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2D conformal field theory used in string theory
In physics, the Polyakov action is an action of the two-dimensional conformal field theory describing the worldsheet of a string in string theory. It was
Polyakov_action
Russian theoretical physicist
Alexander Markovich Polyakov (Russian: Алекса́ндр Ма́ркович Поляко́в; born 27 September 1945) is a Russian theoretical physicist, formerly at the Landau
Alexander Polyakov (physicist)
Alexander_Polyakov_(physicist)
Physical quantity of dimension energy × time
least action Principle of maximum entropy Some actions: Bagger–Lambert–Gustavsson action Einstein–Hilbert action Nambu–Goto action Polyakov action Neuenschwander
Action_(physics)
26-dimensional string theory
can be said to be defined by the path integral quantization of the Polyakov action: I 0 [ g , X ] = T 8 π ∫ M d 2 ξ g g m n ∂ m x μ ∂ n x ν G μ ν ( x
Bosonic_string_theory
Invariant action in bosonic string theory
analog correction is provided by the Polyakov action, which is classically equivalent to the Nambu–Goto action, but gives the 'correct' quantum theory
Nambu–Goto_action
Theories in particle physics and cosmology
superstring revolution String theory landscape Theory Nambu–Goto action Polyakov action Bosonic string theory Superstring theory Type I string Type II string
Brane_cosmology
Mathematical concept
< 2 π {\displaystyle 0\leq \sigma <2\pi } . In order to define the Polyakov action, the world-sheet is equipped with a world-sheet metric g {\displaystyle
Worldsheet
Principle in theoretical physics
elaborated in articles by Steven Gubser, Igor Klebanov, and Alexander Markovich Polyakov, and by Edward Witten. By 2015, Maldacena's article had over 10,000 citations
Holographic_principle
Extended physical object in string theory
superstring revolution String theory landscape Theory Nambu–Goto action Polyakov action Bosonic string theory Superstring theory Type I string Type II string
Brane
Hypothetical elementary particle that mediates gravity
superstring revolution String theory landscape Theory Nambu–Goto action Polyakov action Bosonic string theory Superstring theory Type I string Type II string
Graviton
Riemannian manifold with SU(n) holonomy
quotients of the quintic by various Z 5 {\displaystyle \mathbb {Z} _{5}} actions are also Calabi–Yau and have received a lot of attention in the literature
Calabi–Yau_manifold
Hypothetical physical entity
superstring revolution String theory landscape Theory Nambu–Goto action Polyakov action Bosonic string theory Superstring theory Type I string Type II string
String_(physics)
Collection of possible string theory vacua
superstring revolution String theory landscape Theory Nambu–Goto action Polyakov action Bosonic string theory Superstring theory Type I string Type II string
String_theory_landscape
Secondary characteristic classes of 3-manifolds
superstring revolution String theory landscape Theory Nambu–Goto action Polyakov action Bosonic string theory Superstring theory Type I string Type II string
Chern–Simons_form
Class of quantum field theory models
model Little Higgs Skyrmion, a soliton in non-linear sigma models Polyakov action WZW model Fubini–Study metric, a metric often used with non-linear
Non-linear_sigma_model
Type of smooth complex surface of kodaira dimension 0
Kummer surface is the quotient of a two-dimensional abelian variety A by the action a ↦ − a {\displaystyle a\mapsto -a} . This results in 16 singularities,
K3_surface
Topics referred to by the same term
brand Polyakov action, two-dimensional action of a conformal field theory describing the worldsheet of a string in string theory 't Hooft–Polyakov monopole
Polyakov_(disambiguation)
Local rescaling of a metric tensor
symmetry in conformal field theory. It is, for example, a symmetry of the Polyakov action. When quantum mechanical effects break the conformal invariance of
Weyl_transformation
Compact astronomical body
superstring revolution String theory landscape Theory Nambu–Goto action Polyakov action Bosonic string theory Superstring theory Type I string Type II string
Black_hole
bosonic string theory. String theory is formulated in terms of the Polyakov action, which describes how strings move through space and time. Like springs
History_of_string_theory
string theory. These theories are without supersymmetry. Polyakov action Nambu-Goto action Bosonic string theory Kondo model (s-d model) Minimal model
List of quantum field theories
List_of_quantum_field_theories
Manifold with Riemannian, complex and symplectic structure
\mathbb {CP} ^{n}} (up to a positive multiple) that is invariant under the action of U ( n + 1 ) {\displaystyle \mathrm {U} (n+1)} on C P n {\displaystyle
Kähler_manifold
Process in particle physics
superstring revolution String theory landscape Theory Nambu–Goto action Polyakov action Bosonic string theory Superstring theory Type I string Type II string
Tachyon_condensation
Theory of subatomic structure
elaborated in articles by Steven Gubser, Igor Klebanov, and Alexander Markovich Polyakov, and by Edward Witten. By 2010, Maldacena's article had over 7000 citations
String_theory
Quantum mechanical model based on mathematical matrices
superstring revolution String theory landscape Theory Nambu–Goto action Polyakov action Bosonic string theory Superstring theory Type I string Type II string
Matrix_theory_(physics)
Eight-dimensional Riemannian manifold
superstring revolution String theory landscape Theory Nambu–Goto action Polyakov action Bosonic string theory Superstring theory Type I string Type II string
Spin(7)-manifold
Hypothetical faster-than-light particle
superstring revolution String theory landscape Theory Nambu–Goto action Polyakov action Bosonic string theory Superstring theory Type I string Type II string
Tachyon
Duality between theories of gravity on anti-de Sitter space and conformal field theories
on in two articles, one by Steven Gubser, Igor Klebanov and Alexander Polyakov, and another by Edward Witten. By 2015, Maldacena's article had over 10
AdS/CFT_correspondence
Framework of superstring theory
superstring revolution String theory landscape Theory Nambu–Goto action Polyakov action Bosonic string theory Superstring theory Type I string Type II string
M-theory
Type of geometry in mathematics
superstring revolution String theory landscape Theory Nambu–Goto action Polyakov action Bosonic string theory Superstring theory Type I string Type II string
Ricci-flat_manifold
Theory of strings with supersymmetry
fields the D-branes can be included in the action by adding an extra U(1) vector field to the string action. ∂ z → ∂ z + i A z ( z , z ¯ ) {\displaystyle
Superstring_theory
248-dimensional exceptional simple Lie group
superstring revolution String theory landscape Theory Nambu–Goto action Polyakov action Bosonic string theory Superstring theory Type I string Type II string
E8_(mathematics)
Unobservable spacetime curves needed to describe Dirac monopoles
superstring revolution String theory landscape Theory Nambu–Goto action Polyakov action Bosonic string theory Superstring theory Type I string Type II string
Dirac_string
Generalization of a manifold
superstring revolution String theory landscape Theory Nambu–Goto action Polyakov action Bosonic string theory Superstring theory Type I string Type II string
Conifold
Speculative feature of the early universe
one-dimensional objects and obey the Nambu–Goto action, which is classically equivalent to the Polyakov action that defines the bosonic sector of superstring
Cosmic_string
Branch of string theory
superstring revolution String theory landscape Theory Nambu–Goto action Polyakov action Bosonic string theory Superstring theory Type I string Type II string
F-theory
Peruvian theoretical physicist (b. 1954)
superstring revolution String theory landscape Theory Nambu–Goto action Polyakov action Bosonic string theory Superstring theory Type I string Type II string
Barton_Zwiebach
Geometric space whose points represent algebro-geometric objects of some fixed kind
back from the rigidified problem to the original by taking quotient by the action of G, and the problem of constructing the moduli space becomes that of finding
Moduli_space
Lower energy limit in string theory
gauge theories. Specifically, it was shown that the mass of a 't Hooft-Polyakov monopole and a Julia-Zee dyon is bounded from below by a quantity proportional
Bogomol'nyi–Prasad–Sommerfield bound
Bogomol'nyi–Prasad–Sommerfield_bound
Unified field theory
Einstein–Hilbert action for the bundle, as a whole. The equations of motion, the Euler–Lagrange equations, can be then obtained by considering where the action is stationary
Kaluza–Klein_theory
Breakdown of conformal symmetry at the quantum level
ISBN 0-521-63303-6. Vol. 2: Superstring theory and beyond. ISBN 0-521-63304-4. Polyakov, A.M. (1981). "Quantum geometry of bosonic strings". Physics Letters B
Conformal_anomaly
Solitons in Euclidean spacetime
't Hooft–Polyakov monopoles play the role of instantons. In his 1977 paper Quark Confinement and Topology of Gauge Groups, Alexander Polyakov demonstrated
Instanton
Set of equations that describe superstring theory in a non-perturbative framework
superstring revolution String theory landscape Theory Nambu–Goto action Polyakov action Bosonic string theory Superstring theory Type I string Type II string
Matrix_string_theory
Type of 2D conformal field theory
the existence of this symmetry is through repeated application of the Polyakov–Wiegmann identity regarding products of G {\displaystyle G} -valued fields:
Wess–Zumino–Witten_model
Simple Lie group; the automorphism group of the octonions
representation is the adjoint representation, and the 7-dimensional one is action of G2 on the imaginary octonions. There are two non-isomorphic irreducible
G2_(mathematics)
Italian physicist (1923–2014)
Gold Medal 2005 Enrico Fermi Prize of the Italian Physical Society Polyakov action Chang, Kenneth (5 July 2014). "Bruno Zumino Dies at 91; Sought to Tie
Bruno_Zumino
EPR Paradox in quantum physics Alexander Polyakov, developed the concepts of Polyakov action, 't Hooft–Polyakov monopole and BPST instanton Isaak Pomeranchuk
List_of_Russian_scientists
Type of Lie algebra of interest in physics
superstring revolution String theory landscape Theory Nambu–Goto action Polyakov action Bosonic string theory Superstring theory Type I string Type II string
Loop_algebra
Algebraic structure used in theoretical physics
superstring revolution String theory landscape Theory Nambu–Goto action Polyakov action Bosonic string theory Superstring theory Type I string Type II string
Supergroup_(physics)
In physics and geometry: conjectured relation between pairs of Calabi–Yau manifolds
Kontsevich, Maxim (1995a), "Enumeration of Rational Curves Via Torus Actions", The Moduli Space of Curves, Birkhäuser, p. 335, arXiv:hep-th/9405035
Mirror symmetry (string theory)
Mirror_symmetry_(string_theory)
Two-dimensional conformal field theory
dilaton background requires it to take an exponential solution. The Polyakov action in this background is then identical to Liouville field theory, with
Liouville_field_theory
American physicist (1931–2023)
died in Pasadena, California on April 21, 2023, at the age of 92. Polyakov action Stanley Deser, John Simon Guggenheim Foundation Chang, Kenneth (May
Stanley_Deser
EPR Paradox in quantum physics Alexander Polyakov, developed the concepts of Polyakov action, 't Hooft–Polyakov monopole and BPST instanton Isaak Pomeranchuk
List_of_Russian_physicists
52-dimensional exceptional simple Lie group
1152. It has minimal faithful degree μ(G) = 24, which is realized by the action on the 24-cell. The group has ID (1152,157478) in the small groups library
F4_(mathematics)
Seven-dimensional Riemannian manifold
superstring revolution String theory landscape Theory Nambu–Goto action Polyakov action Bosonic string theory Superstring theory Type I string Type II string
G2_manifold
Symmetry between bosons and fermions
superstring revolution String theory landscape Theory Nambu–Goto action Polyakov action Bosonic string theory Superstring theory Type I string Type II string
Supersymmetry
Aspect of theoretical physics
superstring revolution String theory landscape Theory Nambu–Goto action Polyakov action Bosonic string theory Superstring theory Type I string Type II string
Type_I_string_theory
Extended objects found in string theory
velocities the open strings may be described by a low-energy effective action that contains two complex scalar fields that are coupled via a term ϕ 2
D-brane
Base space for supersymmetric theories
^{*}Q+\epsilon Q^{\dagger })\Phi .} We can evaluate this variation using the action of Q {\displaystyle Q} on the superfields [ Q , Φ ] = ( ∂ ∂ θ − i θ ∗ ∂
Superspace
Object in six-dimensional spacetime
superstring revolution String theory landscape Theory Nambu–Goto action Polyakov action Bosonic string theory Superstring theory Type I string Type II string
NS5-brane
superstring revolution String theory landscape Theory Nambu–Goto action Polyakov action Bosonic string theory Superstring theory Type I string Type II string
Hořava–Witten_theory
Candidate "Theory of Everything"
superstring revolution String theory landscape Theory Nambu–Goto action Polyakov action Bosonic string theory Superstring theory Type I string Type II string
Introduction_to_M-theory
Algebra used in 2D conformal field theories and string theory
the course of this construction, one employs a Fock space that admits an action of vertex operators attached to elements of a lattice. Borcherds formulated
Vertex_operator_algebra
Hypothetical particle
testing the theory through cosmology and experimentation. The dilaton-gravity action is ∫ d D x − g [ 1 2 κ ( Φ R − ω [ Φ ] g μ ν ∂ μ Φ ∂ ν Φ Φ ) − V [ Φ ] ]
Dilaton
Lie algebra, usually infinite-dimensional
superstring revolution String theory landscape Theory Nambu–Goto action Polyakov action Bosonic string theory Superstring theory Type I string Type II string
Kac–Moody_algebra
Two-form field
to the action while the Kalb–Ramond field must be integrated over the two-dimensional worldsheet of the string. In particular, while the action for a charged
Kalb–Ramond_field
Branch of string theory
equations entering into string cosmology. The starting point is the Polyakov action, which can be written as S 2 = 1 4 π α ′ ∫ d 2 z γ [ γ a b G μ ν (
String_cosmology
Conformal field theory on a 2D spacetime
differential equation called a Belavin–Polyakov–Zamolodchikov equation after Alexander Belavin, Alexander Polyakov and Alexander Zamolodchikov. The order
Two-dimensional conformal field theory
Two-dimensional_conformal_field_theory
Equivalence of two physical theories
superstring revolution String theory landscape Theory Nambu–Goto action Polyakov action Bosonic string theory Superstring theory Type I string Type II string
S-duality
Aspect of theoretical physics
superstring revolution String theory landscape Theory Nambu–Goto action Polyakov action Bosonic string theory Superstring theory Type I string Type II string
Type_II_string_theory
Equivalence of two physical theories
superstring revolution String theory landscape Theory Nambu–Goto action Polyakov action Bosonic string theory Superstring theory Type I string Type II string
T-duality
Algebraic structure used in theoretical physics
{g}}_{1}} is a g 0 {\displaystyle {\mathfrak {g}}_{0}} -module for the action a d a : b → [ a , b ] , a ∈ g 0 , b , [ a , b ] ∈ g 1 {\displaystyle \mathrm
Lie_superalgebra
Type of Riemannian manifold
instantons. The Gibbons–Hawking ansatz gives examples invariant under a circle action. Many examples of noncompact hyperkähler manifolds arise as moduli spaces
Hyperkähler_manifold
Asymmetry of classical and quantum action
or quantum anomaly is the failure of a symmetry of a theory's classical action to be a symmetry of any regularization of the full quantum theory. In classical
Anomaly_(physics)
Russian businessman
Samuel (Shmuel) Polyakov (also Poliakoff, Poliakov, Russian: Самуил Соломонович Поляков) was a Russian businessman, informally known as the "most famous
Samuel_Polyakov
This is a list of string theory topics. Strings Nambu–Goto action Polyakov action Bosonic string theory Superstring theory Type I string Type II string
List_of_string_theory_topics
Algebra combining both supersymmetry and conformal symmetry
superstring revolution String theory landscape Theory Nambu–Goto action Polyakov action Bosonic string theory Superstring theory Type I string Type II string
Superconformal_algebra
Conjectured duality combining S-duality and T-duality
superstring revolution String theory landscape Theory Nambu–Goto action Polyakov action Bosonic string theory Superstring theory Type I string Type II string
U-duality
Strong-weak duality in supersymmetric theories of theoretical physics
took a step forward in 1974 when Gerard 't Hooft and Alexander Markovich Polyakov independently constructed monopoles not as quantized point particles, but
Montonen–Olive_duality
superstring revolution String theory landscape Theory Nambu–Goto action Polyakov action Bosonic string theory Superstring theory Type I string Type II string
S-brane
Generalized manifold
discontinuous action of a possibly infinite group of diffeomorphisms with finite isotropy subgroups. In particular this applies to any action of a finite
Orbifold
Formalism in string theory
D'Hoker, Eric; Giddings, Steven B. (1987). "Unitarity of the closed bosonic Polyakov string". Nuclear Physics B. 291. Elsevier BV: 90–112. Bibcode:1987NuPhB
String_field_theory
EPR Paradox in quantum physics Alexander Polyakov, developed the concepts of Polyakov action, 't Hooft–Polyakov monopole and BPST instanton Isaak Pomeranchuk
List_of_Russian_people
133-dimensional exceptional simple Lie group
product of the diagonal automorphism group Z/gcd(2, q−1)Z (given by the action of E7,ad(q)) and the group of field automorphisms (i.e., cyclic of order
E7_(mathematics)
Mathematical theory
H ) {\displaystyle Fred({\mathcal {H}})} which are equivariant under an action of P U ( H ) {\displaystyle PU({\mathcal {H}})} is equivalent to the original
Twisted_K-theory
Diagrams describing the matter content
superstring revolution String theory landscape Theory Nambu–Goto action Polyakov action Bosonic string theory Superstring theory Type I string Type II string
Quiver_diagram
78-dimensional exceptional simple Lie group
fundamental group Z/3Z in the sense of algebraic geometry, with Galois action as on the third roots of unity; this means that they admit exactly one triple
E6_(mathematics)
Modern theory of gravitation that combines supersymmetry and general relativity
Eugène Cremmer, Bernard Julia and Joël Scherk (CJS) found the classical action for an 11-dimensional supergravity theory. This remains today the only known
Supergravity
Generalization of a black hole to higher dimensions
superstring revolution String theory landscape Theory Nambu–Goto action Polyakov action Bosonic string theory Superstring theory Type I string Type II string
Black_brane
Superstring quantization approach
also be acquired through this formalism by using a different worldsheet action. There are various ways to quantize the string within this framework including
RNS_formalism
Italian theoretical physicist
by Alexander Polyakov who used the effect of Di Vecchia and colleagues for the quantization of strings, then known as the Polyakov action. In the 2000s
Paolo_Di_Vecchia
Application of K-theory in string theory
transformations, but due to the presence of Chern-Simons terms in the supergravity actions these large gauge transformations transform not only the p-form potentials
K-theory_(physics)
Brane in eleven-dimensional supergravity
superstring revolution String theory landscape Theory Nambu–Goto action Polyakov action Bosonic string theory Superstring theory Type I string Type II string
M2-brane
Physics concept of subatomic structure
superstring revolution String theory landscape Theory Nambu–Goto action Polyakov action Bosonic string theory Superstring theory Type I string Type II string
Heterotic_string_theory
Property of a differential manifold that includes complex structures
of Weyl spinors. Vectors have an action on differential forms given by the interior product. One-forms have an action on forms given by the wedge product
Generalized_complex_structure
Pontecorvo–Maki–Nakagawa–Sakata matrix for neutrino mixing Polyakov action A modification of the Nambu–Goto action for strings that eliminates the square root. PQ
Glossary_of_string_theory
Theory in physics
ISBN 0-521-63303-6. Vol. 2: Superstring theory and beyond. ISBN 0-521-63304-4. Polyakov, A.M. (1981). "Quantum geometry of bosonic strings". Physics Letters B
Non-critical_string_theory
Concept in theoretical physics
{\displaystyle \sigma (\Omega )={\bar {\Omega }}} The locus where the orientifold action reduces to the change of the string orientation is called the orientifold
Orientifold
Hypothetical particle dual to the photon
⋆ {\displaystyle \star } is the Hodge star. This implies the following action in the spacetime manifold M {\displaystyle M} : S = ∫ M [ 1 2 F ∧ ⋆ F −
Dual_photon
1974 spy novel by John le Carré
reports to a Soviet official stationed at the embassy in London called Polyakov. Shortly after Tarr relayed this to the Circus, Irina was forcibly returned
Tinker_Tailor_Soldier_Spy
POLYAKOV ACTION
POLYAKOV ACTION
Girl/Female
Tamil
Action
Boy/Male
Tamil
Karmendra | கரà¯à®®à¯‡à®¨à¯à®¤à¯à®°
God of action
Karmendra | கரà¯à®®à¯‡à®¨à¯à®¤à¯à®°
Girl/Female
Hindu
Famous action
Boy/Male
Sikh
One whose actions are gem-like, Friendly gem
Girl/Female
Indian
Determined action
Boy/Male
Tamil
Action
Girl/Female
Tamil
Action, A work of art
Boy/Male
Hindu
Deed, Action
Girl/Female
Tamil
Kirtika | கிரà¯à®¤à¯€à®•à®¾, கீரà¯à®¤à¯€à®•à®¾Â
Famous action
Kirtika | கிரà¯à®¤à¯€à®•à®¾, கீரà¯à®¤à¯€à®•à®¾Â
Girl/Female
Muslim
Determined action
Girl/Female
Tamil
Satkrithi | ஸதà¯à®•à¯à®°à¯€à®¤à¯€
Good action
Satkrithi | ஸதà¯à®•à¯à®°à¯€à®¤à¯€
Boy/Male
Hindu
Action
Boy/Male
Tamil
Deed, Action
Girl/Female
Hindu
Action, A work of art
Girl/Female
Tamil
Action, A work of art
Girl/Female
Tamil
Kirthika | கிரà¯à®¤à¯€à®•à®¾, கீரà¯à®¤à¯€à®•à®¾Â
Famous action
Kirthika | கிரà¯à®¤à¯€à®•à®¾, கீரà¯à®¤à¯€à®•à®¾Â
Boy/Male
Hindu
Deed, Action
Girl/Female
Hindu
Famous action
Boy/Male
Muslim
Deed, Action
Girl/Female
Hindu
Action, A work of art
POLYAKOV ACTION
POLYAKOV ACTION
Girl/Female
Tamil
Devarshee | தேவரà¯à®·à®¿
A God like saint. Devarshee was used for naradmuni because he was the son of Brahma and was the saint of Vishnu Bhagwan, Goddess Lakshmi
Boy/Male
Bengali, Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Oriya, Sanskrit, Sindhi, Telugu, Traditional
Blessing of the Goddess
Surname or Lastname
English
English : variant spelling of Kimberley.
Female
English
 Short form of English Olivia, probably LIVIA means "elf army." Compare with another form of Livia.
Girl/Female
Muslim/Islamic
Intelligent
Surname or Lastname
English
English : habitational name from Bartley in Hampshire, or from Bartley Green in the West Midlands, both of which are named with Old English be(o)rc ‘birch’ + lēah ‘woodland clearing’; compare Barclay.Americanized spelling of German (Swabian) Bartle and the Swiss cognate Bartli.The surname Bartley was brought to VA from Northumberland in 1724.
Male
Spanish
Masculine form of Spanish Candelaria, CANDELARIO means "candle."
Girl/Female
British, English
Park with Deer
Girl/Female
German, Kurdish, Teutonic
Protected; Beloved Warrior
Boy/Male
German, Greek, Gujarati, Hindu, Indian, Kannada
From the Forest; Wood or Forest
POLYAKOV ACTION
POLYAKOV ACTION
POLYAKOV ACTION
POLYAKOV ACTION
POLYAKOV ACTION
n.
A frequented track; habitual place of action; sphere; as, the walk of the historian.
n.
A process or condition of acting or moving, as opposed to rest; the doing of something; exertion of power or force, as when one body acts on another; the effect of power exerted on one body by another; agency; activity; operation; as, the action of heat; a man of action.
a.
Void of action.
n.
Action by, or originating in, one's self or itself.
n.
An engagement between troops in war, whether on land or water; a battle; a fight; as, a general action, a partial action.
n.
Conduct; course of action; behavior.
adv.
In an actionable manner.
n.
Alt. of Actionist
v. i.
To be in action or motion; to move; to get along; to progress; to stir.
n.
The science which treats of phenomena due to plutonic action, as in volcanoes, hot springs, etc.
n.
A right of action; as, the law gives an action for every claim.
n.
The revival of an action.
n.
Any one of the active processes going on in an organism; the performance of a function; as, the action of the heart, the muscles, or the gastric juice.
a.
That may be the subject of an action or suit at law; as, to call a man a thief is actionable.
n.
Effective motion; also, mechanism; as, the breech action of a gun.
n.
Movement; as, the horse has a spirited action.
n.
The manner or action of a wag; mischievous merriment; sportive trick or gayety; good-humored sarcasm; pleasantry; jocularity; as, the waggery of a schoolboy.
v. t.
To excite; to rouse; to move to action; to awaken.
v. t.
To put in motion or action; to arouse; to excite.
n.
a state of opposition or contest; an act of opposition; an inimical contest, act, or action; enmity; hostility.