channel where one spatial dimension is compactifed on a circle of circumference β and we confirm this modularity for free massive theories for both bosons and fermions for D = 2, 3, 4. We list various properties of ρvac that are generally required, for instance ρvac = 0 for conformal field theories, and others, such as the constraint that ρvac has opposite signs for free bosons verses fermions of the same mass, which is related to constraints from supersymmetry. Using the Thermodynamic Bethe Ansatz we compute ρvac exactly for 2 classes of integrable QFT’s in 2D and interpreting some previously known results. We apply our definition of ρvac to Lattice QCD data with two light quarks (up and down) and one additional massive flavor (the strange quark), and find it is negative, ρvac ≈ − (200 MeV)4. Finally we make some remarks on the Cosmological Constant Problem since ρvac is central to any discussion of it.

Large-N integrated correlators in N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 4 SYM: when resurgence meets modularity

Exact expressions for certain integrated correlators of four half-BPS operators in N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 4 supersymmetric Yang-Mills theory with gauge group SU(N) have been recently obtained thanks to a beautiful interplay between supersymmetric localisation and modular invariance. The large-N expansion at fixed Yang-Mills coupling of such integrated correlators produces an asymptotic series of perturbative terms, holographically related to higher derivative interactions in the low energy expansion of the type IIB effective action, as well as exponentially suppressed corrections at large N, interpreted as contributions from coincident (p, q)-string world-sheet instantons. In this work we define a manifestly modular invariant Borel resummation of the perturbative large-N expansion of these integrated correlators, from which we extract the exact non-perturbative large-N sectors via resurgence analysis. Furthermore, we show that in the ’t Hooft limit such modular invariant non-perturbative completions reduce to known resurgent genus expansions. Finally, we clarify how the same non-perturbative data is encoded in the decomposition of the integrated correlators based on SL(2, ℤ) spectral theory.

Modular invariants of a vector and a covector for some elementary abelian p-groups

Let F p be the prime field of order p > 0 and G be an elementary abelian p-group. For some n-dimensional cohyperplane G-representations V over F p , we show that F p [ V ⊕ V * ] G , the invariant ring of a vector and a covector is a complete intersection by exhibiting an explicit generating set (in fact, a SAGBI basis) and exposing all relations among the generators.

Modular Invariance of (Logarithmic) Intertwining Operators

Let V be a C2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C_2$$\\end{document}-cofinite vertex operator algebra without nonzero elements of negative weights. We prove the conjecture that the spaces spanned by analytic extensions of pseudo-q-traces (q=e2πiτ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$q=e^{2\\pi i\ au }$$\\end{document}) shifted by -c24\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$-\\frac{c}{24}$$\\end{document} of products of geometrically-modified (logarithmic) intertwining operators among grading-restricted generalized V-modules are invariant under modular transformations. The convergence and analytic extension result needed to formulate this conjecture and some consequences on such shifted pseudo-q-traces were proved by Fiordalisi (Logarithmic intertwining operator and genus-one correlation functions, 2015) and Fiordalisi (Commun Contemp Math 18:1650026, 2016) using the method developed in Huang (Commun Contemp Math 7:649–706, 2005). The method that we use to prove this conjecture is based on the theory of the associative algebras AN(V)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$A^{N}(V)$$\\end{document} for N∈N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$N\\in \\mathbb {N}$$\\end{document}, their graded modules and their bimodules introduced and studied by the author in Huang (Associative algebras and the representation theory of grading-restricted vertex algebras, 2020) and Huang (Commun Math Phys 396:1–44, 2022). This modular invariance result gives a construction of C2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C_2$$\\end{document}-cofinite genus-one logarithmic conformal field theories from the corresponding genus-zero logarithmic conformal field theories.

Standard and Non-Standard Aspects of Neutrino Physics

This review provides a succinct overview of the basic aspects of neutrino physics. The topics covered include neutrinos in the standard model and the three-neutrino mixing scheme; the current status of neutrino oscillation measurements and what remains to be determined; the seesaw mechanisms for neutrino mass generation and the associated phenomenology, including the leptogenesis mechanism to explain the observed matter–antimatter asymmetry of the Universe; and models for the origin of the pattern of neutrino mixing and lepton masses based on discrete flavour symmetries and modular invariance.

Modular Invariance Approach to the Flavour Problem (From Bottom Up)

Modular Invariance Approach to the Flavour Problem (From Bottom Up)

Manifest modular invariance in the near-critical Ising model

Using recent results in mathematics, I point out that free energies and scale-dependent central charges away from criticality can be represented in compact form where modular invariance is manifest. The main example is the near-critical Ising model on a thermal torus, but the methods are not restricted to modular symmetry, and apply to automorphic symmetries more generally. One application is finite-size effects.

Nonrelativistic Dirac fermions on the torus

Two dimensional conformal field theories have been extensively studied in the past. When considered on the torus, they are strongly constrained by modular invariance. However, introducing relevant deformations or chemical potentials pushes these theories away from criticality, where many of their aspects are still poorly understood. In this note we make a step towards filling this gap, by analyzing the theory of a Dirac fermion on the torus, deformed by a mass term and a chemical potential for the particle number symmetry. The theory breaks conformal and Lorentz invariance, and we study its spectrum and partition function. We also focus on two limits that are interesting on their own right: a massless relativistic fermion with nonzero chemical potential (a simple model for CFTs at finite density), and nonrelativistic Schrodinger fermions (of relevance in condensed matter systems). Taking inspiration from recent developments in massive modular forms, we obtain a representation of the torus free energy based on Fourier-transforming over a twisted boundary condition. This dual representation fullfills many properties analogous to modular invariance in CFTs. In particular, we use this result to derive Cardy-like formulas for the high energy density of states of these theories.

Tachyons and Misaligned Supersymmetry in orientifold vacua

We extend the analysis of Misaligned Supersymmetry to orientifold vacua. The lack of modular invariance in the Klein bottle, annulus and Möbius strip amplitudes implies that the IR properties of the physical spectrum are related to the UV behaviour of closed strings freely propagating between D-branes and O-planes, and vice versa. We thus build sector averaged sums 〈d(n)〉 associated to both representations of each Riemann surface. We find that the 〈d(n)〉’s associated to the loop channel control the coupling of closed-string tachyons to D-branes and/or O-planes, and therefore their vanishing is a necessary, but not sufficient, condition for stability in the closed sector. Similarly, the sector averaged sums associated to the tree-level amplitudes encode the presence of tachyons in the physical spectrum, and their vanishing is a sufficient, but not necessary, condition for classical stability. Using this technology, it is difficult, if not impossible, to formulate necessary and sufficient criteria for the absence of tachyons since, in orientifold vacua, they can be removed from the spectrum by the orientifold projection. Although in these cases one would naively expect a cancellation between 〈d(n)〉’s associated to different amplitudes, this does not occur in practice. We provide several examples in ten and lower dimensions supporting our results.

On K3 surfaces of Picard rank 14

AbstractWe study complex algebraic K3 surfaces with finite automorphism groups and polarized by rank 14, 2‐elementary lattices. Three such lattices exist, namely, , , and . As part of our study, we provide birational models for these surfaces as quartic projective hypersurfaces and describe the associated coarse moduli spaces in terms of suitable modular invariants. Additionally, we explore the connection between these families and dual K3 families related via the Nikulin construction.

String Theory Bounds on the Cosmological Constant, the Higgs Mass, and the Quark and Lepton Masses

In this paper, we elaborate on the new understanding of the cosmological constant and the gauge hierarchy problems in the context of string theory in its metastring formulation, based on the concepts of modular spacetime and Born geometry. The interplay of phase space (and Born geometry), the Bekenstein bound, the mixing between ultraviolet (UV) and infrared (IR) physics and modular invariance in string theory is emphasized. This new viewpoint is fundamentally rooted in quantum contextuality and not in statistical observer bias (anthropic principle). We also discuss the extension of this point of view to the problem of masses of quarks and leptons and their respective mixing matrices.

Symmetries and spectral statistics in chaotic conformal field theories

We discuss spectral correlations in coarse-grained chaotic two-dimensional CFTs with large central charge. We study a partition function describing the dense part of the spectrum of primary states in a way that disentangles the chaotic properties of the spectrum from those which are a consequence of Virasoro symmetry and modular invariance. We argue that random matrix universality in the near-extremal limit is an independent feature of each spin sector separately; this is a non-trivial statement because the exact spectrum is fully determined by only the spectrum of spin zero primaries and those of a single non-zero spin (“spectral determinacy”). We then describe an argument analogous to the one leading to Cardy’s formula for the averaged density of states, but in our case applying it to spectral correlations: assuming statistical universalities in the near-extremal spectrum in all spin sectors, we find similar random matrix universality in a large spin regime far from extremality.

Modular invariance and the QCD angle

String compactifications on an orbi-folded torus with complex structure give rise to chiral fermions, spontaneously broken CP, modular invariance. We show that this allows simple effective theories of flavour and CP where: i) the QCD angle vanishes; ii) the CKM phase is large; iii) quark and lepton masses and mixings can be reproduced up to order one coefficients. We implement such general paradigm in supersymmetry or supergravity, with modular forms or functions, with or without heavy colored states.

Casimir energy and modularity in higher-dimensional conformal field theories

An important problem in Quantum Field Theory (QFT) is to understand the structures of observables on spacetime manifolds of nontrivial topology. Such observables arise naturally when studying physical systems at finite temperature and/or finite volume and encode subtle properties of the underlying microscopic theory that are often obscure on the flat spacetime. Locality of the QFT implies that these observables can be constructed from more basic building blocks by cutting-and-gluing along a spatial slice, where a crucial ingredient is the Hilbert space on the spatial manifold. In Conformal Field Theory (CFT), thanks to the operator-state correspondence, we have a non-perturbative understanding of the Hilbert space on a spatial sphere. However it remains a challenge to consider more general spatial manifolds. Here we study CFTs in spacetime dimensions d > 2 on the spatial manifold T2 × ℝd−3 which is one of the simplest manifolds beyond the spherical topology. We focus on the ground state in this Hilbert space and analyze universal properties of the ground state energy, also commonly known as the Casimir energy, which is a nontrivial function of the complex structure moduli τ of the torus. The Casimir energy is subject to constraints from modular invariance on the torus which we spell out using PSL(2, ℤ) spectral theory. Moreover we derive a simple universal formula for the Casimir energy in the thin torus limit using the effective field theory (EFT) from Kaluza-Klein reduction of the CFT, with exponentially small corrections from worldline instantons. We illustrate our formula with explicit examples from well-known CFTs including the critical O(N) model in d = 3 and holographic CFTs in d ≥ 3.

Fake Z

Recently introduced connections between quantum codes and Narain CFTs provide a simple ansatz to express a modular-invariant function Zleft(tau, overline{tau}right) in terms of a multivariate polynomial satisfying certain additional properties. These properties include algebraic identities, which ensure modular invariance of Zleft(tau, overline{tau}right) , and positivity and integrality of coefficients, which imply positivity and integrality of the \U0001d532(1)n × \U0001d532(1)n character expansion of Zleft(tau, overline{tau}right) . Such polynomials come naturally from codes, in the sense that each code of a certain type gives rise to the so-called enumerator polynomial, which automatically satisfies all necessary properties, while the resulting Zleft(tau, overline{tau}right) is the partition function of the code CFT — the Narain theory unambiguously constructed from the code. Yet there are also “fake” polynomials satisfying all necessary properties, that are not associated with any code. They lead to Zleft(tau, overline{tau}right) satisfying all modular bootstrap constraints (modular invariance and positivity and integrality of character expansion), but whether they are partition functions of any actual CFT is unclear. We consider the group of the six simplest fake polynomials and denounce the corresponding Z’s as fake: we show that none of them is the torus partition function of any Narain theory. Moreover, four of them are not partition functions of any unitary 2d CFT; our analysis for other two is inconclusive. Our findings point to an obvious limitation of the modular bootstrap approach: not every solution of the full set of torus modular bootstrap constraints is due to an actual CFT. In the paper we consider six simple examples, keeping in mind that thousands more can be constructed.