<p>We derive a Landau field theory for a lattice higher gauge theory defined on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>p</mi></math>-dimensional open cells (i.e., sites, links, faces, cubes, etc.), and study its continuum-limit and phases. In this approach, the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>p</mi></math>-dimensional Wilson-surface operator of the higher gauge theory is promoted to a fundamental functional field that is charged under the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>p</mi></math>-form global symmetry. By explicitly solving the functional equation of motion, we show that the classical solution exhibits the area (perimeter) law in the strong (weak) gauge coupling limit. In the deconfined phase, we also construct topological defects for both U(1) and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msub><mrow><mi mathvariant="double-struck">Z</mi></mrow><mrow><mi>N</mi></mrow></msub></mrow></math> higher-form gauge theories, as analogs of vortex and domain-wall solutions in conventional field theories with 0-form global symmetries. Besides, we examine low-energy effective theory by identifying the phase modulations of the functional field as low-energy modes, and discuss the Coleman-Mermin-Wagner theorem for higher-form global symmetries. Finally, we discuss infrared duality among Landau field theories, which originates from Kramers-Wannier duality in lattice higher-form gauge theories.</p>

South Korea

Kiyoharu Kawana

Landau theory for lattice higher-form gauge theories and the Kramers-Wannier duality

We study phases and propagation of closed $\textit{p}$-brane within the framework of effective field theory with higher-form global symmetries, i.e., $\textit{brane-field theory}$. We extend our previous studies by including the kinetic term of the center-of-mass motion as well as the kinetic term for the relative motions constructed by the area derivatives. This inclusion gives rise to another scalar Nambu-Goldstone mode in the broken phase, enriching the phase structures of $\textit{p}$-brane. For example, when the higher-form global symmetries are discrete ones, we show that the low-energy effective theory in the broken phase is described by a topological field theory of the axion $\textit{φ}$($\textit{X}$) and $\textit{p}$-form field $\textit{A}$ ($\textit{X}$) with multiple (emergent) higher-form global symmetries. After the mean-field analysis, we investigate the propagation of $\textit{p}$-brane in the present framework. We find the (functional) plane-wave solutions for the kinetic terms and derive a path-integral representation of the brane propagator. This representation motivates us to study the brane propagation within the Born-Oppenheimer approximation, where the volume of $\textit{p}$-brane is treated as constant. In the volume-less limit (i.e. point-particle limit), the propagator reduces to the ordinary propagator of relativistic particle, whereas it describes the propagation of the area elements in the large-volume limit. Correspondingly, it is shown that the Hausdorff dimension of $\textit{p}$-brane varies from 2 to 2($\textit{p}$ + 1) as we increase the $\textit{p}$-brane volume within the Born-Oppenheimer approximation. Although these results are quite intriguing, we also point out that the Born-Oppenheimer approximation is invalid in the point-particle limit, highlighting the quantum nature of $\textit{p}$-brane as an extended object in spacetime.

Phases and propagation of closed $\textit{p}$-brane

We provide an improved definition of new conserved quantities derived from the energy–momentum tensor in curved spacetime by introducing an additional scalar function. We find that the conserved current and the associated conserved charge become geometric under a certain initial condition of the scalar function, and show that such a conserved geometric current generally exists in curved spacetime. Furthermore, we demonstrate that the geometric conserved current agrees with the entropy current in an effective theory of a perfect fluid, thus the conserved charge is the total entropy of the system. While the geometric charge can be regarded as the entropy for a nondissipative fluid, its physical meaning should be investigated for more general cases.

Japan

Sinya Aoki

Yoshimasa Hidaka

Kengo Shimada

Geometric conservation in curved spacetime and entropy

We discuss the classical continuum limit of the string field theory dual to the <inline-formula><tex-math id='TM0001' notation='LaTeX'>$\mathrm{SU}(N)$</tex-math></inline-formula> lattice gauge theory and investigate various fundamental phenomena in the continuum theory at the mean-field level. Our construction of the continuum theory is based on the concept of <italic>area derivative</italic>, which can be regarded as a generalization of the ordinary derivative <inline-formula><tex-math id='TM0002' notation='LaTeX'>$\partial /\partial x^\mu$</tex-math></inline-formula> to operators acting on functional fields <inline-formula><tex-math id='TM0003' notation='LaTeX'>$\phi [C]$</tex-math></inline-formula> on the loop space. The resultant continuum theory has a <inline-formula><tex-math id='TM0004' notation='LaTeX'>$\mathbb {Z}_N^{}$</tex-math></inline-formula> 1-form global symmetry, which originates in the <inline-formula><tex-math id='TM0005' notation='LaTeX'>$\mathbb {Z}_N^{}$</tex-math></inline-formula> center symmetry in the gauge theory. We find that the confined and deconfined phases of the gauge theory are identified by the unbroken and broken phases of the <inline-formula><tex-math id='TM0006' notation='LaTeX'>$\mathbb {Z}_N^{}$</tex-math></inline-formula> symmetry, respectively, by showing the Area/Perimeter law of the classical solution. In the broken phase, the low-energy effective theory is described by a <inline-formula><tex-math id='TM0007' notation='LaTeX'>$\mathrm{BF}$</tex-math></inline-formula>-type topological field theory and has an emergent <inline-formula><tex-math id='TM0008' notation='LaTeX'>$\mathbb {Z}_N^{}$</tex-math></inline-formula> <inline-formula><tex-math id='TM0009' notation='LaTeX'>$(D-2)$</tex-math></inline-formula>-form global symmetry. The existence of the emergent symmetry is deeply related to <inline-formula><tex-math id='TM0010' notation='LaTeX'>$(D-2)$</tex-math></inline-formula>-dimensional topological configurations (i.e. center vortex for <inline-formula><tex-math id='TM0011' notation='LaTeX'>$D=4$</tex-math></inline-formula>), and we explicitly construct such a topological defect in the continuum theory. Finally, we also comment on the upper and lower critical dimensions of the gauge theory/string field theory.

Classical Continuum Limit of the String Field Theory Dual to Lattice Gauge Theory

In models with non-minimal Higgs sectors, enforcing (near) Higgs alignment, necessary to prevent significant deviations in the Higgs boson coupling from the standard model prediction, causes a serious fine-tuning problem. We demonstrate that the Higgs alignment is naturally deduced from the multicritical point principle (MPP) in the general two Higgs doublet model while keeping non-zero CP-violating phases. Furthermore, we discuss the possibility of realizing the Yukawa alignment from the MPP, which is necessary to prevent flavor-changing neutral currents mediated by Higgs bosons at tree level, and find that it seems difficult to realize it by the MPP due to the non-diagonal structure of the CKM matrix.

Taiwan

Hikaru Kawai

Kin-ya Oda

Kei Yagyu

Higgs alignment from multicritical-point principle in two Higgs doublet models

We propose a field theory of closed $\textit{p}$-brane $\textit{C}$ interacting with a ($\textit{p}$ + 1)-form gauge field $\textit{A}$ . This is a generalization of the Ginzburg-Landau theory (Abelian-Higgs model) for superconducting particles to higher-dimensional superconducting branes. A higher-form gauge invariant action is constructed by utilizing the Area derivative, which is a higher-dimensional generalization of the ordinary derivative. We find that the fundamental phenomena of superconductivity, such as the Meisser effect, topological defects, topological order, are naturally extended in the brane-field theory. We explicitly construct a topologically non-trivial static configuration that is characterized by the first homotopy group. Then, we calculate the low-energy effective theory in the presence of the topological defect and find that it is described by a BF-type topological field theory coupled with the world-volume of the topological defect. We also discuss a potential duality between the superconducting brane-field model and a brane-field model with a global U(1) higher-form symmetry as a generalization of the Particle-Vortex duality.

Field theory for superconducting branes and generalized particle-vortex duality

We investigate a structure of a 4-dimensional bulk space constructed from the O(N) invariant critical φ 4 model in 3-dimension using the conformal smearing. We calculate a bulk metric corresponding to the information metric and the bulk-to-boundary propagator for a composite scalar field φ 2 in the large N expansion. We show that the bulk metric describes an asymptotic AdS space at both UV (near boundary) and IR (deep in the bulk) limits, which correspond to the asymptotic free UV fixed point and the Wilson-Fisher IR fixed point of the 3-dimensional φ 4 model, respectively. The bulk-to-boundary scalar propagator, on the other hand, encodes   <math> <msub> <mtext>∆</mtext> <msup> <mi>φ</mi> <mn>2</mn> </msup> </msub> </math>  $$ {\Delta }_{\varphi^2} $$  (the conformal dimension of φ 2) into its z (a coordinate in the extra direction of the AdS space) dependence. Namely it correctly reproduces not only   <math> <msub> <mtext>∆</mtext> <msup> <mi>φ</mi> <mn>2</mn> </msup> </msub> </math>  $$ {\Delta }_{\varphi^2} $$  = 1 at UV fixed point but also   <math> <msub> <mtext>∆</mtext> <msup> <mi>φ</mi> <mn>2</mn> </msup> </msub> </math>  $$ {\Delta }_{\varphi^2} $$  = 2 at the IR fixed point for the boundary theory. Moreover, we confirm consistency with the GKP-Witten relation in the interacting theory that the coefficient of the   <math> <msup> <mi>z</mi> <msub> <mtext>∆</mtext> <msup> <mi>φ</mi> <mn>2</mn> </msup> </msub> </msup> </math>  $$ {z}^{\Delta _{\varphi^2}} $$  term in z → 0 limit agrees exactly with the two-point function of φ 2 including an effect of the φ 4 interaction.

AdS/CFT correspondence for the O(N) invariant critical φ 4 model in 3-dimensions by the conformal smearing

Q-balls are non-topological solitons in field theories whose stability is typically guaranteed by the existence of a global conserved charge. A classic realization is the Friedberg-Lee-Sirlin (FLS) Q-ball in a two-scalar system where a real scalar $\textit{χ}$ triggers symmetry breaking and confines a complex scalar Φ with a global U(1) symmetry. A quartic interaction $\textit{κχ}$ |Φ| with $\textit{κ}$ > 0 is usually considered to produce a nontrivial Q-ball configuration, and this repulsive force contributes to its stability. On the other hand, the attractive cubic interaction Λ$\textit{χ}$|Φ| is generally allowed in a renormalizable theory and could induce an instability. In this paper, we study the behavior of the Q-ball under the influence of this attractive force which has been overlooked. We find approximate Q-ball solutions in the limit of weak and moderate force couplings using the thin-wall and thick-wall approximations respectively. Our analytical results are consistent with numerical simulations and predict the parameter dependencies of the maximum charge. A crucial difference with the ordinary FLS Q-ball is the existence of the maximum charge beyond which the Q-ball solution is classically unstable. Such a limitation of the charge fundamentally affects Q-ball formation in the early Universe and could plausibly lead to the formation of primordial black holes.

Germany

Yu Hamada

TaeHun Kim

Philip Lu

Q-balls in the presence of attractive force

In ordinary thermodynamics, around first-order phase transitions, the intensive parameters such as temperature and pressure are automatically fixed to the phase transition point when one controls the extensive parameters such as total volume and total energy. From the microscopic point of view, the extensive parameters are more fundamental than the intensive parameters. Analogously, in conventional quantum field theory (QFT), coupling constants (including masses) in the path integral correspond to intensive parameters in the partition function of the canonical formulation. Therefore, it is natural to expect that, in a more fundamental formulation of QFT, coupling constants are dynamically fixed a posteriori, just as the intensive parameter in the microcanonical formulation. Here, we demonstrate that the automatic tuning of the coupling constants is realized at a quantum phase transition point at zero temperature, even when the transition is of higher order, due to the Lorentzian nature of the path integral. This naturally provides a basic foundation for the multicritical point principle. As a concrete toy model for solving the Higgs hierarchy problem, we study how the mass parameter is fixed in the <math><msup><mi>ϕ</mi><mn>4</mn></msup></math> theory at the one-loop level in the microcanonical or further generalized formulation of QFT. We find that there are two critical points for the renormalized mass: zero and of the order of ultraviolet cutoff. In the former, the Higgs mass is automatically tuned to be zero and thus its fine-tuning problem is solved. We also show that the quadratic divergence is absent in a more realistic two-scalar model that realizes the dimensional transmutation. Additionally, we explore the possibility of fixing quartic coupling in <math><msup><mi>ϕ</mi><mn>4</mn></msup></math> theory and find that it can be fixed to a finite value.

Quantum phase transition and absence of quadratic divergence in generalized quantum field theories

We construct a bulk spacetime from a boundary conformal field theory, <math><mi>O</mi><mo>(</mo><mi>N</mi><mo>)</mo></math> free scalar model, at finite temperature using a smearing technique, called a conformal flow. The bulk metric is constructed from the thermal two-point function of the smeared boundary elementary field in such a way that it can be interpreted as an information metric associated with the boundary thermofield double state. Near the boundary (UV region), an asymptotically anti–de Sitter (AdS) spacetime is obtained with a leading order perturbation of scalar mode. Based on the falloff behavior of the perturbations and the <math><mi>O</mi><mo>(</mo><mi>N</mi><mo>)</mo></math> symmetry in the conformal field theory, we argue that the corresponding bulk theory is a modified gravity with scalar mode such as <math><mi>f</mi><mo>(</mo><mi>R</mi><mo>)</mo></math> gravity rather than Einstein’s general relativity coupled minimally to matter fields. Moving to the Einstein frame, we show that the metric is asymptotically the same as the AdS black brane solution. On the other hand, deep in the bulk (IR region), the spacetime turns out to be conformally equivalent to the near horizon limit of AdS extremal black brane, though it is no longer a solution of <math><mi>f</mi><mo>(</mo><mi>R</mi><mo>)</mo></math> gravity, and hence more general classes of modified gravity need to be considered.