Past String Theory Seminar

The Bethe-Gauge Correspondence (BGC) of Nekrasov and Shatashvili, linking quantum integrable spin chains to two-dimensional supersymmetric gauge theories with N=2 supersymmetry, stands out as a significant instance of the deep connection between supersymmetric gauge theories and integrable models. In this talk, I will delve into this correspondence and its origins for superspin chains. To achieve this, I will first elucidate the Bethe Side and its corresponding Gauge Side of the BGC. Subsequently, it becomes evident that the BGC can be naturally realized within String Theory. I will initially outline the brane configuration for the realization of the Gauge Side. Through the use of string dualities, this brane configuration will be mapped to another, embodying the Bethe Side of the correspondence. The 4D Chern-Simons theory plays a crucial role in this latter duality frame, elucidating the integrability of the Bethe Side. Lastly, I will elaborate on computing the main object of interest for integrable superspin chains—the R-matrix—from the BGC. While this provides a rather comprehensive picture of the correspondence, some important questions remain for further clarification. I will summarize some of the most interesting ones at the end of the talk.

In this talk, I will discuss correlation functions in 6d (2, 0) theories of two 1/2-BPS operators inserted away from a 1/2-BPS surface defect. In the large central charge limit the leading connected contribution corresponds to sums of tree-level Witten diagram in AdS7×S4 in the presence of an AdS3 defect. I will show that these correlators can be uniquely determined by imposing only superconformal symmetry and consistency conditions, eschewing the details of the complicated effective Lagrangian. I will present the explicit result of all such two-point functions, which exhibits remarkable hidden simplicity.

 I will explore subtle aspects of rank-one 4d N=2 supersymmetric QFTs through their low-energy Coulomb-branch physics. This low-energy Lagrangian is famously encoded in the Seiberg-Witten (SW) curve, which is a one-parameter family of elliptic curves. Less widely appreciated is the fact that various properties of the QFTs, including properties that cannot be read off from the Lagrangian, are nonetheless encoded into the SW curve, in particular in its Mordell-Weil group. This includes the global form of the flavour group, the one-form symmetries under which defect lines are charged, and the "global form" of the theory. In particular, I will discuss in detail the difference between the pure SU(2) and the pure SO(3) N=2 SYM theories from this perspective. I will also comment on 5d SCFTs compactified on a circle in this context.

I will discuss a KLT relation of closed string amplitudes into open string amplitudes for closed string states carrying winding and momentum in toroidal compactifications. The goal is to introduce an interesting D-brane set up in the target space in order to accommodate both quantum numbers of the closed string. I will then discuss KLT factorization of amplitudes for winding closed strings in the presence of a critical Kalb-Ramond field and the relevance of this work for nonrelativistic string theory when taking the zero Regge limit. 

I will review some aspects of gravity in asymptotically flat spacetime and mention important challenges to obtain a holographic description in this framework. I will then present two important approaches towards flat space holography, namely Carrollian and celestial holography, and explain how they are related to each other. Similarities and differences between flat and anti-de Sitter spacetimes will be emphasized throughout the talk. 
 

Recent progress in AdS/CFT has provided a good understanding of how the bulk spacetime is encoded in the entanglement structure of the boundary CFT. However, little is known about how spacetime emerges directly from the bulk quantum theory. We address this question in AdS3 pure gravity, which we formulate as a topological quantum field theory. We explain how gravitational entropy can be viewed as bulk entanglement entropy of gravitational edge modes.  These edge modes transform under a quantum group symmetry. This suggests an effective description of bulk microstates in terms of collective, anyonic degrees of freedom whose entanglement leads to the emergence of the bulk spacetime.  Time permitting we will discuss a proposal for how our bulk TQFT arises from an ensemble of approximate CFT’s, generalizing the relation between JT gravity and random matrix ensemble.

I will talk about supersymmetric quantum field theories with 8 supercharges in dimensions 3-6. After a brief introduction I will mostly speak about the moduli space of vacua of such theories, and in particular their Higgs branches, which are so called symplectic singularities (or mild generalisations thereof). Powerful theorems from mathematics say that a singular Higgs branch is stratified into a disjoint union of smooth open subsets, so called symplectic leaves. This stratification matches exactly the pattern of partial Higgsings of the theory in question. After introducing the stratification and explaining its physical interpretation, I will show how brane systems and so called magnetic quivers can be used to compute it.

We review the construction of non-invertible duality defects in various dimensions. We explain how they can be preserved along RG flows and how their realization on gapped phases contains their 't Hooft anomalies. We finally give a presentation of the anomalies from the Symmetry TFT. Time permitting I will discuss some possible future applications.

We shall consider a crossing equation of the Euclidean conformal group in terms of conformal partial waves and in particular, a position independent representation of this equation. We shall briefly discuss the relevance of this equation to the problem of cosmological bootstrap. Thereafter, we shall sketch the derivation of the Biedenharn-Eliiot identity (a pentagon identity) for the 6j symbols of the conformal group and show how this provides us with an exact solution to said crossing equation. For the conformal group (which is non-compact), this involves some careful bookkeeping of the spinning representations. Finally, we shall discuss some consistency checks on the result obtained, and some open questions. 

Scattering amplitudes in string theory are written as formal integrals of correlations functions over the moduli space of punctured Riemann surfaces. It's well-known, albeit not often emphasized, that this prescription is only approximately correct because of the ambiguities in defining the integration domain. In this talk, we propose a resolution of this problem for one-loop open-string amplitudes and present their first evaluation at finite energy and scattering angle. Our method involves a deformation of the integration contour over the modular parameter to a fractal contour introduced by Rademacher in the context of analytic number theory. This procedure leads to explicit and practical formulas for the one-loop planar and non-planar type-I superstring four-point amplitudes, amenable to numerical evaluation. We plot the amplitudes as a function of the Mandelstam invariants and directly verify long-standing conjectures about their behavior at high energies.

Scattering amplitudes in string theory are written as formal integrals of correlations functions over the moduli space of punctured Riemann surfaces. It's well-known, albeit not often emphasized, that this prescription is only approximately correct because of the ambiguities in defining the integration domain. In this talk, we propose a resolution of this problem for one-loop open-string amplitudes and present their first evaluation at finite energy and scattering angle. Our method involves a deformation of the integration contour over the modular parameter to a fractal contour introduced by Rademacher in the context of analytic number theory. This procedure leads to explicit and practical formulas for the one-loop planar and non-planar type-I superstring four-point amplitudes, amenable to numerical evaluation. We plot the amplitudes as a function of the Mandelstam invariants and directly verify long-standing conjectures about their behavior at high energies.

Recently, a relation was introduced connecting codes of various types with the space of abelian (Narain) 2d CFTs. We extend this relation to provide holographic description of code CFTs in terms of abelian Chern-Simons theory in the bulk. For codes over the alphabet Z_p corresponding bulk theory is, schematically, U(1)_p times U(1)_{-p} where p stands for the level. Furthermore, CFT partition function averaged over all code theories for the codes of a given type is holographically given by the Chern-Simons partition function summed over all possible 3d geometries. This provides an explicit and controllable example of holographic correspondence where a finite ensemble of CFTs is dual to "topological/CS gravity" in the bulk. The parameter p controls the size of the ensemble and "how topological" the bulk theory is. Say, for p=1 any given Narain CFT is described holographically in terms of U(1)_1^n times U(1)_{-1}^n Chern-Simons, which does not distinguish between different 3d geometries (and hence can be evaluated on any of them). When p approaches infinity, the ensemble of code theories covers the whole Narain moduli space with the bulk theory becoming "U(1)-gravity" proposed by Maloney-Witten and Afkhami-Jeddi et al.

We argue that the existence of solitons in theories in which local symmetries are spontaneously broken requires spacetime to be enlarged by additional coordinates that are associated with large local transformations. In the context of gravity theories the usual coordinates of spacetime can be thought of arising in this way. E theory automatically contains such an enlarged spacetime. We propose that spacetime appears in an underlying theory when the local symmetries are spontaneously broken.

There has been recent advances in understanding the microscopic origin of the Bekenstein-Hawking entropy of supersymmetric ant de Sitter (AdS) black holes using holography and localization applied to the dual quantum field theory. In this talk, after a brief overview of the general picture, I will propose a BPS partition function -- based on gluing elementary objects called gravitational blocks -- for known AdS black holes with arbitrary rotation and generic magnetic and electric charges. I will then show that the attractor equations and the Bekenstein-Hawking entropy can be obtained from an extremization principle.

Random unitary circuits (RUCs) have served as important sources of insights in studying operator dynamics. While the simplicity of RUCs allows us to understand the nature of operator growth in a quantitative way, randomness of the dynamics in time prevents them to capture certain aspects of operator dynamics. To explore these aspects, in this talk, I consider the operator dynamics of a minimal Floquet many-body circuit whose time-evolution operator is fixed at each time step. In particular, I compute the partial spectral form factor of the model and show that it displays nontrivial universal physics due to operator dynamics. I then discuss the out-of-ordered correlator of the system, which turns out to capture the main feature of it in a generic chaotic many-body system, even in the infinite on-site Hilbert space dimension limit.

I will discuss various possible ways a global symmetry can act on operators in a quantum field theory. The possible actions on q-dimensional operators are referred to as q-charges of the symmetry. Crucially, there exist generalized higher-charges already for an ordinary global symmetry described by a group G. The usual charges are 0-charges, describing the action of the symmetry group G on point-like local operators, which are well-known to correspond to representations of G. We find that there is a neat generalization of this fact to higher-charges: i.e. q-charges are (q+1)-representations of G. I will also discuss q-charges for generalized global symmetries, including not only invertible higher-form and higher-group symmetries, but also non-invertible categorical symmetries. This talk is based on a recent (arXiv: 2304.02660) and upcoming works with Sakura Schafer-Nameki.

I will discuss how to place conformal boundary conditions on free higher derivative theories of scalars and spinors as well as the zoo of boundary renormalization group flows that connect the different boundary conditions.  Historically, there are connections to Poisson’s ratio and classical equations governing the bending of thin steel plates.  As these higher derivative theories are often invoked in the context of cosmology, there may be cosmological applications for the boundary conditions discussed here.
 

In this talk I will discuss a novel mechanism  that couples matter fields to three-dimensional de Sitter quantum gravity. This construction is based on the Chern-Simons formulation of three-dimensional Euclidean gravity, and it centers on a collection of Wilson loops winding around Euclidean de Sitter space. We coin this object a Wilson spool.  To construct the spool, we build novel representations of su(2). To evaluate the spool, we adapt and exploit several known exact results in Chern-Simons theory. Our proposal correctly reproduces the one-loop determinant of a free massive scalar field on S^3 as G_N->0. Moreover, allowing for quantum metric fluctuations, it can be systematically evaluated to any order in perturbation theory.   

We propose new G2-holonomy manifolds, which geometrize the Gaiotto-Kim 4d N = 1 duality
domain walls of 5d N = 1 theories. These domain walls interpolate between different extended
Coulomb branch phases of a given 5d superconformal field theory. Our starting point is the
geometric realization of such a 5d superconformal field theory and its extended Coulomb
branch in terms of M-theory on a non-compact singular Calabi-Yau three-fold and its Kahler
cone. We construct the 7-manifold that realizes the domain wall in M-theory by fibering the
Calabi-Yau three-fold over a real line, whilst varying its Kahler parameters as prescribed by
the domain wall construction. In particular this requires the Calabi-Yau fiber to pass through
a canonical singularity at the locus of the domain wall. Due to the 4d N = 1 supersymmetry
that is preserved on the domain wall, we expect the resulting 7-manifold to have holonomy G2.
Indeed, for simple domain wall theories, this construction results in 7-manifolds, which are
known to admit torsion-free G2-holonomy metrics. We develop several generalizations to new
7-manifolds, which realize domain walls in 5d SQCD theories.

I will present results from arXiv:2202.13924, where we studied the difference between integrable and chaotic motion in quantum theory as manifested by the complexity of the corresponding evolution operators. The notion of complexity of interest to us will be Nielsen’s complexity applied to the time-dependent evolution operator of the quantum systems. I will review Nielsen’s complexity, discuss the difficulties associated with this definition and introduce a simplified approach which appears to retain non-trivial information about the integrable properties of the dynamical systems.

A 2d SCFT given as a non-linear sigma model of a Ricci-flat Kahler target 

space is not a rational CFT in general; only special points in the moduli 

space of the target-space metric, the 2d SCFTs are rational. 

Gukov-Vafa's paper in 2002 hinted at a possibility that such special points 

may be characterized by the property "complex multiplication" of the target space, 

which has its origin in number theory. We revisit the idea, refine the Conjecture, 

and prove it in the case the target space is T^4. 
 

This presentation is based on arXiv:2205.10299 and 2212.13028 .

5d Superconformal Field Theories (SCFTs) are intrinsically strongly-coupled UV fixed points, whose realization hinges on string theoretic methods: they can be constructed by compactifying M-theory on local Calabi-Yau threefold singularities or alternatively from the world-volume of 5-brane-webs in type IIB string theory. There is a correspondence between 5-brane-webs and toric Calabi-Yau threefolds, however this breaks down when multiple 5-branes are allowed to end on a single 7-brane. In this talk, we extend this connection and provide a geometric realization of brane configurations including 7-branes. Along the way, we also review techniques developed in the past few years to describe the Higgs branch of these 5d SCFTs, including magnetic quivers and Hasse diagram for symplectic singularities. 

In her recent work, Mina Aganagic proposed novel perspectives on computing knot homologies associated with any simple Lie algebra. One of her proposals relies on counting intersection points between Lagrangians in Landau-Ginsburg models on symmetric powers of Riemann surfaces. In my talk, I am going to present a concrete algebraic algorithm for finding such intersection points, turning the proposal into an actual calculational tool. I am going to illustrate the construction on the example of the sl_2 invariant for the Hopf link. I am also going to comment on the extension of the story to homological invariants associated to gl(m|n) super Lie algebras, solving this long-standing problem. The talk is based on our work in progress with Mina Aganagic and Elise LePage.

When do two quantum field theories describe the same physics? I will discuss some approaches to this question in the context of superconformal field theories in four and six dimensions. First, I will discuss the construction of 6d (1,0) SCFTs from the perspective of the "atomic classification", focussing on an oft-overlooked subtlety whereby distinct SCFTs in fact share an effective description on the generic point of the tensor branch. We will see how to determine the difference in the Higgs branch operator spectrum from the atomic perspective, and how that agrees with a dual class S perspective. I will explain how other 4d N=2 SCFTs, which a priori look like distinct theories, can be shown to describe the same physics, as they arise as torus-compactifications of identical 6d theories.

In recent progress on the black hole information paradox, Page curves consistent with unitarity have been obtained in 2d models and in bottom-up braneworld models using the notion of double holography. In this talk we discuss top-down models realizing 4d black holes coupled to a bath in Type IIB string theory and obtain Page curves. We make the ideas behind double holography precise in these models and address causality puzzles which have arisen in the bottom-up models, leading to a refinement of their interpretation.
 

We study supersymmetric gauge theories with 8 supercharges in d=3,4,5,6. For these theories, one can perform Higgsings by turning on VEVs of scalar fields. However, this process can often be difficult when dealing with superconformal field theories (SCFTs) where the Lagrangian is often not known. Using techniques of magnetic quivers and a new algorithm we call "Inverted Quiver Subtraction", we show how one can easily obtain the SCFT(s) after Higgsing. This technique can be equally well applied to SCFTs in d=3,4,5,6. 

Conformal Field Theories (CFT) are believed to be exactly solvable once their primary scaling fields and their 3-point functions are known. This input is called the spectrum and structure constants of the CFT respectively. I will review recent work where this conformal bootstrap program can be rigorously carried out for the case of Liouville CFT, a theory that plays a fundamental role in 2d random surface theory and many other fields in physics and mathematics. Liouville CFT has a probabilistic formulation on an arbitrary Riemann surface and the bootstrap formula can be seen as a "quantization" of the plumbing construction of surfaces with marked points axiomatically discussed earlier by Graeme Segal. Joint work with Colin Guillarmou, Remi Rhodes and Vincent Vargas.

After reviewing different aspects of thermalization and chaos in holographic quantum systems, I will argue that universal aspects can be captured using an effective field theory framework that shares similarities with hydrodynamics. Focusing on the quantum butterfly effect, I will explain how to develop a simple effective theory of the 'scramblon' from path integral considerations. I will also discuss applications of this formalism to shockwave scattering in black hole backgrounds in AdS/CFT.

Only a decade ago the detection of gravitational waves seemed like a fantasy to most, and merely a handful of 
people in the world believed in the validity and even great potential of using the powerful framework of EFT, and 
more generally -- advances in QFT to study gravity theory for real-world gravitational waves. I will present the 
significant advancement accomplished uniquely via the tower of EFTs with the EFT of spinning gravitating objects, 
and the incorporation of QFT advances, which my work has pioneered since those days. Today, only 6 years after 
the official birth of precision gravity with a rapidly growing influx of gravitational-wave data, and a decade of great 
theoretical progress, the power and insight of using modern QFT for real-world gravity have become incontestable.

Argyres—Douglas (AD) theories are 4d N=2 SCFTs which have some unusual features, and until recently, explicit holographic duals of these theories were unknown. We will consider a concrete class of these theories obtained by wrapping the 6d N=(2,0) ADE theories on a (twice) punctured sphere: one irregular and one regular puncture, and construct their holographic duals. The novel aspects of these solutions require a relaxation of the regularity conditions of the usual Gaiotto—Maldacena framework and to allow for brane singularities. We show how to construct the dictionary between the AdS(5) solutions and the field theory and match observables between the two. If time allows, I will comment on some on-going work about further compactifying the AD theories on spindles, or the 6d theories on four-dimensional orbifolds. 

I will describe a procedure, known as holomorphic twist, to isolate protected quantities in supersymmetric quantum field theories. The resulting theories are holomorphic, interacting and have infinite dimensional symmetries, analogous to the holomorphic half of a 2D CFT. I will explain how to study quantum corrections to these symmetries and other  higher operations.
As a surprise, we find a novel UV manifestation of
confinement, dubbed "holomorphic confinement," in the example of pure
SU(N) super Yang-Mills.

Quantum field theories (QFTs) in d dimensions that posses a (d-1)-form symmetry are conjectured to decompose into disjoint “universes”, each of which is itself a (local and unitary) QFT. I will give an overview of our current understanding of decomposition, and then discuss how this phenomenon occurs in the fusion of condensation defects of certain 3d QFTs. This gives a “microscopic” explanation of why in these instances, the fusion coefficient can be taken as an integer rather than a general TQFT.

According to the correspondence principle, classical physics emerges in the limit of large quantum numbers. We examine three examples of the semiclassical description of conformal field theory data: large charge boundary operators in the O(2) model, large spin impurities in the free triplet scalar field theory and large charge Wilson lines in QED. By simultaneously taking the coupling to zero and quantum numbers to infinity, we can connect the microscopic to the emergent classical description smoothly.

Liouville conformal field theory models two-dimensional gravity with a cosmological constant and conformal matter. In its timelike regime, it reproduces the characteristic negative kinetic term of the conformal factor of the metric in the Einstein-Hilbert action, the sign which infamously makes the gravity path integral ill-defined. In this talk, I will first discuss the perturbative computation of the timelike Liouville partition function around the sphere saddle and propose an all-orders result. I will then turn to the disk and present the bulk 1-point functions of this CFT, and discuss possible interpretations in terms of boundary conditions.

I will discuss the free compact boson CFT thrown out of equilibrium by marginal deformations, modeled by quenching or periodically driving the compactification radius of the free boson between two different values. All the dynamics will be shown to be crucially dependent on the ratio of the compactification radii via the Zamolodchikov distance in the space of marginal deformations. I will present various exact analytic results for the Loschmidt echo and the time evolution of energy density for both the quench and the periodic drive. Finally, I will present a non-perturbative computation of the  Rényi divergence, an information-theoretic distance measure, between two marginally deformed thermal density matrices.

The talk will be based on the recent preprint: arXiv:2206.11287

I will discuss the large N behavior of partition functions of the ABJM theory on compact Euclidean manifolds. I will pay particular attention to the S^3 free energy and the topologically twisted index for which I will present closed form expressions valid to all order in the large N expansion. These results have important implications for holography and the microscopic entropy counting of AdS_4 black holes which I will discuss. I will also briefly discuss generalizations to other SCFTs arising from M2-branes.

The line operators in a 2+1D TQFT form an algebraic structure called a modular tensor category (MTC). There is a natural action of a Galois group on MTCs which maps a given TQFT to other 'Galois conjugate' TQFTs. I will describe this Galois action and give several examples of Galois conjugate TQFTs. Galois action on a unitary TQFT can result in a non-unitary TQFT. I will derive a sufficient condition under which unitarity is preserved. Finally, I will describe the invariance of 0-form and 1-form symmetries of TQFTs under Galois action.    

Topological defects in quantum field theory can be understood as a generalised notion of symmetry, where the operation is not required to be invertible. Duality transformations are an important example of this. By considering defects of various dimensions, one is naturally led to more complicated algebraic structures than just groups. So-called 2-groups are a first instance, which arise from invertible defects of codimension 1 and 2. Without invertibility one arrives at so-called "fusion categories”. I would like to explain how one can "gauge" such non-invertible symmetries in the case of topological field theories, and I will focus on results in two and three dimensions. This talk is based on joint work with Nils Carqueville, Vincentas Mulevicius, Gregor Schaumann, and Daniel Scherl.

We consider the four-point correlator of the stress-energy tensor in N=4 SYM, to leading order in inverse powers of the central charge, but including all order corrections in 1/lambda. This corresponds to the AdS version of the Virasoro-Shapiro amplitude to all orders in the small alpha'/low energy expansion. Using dispersion relations in Mellin space, we derive an infinite set of sum rules. These sum rules strongly constrain the form of the amplitude, and determine all coefficients in the low energy expansion in terms of the CFT data for heavy string operators, in principle available from integrability. For the first set of corrections to the flat space amplitude we find a unique solution consistent with the results from integrability and localisation.

I will review aspects of the theory of symmetry-protected topological phases, focusing on the case of one-dimensional quantum chains. Important concepts include the bulk-boundary correspondence, with bulk topological invariants leading to interesting boundary phenomena. I will discuss topological invariants and associated boundary phenomena in the case that the system is gapless and described at low energies by a conformal field theory. Based on work with Ruben Verresen, Ryan Thorngren and Frank Pollmann.

There have been several proposals of scale-separated AdS vacua in the literature. All known examples arise from the effective field theory of flux compactifications with low supersymmetry, and there are often doubts about their consistency as 10 or 11d backgrounds in string theory. These issues can often be tackled in the bulk theory, or by analysis of the dual CFT via holography. I will review the most common issues, and focus the analysis on the recently constructed family of 3d scale-separated AdS vacua, which is dual to a two-dimensional CFT, emphasizing the discrete symmetry structure of the model in comparison to DGKT. Finally, I will comment on the tantalizing observation of integer operator dimensions in DGKT-like vacua, and comment on possible places to look for consistency issues in these models.

Holographic correlation functions are under good analytic control when none of the single trace operators live in long multiplets. This is famously the case for SCFTs with sixteen supercharges but it is also possible to construct examples with eight supercharges by exploiting space filling branes in AdS. In particular, one can study 4d N=2 theories which are related to each other by an S-fold in much the same way that N=3 theories are related to N=4 Super Yang-Mills. I will describe how modern methods provide a window into their correlation functions with an emphasis on anomalous dimensions. To compare the different S-folds we will need to go to one loop, and to go to one loop we will need to account for operator mixing. This provides an example of resolving degeneracy by resolving degeneracy.

Great progress has been made recently in exploiting categorical/topological/higher symmetries in quantum field theory. I will explain how the same structure is realised directly in the lattice models of statistical mechanics, generalizing Kramers-Wannier duality to a wide class of models. In particular, I will give an overview of my work with Aasen and Mong on using fusion categories to find and analyse lattice topological defects in two and 1+1 dimensions.  These defects possess a variety of remarkable properties. Not only is the partition function is independent of deformations of their path, but they can branch and fuse in a topologically invariant fashion.  The universal behaviour under Dehn twists gives exact results for scaling dimensions, while gluing a topological defect to a boundary allows universal ratios of the boundary g-factor to be computed exactly on the lattice.  I also will describe how terminating defect lines allows the construction of fractional-spin conserved currents, giving a linear method for Baxterization, I.e. constructing integrable models from a braided tensor category.

 I will present a scenario where the early universe is in a topological phase of gravity.  I will discuss a number of analogies which motivate considering gravity in such a phase. Cosmological puzzles such as the horizon problem provide a phenomenological connection to this phase and can be explained in terms of its topological nature. To obtain phenomenological estimates, a concrete realization of this scenario using Witten's four dimensional topological gravity will be used. In this model, the CMB power spectrum can be estimated by certain conformal anomaly coefficients. A qualitative prediction of this phase is the absence of tensor modes in cosmological fluctuations.