4D Theory Research Articles

(2,0) theory on S5 × S1 and quantum M2 branes

The superconformal index Z of the 6d (2,0) theory on S5×S1 (which is related to the localization partition function of 5d SYM on S5) should be captured at large N by the quantum M2 brane theory in the dual M-theory background. Generalizing the type IIA string theory limit of this relation discussed in arXiv:2111.15493 and arXiv:2304.12340, we consider semiclassically quantized M2 branes in a half-supersymmetric 11d background which is a twisted product of thermal AdS7 and S4. We show that the leading non-perturbative term at large N is reproduced precisely by the 1-loop partition function of an “instanton” M2 brane wrapped on S1×S2 with S2⊂S4. Similarly, the (2,0) theory analog of the BPS Wilson loop expectation value is reproduced by the partition function of a “defect” M2 brane wrapped on thermal AdS⊂3 AdS7. We comment on a curious analogy of these results with similar computations in arXiv:2303.15207 and arXiv:2307.14112 of the partition function of quantum M2 branes in AdS×4S7/Zk which reproduced the corresponding localization expressions in the ABJM 3d gauge theory.

Analysis of Entrepreneurial Aspects of Appropriate Technology Training Models to Create Farmer Business Opportunities

The objective of this study was to examine the entrepreneurial dimensions associated with the utilization of suitable technology in order to generate commercial prospects for the supported partner villages of maize farmers. The study participants consisted of a sample of 23 agricultural practitioners who underwent interviews and completed a structured questionnaire. The hypothesis suggests that entrepreneurial elements have a beneficial influence on the creation of company possibilities. The aforementioned elements encompass an autonomous mentality, work attitude, propensity for risk-taking, and inclination towards cooperation. The collection of data was conducted through the methods of observation, interviews, and documentation. The research being conducted falls under the category of research and development, commonly referred to as R&D, and is based on Thiagarajan's 4D theory notion, which encompasses the stages of Define, Design, Develop, and Disseminate. The data underwent examination using descriptive statistics and qualitative analytical techniques. Based on the statistical analysis conducted on the four dimensions of entrepreneurship under investigation, it is evident that all of these dimensions exert an impact on the capacity to generate opportunities. Nevertheless, the most significant factor contributing to the creation of business possibilities is the work attitude, accounting for 39% of the overall influence. The present study highlights the expressed desire of farmers to get training in entrepreneurship, with the aim of enhancing their entrepreneurial competencies.

Averaging over codes and an SU(2) modular bootstrap

Error-correcting codes are known to define chiral 2d lattice CFTs where all the U(1) symmetries are enhanced to SU(2). In this paper, we extend this construction to a broader class of length-n codes which define full (non-chiral) CFTs with SU(2)n symmetry, where n = c + overline{c} . We show that codes give a natural discrete ensemble of 2d theories in which one can compute averaged observables. The partition functions obtained from averaging over all codes weighted equally is found to be given by the sum over modular images of the vacuum character of the full extended symmetry group, and in this case the number of modular images is finite. This averaged partition function has a large gap, scaling linearly with n, in primaries of the full SU(2)n symmetry group. Using the sum over modular images, we conjecture the form of the genus-2 partition function. This exhibits the connected contributions to disconnected boundaries characteristic of wormhole solutions in a bulk dual.

Remarks on Berry connection in QFT, anomalies, and applications

Berry connection has been recently generalized to higher-dimensional QFT, where it can be thought of as a topological term in the effective action for background couplings. Via the inflow, this term corresponds to the boundary anomaly in the space of couplings, another notion recently introduced in the literature. In this note we address the question of whether the old-fashioned Berry connection (for time-dependent couplings) still makes sense in a QFT on \Sigma^{(d)}× \mathbb{R}Σ(d)×ℝ, where \Sigma^{(d)}Σ(d) is a dd-dimensional compact space and \mathbb{R}ℝ is time. Compactness of \Sigma^{(d)}Σ(d) relieves us of the IR divergences, so we only have to address the UV issues. We describe a number of cases when the Berry connection is well defined (which includes the tt^*tt* equations), and when it is not. We also mention a relation to the boundary anomalies and boundary states on the Euclidean \Sigma^{(d)} × \mathbb{R}_{≥ 0}Σ(d)×ℝ≥0. We then work out the examples of a free 3D Dirac fermion and a 3D \mathcal{N}=2𝒩=2 chiral multiplet. Finally, we consider 3D theories on \mathbb{T}^2× \mathbb{R}𝕋2×ℝ, where the space \mathbb{T}^2𝕋2 is a two-torus, and apply our machinery to clarify some aspects of the relation between 3D SUSY vacua and elliptic cohomology. We also comment on the generalization to higher genus.

Fermionic higher-form symmetries

In this paper, we explore a new type of global symmetries - the fermionic higher-form symmetries. They are generated by topological operators with fermionic parameter, which act on fermionic extended objects. We present a set of field theory examples with fermionic higher-form symmetries, which are constructed from fermionic tensor fields. They include the free fermionic tensor theories, a new type of fermionic topological quantum field theories, as well as the exotic 6d (4,0) theory. We also discuss the gauging and breaking of such global symmetries and the relation to the no global symmetry swampland conjecture.

Theta -diagram technique for {mathcal {N}}=1, d=4 superfields

We describe a diagrammatic procedure to carry out the Grassmann integration in super-Feynman diagrams of 4d theories expressed in terms of {mathcal {N}}=1 superfields. This method is alternative to the well known D-algebra approach. We develop it in detail for theories containing vector, chiral and anti-chiral superfields, with the type of interactions which occur in {mathcal {N}}=2 SYM theories with massless matter, but it would be possible to extend it to other cases. The main advantage is that this method is algorithmic; we implemented it as a Mathematica program that, given the description of a super Feynman diagram in momentum space, returns directly the polynomial in the momenta produced by the Grassmann integration.

On the 6d origin of non-invertible symmetries in 4d

It is well-known that six-dimensional superconformal field theories can be exploited to unravel interesting features of lower-dimensional theories obtained via compactifications. In this short note we discuss a new application of 6d (2,0) theories in constructing 4d theories with Kramers-Wannier-like non-invertible symmetries. Our methods allow to recover previously known results, as well as to exhibit infinitely many new examples of four dimensional theories with “M-ality” defects (arising from operations of order M generalizing dualities). In particular, we obtain examples of order M = pk, where p > 1 is a prime number and k is a positive integer.

On the master space for brane brick models

We systematically study the master space of brane brick models that represent a large class of 2d (0, 2) quiver gauge theories. These 2d (0, 2) theories are worldvolume theories of D1-branes that probe singular toric Calabi-Yau 4-folds. The master space is the freely generated space of chiral fields subject to the J- and E-terms and the non-abelian part of the gauge symmetry. We investigate several properties of the master space for abelian brane brick models with U(1) gauge groups. For example, we calculate the Hilbert series, which allows us by using the plethystic programme to identify the generators and defining relations of the master space. By studying several explicit examples, we also show that the Hilbert series of the master space can be expressed in terms of characters of irreducible representations of the full global symmetry of the master space.

Photon region and shadow of a rotating 5D black string

To explore the possible clues for the extra dimension from the Event Horizon Telescope (EHT) observations, we study the shadow of the rotating 5D black string in General Relativity (GR). Instead of investigating the shadow in the effective 4D theory, we concern the motion of photons along the extra dimension z with a conserved momentum P_z, which appears as an effective mass in the geodesic equations of photons. The existence of P_z enlarges the photon regions and the shadow of the rotating 5D black string while it has slight impact on the distortion. The EHT observations of M87* and SgrA* can rule out the black string model with an infinite length along the extra dimension, and support the hypothesis that the extra dimension is compact to avoid the Gregory-Laflamme (GL) instability, where the length of the black string/the compact extra dimension can be constrained as 2.03125~textrm{mm} lesssim ell lesssim 2.6~textrm{mm} and 2.28070~textrm{mm} lesssim ell lesssim 2.6~textrm{mm} respectively.

Chern-Simons-Trinion theories: One-form symmetries and superconformal indices

We study 3d theories containing 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} = 3 Chern-Simons vector multiplets coupled to the SU(N)3 flavour symmetry of 3d TN theories with Chern-Simons levels k1, k2 and k3. It was formerly pointed out that these theories flow to infrared SCFTs with enhanced 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 supersymmetry when 1/k1 + 1/k2 + 1/k3 = 0. We examine superconformal indices of these theories which reveal that supersymmetry of the infrared SCFTs may get enhanced to 4 ≤ 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} ≤ 6 if such a condition is satisfied. Moreover, even if the Chern-Simons levels do not obey the aforementioned condition, we find that there is still an infinite family of theories that flows to infrared SCFTs with 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 supersymmetry. The ’t Hooft anomalies of the one-form symmetries of these theories are analysed. As a by-product, we observe that there is generally a decoupled topological sector in the infrared. When the infrared SCFTs have 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 supersymmetry, we also study the Higgs and Coulomb branch limits of the indices which provide geometric information of the moduli space of the theories in question in terms of the Hilbert series.

State dependence of Krylov complexity in 2d CFTs

We compute the Krylov Complexity of a light operator O\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{O} $$\\end{document}L in an eigenstate of a 2d CFT at large central charge c. The eigenstate corresponds to a primary operator O\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{O} $$\\end{document}H under the state-operator correspondence. We observe that the behaviour of K-complexity is different (either bounded or exponential) depending on whether the scaling dimension of O\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{O} $$\\end{document}H is below or above the critical dimension hH = c/24, marked by the 1st order Hawking-Page phase transition point in the dual AdS3 geometry. Based on this feature, we hypothesize that the notions of operator growth and K-complexity for primary operators in 2d CFTs are closely related to the underlying entanglement structure of the state in which they are computed, thereby demonstrating explicitly their state-dependent nature. To provide further evidence for our hypothesis, we perform an analogous computation of K-complexity in a model of free massless scalar field theory in 2d, and in the integrable 2d Ising CFT, where there is no such transition in the spectrum of states.

Supersymmetric reduction of 3D theories on a line segment

Supersymmetric reduction of 3D theories on a line segment

Aspects of dynamical cobordism in AdS/CFT

The cobordism conjecture implies that consistent theories of Quantum Gravity must admit the introduction of boundaries. We study the dynamical realization of the cobordism conjecture in type IIB in AdS5 × S5, using the existing gravity duals of 4d \U0001d4a9 = 4 SYM with Gaiotto-Witten superconformal boundary conditions (near-horizon limits of D3-branes ending on NS5- and D5-branes). We show that these configurations are, from the 5d perspective, dynamical cobordism solutions which start from an asymptotic AdS5 vacuum and evolve until they hit an end of the world (ETW) brane with AdS4 worldvolume. The latter displays localization of gravity, and provide a completion of the Karch-Randall (KR) AdS branes, in which the backreaction of running scalars replace the KR cusp in the warp factor with a smooth bump. The dynamical scalars are either in the SO(6) invariant AdS5 bulk sector (e.g. describing the S5 size and its shrinking at the cobordism boundary) or brane localized (e.g. the SO(6) SO(3) × SO(3) squashing due to boundary conditions). We introduce a novel double scaling limit which zooms into the ETW brane and makes localization of gravity manifest, and which shows a tantalizing relation with wedge holography. We extend the picture to AdS5 theories with less (super)symmetry, via orbifolds and S-folds, leading to dynamical cobordisms for gravity duals of 4d theories with \U0001d4a9 = 2 and \U0001d4a9 = 3 supersymmetry.

On the brane expansion of the Schur index

We consider the Schur index of mathcal{N} = 4 U(N) SYM theory in 4d and its holographic giant graviton-type expansion at finite N. We compute the world-volume brane superconformal index by a recently proposed definition of the gauge holonomy integral as a multivariate residue. This is evaluated by a novel deformation algorithm that avoids Gröbner basis methods. Various terms of the brane expansion are computed and their sum is shown to be free of wall-crossing singularities to the order we explored. The relation between the brane expansion and previous giant graviton-type represenations of the Schur index is clarified.

Learning trivializing flows

The recent introduction of machine learning techniques, especially normalizing flows, for the sampling of lattice gauge theories has shed some hope on improving the sampling efficiency of the traditional hybrid Monte Carlo (HMC) algorithm. In this work we study a modified HMC algorithm that draws on the seminal work on trivializing flows by Lüscher. Autocorrelations are reduced by sampling from a simpler action that is related to the original action by an invertible mapping realised through Normalizing Flows models with a minimal set of training parameters. We test the algorithm in a phi ^{4} theory in 2D where we observe reduced autocorrelation times compared with HMC, and demonstrate that the training can be done at small unphysical volumes and used in physical conditions. We also study the scaling of the algorithm towards the continuum limit under various assumptions on the network architecture.

Affine characters at negative level and elliptic genera of non-critical strings

We study the elliptic genera of the non-critical strings of six dimensional superconformal field theories from the point of view of the strings’ worldsheet theory. We formulate a general ansatz for these in terms of characters of the affine Lie algebra associated to the 6d gauge group at negative level, and provide ample evidence for the validity of this ansatz for 6d theories obtained via F-theory compactification on elliptically fibered Calabi-Yau manifolds over a Hirzebruch base. We obtain novel closed form results for many elliptic genera in terms of our ansatz, and show that our results specialize consistently when moving along Higgsing trees.

E-strings, F4, and D4 triality

We study the E-string theory on ℝ4 × T2 with Wilson lines. We consider two examples where interesting automorphisms arise. In the first example, the spectrum is invariant under the F4 Weyl group acting on the Wilson line parameters. We obtain the Seiberg-Witten curve expressed in terms of Weyl invariant F4 Jacobi forms. We also clarify how it is related to the thermodynamic limit of the Nekrasov-type formula. In the second example, the spectrum is invariant under the D4 triality combined with modular transformations, the automorphism originally found in the 4d \U0001d4a9 = 2 supersymmetric SU(2) gauge theory with four massive flavors. We introduce the notion of triality invariant Jacobi forms and present the Seiberg-Witten curve in terms of them. We show that this Seiberg-Witten curve reduces precisely to that of the 4d theory with four flavors in the limit of T2 shrinking to zero size.

Bootstrapping string dynamics in the 6d \U0001d4a9 = (2, 0) theories

We present two complementary approaches to calculating the 2-point function of stress tensors in the presence of a 1/2 BPS surface defect of the 6d \U0001d4a9 = (2, 0) theories. First, we use analytical bootstrap techniques at large N to obtain the first nontrivial correction to this correlator, from which we extract the defect CFT (dCFT) data characterising the 2d dCFT of the 1/2 BPS plane. Along the way we derive a supersymmetric inversion formula, obtain the relevant superconformal blocks and check that crossing symmetry is satisfied. Notably our result features a holomorphic function whose appearance is related to the chiral algebra construction of Beem, Rastelli and van Rees. Second, we use that chiral algebra description to obtain exact results for the BPS sector of the dCFT, valid at any N and for any choice of surface operator. These results provide a window into the dynamics of strings of the mysterious 6d theories.

Effective approach to the Antoniadis-Mottola model: quantum decoupling of the higher derivative terms

We explore the decoupling of massive ghost mode in the 4D (four-dimensional) theory of the conformal factor of the metric. The model was introduced by Antoniadis and Mottola in [1] and can be regarded as a close analog of the fourth-derivative quantum gravity. The analysis of the derived one-loop nonlocal form factors includes their asymptotic behavior in the UV and IR limits. In the UV (high energy) domain, our results reproduce the Minimal Subtraction scheme-based beta functions of [1]. In the IR (i.e., at low energies), the diagrams with massive ghost internal lines collapse into tadpole-type graphs without nonlocal contributions and become irrelevant. On the other hand, those structures that contribute to the running of parameters of the action and survive in the IR, are well-correlated with the divergent part (or the leading in UV contributions to the form factors), coming from the effective low-energy theory of the conformal factor. This effective theory describes only the light propagating mode. Finally, we discuss whether these results may shed light on the possible running of the cosmological constant at low energies.

One-dimensional viscoelastic von Kármán theories derived from nonlinear thin-walled beams

We derive an effective one-dimensional limit from a three-dimensional Kelvin–Voigt model for viscoelastic thin-walled beams, in which the elastic and the viscous stress tensor comply with a frame-indifference principle. The limiting system of equations comprises stretching, bending, and twisting both in the elastic and the viscous stress. It coincides with the model already identified via Friedrich and Kružík (Arch Ration Mech Anal 238:489–540, 2020) and Friedrich and Machill (Nonlinear Differ Equ Appl NoDEA 29, Article number: 11, 2022) by a successive dimension reduction, first from 3D to a 2D theory for von Kármán plates and then from 2D to a 1D theory for ribbons. In the present paper, we complement the previous analysis by showing that the limit can also be obtained by sending the height and width of the beam to zero simultaneously. Our arguments rely on the static Gamma -convergence in Freddi et al. (Math Models Methods Appl Sci 23:743–775, 2013), on the abstract theory of metric gradient flows (Ambrosio et al. in Gradient flows in metric spaces and in the space of probability measures. Lectures mathematics, ETH Zürich, Birkhäuser, Basel, 2005), and on evolutionary Gamma -convergence (Sandier and Serfaty in Commun Pure Appl Math 57:1627–1672, 2004).