We revisit the scalar $O(N)$ model in the dimension range $4<d<6$ and study the effects caused by its metastability. As shown in previous work, this model formally possesses a fixed point where, perturbatively in the $1/N$ expansion, the operator scaling dimensions are real and above the unitarity bound. Here, we further show that these scaling dimensions do acquire small imaginary parts due to the instanton effects. In $d$ dimensions and for large $N$, we find that they are of order $e^{-N f(d)}$, where, remarkably, the function $f(d)$ equals the sphere free energy of a conformal scalar in $d-2$ dimensions. The non-perturbatively small imaginary parts also appear in other observables, such as the sphere free energy and two and three-point function coefficients, and we present some of their calculations. Therefore, at sufficiently large $N$, the $O(N)$ models in $4<d<6$ may be thought of as complex CFTs. When $N$ is large enough for the imaginary parts to be numerically negligible, the five-dimensional $O(N)$ models may be studied using the techniques of numerical bootstrap.
Small instantons can play an important role in Yang-Mills theories whose gauge couplings are sizeable at small distances. An interesting class of theories where this could occur is in weakly-gauged holographic models (dual to Yang-Mills theories interacting with strongly-coupled CFTs), since gauge couplings are indeed enhanced towards the UV boundary of the 5D AdS space. However, contrary to expectations, we show that small instantons in these non-asymptotically-free models are highly suppressed and ineffective. This is due to the conservation of topological charge that forbids instantons to b…
M5-branes on an ADE singularity are described by certain six-dimensional "conformal matter" superconformal field theories. Their Higgs moduli spaces contain information about various dynamical processes for the M5s; however, they are not directly accessible due to the lack of a Lagrangian formulation. Using anomaly matching, we compute their dimensions. The result implies that M5 fractions can recombine in several different ways, where the M5s are leaving behind frozen versions of the singularity. The anomaly polynomial gives hints about the nature of the freezing. We also check the Higgs dime…
Mani L. Bhaumik Institute for Theoretical Physics, Department of Physics and Astronomy, University of California, Los Angeles（加州大学洛杉矶分校曼尼·L·巴乌米克理论物理研究所物理学与天文学系）
We discuss the connections between the complex SYK model at the conformal limit and warped conformal field theories. Both theories have an $SL(2,R) \times U(1)$ 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.
In the search of YM instanton sheaves with topological charge two, the rank of beta matrix in the monad construction can be dropped from the bundle case with rank(beta)= 2 to either rank(beta) = 1 [4] or 0 on some points of CP^3 of the sheaf cases. In this paper, we first show that the sheaf case with rank(beta)= 0 does not exist for the previous construction of SU(2) complex YM instantons [3]. We then show that in the new "extended complex YM instantons" discovered in this paper, rank(beta) can be either 2 on the whole CP^3 (bundle) with some given ADHM data or 1, 0 on some points of CP^3 wit…
We study complex CFTs describing fixed points of the two-dimensional $Q$-state Potts model with $Q>4$. Their existence is closely related to the weak first-order phase transition and walking RG behavior present in the real Potts model at $Q>4$. The Potts model, apart from its own significance, serves as an ideal playground for testing this very general relation. Cluster formulation provides nonperturbative definition for a continuous range of parameter $Q$, while Coulomb gas description and connection to minimal models provide some conformal data of the complex CFTs. We use one and two-loop co…
We discuss walking behavior in gauge theories and weak first-order phase transitions in statistical physics. Despite appearing in very different systems (QCD below the conformal window, the Potts model, deconfined criticality) these two phenomena both imply approximate scale invariance in a range of energies and have the same RG interpretation: a flow passing between pairs of fixed point at complex coupling. We discuss what distinguishes a real theory from a complex theory and call these fixed points complex CFTs. By using conformal perturbation theory we show how observables of the walking th…
We develop an analytic approach to the four-point crossing equation in CFT, for general spacetime dimension. In a unitary CFT, the crossing equation (for, say, the s- and t-channel expansions) can be thought of as a vector equation in an infinite-dimensional space of complex analytic functions in two variables, which satisfy a boundedness condition in the u-channel Regge limit. We identify a useful basis for this space of functions, consisting of the set of s- and t-channel conformal blocks of double-twist operators in mean field theory. We describe two independent algorithms to construct the …
We prove the instability of $d$-dimensional conformal field theories (CFTs) having in the operator-product expansion of two fundamental fields a primary operator of scaling dimension $h=d/2+i r$, with non-vanishing $ r\in\mathbb{R}$. From an AdS/CFT point of view, this corresponds to a well-known tachyonic instability, associated to a violation of the Breitenlohner-Freedman bound in AdS${}_{d+1}$; we derive it here directly for generic $d$-dimensional CFTs that can be obtained as limits of multiscalar quantum field theories, by applying the harmonic analysis for the Euclidean conformal group t…
Relaxation Process During Complex Time Evolution In Two-Dimensional Integrable and Chaotic CFTs
University of Chinese Academy of Sciences（中国科学院大学）；RIKEN Interdisciplinary Theoretical and Mathematical Sciences (iTHEMS)（理研交叉理论与数学科学研究所(iTHEMS)）；Asia Pacific Center for Theoretical Physics（亚太理论物理中心）；University of Ljubljana（卢布尔雅那大学）；Georgia Institute of Technology（佐治亚理工学院）
We apply the OPE inversion formula to thermal two-point functions of bosonic and fermionic CFTs in general odd dimensions. This allows us to analyze in detail the operator spectrum of these theories. We find that nontrivial thermal CFTs arise when the thermal mass satisfies an algebraic transcendental equation that ensures the absence of an infinite set of operators from the spectrum. The solutions of these gap equations for general odd dimensions are in general complex numbers and follow a particular pattern. We argue that this pattern unveils the large-$N$ vacuum structure of the correspondi…
We study the famous example of weakly first order phase transitions in the 1+1D quantum Q-state Potts model at Q>4. We numerically show that these weakly first order transitions have approximately conformal invariance. Specifically, we find entanglement entropy on considerably large system sizes fits perfectly with the universal scaling law of this quantity in the conformal field theories (CFTs). This supports that the weakly first order transitions is proximate to complex fixed points, which are described by recent conjectured complex CFTs. Moreover, the central charge extracted from this fit…
Schwinger pair production is analyzed in a BPST instanton background and in its SL$(2,\mathbb{C})$ complex extension for complex scalar particles. A non-Abelian extension of the worldline instanton method is utilized, wherein Wong's equations in a coherent state picture adopted for SL$(2,\mathbb{C})$ are solved in Euclidean spacetime. While pair production is not predicted in the BPST instanton, a complex extension of the BPST instanton, existing as parallel fields in Minkowski spacetime, is shown to decay via the Schwinger effect.
We study fixed points with N scalar fields in $4 - \varepsilon$ dimensions to leading order in $\varepsilon$ using a bottom-up approach. We do so by analyzing O(N) invariants of the quartic coupling $λ_{ijkl}$ that describes such CFTs. In particular, we show that $λ_{iijj}$ and $λ_{ijkl}^2$ are restricted to a specific domain, refining a result by Rychkov and Stergiou. We also study averages of one-loop anomalous dimensions of composite operators without gradients. In many cases, we are able to show that the O(N) fixed point maximizes such averages. In the final part of this work, we generaliz…
We analyze two-dimensional nonlinear sigma models at nonzero chemical potentials, which are governed by a complex action. In the spirit of contour deformations (thimbles) we extend the fields into the complex plane, which allows to incorporate the chemical potentials mu as twisted boundary conditions. We write down the equations of motion and find exact BPS-like solutions in terms of pairs of (anti)holomorphic functions, in particular generalizations of unit charge and fractional instantons to generic mu. The decay of these solutions is controled by the imaginary part of mu and a vanishing ima…
Institute for Advanced Study, Kyushu University（九州大学高等研究院）；Department of Physics, Kyushu University（九州大学物理系）；RIKEN Interdisciplinary Theoretical and Mathematical Sciences (iTHEMS)（理化学研究所跨学科理论与数学科学中心）
Instantons on the Taub-NUT space are related to `bow solutions' via a generalization of the ADHM-Nahm transform. Both are related to complex geometry, either via the twistor transform or via the Kobayashi-Hitchin correspondence. We explore various aspects of this complex geometry, exhibiting equivalences. For both the instanton and the bow solution we produce two monads encoding each of them respectively. Identifying these monads we establish the one-to-one correspondence between the instanton and the bow solution.
We construct the moduli space of Spin(7)-instantons on a hermitian complex vector bundle over a closed 8-dimensional manifold endowed with a (possibly non-integrable) Spin(7)-structure. We find suitable perturbations that achieve regularity of the moduli space, so that it is smooth and of the expected dimension over the irreducible locus.
Flux Vacua Near the Boundary of Large Complex Structure
Harish-Chandra Research Institute（哈里什-昌德拉研究所）；Homi Bhabha National Institute（霍米·巴巴国立研究所）；Dipartimento di Fisica e Astronomia, Università di Bologna（博洛尼亚大学物理与天文学系）；INFN, Sezione di Bologna（意大利国家核物理研究所博洛尼亚分部）；Leinweber Institute for Theoretical Physics, Randall Laboratory of Physics, University of Michigan（密歇根大学莱恩韦伯理论物理研究所兰德尔物理实验室）；New York University Abu Dhabi（纽约大学阿布扎比分校）
We extend the use of Configurational Information Measures (CIMSs) to instantons and vacuum decay in arbitrary spatial dimensions. We find that both the complexity and the information content in the shape of instanton solutions have distinct regions of behavior in parameter space, discriminating between qualitative thin and thick wall profiles. For Euclidean spaces of dimension $D$, we show that for $D \gt 6$ instantons undergo a qualitative change of behavior from their lower- dimensional counterparts, indicating that $D=6$ is a critical dimension. We also find a scaling law relating the diffe…
Worldline instantons have previously been used to study the probability of Schwinger pair production (both the exponential and pre-exponential parts) and photon-stimulated pair production (the exponential part). Previous studies obtained the pair-production probability on the probability level by using unitarity, i.e. the imaginary part of the effective action for Schwinger pair production or the imaginary part of the polarization tensor for photon-stimulated pair production. The corresponding instantons are closed loops in the complex plane. Here we show how to use instantons on the amplitude…
This work is a continuation of our previous paper arXiv:1812.06473 where we have constructed ${\cal N}=2$ supersymmetric Yang-Mills theory on 4D manifolds with a Killing vector field with isolated fixed points. In this work we expand on the mathematical aspects of the theory, with a particular focus on its nature as a cohomological field theory. The well-known Donaldson-Witten theory is a twisted version of ${\cal N}=2$ SYM and can also be constructed using the Atiyah-Jeffrey construction. This theory is concerned with the moduli space of anti-self-dual gauge connections, with a deformation th…
Complex Worldline Instantons in (Anti-)de Sitter Space
University of Chicago（芝加哥大学）；Leinweber Institute for Theoretical Physics, University of Chicago（芝加哥大学莱因韦伯理论物理研究所）；Kavli Institute for Cosmological Physics, University of Chicago（芝加哥大学卡弗里宇宙物理学研究所）；Jefferson Physical Laboratory, Harvard University（哈佛大学杰斐逊物理实验室）；Department of Astronomy & Astrophysics, University of Chicago（芝加哥大学天文学与天体物理学系）
