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theoretical physics Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain, and predict List of natural phenomena, natural phenomena. This is in contrast to experimental p ...
, a chiral anomaly is the anomalous nonconservation of a
chiral Chirality () is a property of asymmetry important in several branches of science. The word ''chirality'' is derived from the Greek language, Greek (''kheir''), "hand", a familiar chiral object. An object or a system is ''chiral'' if it is dist ...
current. In everyday terms, it is analogous to a sealed box that contained equal numbers of left and right-handed bolts, but when opened was found to have more left than right, or vice versa. Such events are expected to be prohibited according to classical
conservation law In physics, a conservation law states that a particular measurable property of an isolated physical system does not change as the system evolves over time. Exact conservation laws include conservation of mass-energy, conservation of linear momen ...
s, but it is known there must be ways they can be broken, because we have evidence of charge–parity non-conservation ("CP violation"). It is possible that other imbalances have been caused by breaking of a ''chiral law'' of this kind. Many physicists suspect that the fact that the observable universe contains more matter than antimatter is caused by a chiral anomaly. Research into
chiral symmetry breaking In particle physics, chiral symmetry breaking generally refers to the dynamical spontaneous breaking of a chiral symmetry associated with massless fermions. This is usually associated with a gauge theory such as quantum chromodynamics, the quant ...
laws is a major endeavor in particle physics research at this time.


Informal introduction

The chiral anomaly originally referred to the anomalous
decay rate Radioactive decay (also known as nuclear decay, radioactivity, radioactive disintegration, or nuclear disintegration) is the process by which an unstable atomic nucleus loses energy by radiation. A material containing unstable nuclei is conside ...
of the
neutral Neutral or neutrality may refer to: Mathematics and natural science Biology * Neutral organisms, in ecology, those that obey the unified neutral theory of biodiversity Chemistry and physics * Neutralization (chemistry), a chemical reaction in ...
pion In particle physics, a pion (, ) or pi meson, denoted with the Greek alphabet, Greek letter pi (letter), pi (), is any of three subatomic particles: , , and . Each pion consists of a quark and an antiquark and is therefore a meson. Pions are the ...
, as computed in the current algebra of the chiral model. These calculations suggested that the decay of the pion was suppressed, clearly contradicting experimental results. The nature of the anomalous calculations was first explained in 1969 by Stephen L. Adler and
John Stewart Bell John Stewart Bell (28 July 1928 – 1 October 1990) was a physicist from Northern Ireland and the originator of Bell's theorem, an important theorem in quantum mechanics, quantum physics regarding hidden-variable theory, hidden-variable theor ...
& Roman Jackiw. This is now termed the Adler–Bell–Jackiw anomaly of
quantum electrodynamics In particle physics, quantum electrodynamics (QED) is the Theory of relativity, relativistic quantum field theory of electrodynamics. In essence, it describes how light and matter interact and is the first theory where full agreement between quant ...
. This is a symmetry of classical
electrodynamics In physics, electromagnetism is an interaction that occurs between particles with electric charge via electromagnetic fields. The electromagnetic force is one of the four fundamental forces of nature. It is the dominant force in the interacti ...
that is violated by quantum corrections. The Adler–Bell–Jackiw anomaly arises in the following way. If one considers the classical (non-quantized) theory of
electromagnetism In physics, electromagnetism is an interaction that occurs between particles with electric charge via electromagnetic fields. The electromagnetic force is one of the four fundamental forces of nature. It is the dominant force in the interacti ...
coupled to massless
fermion In particle physics, a fermion is a subatomic particle that follows Fermi–Dirac statistics. Fermions have a half-integer spin (spin 1/2, spin , Spin (physics)#Higher spins, spin , etc.) and obey the Pauli exclusion principle. These particles i ...
s (electrically charged
Dirac spinor In quantum field theory, the Dirac spinor is the spinor that describes all known fundamental particles that are fermions, with the possible exception of neutrinos. It appears in the plane-wave solution to the Dirac equation, and is a certain comb ...
s solving the
Dirac equation In particle physics, the Dirac equation is a relativistic wave equation derived by British physicist Paul Dirac in 1928. In its free form, or including electromagnetic interactions, it describes all spin-1/2 massive particles, called "Dirac ...
), one expects to have not just one but two
conserved current In physics a conserved current is a current, j^\mu, that satisfies the continuity equation \partial_\mu j^\mu=0. The continuity equation represents a conservation law, hence the name. Indeed, integrating the continuity equation over a volume V, ...
s: the ordinary electrical current (the vector current), described by the Dirac field j^\mu = \overline\psi\gamma^\mu\psi as well as an axial current j_5^\mu = \overline\psi\gamma^5\gamma^\mu\psi~. When moving from the classical theory to the quantum theory, one may compute the quantum corrections to these currents; to first order, these are the one-loop
Feynman diagram In theoretical physics, a Feynman diagram is a pictorial representation of the mathematical expressions describing the behavior and interaction of subatomic particles. The scheme is named after American physicist Richard Feynman, who introduced ...
s. These are famously divergent, and require a
regularization Regularization may refer to: * Regularization (linguistics) * Regularization (mathematics) * Regularization (physics) * Regularization (solid modeling) * Regularization Law, an Israeli law intended to retroactively legalize settlements See also ...
to be applied, to obtain the renormalized amplitudes. In order for the
renormalization Renormalization is a collection of techniques in quantum field theory, statistical field theory, and the theory of self-similar geometric structures, that is used to treat infinities arising in calculated quantities by altering values of the ...
to be meaningful, coherent and consistent, the regularized diagrams must obey the same symmetries as the zero-loop (classical) amplitudes. This is the case for the vector current, but not the axial current: it cannot be regularized in such a way as to preserve the axial symmetry. The axial symmetry of classical electrodynamics is broken by quantum corrections. Formally, the Ward–Takahashi identities of the quantum theory follow from the
gauge symmetry In physics, a gauge theory is a type of field theory in which the Lagrangian, and hence the dynamics of the system itself, does not change under local transformations according to certain smooth families of operations (Lie groups). Formally, t ...
of the electromagnetic field; the corresponding identities for the axial current are broken. At the time that the Adler–Bell–Jackiw anomaly was being explored in physics, there were related developments in
differential geometry Differential geometry is a Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of Calculus, single variable calculus, vector calculus, lin ...
that appeared to involve the same kinds of expressions. These were not in any way related to quantum corrections of any sort, but rather were the exploration of the global structure of
fiber bundle In mathematics, and particularly topology, a fiber bundle ( ''Commonwealth English'': fibre bundle) is a space that is a product space, but may have a different topological structure. Specifically, the similarity between a space E and a pr ...
s, and specifically, of the
Dirac operator In mathematics and in quantum mechanics, a Dirac operator is a first-order differential operator that is a formal square root, or half-iterate, of a second-order differential operator such as a Laplacian. It was introduced in 1847 by William Ham ...
s on
spin structure In differential geometry, a spin structure on an orientable Riemannian manifold allows one to define associated spinor bundles, giving rise to the notion of a spinor in differential geometry. Spin structures have wide applications to mathemati ...
s having
curvature form In differential geometry, the curvature form describes curvature of a connection on a principal bundle. The Riemann curvature tensor in Riemannian geometry can be considered as a special case. Definition Let ''G'' be a Lie group with Lie algebra ...
s resembling that of the
electromagnetic tensor In electromagnetism, the electromagnetic tensor or electromagnetic field tensor (sometimes called the field strength tensor, Faraday tensor or Maxwell bivector) is a mathematical object that describes the electromagnetic field in spacetime. Th ...
, both in four and three dimensions (the
Chern–Simons theory The Chern–Simons theory is a 3-dimensional topological quantum field theory of Schwarz type. It was discovered first by mathematical physicist Albert Schwarz. It is named after mathematicians Shiing-Shen Chern and James Harris Simons, who intr ...
). After considerable back and forth, it became clear that the structure of the anomaly could be described with bundles with a non-trivial
homotopy group In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, denoted \pi_1(X), which records information about loops in a space. Intuitively, homo ...
, or, in physics lingo, in terms of
instanton An instanton (or pseudoparticle) is a notion appearing in theoretical and mathematical physics. An instanton is a classical solution to equations of motion with a finite, non-zero action, either in quantum mechanics or in quantum field theory. M ...
s. Instantons are a form of topological soliton; they are a solution to the ''classical'' field theory, having the property that they are stable and cannot decay (into
plane wave In physics Physics is the scientific study of matter, its Elementary particle, fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of ...
s, for example). Put differently: conventional field theory is built on the idea of a
vacuum A vacuum (: vacuums or vacua) is space devoid of matter. The word is derived from the Latin adjective (neuter ) meaning "vacant" or "void". An approximation to such vacuum is a region with a gaseous pressure much less than atmospheric pressur ...
– roughly speaking, a flat empty space. Classically, this is the "trivial" solution; all fields vanish. However, one can also arrange the (classical) fields in such a way that they have a non-trivial global configuration. These non-trivial configurations are also candidates for the vacuum, for empty space; yet they are no longer flat or trivial; they contain a twist, the instanton. The quantum theory is able to interact with these configurations; when it does so, it manifests as the chiral anomaly. In mathematics, non-trivial configurations are found during the study of
Dirac operator In mathematics and in quantum mechanics, a Dirac operator is a first-order differential operator that is a formal square root, or half-iterate, of a second-order differential operator such as a Laplacian. It was introduced in 1847 by William Ham ...
s in their fully generalized setting, namely, on
Riemannian manifold In differential geometry, a Riemannian manifold is a geometric space on which many geometric notions such as distance, angles, length, volume, and curvature are defined. Euclidean space, the N-sphere, n-sphere, hyperbolic space, and smooth surf ...
s in arbitrary dimensions. Mathematical tasks include finding and classifying structures and configurations. Famous results include the
Atiyah–Singer index theorem In differential geometry, the Atiyah–Singer index theorem, proved by Michael Atiyah and Isadore Singer (1963), states that for an elliptic differential operator on a compact manifold, the analytical index (related to the dimension of the space ...
for Dirac operators. Roughly speaking, the symmetries of
Minkowski spacetime In physics, Minkowski space (or Minkowski spacetime) () is the main mathematical description of spacetime in the absence of gravitation. It combines inertial space and time manifolds into a four-dimensional model. The model helps show how a s ...
,
Lorentz invariance In a relativistic theory of physics, a Lorentz scalar is a scalar expression whose value is invariant under any Lorentz transformation. A Lorentz scalar may be generated from, e.g., the scalar product of vectors, or by contracting tensors. While ...
,
Laplacian In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a scalar function on Euclidean space. It is usually denoted by the symbols \nabla\cdot\nabla, \nabla^2 (where \nabla is th ...
s, Dirac operators and the U(1)xSU(2)xSU(3)
fiber bundle In mathematics, and particularly topology, a fiber bundle ( ''Commonwealth English'': fibre bundle) is a space that is a product space, but may have a different topological structure. Specifically, the similarity between a space E and a pr ...
s can be taken to be a special case of a far more general setting in
differential geometry Differential geometry is a Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of Calculus, single variable calculus, vector calculus, lin ...
; the exploration of the various possibilities accounts for much of the excitement in theories such as
string theory In physics, string theory is a theoretical framework in which the point-like particles of particle physics are replaced by one-dimensional objects called strings. String theory describes how these strings propagate through space and intera ...
; the richness of possibilities accounts for a certain perception of lack of progress. The Adler–Bell–Jackiw anomaly is seen experimentally, in the sense that it describes the decay of the
neutral pion In particle physics, a pion (, ) or pi meson, denoted with the Greek letter pi (), is any of three subatomic particles: , , and . Each pion consists of a quark and an antiquark and is therefore a meson. Pions are the lightest mesons and, more g ...
, and specifically, the width of the decay of the neutral pion into two
photon A photon () is an elementary particle that is a quantum of the electromagnetic field, including electromagnetic radiation such as light and radio waves, and the force carrier for the electromagnetic force. Photons are massless particles that can ...
s. The neutral pion itself was discovered in the 1940s; its decay rate (width) was correctly estimated by J. Steinberger in 1949. The correct form of the anomalous divergence of the axial current is obtained by Schwinger in 1951 in a 2D model of electromagnetism and massless fermions. That the decay of the neutral pion is suppressed in the current algebra analysis of the chiral model is obtained by Sutherland and Veltman in 1967. An analysis and resolution of this anomalous result is provided by Adler and Bell & Jackiw in 1969. A general structure of the anomalies is discussed by Bardeen in 1969. The
quark model In particle physics, the quark model is a classification scheme for hadrons in terms of their valence quarks—the quarks and antiquarks that give rise to the quantum numbers of the hadrons. The quark model underlies "flavor SU(3)", or the Eig ...
of the pion indicates it is a bound state of a quark and an anti-quark. However, the
quantum number In quantum physics and chemistry, quantum numbers are quantities that characterize the possible states of the system. To fully specify the state of the electron in a hydrogen atom, four quantum numbers are needed. The traditional set of quantu ...
s, including parity and angular momentum, taken to be conserved, prohibit the decay of the pion, at least in the zero-loop calculations (quite simply, the amplitudes vanish.) If the quarks are assumed to be massive, not massless, then a
chirality Chirality () is a property of asymmetry important in several branches of science. The word ''chirality'' is derived from the Greek (''kheir''), "hand", a familiar chiral object. An object or a system is ''chiral'' if it is distinguishable fro ...
-violating decay is allowed; however, it is not of the correct size. (Chirality is not a
constant of motion In mechanics, a constant of motion is a physical quantity conserved throughout the motion, imposing in effect a constraint on the motion. However, it is a ''mathematical'' constraint, the natural consequence of the equations of motion, rather tha ...
of massive spinors; they will change handedness as they propagate, and so mass is itself a chiral symmetry-breaking term. The contribution of the mass is given by the Sutherland and Veltman result; it is termed "PCAC", the partially conserved axial current.) The Adler–Bell–Jackiw analysis provided in 1969 (as well as the earlier forms by Steinberger and Schwinger), do provide the correct decay width for the neutral pion. Besides explaining the decay of the pion, it has a second very important role. The one loop amplitude includes a factor that counts the grand total number of leptons that can circulate in the loop. In order to get the correct decay width, one must have exactly three generations of quarks, and not four or more. In this way, it plays an important role in constraining the
Standard model The Standard Model of particle physics is the Scientific theory, theory describing three of the four known fundamental forces (electromagnetism, electromagnetic, weak interaction, weak and strong interactions – excluding gravity) in the unive ...
. It provides a direct physical prediction of the number of quarks that can exist in nature. Current day research is focused on similar phenomena in different settings, including non-trivial topological configurations of the
electroweak theory In particle physics, the electroweak interaction or electroweak force is the unified description of two of the fundamental interactions of nature: electromagnetism (electromagnetic interaction) and the weak interaction. Although these two forc ...
, that is, the
sphaleron A sphaleron ( "slippery") is a static (time-independent) solution to the electroweak field equations of the Standard Model of particle physics, and is involved in certain hypothetical processes that violate baryon and lepton numbers. Such proces ...
s. Other applications include the hypothetical non-conservation of
baryon number In particle physics, the baryon number (B) is an additive quantum number of a system. It is defined as B = \frac(n_\text - n_), where is the number of quarks, and is the number of antiquarks. Baryons (three quarks) have B = +1, mesons (one q ...
in GUTs and other theories.


General discussion

In some theories of
fermion In particle physics, a fermion is a subatomic particle that follows Fermi–Dirac statistics. Fermions have a half-integer spin (spin 1/2, spin , Spin (physics)#Higher spins, spin , etc.) and obey the Pauli exclusion principle. These particles i ...
s with
chiral symmetry A chiral phenomenon is one that is not identical to its mirror image (see the article on mathematical chirality). The spin of a particle may be used to define a handedness, or helicity, for that particle, which, in the case of a massless particl ...
, the quantization may lead to the breaking of this (global) chiral symmetry. In that case, the charge associated with the chiral symmetry is not conserved. The non-conservation happens in a process of tunneling from one
vacuum A vacuum (: vacuums or vacua) is space devoid of matter. The word is derived from the Latin adjective (neuter ) meaning "vacant" or "void". An approximation to such vacuum is a region with a gaseous pressure much less than atmospheric pressur ...
to another. Such a process is called an
instanton An instanton (or pseudoparticle) is a notion appearing in theoretical and mathematical physics. An instanton is a classical solution to equations of motion with a finite, non-zero action, either in quantum mechanics or in quantum field theory. M ...
. In the case of a symmetry related to the conservation of a fermionic particle number, one may understand the creation of such particles as follows. The definition of a particle is different in the two vacuum states between which the tunneling occurs; therefore a state of no particles in one vacuum corresponds to a state with some particles in the other vacuum. In particular, there is a
Dirac sea The Dirac sea is a theoretical model of the electron vacuum as an infinite sea of electrons with negative energy, now called '' positrons''. It was first postulated by the British physicist Paul Dirac in 1930 to explain the anomalous negative-en ...
of fermions and, when such a tunneling happens, it causes the
energy level A quantum mechanics, quantum mechanical system or particle that is bound state, bound—that is, confined spatially—can only take on certain discrete values of energy, called energy levels. This contrasts with classical mechanics, classical pa ...
s of the sea fermions to gradually shift upwards for the particles and downwards for the anti-particles, or vice versa. This means particles which once belonged to the Dirac sea become real (positive energy) particles and particle creation happens. Technically, in the
path integral formulation The path integral formulation is a description in quantum mechanics that generalizes the stationary action principle of classical mechanics. It replaces the classical notion of a single, unique classical trajectory for a system with a sum, or ...
, an anomalous symmetry is a symmetry of the
action Action may refer to: * Action (philosophy), something which is done by a person * Action principles the heart of fundamental physics * Action (narrative), a literary mode * Action fiction, a type of genre fiction * Action game, a genre of video gam ...
\mathcal A, but not of the measure and therefore ''not'' of the generating functional :\mathcal Z=\int\! of the quantized theory ( is Planck's action-quantum divided by 2). The measure d\mu consists of a part depending on the fermion field mathrm\psi/math> and a part depending on its complex conjugate mathrm\bar/math>. The transformations of both parts under a chiral symmetry do not cancel in general. Note that if \psi is a
Dirac fermion In physics, a Dirac fermion is a spin-½ particle (a fermion) which is different from its antiparticle. A vast majority of fermions fall under this category. Description In particle physics, all fermions in the standard model have distinct antipar ...
, then the chiral symmetry can be written as \psi \rightarrow e^\psi where \gamma^5 is the chiral gamma matrix acting on \psi. From the formula for \mathcal Z one also sees explicitly that in the
classical limit The classical limit or correspondence limit is the ability of a physical theory to approximate or "recover" classical mechanics when considered over special values of its parameters. The classical limit is used with physical theories that predict n ...
, anomalies don't come into play, since in this limit only the extrema of \mathcal A remain relevant. The anomaly is proportional to the instanton number of a gauge field to which the fermions are coupled. (Note that the gauge symmetry is always non-anomalous and is exactly respected, as is required for the theory to be consistent.)


Calculation

The chiral anomaly can be calculated exactly by
one-loop Feynman diagram In physics, a one-loop Feynman diagram is a connected Feynman diagram with only one cycle ( unicyclic). Such a diagram can be obtained from a connected tree diagram by taking two external lines of the same type and joining them together into a ...
s, e.g. Steinberger's "triangle diagram", contributing to the
pion In particle physics, a pion (, ) or pi meson, denoted with the Greek alphabet, Greek letter pi (letter), pi (), is any of three subatomic particles: , , and . Each pion consists of a quark and an antiquark and is therefore a meson. Pions are the ...
decays, and \pi^0\to e^+e^-\gamma. The amplitude for this process can be calculated directly from the change in the measure of the fermionic fields under the chiral transformation. Wess and Zumino developed a set of conditions on how the partition function ought to behave under
gauge transformation In the physics of gauge theory, gauge theories, gauge fixing (also called choosing a gauge) denotes a mathematical procedure for coping with redundant Degrees of freedom (physics and chemistry), degrees of freedom in field (physics), field variab ...
s called the Wess–Zumino consistency condition. Fujikawa derived this anomaly using the correspondence between
functional determinant In functional analysis, a branch of mathematics, it is sometimes possible to generalize the notion of the determinant of a square matrix of finite order (representing a linear transformation from a finite-dimensional vector space to itself) to the ...
s and the partition function using the
Atiyah–Singer index theorem In differential geometry, the Atiyah–Singer index theorem, proved by Michael Atiyah and Isadore Singer (1963), states that for an elliptic differential operator on a compact manifold, the analytical index (related to the dimension of the space ...
. See Fujikawa's method.


An example: baryon number non-conservation

The Standard Model of
electroweak In particle physics, the electroweak interaction or electroweak force is the unified description of two of the fundamental interactions of nature: electromagnetism (electromagnetic interaction) and the weak interaction. Although these two forc ...
interactions has all the necessary ingredients for successful
baryogenesis In physical cosmology, baryogenesis (also known as baryosynthesis) is the physical process that is hypothesized to have taken place during the early universe to produce baryonic asymmetry, the observation that only matter (baryons) and not anti ...
, although these interactions have never been observed and may be insufficient to explain the total
baryon number In particle physics, the baryon number (B) is an additive quantum number of a system. It is defined as B = \frac(n_\text - n_), where is the number of quarks, and is the number of antiquarks. Baryons (three quarks) have B = +1, mesons (one q ...
of the observed universe if the initial baryon number of the universe at the time of the Big Bang is zero. Beyond the violation of
charge conjugation In physics, charge conjugation is a transformation that switches all particles with their corresponding antiparticles, thus changing the sign of all charges: not only electric charge but also the charges relevant to other forces. The term C- ...
C and
CP violation In particle physics, CP violation is a violation of CP-symmetry (or charge conjugation parity symmetry): the combination of C-symmetry (charge conjugation symmetry) and P-symmetry ( parity symmetry). CP-symmetry states that the laws of physics s ...
CP (charge+parity), baryonic charge violation appears through the Adler–Bell–Jackiw anomaly of the U(1) group. Baryons are not conserved by the usual electroweak interactions due to quantum chiral anomaly. The classic electroweak Lagrangian conserves
baryon In particle physics, a baryon is a type of composite particle, composite subatomic particle that contains an odd number of valence quarks, conventionally three. proton, Protons and neutron, neutrons are examples of baryons; because baryons are ...
ic charge. Quarks always enter in bilinear combinations q\bar q, so that a quark can disappear only in collision with an antiquark. In other words, the classical baryonic current J_\mu^B is conserved: :\partial^\mu J_\mu^B = \sum_j \partial^\mu(\bar q_j \gamma_\mu q_j) = 0. However, quantum corrections known as the
sphaleron A sphaleron ( "slippery") is a static (time-independent) solution to the electroweak field equations of the Standard Model of particle physics, and is involved in certain hypothetical processes that violate baryon and lepton numbers. Such proces ...
destroy this
conservation law In physics, a conservation law states that a particular measurable property of an isolated physical system does not change as the system evolves over time. Exact conservation laws include conservation of mass-energy, conservation of linear momen ...
: instead of zero in the right hand side of this equation, there is a non-vanishing quantum term, :\partial^\mu J_\mu^B = \frac G^ \tilde_^a, where is a numerical constant vanishing for ℏ =0, :\tilde_^a = \frac \epsilon_ G^, and the gauge field strength G_^a is given by the expression :G_^a = \partial_\mu A_\nu^a - \partial_\nu A_\mu^a + g f^_ A_\mu^b A_\nu^c ~. Electroweak sphalerons can only change the baryon and/or lepton number by 3 or multiples of 3 (collision of three baryons into three leptons/antileptons and vice versa). An important fact is that the anomalous current non-conservation is proportional to the total derivative of a vector operator, G^\tilde_^a = \partial^\mu K_\mu (this is non-vanishing due to
instanton An instanton (or pseudoparticle) is a notion appearing in theoretical and mathematical physics. An instanton is a classical solution to equations of motion with a finite, non-zero action, either in quantum mechanics or in quantum field theory. M ...
configurations of the gauge field, which are pure gauge at the infinity), where the anomalous current K_\mu is :K_\mu = 2\epsilon_ \left( A^ \partial^\alpha A^ + \frac f^ A^ A^ A^ \right), which is the
Hodge dual In mathematics, the Hodge star operator or Hodge star is a linear map defined on the exterior algebra of a finite-dimensional oriented vector space endowed with a nondegenerate symmetric bilinear form. Applying the operator to an element of the ...
of the Chern–Simons 3-form.


Geometric form

In the language of
differential form In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many applications ...
s, to any self-dual curvature form F_A we may assign the abelian 4-form \langle F_A\wedge F_A\rangle:=\operatorname\left(F_A\wedge F_A\right). Chern–Weil theory shows that this 4-form is locally ''but not globally'' exact, with potential given by the Chern–Simons 3-form locally: :d\mathrm(A)=\langle F_A\wedge F_A\rangle. Again, this is true only on a single chart, and is false for the global form \langle F_\nabla\wedge F_\nabla\rangle unless the instanton number vanishes. To proceed further, we attach a "point at infinity" onto \mathbb^4 to yield S^4, and use the
clutching construction In topology, a branch of mathematics, the clutching construction is a way of constructing fiber bundles, particularly vector bundles on spheres. Definition Consider the sphere S^n as the union of the upper and lower hemispheres D^n_+ and D^n_- alo ...
to chart principal A-bundles, with one chart on the neighborhood of and a second on S^4-k. The thickening around , where these charts intersect, is trivial, so their intersection is essentially S^3. Thus instantons are classified by the third
homotopy group In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, denoted \pi_1(X), which records information about loops in a space. Intuitively, homo ...
\pi_3(A), which for A = \mathrm\cong S^3 is simply the third 3-sphere group \pi_3(S^3)=\mathbb. The divergence of the baryon number current is (ignoring numerical constants) :\mathbf\star j_b = \langle F_\nabla\wedge F_\nabla\rangle, and the instanton number is :\int_ \langle F_\nabla\wedge F_\nabla\rangle\in\mathbb.


See also

*
Anomaly (physics) In quantum physics an anomaly 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 physics, a classical anomaly is the failure of a symm ...
* Chiral magnetic effect *
Global anomaly Primary Examples In theoretical physics, a global anomaly is a type of anomaly: in this particular case, it is a quantum effect that invalidates a large gauge transformation that would otherwise be preserved in the classical theory. This lea ...
* Gravitational anomaly * strong CP problem#Strong CP problem, Strong CP problem


References


Further reading


Published articles

* * * * * * *


Textbooks

* *


Preprints

*{{cite arXiv , last1=Yang , first1=J.-F. , year=2003 , title=Trace and chiral anomalies in QED and their underlying theory interpretation , eprint=hep-ph/0309311 Anomalies (physics) Quantum chromodynamics Standard Model Conservation laws