We consider actions that are general functions of the world sheet/world volume metric and the induced metric for extended objects embedded in spacetime as Riemannian manifolds, area-metric manifolds, and volume-metric manifolds. For strings on a Riemannian spacetime, we consider general actions respecting volume-preserving diffeomorphisms (VPD), general diffeomorphisms, and diffeomorphisms with Weyl symmetry, respectivelyn. Well-known Schild, Nambu-Goto, and Polyakov actions are included as special cases. We reach two main conclusions: (1) When actions are functions of both the worldsheet metric and induced metrics, all nontrivial self-consistent actions are classically equivalent. (2) As a physical constraint on the classical action, VPD symmetry is as strong as the full diffeomorphism symmetry. The discussion is then extended to strings in spacetime manifolds equipped with the area or volume metrics. Then, we further consider higher-dimensional extended objects in spacetime defined with area or volume metrics, and show the equivalence between the generalized Schild actions and the generalized Nambu-Goto action. We prove a general theorem on VPD that explains this equivalence. Incidentally, while only the area metric is needed to define the string world sheet action, we show that the Polyakov action with an area-metric perturbation cannot describe critical strings without other interaction terms.

Taiwan

Pei-Ming Ho

Japan

Hikaru Kawai

Henry Liao

General actions of extended objects and volume-preserving diffeomorphism

The matrix models are non-perturbative formulations of string theory, from which many believe that spacetime arises. The matrix fluctuations around the spacetime thus created should represent both matter and gravitational fields. In this paper, we discuss how the gravitational field emerges from the IIB matrix model. In particular, we consider how diffeomorphism invariance arises and how unitarity is guaranteed in this theory. Specifically, we consider matrices as bilocal fields and discuss how the Lorentz-invariant vacuum and low-energy excitations around it can be expressed. We then discuss how the conditions for the theory to be unitary can be written in terms of bilocal fields. We argue that in the low-energy limit, the bilocal fields are reduced to local fields consisting of a finite number of massless fields and an infinite number of massive fields, satisfying unitarity.

Austria

Harold Steinacker

General Relativity in IIB matrix model

The Ishibashi-Kawai-Kitazawa-Tsuchiya matrix model, from the holographic perspective, arises at the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>p</mi><mo>=</mo><mo>−</mo><mn>1</mn></math> end point of the family of dualities relating type II supergravities on near-horizon <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi mathvariant="normal">D</mi><mi>p</mi></mrow></math> brane geometries to (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>p</mi><mo>+</mo><mn>1</mn></mrow></math>)-dimensional super Yang-Mills theories with 16 supercharges. In this work, we detail and expand results reported in a recent paper by establishing the holographic dictionary between gauge-invariant operators in the lowest Bogomol’nyi-Prasad-Sommerfield multiplet of the matrix model and the corresponding Kaluza-Klein fluctuations of Euclidean IIB supergravity compactified on the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi mathvariant="normal">D</mi><mo stretchy="false">(</mo><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></math> instanton background. The full nonlinear dynamics of these fluctuations can be encoded in a one-dimensional maximal gauged supergravity, which we construct explicitly. We provide its complete Lagrangian, up to second order in fermions, together with the corresponding supersymmetry transformations. We further discuss real forms of the theory for noncompact gauge groups and their embeddings into ten-dimensional supergravities with Lorentzian signature. From the analysis of the one-dimensional Killing spinor equations, we derive different classes of half-supersymmetric solutions, and discuss their uplifts as well as their relations to known solutions.

France

Franz Ciceri

Henning Samtleben

Supergravity dual for Ishibashi-Kawai-Kitazawa-Tsuchiya holography

We study the quantum aspects of the conformal gravity in four dimensions, specifically addressing a known discrepancy in beta functions between general quadratic curvature theories and conformal gravity, which corresponds to two scalar degrees of freedom. We demonstrate that this mismatch is resolved by carefully introducing gauge-fixing and ghost terms via the BRST symmetry, which effectively adds the two scalar modes. Drawing lessons from two-dimensional quantum gravity and Liouville theory, we proceed to integrate the four-dimensional trace anomaly to derive a consistent Liouville action, which is given by a free-field action for the conformal mode with a consistent conformal anomaly. Finally we give the condition that the BRST transformation is anomaly free.

China

Nobuyoshi Ohta

Noncritical conformal gravity and four-dimensional Liouville theory

We investigate the twisted Eguchi-Kawai (TEK) reduced model of four-dimensional <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>S</mi><mi>U</mi><mo stretchy="false">(</mo><mi>N</mi><mo stretchy="false">)</mo></math> gauge theory in the presence of two-flavor adjoint fermions (adjoint TEK model). Using Monte Carlo simulations with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>N</mi><mo>=</mo><mn>121</mn></math>, twist parameter <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>k</mi><mo>=</mo><mn>1</mn></math>, hopping parameter <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>κ</mi><mo>=</mo><mn>0.01</mn><mi>–</mi><mn>0.03</mn></mrow></math> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mo>≪</mo><msub><mi>κ</mi><mi>c</mi></msub></math>) and inverse ’t Hooft coupling <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>b</mi><mo>=</mo><mn>0.30</mn><mi>–</mi><mn>0.45</mn></math>, we show that heavy adjoint fermions stabilize the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mo stretchy="false">(</mo><msub><mi mathvariant="double-struck">Z</mi><mi>N</mi></msub><msup><mo stretchy="false">)</mo><mn>4</mn></msup></math> center-symmetric vacuum even for the minimal twist satisfying <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>k</mi><mo>/</mo><msqrt><mi>N</mi></msqrt><mo>&lt;</mo><mn>1</mn><mo>/</mo><mn>9</mn></math>, where the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mo stretchy="false">(</mo><msub><mi mathvariant="double-struck">Z</mi><mi>N</mi></msub><msup><mo stretchy="false">)</mo><mn>4</mn></msup></math> symmetry is spontaneously broken in the absence of adjoint fermions. This result also suggests that the adjoint TEK model with the minimal twist is equivalent to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>S</mi><mi>U</mi><mo stretchy="false">(</mo><mi>N</mi><mo stretchy="false">)</mo></math> gauge theory over a broader <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mo stretchy="false">(</mo><mi>κ</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo></math> parameter region than the adjoint EK model without twist. We further extend our analysis to a partially reduced model to realize a geometry akin to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi mathvariant="double-struck">R</mi><mn>3</mn></msup><mo>×</mo><msup><mi>S</mi><mn>1</mn></msup></math> and study the theory under <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>S</mi><mn>1</mn></msup></math> compactification with periodic adjoint fermions. Numerical simulations with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>N</mi><mo>=</mo><mn>16</mn><mi>–</mi><mn>49</mn></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>b</mi><mo>=</mo><mn>0.30</mn><mi>–</mi><mn>0.46</mn></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>κ</mi><mo>=</mo><mn>0.03</mn><mi>–</mi><mn>0.16</mn></math> are consistent with the adiabatic continuity conjecture; with periodic adjoint fermions, the theory remains in a center-symmetric (confined) phase as the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>S</mi><mn>1</mn></msup></math> circle size is reduced, in contrast to the deconfining transition observed in the pure TEK model or in the TEK model with antiperiodic adjoint fermions. We present the Polyakov loop measurements and consistency checks supporting these findings.

Yudai Hamada

Tatsuhiro Misumi

<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi mathvariant="double-struck">Z</mi><mi>N</mi></msub></math> stability and continuity in the twisted Eguchi-Kawai model with two-flavor adjoint fermions

A new perspective on the inverse string theory Kawai-Lewellen-Tye (KLT) kernel is provided which establishes the universality of scattering amplitudes in the biadjoint scalar theory, pions in the nonlinear sigma model (NLSM), and mixed amplitudes (NLSM+<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>ϕ</mi><mn>3</mn></msup></math>) recently studied in the literature. We show that all these amplitudes can be viewed as equivalent, arising from a single function, the inverse string theory KLT kernel, evaluated at different kinematic points. In this way cubic colored scalars and pions become interchangeable through a procedure we call the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>α</mi><mo>′</mo></msup></math> shift. The latter complements the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>δ</mi></math> shift proposed by Arkani-Hamed et al., and demonstrates an inherent equivalence of scattering amplitudes in different quantum field theories by embedding them in a common stringy framework.

Czechia

Christoph Bartsch

Karol Kampf

Jiří Novotný

United States

Jaroslav Trnka

Universality of colored scalars from the stringy Kawai-Lewellen-Tye kernel

Holographic dualities that relate type II strings on near-horizon <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi mathvariant="normal">D</mi><mi>p</mi></mrow></math> brane geometries to super Yang-Mills theories with sixteen supercharges in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>p</mi><mo>+</mo><mn>1</mn></math> dimensions provide nonconformal generalizations of the famous AdS/CFT correspondence. For the extremal case <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>p</mi><mo>=</mo><mo>−</mo><mn>1</mn></math>, this suggests a holographic duality for the Ishibashi-Kawai-Kitazawa-Tsuchiya (IKKT) matrix model—super Yang-Mills theory reduced to zero dimensions. Despite intriguing and highly nontrivial results in the IKKT model, this duality has remained largely unexplored so far. In this Letter, we consider the lowest supermultiplet of gauge invariant operators of the model and identify its states with the lowest Kaluza-Klein fluctuations of (Euclidean) IIB supergravity on the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi mathvariant="normal">D</mi><mo stretchy="false">(</mo><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></math> instanton background. We construct its holographic bulk realization as a one-dimensional maximal supergravity with 32 supercharges and local SO(10) invariance, capturing the full nonlinear dynamics. Analyzing the bulk Killing spinor equations, we construct the most general half-supersymmetric solutions, which typically break SO(10). We present their uplifts to IIB supergravity, and furthermore to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>p</mi><mi>p</mi></mrow></math> waves in twelve dimensions. These results provide the minimal setup for conducting precision tests of holography involving Einstein gravity.

Holography for the Ishibashi-Kawai-Kitazawa-Tsuchiya Matrix Model

Recently it is asserted that the standard beta function does not describe the correct running of the coupling constant in some theories. We show that the problem arises from the assumption <math altimg='si1.svg'><mrow><mi>&#956;</mi><mo linebreak='goodbreak'>=</mo><mi>p</mi></mrow></math> ( <math altimg='si2.svg'><mi>&#956;</mi></math> is a renormalization point) and that a suitable choice of <math altimg='si2.svg'><mi>&#956;</mi></math> gives the correct running. It is also claimed that neither the cosmological constant nor Newton coupling run. We argue that running can be discussed when we consider the curved spacetime.

Standard running, “physical running”, cosmological constant and Newton coupling

We discuss the effect of wave function renormalization (WFR) in asymptotically safe gravity. We show that there are two WFR-invariant quantities, and the renormalization (RG) equations may be written entirely in terms of these quantities. The same set of RG equations can be obtained whether we fix the vacuum energy or Newton coupling along the RG trajectory. The flow of the Newton constant and the vacuum energy is also discussed in detail. In particular we discuss how the vacuum energy behaves near the singular barrier in the low energy.

Wave function renormalization in asymptotically safe quantum gravity

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.