Dirac field

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### Context
#### Fields and quanta
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[[!include fields and quanta - table]]
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#### Spin geometry
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[[!include higher spin geometry - contents]]
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#Contents#
* table of contents
{:toc}

## Idea

In [[physics]], specifically in [[field theory]], a _Dirac field_ is an [[electromagnetism|electromagentially]] charged and possibly [[mass|massive]] a _[[fermion|fermionic]] [[spinor]] field_.

This includes [[fermions]] in the [[standard model of particle physics]], such as [[electrons]], [[quarks]], [[neutrinos]] and [[muons]], after the [[Higgs mechanism]] has equipped them with [[mass]].

Mathematically, the Dirac field is the [[field (physics)|field]] whose [[field bundle]] is a [[spinor bundle]] with typical [[fiber]] a [[Dirac representation]] in [[supergeometry|odd super-degree]] and whose [[equation of motion]] is a [[Dirac equation]].

Notice that there are other [[spinor]] [[field (physics)|fields]] which would not be called "Dirac fields", such as those transforming in a [[Majorana representation]] or a [[Weyl representation]], or for example the [[gravitino]] field which would be called a _[[Rarita-Schwinger field]]_.

## Properties

### Free Dirac field in Couloumb background
 {#FreeDiracFieldInCoulombBackground}

In [[solid state physics]], the [[electrons]] described by the [[Dirac equation]] are encountered in the [[electromagnetic field|electromagnetic]] [[background field]] that is sourced by [[atomic nuclei]] such as those on the lattice sites of a [[crystal]].

In a common approximation used in [[condensed matter theory]] for computing [[ground state]]-properties of [[crystals]], the nuclei are typically treated as motionless (due to their large [[mass]] compared to that of the electrons), hence sourcing a static Coulomb background field, while the electromagnetic interaction of the electrons with each other is neglected. In this approximation one is dealing with a "[[free field|free]] [[Dirac field]] in a static [[electromagnetic field|Coulomb]] [[background field|background]]".

> (graphics from [[schreiber:Anyonic topological order in TED K-theory|SS22]], cf. [Rößler 2004, Fig. 2.1](solid+state+physics#Roessler04))

The [[vacua]]/[[ground state]] of the [[free field|free]] [[Dirac field]] in such a "classical Coulomb background" (review in [Thaller 92, Sec. 10](#Thaller92)) have been computed in [Klaus & Scharf 77](#KlausScharf77) and found to be characterized by [[Fredholm operators]] (amplified in [Carey, Hurst & O'Brien 82](#CareyHurstOBrien82)) whose [[kernel]]/[[cokernel]] are the [[Hilbert spaces]] of single [[electron]]/[[positron]] [[quantum states]] which are occupied in the charged vacuum, hence whose [[Fredholm index]] is the total [[charge]] in this vacuum.

> (graphics from SS 22, Sc 22)

## Related concepts

* [[Dirac spinor]]

* [[Dirac conjugate]]

* [[Dirac operator]], [[Dirac equation]]

* [[Dirac current]]

* [[electron propagator]]

## Reference

### General

A concise collection of the Dirac field over [[Minkowski spacetime]] is in

* {#DermisekI9} [[Radovan Dermisek]], _qft I-9_, 2009 ([pdf](http://www.physics.indiana.edu/~dermisek/QFT_09/qft-I-9-4p.pdf), [[DermisekDiracField.pdf:file]])

with background on [[spinor]] [[field (physics)|fields]] in

* {#DermisekI8} [[Radovan Dermisek]], _qft I-8_, 2009 ([pdf](http://www.physics.indiana.edu/~dermisek/QFT_09/qft-I-8-4p.pdf), [[DermisekSpinorField.pdf:file]])

A textbook account of the interacting Dirac field in the context of [[causal perturbation theory]]:

* {#Scharf95} [[Günter Scharf]], section 2.2 of _[[Finite Quantum Electrodynamics -- The Causal Approach]]_, Berlin: Springer-Verlag, 1995, 2nd edition

* {#Scharf01} [[Günter Scharf]], section 1.8 of _[[Quantum Gauge Theories -- A True Ghost Story]]_, Wiley 2001

Textbook account of the [[free field|free]] Dirac field but possiblyin an [[electromagnetic field|electromagnetic]] [[background field]]:

* {#Thaller92} [[Bernd Thaller]], Section 10 of: *The Dirac Equation*,  Texts and Monographs in Physics, Springer (1992) ([doi:10.1007/978-3-662-02753-0](https://link.springer.com/book/10.1007/978-3-662-02753-0))

### Free Dirac field in Coulomb background

Discussion of the *charged* [[vacua]]/[[ground states]] of the [[free field|free]] Dirac field in a strong [[electromagnetic field|Coulomb]] [[background field]] (such as that sourced by [[atomic nuclei]] on a [[crystal]] lattice):

Identification of the [[vacua]]/[[ground states]] of the [[free field|free]] [[Dirac field]] in a [[electromagnetic field|Coulomb]] [[background field]]:

* {#KlausScharf77} [[Martin Klaus]], [[Günter Scharf]], *The regular external field problem in quantum electrodynamics*, Helv. Phys. Acta **50** (1977) [[doi:10.5169/seals-114890](http://doi.org/10.5169/seals-114890), [hpa-001:1977:50::1019](https://www.e-periodica.ch/cntmng?var=true&pid=hpa-001:1977:50::1019)]

and the observation that these vacua are characterized by [[Fredholm operators]] whose [[index]] is the total charge in the vacuum:

* {#CareyHurstOBrien82} [[Alan Carey]], [[Charles Angas Hurst]], [[Denis O'Brien]], *Automorphisms of the canonical anticommutation relations and index theory*, Journal of Functional Analysis **48** 3 (1982) 360-393 [doi:10.1016/0022-1236(82)90092-1]

This curious observation seems not to have been followed up on much, but some related references are listed in [Thaller 92, p. 319](#Thaller92).

[[!redirects Dirac fields]]


Dirac induction

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###Context###
#### Cohomology
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#### Representation theory
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[[!include representation theory - contents]]
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#Contents#
* table of contents
{:toc}

## Idea

For $G$ a suitable ([[compact Lie group|compact]]) [[Lie group]], the  $G$-[[equivariant K-theory]] of the point is the [[representation ring]] of the group $G$:

$$
  K_G(\ast) \simeq Rep(G)
  \,.
$$

Accordingly the construction of an [[index]] ([[push-forward in generalized cohomology|push-forward]] to the point) in equivariant K-theory is a way of producing $G$-[[representations]] from [[equivariant vector bundles]]. Specifically with $K \hookrightarrow G$ a suitable subgroup for the push-forward from $K$-equivariant to $G$-equivariant K-theory/representations, this method is also called _Dirac induction_ since it is analogous to the construction of _[[induced representations]]_.

Applied to equivariant [[complex line bundles]] on [[coadjoint orbits]] of $G$, Dirac induction is a [[K-theory|K-theoretic]] formulation of the [[orbit method]].

## Properties

### Relation to the orbit method
 {#RelationToTheOrbitMethod}

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###### Proposition

For $G$ a [[compact Lie group]] with [[Lie algebra]] $\mathfrak{g}^\ast$, the [[push-forward in generalized cohomology|push-forward]] in compactly supported [[twisted K-theory|twisted]] $G$-[[equivariant K-theory]] to the point (the $G$-equivariant [[index]]) produces the [[Thom isomorphism]]

$$
  ind_{\mathfrak{g}^\ast}
    \;\colon\;
  K_G^{\sigma + dim G}(\mathfrak{g}^\ast)_{cpt}
   \stackrel{\simeq}{\to}
  K_G^0(\ast) \simeq Rep(G)
  \,.
$$

Moreover, for $i \colon \mathcal{O} \hookrightarrow \mathfrak{g}^\ast$ a regular [[coadjoint orbit]], [[push-forward in generalized cohomology|push-forward]] involves a [[twisted K-theory|twist]] $\sigma$ of the form

$$
  Rep(G) \simeq K_G^0(\ast)
  \stackrel{ind_{\mathcal{O}}}{\leftarrow}
  K_G^{\sigma(\mathcal{O}) + dim(\mathcal{O})}(\mathcal{O})
  \stackrel{i_!}{\to}
  K_G^{\sigma + dim G}(\mathfrak{g}^\ast)_{cpt}
$$

and

1. $i_!$ is surjective

1. $ind_{\mathcal{O}} = ind_{\mathfrak{g}^\ast} \circ i_!$.

=--

This is ([FHT II, (1.27), theorem 1.28](#FHT)). Related results are in ([Hochs 12, section 2.2](#Hochs12)). For more background see at _[[orbit method]]_.

## Related concepts

* [[Schubert calculus]]

## References

The idea of Dirac induction goes back to [[Raoul Bott]]'s formulation in the 1960s of [[index theory]] in the [[equivariant cohomology|equivariant]] context.

* [[Raoul Bott]], _The index theorem for homogeneous differential operators_, In: _Differential and combinatorial topology_ (A symposium in honor of Marston Morse), 1965, Princeton Univ. Press, Princeton, NJ, 167–186
 {#Bott65}

* [[Michael Atiyah]], [[Raoul Bott]], _A Lefschetz fixed point formula for
elliptic complexes. II_. Applications. Ann. of Math. (2), 88:451–
491, 1968.
 {#AtiyahBott68}

A generalization to [[super Lie groups]] is discussed in

* [[Gregory Landweber]], _Twisted representation rings and Dirac induction_, J. Pure Appl. Algebra 206 (2006), no. 1-2, 21-54 ([arXiv:math/0403524](http://arxiv.org/abs/math/0403524))

An inverse to Dirac induction, hence a construction of good equivariant vector bundles that push to a given representation, is discussed in

* [[Daniel Freed]], [[Michael Hopkins]], [[Constantin Teleman]], part II, section 1 of _[[Loop Groups and Twisted K-Theory]]_
 {#FHT}

* [[Peter Hochs]], section 2.2 of _Quantisation of presymplectic manifolds, K-theory and group representations_ ([arXiv:1211.0107](http://arxiv.org/abs/1211.0107))
 {#Hochs12}

The analog of Dirac induction for K-theory replaced by [[elliptic cohomology]] is discussed in

* [[Nora Ganter]], _The elliptic Weyl character formula_ ([arXiv:1206.0528](http://arxiv.org/abs/1206.0528))

[[!redirects Dirac inductions]]


Dirac interaction picture

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#### Algebraic Qunantum Field Theory
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#### Physics
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[[!include physicscontents]]
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# Contents
* table of contents
{: toc}

## Idea

The "interaction picture" in [[quantum physics]] is a way to decompose solutions to the [[Schrödinger equation]] and more generally the construction of [[quantum field theories]] into a [[free field theory]]-part and the [[interaction]] part that acts as a [[perturbation]] of the free theory. Therefore the interaction picture lends itself to the construction of [[perturbative quantum field theory]], and in fact the only mathematically rigorous such construction scheme that is known, namely _[[causal perturbation theory]]_, proceeds this way.

[[dynamics|Dynamics]] in [[physics]] affects both [[observables]] and, dually, [[states]]; this is most well known in [[quantum physics]] but applies equally well to [[classical physics]]. The different "pictures" of physics differ in how the dynamics is explicitly formalized:

* In the [[Schrödinger picture]], states are propagated through [[time]], while observables are held fixed; the axiomatic formalization of this is given by [[cobordism category]] [[representation]]s in [[FQFT]].

* In the [[Heisenberg picture]], the dependence of observables on time (or more generally [[spacetime]]) is encoded, while the state is held fixed; the axiomatic formalization of this is given by the [[Haag–Kastler axioms]] of [[AQFT]].

* The __Dirac (interaction) picture__ is a mixture of these two approaches: dynamics is split into a free (or otherwise solvable) part and an [[interaction]] (often then treated as a [[perturbation theory|perturbation]]); one of these is taken to affect the states, the other the observables.

The pictures are named after those physicists who first used or popularised these approaches to quantum physics.

### In quantum mechanics
 {#InQuantumMechanics}

In [[quantum mechanics]], let $\mathcal{H}$ be some [[Hilbert space]] and let

$$
  H = H_{free} + V
$$

be Hermitian operator, thought of as a [[Hamiltonian]], decomposed as the [[sum]] of a free part ([[kinetic energy]]) and an interaction part ([[potential energy]]).

For example for a  [[non-relativistic particle]] of [[mass]] $m$ propagating on the [[line]] subject to a [[potential energy]] $V_{pot} \colon \mathbb{R} \to \mathbb{R}$, then $\mathcal{H} = L^2(X)$ is the Hilbert space of [[square integrable functions]] and

$$
  H
   =
  \underset{H_{free}}{\underbrace{\tfrac{- \hbar^2}{2m} \frac{\partial^2}{\partial^2 x}}}
    +
  V
  \,,
$$

where $V = V_{pot}(x)$ is the operator of multiplying square integrable functions with the given potential energy function.

Now for

$$
  \array{
    \mathbb{R} &\overset{\vert \psi (-)\rangle }{\longrightarrow}& \mathcal{H}
   \\
   t &\mapsto& \vert \psi(t) \rangle
  }
$$

a one-parameter family of [[quantum states]], the [[Schrödinger equation]] for this state reads

$$
  \frac{d}{d t} \vert \psi(t) \rangle
  \;=\;
  \tfrac{1}{i \hbar} H \vert \psi\rangle
  \,.
$$

It is easy to solve this [[differential equation]] formally via its [[Green function]]: for $\vert \psi \rangle \in \mathcal{H}$ any state, then the unique solution $\vert \psi(-) \rangle$ to the Schrödinger equation subject to $\vert \psi(0) \rangle = \vert \psi \rangle$ is

$$
  \vert \psi(t)\rangle_S \coloneqq \exp( \tfrac{t}{i \hbar} H ) \vert \psi \rangle
  \,.
$$

(One says that this is the solution "in the [[Schrödinger picture]]", whence the subscript.)

However, if $H$ is sufficiently complicated, it may still be very hard to extract from this expression a more explicit formula for $\vert \psi(t) \rangle$, such as, in the example of the free particle on the line, its expression as a function ("[[wave function]]") of $x$ and $t$.

But assume that the analogous expression for $H_{free}$ alone is well understood, hence that the operator

$$
  U_{S,free}(t_1, t_2) \coloneqq \exp\left({\tfrac{t_2 - t_1}{i \hbar} H_{free}}\right)
$$

is sufficiently well understood. The "[[interaction picture]]" is a way to decompose the Schrödinger equation such that its dependence on $V$ gets separated from its dependence on $H_{free}$ in a way that admits to treat $H_{int}$ in [[perturbation theory]].

Namely define analogously

\[
  \label{StateInTheInteractionPicture}
  \begin{aligned}
    \vert \psi(t)\rangle_I
     &\coloneqq
     \exp\left({\tfrac{- t}{i \hbar} H_{free}}\right) \vert \psi(t)\rangle_S
    \\
    & = \exp\left({\tfrac{- t}{i \hbar} H_{free}}\right)
        \exp\left({ \tfrac{+ t}{i \hbar} H} \right)\vert \psi \rangle
    \\
    & = \exp\left({\tfrac{- t}{i \hbar} H_{free}}\right)
        \exp\left({\tfrac{t}{i \hbar} H_{free} + \tfrac{t}{i \hbar} V} \right)\vert \psi \rangle   \end{aligned}
  \,.
\]

This is called the solution of the Schrödinger equation "in the [[interaction picture]]", whence the subscript. Its definition may be read as the result of propagating the actual solution $\vert \psi(-)\rangle_S$ at time $t$ back to time $t = 0$, but using just the free Hamiltonian, hence with "the interaction switched off".

Notice that if the operator $V$ were to commute with $H_{free}$ (which it does not in all relevant examples) then we would simply have $\vert \psi(t)\rangle_I = \exp( \tfrac{t}{i \hbar } V ) \vert \psi\rangle$, hence then the solution (eq:StateInTheInteractionPicture) in the interaction picture would be the result of "propagating" the initial conditions using _only_ the interaction. Now since $V$ may not be assumed to commute with $H_{free}$, the actual form of $\vert \psi(-) \rangle_{I}$ is more complicated. But _infinitesimally_ it remains true that $\vert \psi(-)\rangle_I$ is propagated this way, not by the plain operator $V$, though, but by $V$ viewed in the [[Heisenberg picture]] of the free theory. This is the content of the differential equation (eq:DifferentialEquationInInteractionPicture) below.

But first notice that this will indeed be useful: If an explicit expression for the "state in the [[interaction picture]]" (eq:StateInTheInteractionPicture) is known, then the assumption that also the operator $\exp\left({\tfrac{t}{i \hbar} H_{free}}\right)$ is sufficiently well understood implies that the actual solution

$$
  \vert \psi(t) \rangle_S = \exp\left({\tfrac{t}{i \hbar} H_{free}}\right) \vert \psi(t) \rangle_I
$$

is under control. Hence the question now is how to find $\vert \psi(-)\rangle_I$ given its value at some time $t$. (It is conventional to consider this for $t \to \pm \infty$, see (eq:SMatrixInQuantumMechanics) below.)

Now it is clear from the construction and using the [[product law]] for [[differentiation]], that $\vert \psi(-)\rangle_S$ satisfies the following [[differential equation]]:

\[
  \label{DifferentialEquationInInteractionPicture}
  \frac{d}{d t} \vert \psi(t) \rangle
  \;=\;
  V_I(t) \vert \psi(t)\rangle_I
  \,,
\]

where

$$
  V_I(t)
    \coloneqq
  \exp\left( -\tfrac{t}{i \hbar} H_{free} \right)
    V
  \exp\left( +\tfrac{t}{i \hbar} H_{free} \right)
$$

is known as the [[interaction]] term $V$ "viewed in the interaction picture". But in fact this is just $V$ "viewed in the [[Heisenberg picture]]", but for the _free_ theory. By our running assumption that the free theory is well understood, also $V_I(t)$ is well understood, and hence all that remains now is to find a sufficiently concrete solution to equation (eq:DifferentialEquationInInteractionPicture). This is the heart of working in the interaction picture.

Solutions to equations of the "[[parallel transport]]"-type such as (eq:DifferentialEquationInInteractionPicture) are given by [[time-ordered product|time-ordering]] of Heisenberg picture operators, denoted $T$, applied to the naive exponential solution as above. This is known as the _[[Dyson formula]]_:

$$
  \vert \psi(t)\rangle_I
  \;=\;
  T\left(
    \exp\left(
      \int_{t_0}^t
        V_I(t)
        \tfrac{d t}{i \hbar}
    \right)
  \right)
  \vert \psi(t_0)\rangle
  \,.
$$

Here [[time-ordered product|time-ordering]] means

$$
  T( V_I(t_1) V_I(t_2) )
  \;\coloneqq\;
  \left\{
    \array{
      V_I(t_1) V_I(t_2) &\vert& t_1 \geq t_2
      \\
      V_I(t_2) V_I(t_1) &\vert& t_2 \geq t_2
    }
  \right.
  \,.
$$

(This is abuse of notation: Strictly speaking time ordering acts on the [[tensor algebra]] spanned by the $\{V_I(t)\}_{t \in \mathbb{R}}$ and has to be _followed_ by taking tensor products to actual products. )

In applications to [[scattering]] processes one is interested in prescribing the [[quantum state]]/[[wave function]] far in the past, hence for $t \to - \infty$, and computing its form far in the future, hence for $t \to \infty$.

The operator that sends such "asymptotic ingoing-states" $\vert \psi(-\infty) \rangle_I$ to "asymptotic outgoing states" $\vert \psi(+ \infty) \rangle_I$ is hence the [[limit of a sequence|limit]]

\[
  \label{SMatrixInQuantumMechanics}
  S
  \;\coloneqq\;
  \underset{t \to \infty}{\lim}
  T\left(
    \exp\left(
      \int_{-t}^t
        V_I(t)
        \tfrac{d t}{i \hbar}
    \right)
  \right)
  \,.
\]

This limit (if it exists) is called the _scattering matrix_ or _S-matrix_, for short.

### In quantum field theory

In [[perturbative quantum field theory]] the broad structure of the interaction picture in quantum mechanics ([above](#InQuantumMechanics)) remains a very good guide, but various technical details have to be generalized with due care:

1. The algebra of operators in the [[Heisenberg picture]] of the free theory becomes the _[[Wick algebra]]_ of the [[free field theory]] (taking into account "normal ordering" of field operators) defined on _[[microcausal functionals]]_ built from [[operator-valued distributions]] with constraints on their [[wave front set]].

1. The [[time-ordered products]] in the [[Dyson formula]] have to be refined to [[causal locality|causally ordered]] products and the resulting product at coincident points has to be defined by [[point-extension of distributions]] -- the freedom in making this choice is the [[renormalization]] freedom ("conter-terms").

1. The sharp interaction cutoff in the [[Dyson formula]] that is hidden in the integration over $[t_0,t]$ has to be smoothed out by [[adiabatic switching]] of the interaction (making the whole S-matrix an [[operator-valued distribution]]).

Together these three point are taken care of by the axiomatization of the "[[adiabatic switching|adiabatically switched]] [[S-matrix]]" according to **[[causal perturbation theory]]**.

The analogue of the limit $t \to \infty$ in the construction of the [[S-matrix]] (now: [[adiabatic limit]]) in general does not exist in field theory ("infrared divergencies"). But in fact it need not be taken: The field algebra in a bounded region of [[spacetime]] may be computed with any adiabatic switching that is constant on this region. Moreover, the algebras assigned to regions of spacetime this way satisfy [[causal locality]] by the causal ordering in the construction of the S-matrix. Therefore, even without taking the adiabatic limit in [[causal perturbation theory]] one obtains a field theory in the form of a _[[local net of observables]]_. This is the topic of **[[locally covariant perturbative quantum field theory]]**.

## Related concepts

* [[interaction]]

* [[interacting field algebra]]

* [[interacting field theory]]

* [[S-matrix]]

* [[Møller operator]]

## References

For instance

* [[Eberhard Zeidler]], section 7.19.3 of _Quantum field theory. A bridge between mathematicians and physicists -- volume I_ Springer (2009) ([web](http://www.mis.mpg.de/zeidler/qft.html))

[[!redirects Dirac picture]]
[[!redirects interaction picture]]
[[!redirects Dirac interaction picture]]
[[!redirects Dirac (interaction) picture]]


Dirac manifold

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###Context###
#### $\infty$-Lie theory
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[[!include infinity-Lie theory - contents]]
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#### Symplectic geometry
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[[!include symplectic geometry - contents]]
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#Contents#
* table of contents
{:toc}

## Idea

A _Dirac manifold_ is a [[smooth manifold]] with a [[Dirac structure]] in the sense of ([Courant 90](#Courant)).

(Beware that sometimes "Dirac structure" is used for Dirac brackets in earlier contexts which are related but not formalized in this way.)

An _almost Dirac structure_ on a manifold $X$ is a subbundle $L\subset T X \oplus T^*X$ of the tangent [[Courant algebroid]],  which is [[isotropic submanifold|isotropic]] under a certain symmetric pairing. An almost Dirac structure is a Dirac structure if it satisfies an integrability condition.

## Properties

### Relation to Lagrangian dg-submanifolds

A [[Courant Lie 2-algebroid]] is a [[symplectic Lie n-algebroid]] for $n = 2$. Dirac structures are related to the [[Lagrangian dg-submanifolds]] (see there) of the dg-manifold formally dual to its [[Chevalley-Eilenberg algebra]].

### Relation to D-branes

With suitable identifications Dirac structures characterize [[D-branes]]. This is argued generally in ([Asakawa-Sasa-Watamura](#AsakawaSasaWatamura)).

An example is the canonical [[Cartan-Dirac structure on a Lie group]], which yields the [[conjugacy classes]] of the Lie group as leaves. These are indeed known to be the D-branes of the [[WZW model]] on that Lie group.

## Examples

* [[Cartan-Dirac structure on a Lie group]]

## Related concepts

* [[polarization]]

* [[Poisson structure]]

[[!include infinity-CS theory for binary non-degenerate invariant polynomial - table]]

## References

### General

The original articles are

* [[Ted Courant]], [[Alan Weinstein]], _Beyond Poisson structures_, preprint, Berkeley 1986 [pdf](http://math.berkeley.edu/~alanw/Beyond.pdf)

* [[Irene Dorfman]], _Dirac structures of integrable evolution equations_, Phys. Lett. A 125 (1987), no. 5, 240–246 doi [MR89b:58088](http://www.ams.org/mathscinet-getitem?mr=918697)

* [[Ted Courant]], _Dirac manifolds_, Trans. Amer. Math. Soc. __319__ (1990), no. 2, 631–661 [MR90m:58065](http://www.ams.org/mathscinet-getitem?mr=998124) [doi](http://dx.doi.org/10.2307/2001258); _Tangent Dirac structures_, J. Phys. A 23 (1990), no. 22, 5153–5168 [MR92d:58064](http://www.ams.org/mathscinet-getitem?mr=1085863) [iop](http://stacks.iop.org/0305-4470/23/5153)
 {#Courant}

* [[Irene Dorfman]], _Dirac structures and integrability of non-linear evolution equations_, John Wiley and sons, 1993. xii+176 pp. [MR94j:58081](http://www.ams.org/mathscinet-getitem?mr=1237398)

Lecture notes include section 2 of

* [[Henrique Bursztyn]], [[Alan Weinstein]], _Poisson geometry and Morita equivalence_ ([arXiv:0402347](http://arxiv.org/abs/math/0402347))

The generalization of Dirac structures from base manifolds to base [[Lie groupoids]] ("multiplicative Dirac structure") is discussed in

* [[Cristian Ortiz]], _Multiplicative Dirac structures_ ([arXiv:1212.0176](http://arxiv.org/abs/1212.0176))

Further references include

* [[Anton Alekseev]], Eric Meinrenken, _Dirac structures and Dixmier-Douady bundles_, Int. Math. Res. Not. IMRN 2012, no. 4, 904–956. [MR2889163](http://www.ams.org/mathscinet-getitem?mr=2889163)

### Relation to D-branes
 {#ReferencesRelationToDBranes}

Relating Dirac structures to [[D-branes]]:

* {#AsakawaSasaWatamura} [[Tsuguhiko Asakawa]], Shuhei Sasa, Satoshi Watamura, _D-branes in Generalized Geometry and Dirac-Born-Infeld Action_ [[arXiv:1206.6964](http://arxiv.org/abs/1206.6964)]

Related observations for D-branes in the [[WZW model]] had long been made (unpublished) for the [[Cartan-Dirac structure]] over a Lie group.

[[!redirects Dirac structures]]
[[!redirects Dirac manifolds]]
[[!redirects Dirac structure]]


Dirac measure

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#### Measure and probability theory
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[[!include measure theory - contents]]
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#Contents#
* table of contents
{:toc}

## Idea

A Dirac measure is a [[measure]] whose (unit) mass is concentrated on a single [[point]] $x$ of a [[space]] $X$.

From the point of view of [[probability theory]], a Dirac measure can be seen as the [[law of a random variable|law]] of a [[deterministic random variable]], or more generally one which is [[almost surely]] equal to a point $x$.

See also [[Dirac distribution]] for the analogous concept in the language of [[distributions]].

## Definition

### For measurable spaces

Let $X$ be a [[measurable space]]. Given $x\in X$, the **Dirac measure** $\delta_x$ at $x$ is the [[measure]] defined by
$$
\delta_x(A) \;\coloneqq\;
\begin{cases}
1 & x\in A \\
0 & x\notin A
\end{cases}
$$
for each [[measurable set]] $A\subseteq X$.

### For topological spaces

If $X$ is a [[topological space]], the Dirac measure at $x$ can be also defined as the unique [[Borel measure]] $\delta_x$ which satisfies
$$
\delta_x(U) \;\coloneqq\;
\begin{cases}
1 & x\in U \\
0 & x\notin U
\end{cases}
$$
for each [[open set]] $U\subseteq X$.

Equivalently, it is the [[continuous valuation#extending_valuations_to_measures|extension to a measure]] of the [[continuous valuation#dirac_valuations|Dirac valuations]].

### For locales

(...)

(See also [[correspondence between measure and valuation theory]].)

## Properties

* On [[topological spaces]], Dirac measures are [[Radon measure|Radon]] and [[τ-additive]].

* Every [[continuous valuation#dirac_valuations|Dirac valuation]] on a topological space can be [[continuous valuation#extending_valuations_to_measures|extended]] to a Dirac measure.

* On a [[topological space]] $X$, the [[support]] of the Dirac measure at $x\in X$ is equal to the [[closure]] of $x$. On [[T1]] spaces, this is just the [[singleton]] $\{x\}$.

* The [[pushforward measure]] of a Dirac measure along a [[measurable function]] is again a Dirac measure. This is related to [[natural transformation|naturality]] of the unit map of [[probability and measure monads]].

* Given a Dirac measure $\delta_x$ on a [[measurable space]] $X$ and any [[measure]] $\mu$ on any measurable space $Y$, the [[product measure]] $\delta_x\otimes \mu$ is the unique [[coupling]] of $\delta_x$ and $\mu$.

* The coupling above defines a map $X\times P Y\to P(X\times Y)$ which gives the [[strong monad|strength]] of most [[probability and measure monads]].

## Significance

* The Dirac measures (and the [[continuous valuation#Dirac valuations|Dirac valuations]]) give the unit of all [[probability and measure monads]].

* The probabilistic interpretation is that the Dirac measures are exactly those of [[deterministic]] elements (or almost deterministic), i.e. which are "not truly random".

* In terms of [[random variables]], and somewhat conversely, a [[random element]] of $X$ has the Dirac measure $\delta_x$ as [[law of a random variable|law]] if and only if it is [[almost surely]] equal to $x$.

## See also

* [[Dirac distribution]]

* [[monads of probability, measures, and valuations]]

* [[measure]], [[Radon measure]], [[τ-additive measure]], [[continuous valuation]]

[[!redirects Dirac measures]]

[[!redirects Dirac delta measure]]
[[!redirects Dirac delta measures]]

[[!redirects delta measure]]
[[!redirects delta measures]]

[[!redirects point measure]]
[[!redirects point measures]]

[[!redirects point mass measure]]
[[!redirects point mass measures]]


Dirac monopole

+-- {: .rightHandSide}
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###Context###
#### Physics
+--{: .hide}
[[!include physicscontents]]
=--
#### $\infty$-Chern-Weil theory
+--{: .hide}
[[!include infinity-Chern-Weil theory - contents]]
=--
#### Differential cohomology
+--{: .hide}
[[!include differential cohomology - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A [[circle bundle with connection]] with non-trivial [[first Chern class]], regarded as a [[monopole]].

For more see at *[[flux]]* and at _[[Dirac charge quantization]]_.

## Properties
 {#Properties}

The [[trajectory]] of an [[electron]] in the vicinity of a [[magnetic monopole]] is (e.g. [Ferraro (1956) §306](#Ferraro54)) a helix on the surface of a [[cone (Riemannian geometry)|cone]] in space whose vertex is the location of the magnetic monopole:

## Related concepts

* [[monopole]]

* [[Yang monopole]]

* [[fiber bundles in physics]]

* [['t Hooft operator]]

## References

Textbook accounts:

* {#Ferraro54} V. C. A. Ferraro, §137 in: *Electromagnetic Theory*, The Athlone Press (1954, 1956)

* {#Brylinski93} [[Jean-Luc Brylinski]], §7.1 in: *Loop spaces, characteristic classes and geometric quantization*, Birkhäuser (1993) [[doi:10.1007/978-0-8176-4731-5](https://www.springer.com/gp/book/9780817647308))]

* [[Theodore Frankel]], section 5.5 of _[[The Geometry of Physics - An Introduction]]_, Cambridge University Press (1997) [[doi:10.1017/CBO9781139061377](https://doi.org/10.1017/CBO9781139061377)]

* [[Yakov Shnir]], Part I of: *Magnetic Monopoles*, Springer (2005) [[ISBN:978-3-540-29082-7](https://www.springer.com/gp/book/9783540252771)]

The original argument that magentic [[monopoles]] may appear in [[grand unified theories]]:

* [[Gerard ’t Hooft]], *Magnetic Monopoles in Unified Gauge Theories*, Nucl. Phys. B **79** (1974) 276-284 [doi:10.1016/0550-3213(74)90486-6]

See also:

* [[Tai Tsun Wu]], [[Chen Ning Yang]], _Dirac monopole without strings: monopole harmonics_, Nuclear Physics B107:3 (1976) 365--380, <https://doi.org/10.1016/0550-3213(76)90143-7>

* [[Tai Tsun Wu]], [[Chen Ning Yang]], _Dirac's monopole without strings: Classical Lagrangian theory_, Phys. Rev. D 14, 437 (1976), <https://doi.org/10.1103/PhysRevD.14.437>

Instead of removing the singular locus of the monopole from spacetime (so that [[Maxwell's equations]] without magnetic sources are satisfied on the [[complement]]) one may retain the monopole locus and change the model of the [[electromagnetic field]] from a [[connection on a bundle|connection]] on a [[circle group|U(1)]]-[[principal bundle]] to a suitable *trivialization* of a [[circle 2-bundle with connection]] (a "[[bundle gerbe]]" [[connection on a bundle gerbe|with connection]]):

* [Brylinski 1993, §7.2](#Brylinski93)

* {#Freed00} [[Daniel Freed]], §2 of: *[[Dirac charge quantization and generalized differential cohomology]]*, Surveys in Differential Geometry **7**, Int. Press (2000) 129-194 [[arXiv:hep-th/0011220](http://arxiv.org/abs/hep-th/0011220), [doi:10.4310/SDG.2002.v7.n1.a6](https://dx.doi.org/10.4310/SDG.2002.v7.n1.a6), [spire:537392](https://inspirehep.net/literature/537392)]

* {#ChenGirelli22} [[Hank Chen]], [[Florian Girelli]], §2.5.3 in: *Gauging the Gauge and Anomaly Resolution* [[arXiv:2211.08549](https://arxiv.org/abs/2211.08549)]

[[!redirects Dirac monopoles]]


Dirac operator

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### Context
#### Spin geometry
+-- {: .hide}
[[!include higher spin geometry - contents]]
=--
#### Index theory
+-- {: .hide}
[[!include index theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

### General

For $S \to X$ a [[spinor bundle]] over a [[Riemannian manifold]] $(X,g)$, a _Dirac operator_ on $S$ is an [[differential operator]] on ([[sections]] of) $S$ whose [[principal symbol]] is that of $c \circ d$, where $d$ is the [[exterior derivative]] and $c$ is the [[symbol map]].

More abstractly, for $D$ a Dirac operator, its normalization $D(1+ D^2)^{-1/2}$ is a [[Fredholm operator]], hence defines an element in [[K-homology]].

### Origin and role in Physics

The first relativistic Schrödinger type equation found was Klein-Gordon. At first it did not look that K-G equation could be interpreted physically because of negative energy states and other paradoxes. [[Paul Dirac]] proposed to take a square root of [[Laplace operator]] within the matrix-valued differential operators and obtained a Dirac equation; matrix valued generators involved representations of a Clifford algebra. It also had negative energy solutions, but with half-integer spin interpretation which was appropriate the Pauli exclusion principle together with the Dirac sea picture came at rescue (Klein-Gordon is now also useful with more modern formalisms).

(...)

## Definition

### In components

The tangent bundle of an oriented Riemannian $n$-dimensional manifold $M$ is an $SO(n)$-bundle. Orientation means that the first [[Stiefel-Whitney class]] $w_1(M)$ is zero. If $w_2(M)$ is zero than the $SO(n)$ bundle can be lifted to a $Spin(n)$-bundle. A choice of connection on such a $Spin(n)$-bundle is a $Spin$-structure on $M$. There is a standard $2^{[n/2]}$-dimensional representation of $Spin(n)$-group, so called Spin representation. If $n$ is odd it is irreducible, and if $n$ is even it decomposes into the sum of two irreducible representations of equal dimension $S_+$ and $S_-$. Thus we can associate associated bundles to the original $Spin(n)$ bundle $P$ with respect to these representations. Thus we get the **spinor bundles** $E_\pm := P\times_{Spin(n)} S_\pm\to M$
and $E = E_+\oplus E_-$.

Gamma matrices, which are the representations of the [[Clifford algebra]]
$$
\gamma_a \gamma_b + \gamma_b \gamma_a = -2\delta_{ab} I
$$
$$
\gamma_5 = i^{n(n+1)/2}\gamma_1\cdots\gamma_n, \,\,\,\,\gamma^2_5 = I
$$

thus act on such a space; certain combinations of products of gamma matrices with partial derivatives define a first order Dirac operator $\Gamma(E)\to\Gamma(E_-)$; there are several versions, in mathematics is pretty important the chiral Dirac operator
$$
\Gamma(M,E_+)\to \Gamma(M,E_-)
$$
given by local formula
$$
\sum_a \gamma^a e^\mu_a(x) \nabla_\mu \frac{1+\gamma_5}{2}
$$
where $e^\mu_a(x)$ are orthonormal frames of tangent vectors and $\nabla_\mu$ is the [[covariant derivative]] with respect to the Levi-Civita spin connection. The expression $\frac{1+\gamma_5}{2}$ is the chirality operator.

In Euclidean space the Dirac operator is elliptic, but not in Minkowski space.

The Dirac operator is involved in approaches to the [[Atiyah-Singer index theorem]] about the index of an elliptic operator: namely the index can be easier calculated for Dirac operator and the deformation to the Dirac operator does not change the index. An appropriate version of a Dirac operator is a part of a concept of the [[spectral triple]] in [[noncommutative geometry]] a la [[Alain Connes]].

## Properties

### Eta invariant and functional determinant

The [[eta function]] (see there for more) of a Dirac operator $D$ expresses the [[functional determinant]] of its [[Laplace operator]] $H = D^2$.

### Index and partition function

+-- {: .num_prop}
###### Proposition

Let $(X,g)$ be a [[compact topological space|compact]] [[Riemannian manifold]] and $\mathcal{E}$ a smooth [[super vector bundle]] and indeed a [[Clifford module bundle]] over $X$. Consider a Dirac operator

$$
  D \colon \Gamma(X,\mathcal{E}) \to \Gamma(X, \mathcal{E})
$$

with components (with respect to the $\mathbb{Z}_2$-[[graded vector space|grading]]) to be denoted

$$
  D =
  \left[
    \array{
       0 & D^-
       \\
       D^+ & 0
    }
  \right]
  \,,
$$

where $D^- = (D^+)^\ast$. Then $D^+$ is a [[Fredholm operator]] and its [[index]] is the [[supertrace]] of the [[kernel]] of $D$, as well as of the [[heat kernel]] of $D^2$:

$$
  \begin{aligned}
    ind(D^+) & \coloneqq dim(ker(D^+)) - dim(coker(D^+))
   \\
   & = dim(ker(D^+)) - dim(ker(D^-))
   \\
   & = sTr(ker(D))
   \\
   & = sTr( \exp(-t \, D^2) ) \;\;\; \forall t \gt 0
  \end{aligned}
  \,.
$$

=--

This appears as
([Berline-Getzler-Vergne 04, prop. 3.48, prop. 3.50](BerlineGetzlerVergne04)), based on ([MacKean-Singer 67](#MacKeanSinger67)).

+-- {: .num_remark}
###### Remark

If one thinks of $D^2$ as the time-evolution [[Hamiltonian]] of a system of [[supersymmetric quantum mechanics]] with $D$ the [[supercharge]] on the [[worldline]], then $ker(D)$ is the space of supersymmetric [[quantum states]], $\exp(-t \, D^2)$ is the Euclidean time evolution operator and its [[supertrace]] is the [[partition function]] of the system. Hence we have the translation

*  index = partition function .

=--

## Examples

* [[Kähler-Dirac operator]]

* [[Dolbeault-Dirac operator]]

* [[Spin^c Dirac operator]]

## Related concepts

* [[Dirac spinor]], [[Dirac field]]

* [[Dirac equation]]

* [[spin^c Dirac operator]]

* [[index of a Dirac operator]]

* [[Fredholm module]], [[K-homology]], [[KK-theory]]

* [[Lichnerowicz formula]]

* [[Dirac-Ramond operator]], [[Dirac operator on smooth loop space]]

* [[Feynman slash notation]]

[[!include genera and partition functions - table]]

## References

Textbooks include

* [[H. Blaine Lawson]], [[Marie-Louise Michelsohn]], _[[Spin geometry]]_, Princeton University Press (1989)

* [[Thomas Friedrich]], _Dirac operators in Riemannian geometry_, Graduate studies in mathematics 25, AMS (1997)
 {#Friedrich97}

The relation to [[index theory]] is discussed in

* [[Nicole Berline]], [[Ezra Getzler]], [[Michèle Vergne]], _Heat Kernels and Dirac Operators_, Springer Verlag Berlin (2004)
  {#BerlineGetzlerVergne04}

based on original articles such as

* H. MacKean, [[Isadore Singer]], _Curvature and eigenvalues of the Laplacian_, J. Diff. Geom. 1 (1967)
 {#MacKeanSinger67}

* [[Michael Atiyah]], [[Raoul Bott]], V. K. Patodi, _On the heat equation and the index theorem_, Invent. Math. 19 (1973), 279–330.

See also

* {#Freed87} [[Daniel Freed]], *Geometry of Dirac operators*, (1987) [[pdf](http://www.ma.utexas.edu/users/dafr/DiracNotes.pdf), [[FreedGeometryOfDiracOperators.pdf:file]]]

* C. Nash, _Differential topology and quantum field theory_, Acad. Press 1991.

* [[Eckhard Meinrenken]], _Clifford algebras and Lie groups_, Lecture Notes, University of Toronto, Fall 2009.

* Jing-Song Huang, Pavle Pandžić, J.-S. Huang, P. Pandzic, Dirac Operators in Representation Theory,. Birkhäuser, Boston, 2006, 199 pages; short version _Dirac operators in representation theory_, 48 pp. [pdf](http://www.ims.nus.edu.sg/Programs/liegroups/files/singnotes.pdf)

* J.-S. Huang, Pavle Pandžić, _Dirac cohomology, unitary representations and a proof of a conjecture of Vogan_, J. Amer. Math. Soc. __15__ (2002), 185—202.

* R. Parthasarathy, _Dirac operator and the discrete series_, Ann. of Math. __96__ (1972), 1-30.

[[!redirects Dirac operators]]


Dirac operator on quantum group

Dirac operator are the central ingredient of Alain Connes' approach to noncommutative Spin manifolds (via formalism of [[spectral triple]]s). Examples of Dirac operators on quantum groups and on [[quantum homogeneous space]]s are very important examples.

* [[Ulrich Krähmer]], _Dirac operators on quantum flag manifolds_, Lett. Math. Phys. 67/1 (2004) 49-59, [doi](http://dx.doi.org/10.1023/B:MATH.0000027748.64886.23), [pdf](http://www.maths.gla.ac.uk/~ukraehmer/0305071.pdf), [MR2005b:58009](http://www.ams.org/mathscinet-getitem?mr=2005b:58009)

* Antti J. Harju, _Covariant Dirac operators on quantum groups_, [arxiv/1009.3913](http://arxiv.org/abs/1009.3913)

[[!redirects Dirac operators on quantum group]]
[[!redirects Dirac operators on quantum groups]]


Dirac operator on smooth loop space

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Spin geometry
+-- {: .hide}
[[!include higher spin geometry - contents]]
=--
#### Index theory
+-- {: .hide}
[[!include index theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The traditional definition of a [[Dirac operator]] is formulated for operators acting on [[sections]] of [[spinor bundles]] over [[Riemannian manifolds]], not however directly for bundles over [[infinite dimensional manifolds]].

When the conceptual importance of the [[Dirac-Ramond operator]] in the [[superstring]] [[worldsheet]] [[2d SCFT]] was realized (an operator in a ([[sheaf of vertex operator algebras|sheaf of]]) [[super vertex operator algebra]]) via the relation in the [[large volume limit]] of its [[index]] (suitably regarded) to the [[Witten genus]], then it seemed suggestive that it should be possible to regard the Dirac-Ramond operator as an actual Dirac operator on the infinite-dimensional [[smooth loop space]] of the underlying manifold, and regard its $S^1$-equivariant [[index of a Dirac operator]] in some precise sense.

A Dirac operator on smooth loop space would also serve to make precise the sense in which [[superstring]] quantum dynamics is [[supersymmetric quantum mechanics]] on [[smooth loop space]], an observation that initially motivated the work by [[Witten]] on supersymmetric quantum mechanics.

The definition of Dirac operators on smooth loop spaces is technically tricky, but constructions do exist. It remains however unclear how these constructions relate to the [[Dirac-Ramond operator]] and a rigorous derivation of the [[Witten genus]] as an (equivariant) [[index of a Dirac operator]] along these lines seems to remain open.

## Related concepts

* [[stringor bundle]]

## References

The motivation for constructing Dirac operator on smooth loop spaces is mainly due to the observations about the universal [[elliptic genus]] (the [[Witten genus]]) in

* {#Witten87b} [[Edward Witten]], _The Index Of The Dirac Operator In Loop Space_ Proc. of Conf. on Elliptic Curves and Modular Forms in Algebraic Topology Princeton (1986) ([spire](http://inspirehep.net/record/245523))

### On formal loop space

Since either due to arguing that $S^1$-equivariance localizes the [[index]] of a loop space Dirac operator on constant loops, or else arguing that it sees only the "[[large volume limit]]", it is only the [[formal loop space]] that actually enters the computation of the [[Witten genus]]. Therefore large parts of this literature focus on some version of this [[formal loop space]] while often still speaking of just "loop space". This includes the early articles

* {#Taubes} C.H. Taubes, _$S^1$-actions and elliptic genera_, Communications in mathematical physics **122** (1989), no. 3, pages 455-526, doi:[10.1007/BF01238437](https://dx.doi.org/doi:10.1007/BF01238437).

* {#Brylinski90} [[Jean-Luc Brylinski]], _Representations of loop groups, Dirac operators on loop space, and modular forms_, Topology, 29(4):461–480, (1990) doi:[10.1016/0040-9383(90)90016-D](http://dx.doi.org/10.1016/0040-9383(90%2990016-D)

* J. D. S. Jones and R. Léandre. _A stochastic approach to the Dirac
operator over the free loop space_, Tr.Mat. Inst. Steklova, 217(Prostran.
Petel i Gruppy Diffeomorf.):258–287, 1997

* {#Landweber99} [[Gregory Landweber]], _Dirac operators on loop space_ PhD thesis (Harvard 1999) ([pdf](http://math.bard.edu/greg/LoopDirac.pdf))

Via the [relation of formal loop space to chiral differential operators](formal+loop+space#RelationToChiralDifferentialOperators) similar comments apply modern "geometric" constructions of the [[Witten genus]] in terms of [[sheaves of vertex operator algebras]] of [[chiral differential operators]] and similar.

### On genuine smooth loop space

One construction (and hence possibly the only available) that produces an operator genuinely on the full [[smooth loop space]] over a manifold with [[string structure]] and with [[symbol of a differential operator|symbol]] as it should be for a [[Dirac operator]] is given in

* [[Andrew Stacey]], _The co-Riemannian Structure of Smooth Loop Spaces_ ([arXiv:0809.3108](http://arxiv.org/abs/0809.3108)) and _How to Construct a Dirac Operator in Infinite Dimensions_ ([arXiv:0809.3104](http://arxiv.org/abs/0809.3104))

[[!redirects Dirac operator on a smooth loop space]]

[[!redirects Dirac operators on a smooth loop space]]

[[!redirects Dirac operators on smooth loop space]]

[[!redirects Dirac operators on smooth loop spaces]]

[[!redirects Dirac operator on loop space]]
[[!redirects Dirac operators on loop space]]
[[!redirects Dirac operators on loop spaces]]


Dirac propagator

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Differential geometry
+-- {: .hide}
[[!include synthetic differential geometry - contents]]
=--
#### Algbraic Quantum Field Theory
+--{: .hide}
[[!include AQFT and operator algebra contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

What is called the _Dirac propagator_ is the [[Green functions]] for the [[wave operator]]/[[Klein-Gordon operator]] (hence "[[propagator]]") on a [[globally hyperbolic spacetime]] $(X,e)$ which is the sum of the [[advanced propagator]] $\Delta_A$ and the [[retarded propagator]] $\Delta_R$

$$
  \Delta_D \coloneqq \Delta_R + \Delta_A
$$

Since $\Delta_A(x,y) = \Delta_R(y,x)$, this is symmetric in its arguments,
reflecting the fact that this is the integral kernel for [[time-ordered products]] away from the [[diagonal]].

## Related concepts

[[!include propagators - table]]

## References

What is now called the _Dirac propagator_ was first considered in

* [[Paul Dirac]], _Classical theory of radiating electrons_, Proc. Roy. Soc.  A 167 (1983) 148-169

An overview of the [[Green functions]] of the [[Klein-Gordon operator]], hence of the [[Feynman propagator]], [[advanced propagator]], [[retarded propagator]], [[causal propagator]] etc. is given in

* {#Kocic16} [[Mikica Kocic]], _Invariant Commutation and Propagation Functions Invariant Commutation and Propagation Functions_, 2016 ([[KGPropagatorsOnMinkowskiTable.pdf:file]])

Discussion for general [[globally hyperbolic spacetimes]] includes

* F. Friedlander, _The Wave Equation on a Curved Space-Time_, Cambridge: Cambridge University Press, 1975

* {#BaerGinouxPfaeffle07} [[Christian Bär]], [[Nicolas Ginoux]], [[Frank Pfäffle]], _Wave Equations on Lorentzian Manifolds and Quantization_, ESI Lectures in Mathematics and Physics, European Mathematical Society Publishing House, ISBN 978-3-03719-037-1, March 2007, Softcover ([arXiv:0806.1036](https://arxiv.org/abs/0806.1036))

* {#Ginoux08} [[Nicolas Ginoux]], _Linear wave equations_,  Ch. 3 in [[Christian Bär]], [[Klaus Fredenhagen]], _Quantum Field Theory on Curved Spacetimes: Concepts and Methods_, Lecture Notes in Physics, Vol. 786, Springer, 2009

Review in the context of [[perturbative algebraic quantum field theory]] includes

* [[Katarzyna Rejzner]], sections 4.1 and 6.2.3 of _Perturbative Algebraic Quantum Field Theory_, Mathematical Physics Studies, Springer 2016 ([pdf](https://link.springer.com/book/10.1007%2F978-3-319-25901-7))

[[!redirects Dirac propagators]]


Dirac spinor

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### Context
#### Representation theory
+-- {: .hide}
[[!include representation theory - contents]]#### Representation theory
+-- {: .hide}
[[!include representation theory - contents]]
=--
=--
#### Spin geometry
+-- {: .hide}
[[!include higher spin geometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A type of [[spin representation]].

For the moment see _[here](Majorana+spinor#DiracAndWeylRepresentations)_.

## Related concepts

* [[Dirac field]]

* [[Dirac conjugate]]

* [[Dirac operator]], [[Dirac equation]]

* [[Weyl spinor]], [[Majorana spinor]]

## Literature

See the references at *[[spin representation]]*.

What it would mean if [[neutrinos]] were Dirac spinors:

* Salvador Centelles Chuliá, *Theory and phenomenology of Dirac neutrinos* ([arXiv:2110.15755](https://arxiv.org/abs/2110.15755))

[[!redirects Dirac spinor]]
[[!redirects Dirac spinors]]
[[!redirects Dirac representation]]
[[!redirects Dirac representations]]


Dirac-Born-Infeld action

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Quantum Field Theory
+--{: .hide}
[[!include AQFT and operator algebra contents]]
=--
#### String theory
+-- {: .hide}
[[!include string theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

What is known as _Born-Infeld theory_ ([Born-Infeld 34](#BornInfeld34), often also attributed to [Dirac 62](#Dirac62) and abbreviated "DBI theory") is a [[deformation]] of  the [[theory (physics)|theory]] of [[electromagnetism]] which coincides with ordinary [[electromagnetism]] for small excitations of the [[electromagnetic field]] but is such that there is a maximal value for the [[field strength]] which can never be reached in a physical process.

Just this theory happens to describe the [[Chan-Paton gauge field]] on single [[D-branes]] at low [[energy]], as deduced from [[open string|open]] [[string scattering amplitudes]] ([Fradkin-Tseytlin 85](#FradkinTseytlin85), [Abouelsaood-Callan-Nappi-Yost 87](#AbouelsaoodCallanNappiYost87), [Leigh 89](#Leigh89)).

In this context the [[action functional]] corresponding to Born-Infeld theory arises as the low-energy [[effective action]] on the [[D-branes]], and this is referred to as the _DBI-action_. This is part of the full [[Green-Schwarz action functional]] for super [[D-branes]], being a deformation of the [[Nambu-Goto action]]-summand by the [[field strength]] of the [[Chan-Paton gauge fields]].

On [[intersecting brane|coincident]] [[D-branes]], where one expects [[gauge enhancement]] of the [[Chan-Paton gauge field]] to a [[non-abelian group|non-abelian]] [[gauge group]], a further generalization of the DBI-action to non-abelian [[gauge fields]] is expected to be an analogous [[deformation]] of that of non-abelian [[Yang-Mills theory]].
A widely used proposal is due to [Tseytlin 97](#Tseytlin97), [Myers 99](#Myers99), but a derivation from [[string theory]] of this non-abelian DBI action is lacking; and it is in fact known to be in conflict, beyond the first few orders of correction terms, with effects argued elsewhere in the string theory literature ([Hashimoto-Taylor 97](#HashimotoTaylor97), [Bain 99](#Bain99), [Bergshoeff-Bilal-Roo-Sevrin 01](#BergshoeffBilalRooSevrin01)). The issue remains open.

When the [[D-branes]] in question are interpreted as [[flavor branes]], then the maximal/critical value of the [[electric field]] which arises from the [[DBI-action]] has been argued ([Semenoff-Zarembo 11](#SemenoffZarembo11)) to reflect, via [[holographic QCD]], the [[Schwinger limit]] beyond which the [[vacuum polarization]] caused by the [[electromagnetic field]] leads to [[deconfinement]] of [[quarks]].

## Definition
 {#Definition}

### On $4$-dimensional Minkowski spacetime
  {#On4dMinkowski}

In the simplest situation of flat 4-dimensional [[Minkowski spacetime]] $\mathbb{R}^{3,1}$ and no other [[field (physics)|fields]] besides that of [[electromagnetism]], encoded in a [[Faraday tensor]] [[differential 2-form]]

$$
  F \;=\; F_{a b} d x^a \wedge d x^b
$$

the [[Lagrangian density]] of the Born-Infeld [[action functional]] is

\[
  \label{PlainMinkowskiDBIAction}
  \mathbf{L}_{BI}
  \;=\;
  \sqrt{
    -
    det
    \big(
      (\eta_{\mu\nu})
      +
      \tfrac{1}{T}
      (F_{\mu\nu})
    \big)
  }
  \, dvol
  \,.
\]

Here

* $\eta$ denotes the [[metric tensor]] of [[Minkowski spacetime]], regarded as a $4 \times 4$ [[matrix]]

  $$
    \eta \;=\; (\eta_{a b}) \;=\; diag(-1,1,1,1)
    \,,
  $$

* $F$ is the [[Faraday tensor]], regarded as a $4 \times 4$ [[matrix]]

  $$
    F = (F_{a b})
    \,,
  $$

* $T = \frac{1}{2\pi \, \alpha'} = \frac{1}{2\pi \, \ell_2^2}$ is the [[string tension]],

* $det(\cdots)$ denotes the [[determinant]] of the [[sum]] of these [[matrices]],

* $\sqrt{}$ denotes the [[positive number|positive]] [[square root]],

* $dvol$ denotes a [[volume form]] on [[Minkowski spacetime]], which after a choice of global [[coordinates]] may be taken to be

  \[
    \label{VolumeForm}
    dvol \;=\; d t \wedge d x^1 \wedge d x^2 \wedge d x^3
  \]

In the following, for $\omega_4$ any [[differential 4-form]] on $\mathbb{R}^{3,1}$ we write $\omega_4 / dvol$ for the unique [[smooth function]] $\mathbb{R}^{3,1} \to \mathbb{R}$ such that

$$
  (\omega_4 / dvol) \cdot dvol \;=\; \omega_4
  \,.
$$

+-- {: .num_lemma}
###### Lemma

The [[determinant]] in (eq:PlainMinkowskiDBIAction) evaluates to

\[
  \label{TheDeterminant}
  det( \eta + F )
  \;=\;
  - 1
  -
 \tfrac{1}{2}
 \underset{
   \mathclap{
   {\color{blue}\text{Lagrangian of}}
   \atop
   {\color{blue}\text{electromagnetism}}
   }
 }{
   \underbrace{
     (F \wedge \star F)
  }
  }
  / dvol
  +
    \underset{
      {\color{blue}\text{correction}}
      \atop
      {\color{blue}\text{term}}
    }{
    \underbrace{
      \big(
        \tfrac{1}{2}
        (F\wedge F)
        / \mathrm{dvol}
      \big)^2
    }
    }
  \,,
\]

where

* $F \wedge \star F$ is the 4-form [[wedge product]] of $F$ with its  [[Hodge duality|Hodge dual]], hence is the [[Lagrangian density]] of ordinary [[electromagnetism]],

* $F \wedge F$ is the [[wedge product]] of $F$ with itself.

=--

+-- {: .proof}
###### Proof

We compute as follows:

$$
  \begin{aligned}
  \mathrm{det}
  \big(
    (\eta_{a b})
    +
    (F_{a b})
  \big)
  &
  \; = \phantom{+}
  \tfrac{1}{4!}
  \epsilon^{a_1 a_2 a_3 a_4}
  (\eta_{a_1 b_1} + F_{a_1 b_1})
  (\eta_{a_2 b_2} + F_{a_2 b_2})
  (\eta_{a_3 b_3} + F_{a_3 b_3})
  (\eta_{a_4 b_4} + F_{a_4 b_4})
  \epsilon^{b_1 b_2 b_3 b_4}
  \\
  &
  \; = \phantom{+}
  \underset{
    = \mathrm{det}(\eta) = -1
  }{
  \underbrace{
  \tfrac{1}{4!}
  \epsilon^{a_1 a_2 a_3 a_4}
  \eta_{a_1 b_1}
  \eta_{a_2 b_2}
  \eta_{a_3 b_3}
  \eta_{a_4 b_4}
  \epsilon^{b_1 b_2 b_3 b_4}
  }
  }
  \\
  &
  \phantom{\; =} +
  \underset{
    = 0
  }{
  \underbrace{
  \tfrac{3}{4!}
  \epsilon^{a_1 a_2 a_3 a_4}
  \eta_{a_1 b_1}
  \eta_{a_2 b_2}
  \eta_{a_3 b_3}
  F_{a_4 b_4}
  \epsilon^{b_1 b_2 b_3 b_4}
  }
  }
  \\
  &
  \phantom{\; =} +
  \underset{
    = -\tfrac{2\cdot 6}{4!} F_{a b} F^{a b}
  }{
  \underbrace{
  \tfrac{6}{4!}
  \epsilon^{a_1 a_2 a_3 a_4}
  \eta_{a_1 b_1}
  \eta_{a_2 b_2}
  F_{a_3 b_3}
  F_{a_4 b_4}
  \epsilon^{b_1 b_2 b_3 b_4}
  }
  }
  \\
  &
  \phantom{\; =} +
  \underset{
    = 0
  }{
  \underbrace{
  \tfrac{3}{4!}
  \epsilon^{a_1 a_2 a_3 a_4}
  \eta_{a_1 b_1}
  F_{a_2 b_2}
  F_{a_3 b_3}
  F_{a_4 b_4}
  \epsilon^{b_1 b_2 b_3 b_4}
  }
  }
  \\
  &
  \phantom{\; =} +
  \underset{
    =
    \big(
      \tfrac{1}{2}
      (F \wedge F) / \mathrm{dvol}
    \big)^2
  }{
  \underbrace{
  \tfrac{1}{4!}
  \epsilon^{a_1 a_2 a_3 a_4}
  F_{a_1 b_1}
  F_{a_2 b_2}
  F_{a_3 b_3}
  F_{a_4 b_4}
  \epsilon^{b_1 b_2 b_3 b_4}
  }
  }
  \\
  & \; = \phantom{+}
  -1
    -
  \tfrac{1}{2} (F \wedge \star F) / \mathrm{dvol}
    +
  \big(
    \tfrac{1}{2} (F\wedge F) / \mathrm{dvol}
  \big)^2
  \end{aligned}
$$

In the first line we used the expression of the [[determinant]] via the [[Levi-Civita symbol]] ([here](determinant#eq:DeterminantInTermsOfLCSymbols)) with the [[Einstein summation convention]] being understood throughout.
Then we multiplied out the terms, collecting those with the same number of factors of $\eta$ (of $F$), using that under exchange of the order of factors both [[Levi-Civita symbols]] give a sign, which hence cancel. Of the five terms that appear, the first and the last are themselves the plain [[determinants]] of $\eta$ and of $F$, respectively (again by [that formula](determinant#eq:DeterminantInTermsOfLCSymbols)).

We discuss the identifications of the resulting four summands shown under the braces:

* (first summand) The determinant of $\eta$ equals -1 by definition.

* (second summand) If we exchange indices $(a_i \leftrightarrow b_i)$ the form of this summand remains unchanged, also the factors $\eta_{a_i b_i}$ do not change, since $\eta$ is a [[symmetric matrix]], by definition. But the single factor of $F$ changes by a sign, since the components of a [[differential 2-form]] constitute a [[skew-symmetric matrix]]. In summary this says that the second term is equal to minus itself, and hence has to be [[zero]].

* (third summand) Consider this term first with $\eta$ relaced by the [[identity matrix]] (to be indicated by a [[Kronecker delta]] $(\delta_{a b})$). Observe then that the contraction not involving any factor of $F$ yields

  $$
    \epsilon^{a_1 a_2 a_3 a_4}
      \delta_{a_1 b_1} \delta_{a_2 b_2}
    \epsilon^{b_1 b_2 b_3 b_4}
    \;=\;
    2 \delta^{a_3 a_4}_{b_3 b_4}
    \,,
  $$

  where the symbol on the right is defined to be

  \[
    \label{TheRelativeLCTensor}
    \delta^{a_3 a_4}_{b_3 b_4}
    \;\coloneqq\;
    \left\lbrace
    \array{
      +1 &\vert& a_3 \neq a_4 \;\text{and}\; a_3 = b_3 \;\text{and}\; a_4 = b_4
      \\
      -1 &\vert& a_3 \neq a_4 \;\text{and}\; a_3 = b_4 \;\text{and}\; a_4 = b_3
      \\
      \phantom{+}0 &\vert& \text{otherwise}
    }
    \right.
  \]

  Hence the full expression (with $\eta$ still replaced by $\delta$) is

  $$
    \tfrac{2\cdot 2}{4!}
    \delta^{a_3 a_4}_{b_3 b_4}
    F_{a_3 b_3} F_{a_4 b_4}
    \;=\;
    - \sum_{a, b} F_{a b} F_{a b}
  $$

  where we used that due to the [[skew-symmetric matrix|skew-symmetry]] of $F$ the first case in (eq:TheRelativeLCTensor) does not contribute, only the second case does.

  Now it just remains to translate this back to the situation at hand where we use $\eta$ instead of $\delta$: This just differs by a minus sign in the component with both indices corresponding to the temporal direction, while this is also the case for which raising an index on $F$ picks up a minus sign. Since _either_ of these cases contributes in each summand, there is a global minus sign.

* (fourth summand) Since this involves three factors of $F$ which jointly pick up one minus sign when the indices on each of them are exchanged simultaneously, this vanishes by the same kind of skew-symmetry argument as for the second term.

* (fifth summand) Since this is the [[determinant]] of a [[skew-symmetric matrix]], the [[Pfaffian]]-theorem ([here](Pfaffian#eq:DeterminantIsPfaffianSquared)) says that this term equals the square of the [[Pfaffian]] of $F$, which is (by [this](Pfaffian#eq:PfaffianInTermsOfLCTensor) formula)

  $$
    Pf(F)
    \;=\;
    \tfrac{1}{4 \cdot 4!}
    \epsilon^{a_1 b_1 a_2 b_2} F_{a_1 b_1} F_{a_2 b_2}
  $$

  This is proportial to the coefficient of the [[wedge product]]
  of $F$ with itself, relative to the [[volume form]]:

  $$
    \begin{aligned}
      F \wedge F
      & = \;
      \big(
        \tfrac{1}{2}F_{a_1 b_1} d x^{a_1} \wedge d x^{b_1}
      \big)
      \wedge
      \big(
        \tfrac{1}{2}F_{a_2 b_2} d x^{a_2} \wedge d x^{b_2}
      \big)
      \\
      & = \;
      \tfrac{1}{4} F_{a_1 b_1} F_{a_2 b_2} d x^{a_1} \wedge d x^{b_1} \wedge d x^{a_2} \wedge d x^{b_2}
      \\
      & = \;
      \tfrac{1}{4} F_{a_1 b_1} F_{a_2 b_2}
      \epsilon^{a_1 b_1 a_2 b_2}
      \,
      dvol
      \\
      & =
      4! Pf(F)
      \,
      dvol
    \end{aligned}
  $$

=--

The expression (eq:PlainMinkowskiDBIAction) is supposed to be exact for [[constant function|constant]] [[field strength]] (e..g. [Bachas-Bain-Green 99, above (1.9)](#BachasBainGreen99)), and to pick up [[higher curvature corrections]] for non-constant field strength. The first derivative correction to (eq:PlainMinkowskiDBIAction) is supposed to arise at order $(\partial F)^4$. The explicit expression is given in [Garousi 15 (7)](#Garousi15) (argued there by appealing to [[T-duality]] and [[S-duality]] applied to earlier results on higher curvature corrections in other fields involved).

\linebreak

Consider now the [[Faraday tensor]] $F$ expressed in terms of the [[electric field]] $\vec E$ and [[magnetic field]] $\vec B$ as

$$
  \begin{aligned}
    F_{0 i} & = \phantom{+} E_i
    \\
    F_{i 0} & = - E_i
    \\
    F_{i j} & = \epsilon_{i j k} B^k
  \end{aligned}
$$

Then the general expression (eq:TheDeterminant) for the [[DBI-Lagrangian]] reduces to ([Born-Infeld 34, p. 437](#BornInfeld34), review in [Gibbons 97, (56)](#Gibbons97), [Savvidy 99, (22)](#Savvidy99), [Nastase 15, 9.4](#Nastase15)):

\[
  \label{InTermsOfElectricAndMagneticField}
  \mathbf{L}_{BI}
  \;=\;
  \sqrt{
    - det\left( \eta + \tfrac{1}{T} F \right)
  }
  \,
  dvol_4
  \;=\;
  \sqrt{
    1
    -
    \tfrac{1}{T^2}
    (
    \vec E \cdot \vec E
      -
    \vec B \cdot \vec B
    )
    -
    \tfrac{1}{T^4}
    (\vec B \cdot \vec E)^2
  }
  \,
  dvol_4
\]

### Critical field strength
  {#CriticalFieldStrength}

For the DBI-action (eq:InTermsOfElectricAndMagneticField) to be well-defined, in that the [[square root]] is a [[real number]], hence its argument a non-[[negative number]], requires that

$$
  \begin{aligned}
    &
    -
    \mathrm{det}
    \big(
      (\eta_{\mu \nu})
      +
      \tfrac{1}{T}
      (F_{\mu \nu})
    \big)
    \geq  0
    \\
    & \Leftrightarrow
    \;
  1
  \;-\;
  \tfrac{1}{T^2}
  (E \cdot E - B \cdot B)
  \;-\;
  \tfrac{1}{T^4}
  (B \cdot E)^2
  \;\geq\;
  0
  \\
  & \Leftrightarrow
  \;
  \tfrac{1}{T^2}
  E^2
  -
  \tfrac{1}{T^2}
  B^2
  +
  \tfrac{1}{T^4}
  E^2 B_{\parallel}^2
  \;\leq 1\;
  \\
  & \Leftrightarrow
  \;
  \tfrac{1}{T^2}
  E^2
  \;\leq\;
  \frac{
    1 + \tfrac{1}{T^2} B^2
  }{
    1 + \tfrac{1}{T^2} B_{\parallel}^2
  }
  \\
  & \Leftrightarrow
  \;
  E
  \;\leq\;
  T
  \sqrt{
  \frac{
    T^2 + B^2
  }{
    T^2 + B_{\parallel}^2
  }
  }
  \end{aligned}
$$

where

$$
  B_{\parallel} \coloneqq  \tfrac{1}{\sqrt{E\cdot E}} B \cdot E
$$

is the component of the [[magnetic field]] which is parallel to the [[electric field]].

The resulting maximal electric field strength

$$
  E_{crit}
  \;\coloneqq\;
  T
  \sqrt{
    \frac{
      T^2 + B^2
    }{
      T^2 + B_{\parallel}^2
    }
  }
$$

turns out to be the [[Schwinger limit]] (see [there](Schwinger+effect#CriticalFieldStrength)) beyond which the electric field would cause [[deconfinement|deconfining]] [[quark]]-pair creation ([Hashimoto-Oka-Sonoda 14b, (2.17)](#HashimotoOkaSonoda14b)).

## Related concepts

* [[Myers effect]]

## References

### General

As a proposal for a modification of [[electromagnetism]] in [[spacetime]], the (Dirac-)Born-Infeld (DBI) action originates in

* {#BornInfeld34} [[Max Born]], [[Leopold Infeld]], _Foundations of the New Field Theory_,  Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character, Vol. 144, No. 852 (Mar. 29, 1934), pp. 425-451 ([jstor:2935568](https://www.jstor.org/stable/2935568))

The article by Dirac which came to be commonly cited in this context is

* {#Dirac62} [[Paul Dirac]], _An Extensible Model of the Electron_, Proc. Roy. Soc. A268, (1962) 57-67 ([jstor:2414316](https://www.jstor.org/stable/2414316))

  (which proposes a [[membrane]]-model as a unification of the [[electron]] and the [[muon]])

Broad review and further developments:

* [[Dmitri Sorokin]], *Introductory Notes on Non-linear Electrodynamics and its Applications* ([arXiv:2112.12118](https://arxiv.org/abs/2112.12118))

### For single (abelian) D-branes

As the low energy [[action functional]] for single [[D-branes]] the DBI action is due to

* {#FradkinTseytlin85} [[Efim Fradkin]], [[Arkady Tseytlin]], _Non-linear electrodynamics from quantized strings_, Physics Letters B Volume 163, Issues 1–4, 21 November 1985 (doi:10.1016/0370-2693(85)90205-9)

* {#AbouelsaoodCallanNappiYost87} A. Abouelsaood, [[Curtis Callan]], [[Chiara Nappi]], S. A. Yost, _Open strings in background gauge fields_, Nuclear Physics B Volume 280, 1987, Pages 599-624 (doi:10.1016/0550-3213(87)90164-7)

* {#Leigh89} [[Robert Leigh]], _Dirac-Born-Infeld Action from Dirichlet Sigma Model_, Mod. Phys. Lett. A4 (1989) 2767 ([spire:26399](http://inspirehep.net/record/26399), [doi:10.1142/S0217732389003099](https://doi.org/10.1142/S0217732389003099))

and a full $\kappa$-symmetric [[Green-Schwarz sigma-model]] for [[D-branes]]:

* {#CGNSW96} [[Martin Cederwall]], Alexander von Gussich, [[Bengt Nilsson]], Per Sundell, Anders Westerberg, _The Dirichlet Super-p-Branes in Ten-Dimensional Type IIA and IIB Supergravity_, Nucl.Phys. B490 (1997) 179-201 ([arXiv:hep-th/9611159](http://arxiv.org/abs/hep-th/9611159))

* {#APPS97b} [[Mina Aganagic]], Jaemo Park, Costin Popescu, [[John Schwarz]], _Dual D-Brane Actions_, Nucl. Phys. B496 (1997) 215-230 ([arXiv:hep-th/9702133](https://arxiv.org/abs/hep-th/9702133))

Review:

* {#Gibbons97} [[Gary Gibbons]], _Born-Infeld particles and Dirichlet p-branes_, Nucl. Phys. B514:603-639, 1998 ([arXiv:hep-th/9709027](https://arxiv.org/abs/hep-th/9709027))

* {#Savvidy99} Konstantin G. Savvidy, _Born-Infeld Action in String Theory_, 1999 ([arXiv:hep-th/9906075](https://arxiv.org/abs/hep-th/9906075), [spire:501510](https://inspirehep.net/literature/501510))

* [[Arkady Tseytlin]], _Born-Infeld action, supersymmetry and string theory_, in: [[Mikhail Shifman]] (ed.)  _[[The many faces of the superworld]]_, pp. 417-452, World Scientific (2000) ([arXiv:hep-th/9908105](https://arxiv.org/abs/hep-th/9908105), [doi:10.1142/9789812793850_0025](https://doi.org/10.1142/9789812793850_0025))

* {#Schwarz01} [[John Schwarz]], _Comments on Born-Infeld Theory_, in: [[Atish Dabholkar]], [[Sunil Mukhi]], Spenta R. Wadia (eds.) _[Strings 2001: Proceedings, Strings 2001 Conference](http://inspirehep.net/record/944370)_, Tata Institute of Fundamental Research, Mumbai, India, January 5-10, 2001 ([arXiv:hep-th/0103165](https://arxiv.org/abs/hep-th/0103165), [spire:554347](http://inspirehep.net/record/554347))

* Paul Koerber, _Abelian and Non-abelian D-brane Effective Actions_, Fortsch. Phys. 52 (2004) 871-960 ([arXiv:hep-th/0405227](https://arxiv.org/abs/hep-th/0405227))

* {#Nastase15} [[Horatiu Nastase]], Section 9.2 and 9.4 of: _Introduction to AdS/CFT correspondence_, Cambridge University Press, 2015 ([cds:1984145](http://cds.cern.ch/record/1984145), [doi:10.1017/CBO9781316090954](https://doi.org/10.1017/CBO9781316090954))

Detailed discussion of the relation to the [[Polyakov action]] and the [[Nambu-Goto action]] is in

* {#Nieto01} J. A. Nieto, _Remarks on Weyl invariant p-branes and Dp-branes_ ([arXiv:hep-th/0110227](http://arxiv.org/abs/hep-th/0110227))

Discussion in terms of D-branes as [[leaves]] of [[Dirac structures]] on [[Courant Lie 2-algebroids]] of [[type II geometry]]:

* {#AsakawaSasaWatamura} [[Tsuguhiko Asakawa]], Shuhei Sasa, Satoshi Watamura, _D-branes in Generalized Geometry and Dirac-Born-Infeld Action_ ([arXiv:1206.6964](http://arxiv.org/abs/1206.6964))

See also

* [[Martin Cederwall]], Alexander von Gussich, [[Aleksandar Mikovic]], [[Bengt Nilsson]], Anders Westerberg, _On the Dirac-Born-Infeld Action for D-branes_,  Phys.Lett.B390:148-152, 1997 ([arXiv:hep-th/9606173](https://arxiv.org/abs/hep-th/9606173))

* Ian I. Kogan, Dimitri Polyakov, _DBI Action from Closed Strings and D-brane second Quantization_, Int. J. Mod. Phys. A18 (2003) 1827 ([arXiv:hep-th/0208036](https://arxiv.org/abs/hep-th/0208036))

Discussion of [[loop order|one-loop corrections]]:

* Garrett Goon, Scott Melville, Johannes Noller, _Quantum Corrections to Generic Branes: DBI, NLSM, and More_ ([arXiv:2010.05913](https://arxiv.org/abs/2010.05913))

Derivation of the first DBI-correction from an [[M5-brane]] model via [super-exceptional geometry](exceptional+generalized+geometry#SuperExceptionalGeometryReferences):

* [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], _[[schreiber:super-exceptional M5 flavor|Super-exceptional M5-brane model: Emergence of SU(2)-flavor sector]]_ ([arXiv:2006.00012](https://arxiv.org/abs/2006.00012))

### For coincident (non-abelian) D-branes

Proposals for the generalization of the DBI action to  [[non-abelian group|non-abelian]] [[Chan-Paton gauge fields]] (hence: for [[intersecting branes|coincident]] [[D-branes]]) includes the following:

Via a plain [[trace]]:

* T. Hagiwara, _A non-abelian Born-Infeld Lagrangian_, J. Phys., A14:3059, 1981 ([doi:10.1088/0305-4470/14/11/027](https://iopscience.iop.org/article/10.1088/0305-4470/14/11/027))

Via an antisymmetrized [[trace]]:

* [[Philip Argyres]], [[Chiara Nappi]], _Spin-1 effective actions from open strings_, Nuclear Physics B Volume 330, Issue 1, 22 January 1990, Pages 151-173 Nuclear Physics B (doi:10.1016/0550-3213(90)90305-W)

Via a combination of spacetime and gauge indices:

* Jeong-Hyuck Park, _A Study of a Non-Abelian Generalization of the Born-Infeld Action_, Phys. Lett. B458 (1999) 471-476 ([arXiv:hep-th/9902081](https://arxiv.org/abs/hep-th/9902081))

The now widely accepted proposal via a symmetrized [[trace]] is due to

* {#Tseytlin97} [[Arkady Tseytlin]], _On non-Abelian generalization of Born-Infeld action in string theory_, Nucl.Phys. B501 (1997) 41-52 ([spire:439767](http://inspirehep.net/record/439767))

followed by

* {#Myers99} [[Robert Myers]], _Dielectric-Branes_, JHEP 9912 (1999) 022 ([arXiv:hep-th/9910053](https://arxiv.org/abs/hep-th/9910053))

  (introducing the [[Myers effect]])

The symmetrized trace proposal has become widely accepted.

Review includes:

* {#Chemissany04} W. Chemissany, _On the way of finding the non-Abelian Born-Infeld theory_, 2004 ([spire:1286212](http://inspirehep.net/record/1286212) [[ChemissanyNonAbelianDBI04.pdf:file]])

Issues with this proposal at higher order have been found in

* {#HashimotoTaylor97} [[Akikazu Hashimoto]], [[Washington Taylor]], _Fluctuation Spectra of Tilted and Intersecting D-branes from the Born-Infeld Action_, Nucl. Phys. B503: 193-219, 1997 ([arXiv:hep-th/9703217](https://arxiv.org/abs/hep-th/9703217))

* {#Bain99} P. Bain, _On the non-abelian Born-Infeld action_, In: L. Baulieu , [[Michael Green]], Picco M., Windey P. (eds.) _Progress in String Theory and M-Theory_. NATO Science Series (Series C: Mathematical and Physical Sciences), vol 564. Springer, Dordrecht ([arXiv:hep-th/9909154](https://arxiv.org/abs/hep-th/9909154), [doi:10.1007/978-94-010-0852-5_12](https://doi.org/10.1007/978-94-010-0852-5_12))

* {#BergshoeffBilalRooSevrin01} [[Eric Bergshoeff]], [[Adel Bilal]], [[Mees de Roo]], A. Sevrin, _Supersymmetric non-abelian Born-Infeld revisited_, JHEP 0107, 029 (2001) ([arXiv:hep-th/0105274](https://arxiv.org/abs/hep-th/0105274))

Correction terms have been proposed in

* {#TaylorVanRaamsdonk99} [[Washington Taylor]], [[Mark Van Raamsdonk]], _Multiple D$p$-branes in Weak Background Fields_, Nucl. Phys. B573:703-734, 2000 ([arXiv:hep-th/9910052](https://arxiv.org/abs/hep-th/9910052))

A completely different approach, which defines a theory that is analogous to non-abelian BI via [[TT deformation|$T \bar T$ deformation]] of $D=2$ [[Yang-Mills theory]], is proposed in

* {#BrennanFerkoSethi20} [[T. Daniel Brennan]], [[Christian Ferko]], [[Savdeep Sethi]], *A Non-Abelian Analogue of DBI from $T \bar T$*, SciPost Phys. **8** 052 (2020) [[arXiv:1912.12389](https://arxiv.org/abs/1912.12389), [doi:10.21468/SciPostPhys.8.4.052](https://doi.org/10.21468/SciPostPhys.8.4.052)]

For actual derivation of [[gauge enhancement]] on coincident D-branes see the references [there](enhanced+gauge+symmetry#ReferencesOnCoincidentDBranes).

On [[KK-compactification]] of the non-abelian DBI-action from 10d to 4d:

* Yoshihiko Abe, Tetsutaro Higaki, Tatsuo Kobayashi, Shintaro Takada, Rei Takahashi, *4D effective action from non-Abelian DBI action with magnetic flux background* ([arXiv:2107.11961](https://arxiv.org/abs/2107.11961))

Proposals for non-abelian and [[supersymmetry|supersymmetric]] DBI-actions

for [[D0-branes]]:

* Sudhakar Panda, [[Dmitri Sorokin]], *Supersymmetric and Kappa-invariant Coincident D0-Branes*, 	JHEP 0302 (2003) 055 [[arXivLhep-th/0301065](https://arxiv.org/abs/hep-th/0301065), [doi:10.1088/1126-6708/2003/02/055](https://doi.org/10.1088/1126-6708/2003/02/055)]

* [[Igor Bandos]], Unai D. M. Sarraga, *Complete nonlinear action for supersymmetric multiple D0-brane system*, Phys. Rev. D **106** (2022) 066004 [[doi:10.1103/PhysRevD.106.066004](https://doi.org/10.1103/PhysRevD.106.066004), [arXiv:2204.05973](https://arxiv.org/abs/2204.05973)]

* [[Igor Bandos]], Unai D. M. Sarraga, *Properties of multiple D0-brane system: 11D origin, equations of motion and their solutions* [[arXiv:2212.14829](https://arxiv.org/abs/2212.14829)]

using the [[pure spinor superstring]]:

* Ryota Fujii, Sota Hanazawa, Hiraki Kanehisa, Makoto Sakaguchi, *Supersymmetric Non-abelian DBI Equations from Open Pure Spinor Superstring* [[arXiv:2304.04899](https://arxiv.org/abs/2304.04899)]

### On flavor branes for holographic QCD

Discussion of the [[DBI-action]] for [[flavor brane|flavor branes]] in [[holographic QCD]]:

* [[Vadim Kaplunovsky]], [[Jacob Sonnenschein]], Section 6 of: _Searching for an Attractive Force in Holographic Nuclear Physics_, JHEP 05 (2011) 058 ([arXiv:1003.2621](https://arxiv.org/abs/1003.2621))

[[!include holographic Schwinger effect -- references]]

[[!include DBI-action higher curvature corrections -- references]]

### Single trace observables as weight systems on chord diagrams

Relation of [[single trace observables]] in the [[non-abelian DBI action]] on [[Dp-D(p+2)-brane bound states]] ([hence](Dp-Dp+2-brane+bound+states#ReferencesRelationToMonopoles)
[[Yang-Mills monopoles]]) to [[su(2)]]-[[Lie algebra weight systems]] on [[chord diagrams]] computing [[radii]] averages of [[fuzzy spheres]]:

* {#RamgoolamSpenceThomas04} [[Sanyaje Ramgoolam]], [[Bill Spence]], S. Thomas, Section 3.2 of: _Resolving brane collapse with $1/N$ corrections in non-Abelian DBI_, Nucl. Phys. B703 (2004) 236-276 ([arxiv:hep-th/0405256](https://arxiv.org/abs/hep-th/0405256))

* {#McNamaraPapageorgakis05} [[Simon McNamara]], [[Constantinos Papageorgakis]], [[Sanyaje Ramgoolam]], [[Bill Spence]], Appendix A of: _Finite $N$ effects on the collapse of fuzzy spheres_, JHEP 0605:060, 2006 ([arxiv:hep-th/0512145](https://arxiv.org/abs/hep-th/0512145))

* {#McNamara06} [[Simon McNamara]], Section 4 of: _Twistor Inspired Methods in Perturbative FieldTheory and Fuzzy Funnels_, 2006 ([spire:1351861](http://inspirehep.net/record/1351861), [pdf](https://strings.ph.qmul.ac.uk/sites/default/files/Mcnamaraphd.pdf), [[McNamara06.pdf:file]])

* [[Constantinos Papageorgakis]], p. 161-162 of: _On matrix D-brane dynamics and fuzzy spheres_, 2006 ([[Papageorgakis06.pdf:file]])

### Brane intersections as DBI-spikes/BIons

On [[D1-D3 brane intersections]] as spikes/BIons in the [[D3-brane]] [[DBI action|DBI-theory]]:

* [[Curtis Callan]], [[Juan Maldacena]], _Brane Dynamics From the Born-Infeld Action_, Nucl. Phys. B513 (1998) 198-212 ([arXiv:hep-th/9708147](https://arxiv.org/abs/hep-th/9708147))

* [[Paul Howe]], [[Neil Lambert]], [[Peter West]], _The Self-Dual String Soliton_, Nucl. Phys. B515 (1998) 203-216 ([arXiv:hep-th/9709014](https://arxiv.org/abs/hep-th/9709014))

* [[Gary Gibbons]], _Born-Infeld particles and Dirichlet p-branes_,  	Nucl. Phys. B514: 603-639, 1998 ([arXiv:hep-th/9709027](https://arxiv.org/abs/hep-th/9709027))

* [[Neil Constable]], [[Robert Myers]], Oyvind Tafjord, _The Noncommutative Bion Core_, Phys. Rev. D61 (2000) 106009 ([arXiv:hep-th/9911136](https://arxiv.org/abs/hep-th/9911136))

From the [[M5-brane]]

* {#HoweLambertWest97} [[Paul Howe]], [[Neil Lambert]], [[Peter West]], _The Self-Dual String Soliton_, Nucl. Phys. B515 (1998) 203-216 ([arXiv:hep-th/9709014](https://arxiv.org/abs/hep-th/9709014))

[[!redirects Dirac-Born-Infeld actions]]

[[!redirects Born-Infeld theory]]
[[!redirects Born-Infeld theories]]

[[!redirects DBI action]]
[[!redirects DBI-action]]
[[!redirects DBI actions]]
[[!redirects DBI-actions]]

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[[!redirects non-abelian DBI-action]]
[[!redirects non-abelian DBI actions]]
[[!redirects non-abelian DBI-actions]]

[[!redirects DBI Lagrangian]]
[[!redirects DBI-Lagrangian]]
[[!redirects DBI Lagrangians]]
[[!redirects DBI-Lagrangians]]

[[!redirects DBI field theory]]
[[!redirects DBI field theories]]
[[!redirects DBI-field theory]]
[[!redirects DBI-field theories]]

[[!redirects DBI model]]
[[!redirects DBI-model]]
[[!redirects DBI models]]
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[[!redirects nonabelian DBI-model]]
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[[!redirects nonabelian DBI-models]]

[[!redirects nonabelian DBI field theory]]
[[!redirects nonabelian DBI field theories]]
[[!redirects non-abelian DBI field theory]]
[[!redirects non-abelian DBI field theories]]

[[!redirects nonabelian Born-Infeld action]]
[[!redirects nonabelian Born-Infeld actions]]
[[!redirects non-abelian Born-Infeld action]]
[[!redirects non-abelian Born-Infeld actions]]

[[!redirects nonabelian DBI action]]
[[!redirects nonabelian DBI-action]]

[[!redirects Dirac-Born-Infeld action functional]]
[[!redirects DBI action functional]]

[[!redirects DBI theory]]
[[!redirects DBI theories]]

[[!redirects DBI-theory]]
[[!redirects DBI-theories]]


Dirac-Ramond operator

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Higher spin geometry
+-- {: .hide}
[[!include higher spin geometry - contents]]
=--
#### String theory
+-- {: .hide}
[[!include string theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The 0-mode of the [[supercharge]] in a [[2d SCFT]] (an operator in a ([[sheaf of vertex operator algebras|sheaf of]]) [[super vertex operator algebra]]) behaves like a higher dimensional analog of the operator $D$ in a [[spectral triple]], hence like the supercharge in [[supersymmetric quantum mechanics]] (see the references [there](supersymmetric+quantum+mechanics#InStringTheory)).

Specifically for a [[sigma-model]] [[2d SCFT]] induced from some [[target space]] geometry -- such as the [[worldsheet]]-[[quantum field theory]] of a [[superstring]] propagating on that [[target spacetimes]] -- the Dirac-Ramond operator is a [[2-spectral triple|higher analogue]] of a [[Dirac operator]] on that [[target spacetime]] (roughly like what one would expect of a [[Dirac operator on a smooth loop space]]). This is called the _Dirac-Ramond operator_ ([Ramond 71](#Ramond71)).

The [[index]] of the [[large volume limit]] of the Dirac-Ramond operator is what is now known as the [[Witten genus]] (but in fact the original article [Alvarez-Killingback-Mangano-Windey 87](#AlvarezKillingbackManganoWindey87)
 appeared independently and almost in parallel of Witten's discussion).

## Related concepts

* [[2-spectral triple]]

* [[string structure]], [[twisted string structure]]

* [[Dirac operator on smooth loop space]]

[[!include genera and partition functions - table]]

## References

### General

The original article, in the context of the [[superstring]] of [[string theory]]"

* {#Ramond71} [[Pierre Ramond]], _Dual theory for Free Fermions_, Phys Rev. D3  2415 (1971) ([doi:10.1103/PhysRevD.3.2415](https://doi.org/10.1103/PhysRevD.3.2415))

The Dirac-Ramond operator originates with the early beginning of [[superstring]] models, when they were still called _[[spinning strings]]_ -- see [there](spinning+string#References) for more references.

The concept gained more attention in pure mathematics when it was found that the [[large volume limit]]  of its [[index]], when properly construed, is a universal [[elliptic genus]], now known as the [[Witten genus]]. See [there](elliptic+genus#EllipticGeneraAsBranePartitionFunctionsReferences) for more references.

Articles that explicitly consider the Dirac-Ramond operator in this context:

* {#AlvarezKillingbackManganoWindey87} [[Orlando Alvarez]], T. P. Killingback, Michelangelo Mangano, and Paul Windey, _The Dirac-Ramond operator in string theory and loop space index theorems_, Nuclear Phys. B Proc. Suppl., 1A:189–215, 1987. Nonperturbative methods in field theory (Irvine, CA, 1987)., also: Comm. Math. Phys. Volume 111, Number 1 (1987), 1-160 ([Ecudid](http://projecteuclid.org/euclid.cmp/1104159462))

* {#Landweber99} [[Gregory Landweber]], _Dirac operators on loop space_ PhD thesis (Harvard 1999) ([pdf](http://math.bard.edu/greg/LoopDirac.pdf))

* [[Orlando Alvarez]], [[Paul Windey]], _Analytic index for a family of Dirac-Ramond operators_, Proc. Natl. Acad. Sci. USA, 107(11):4845–4850, 2010

[[!include elliptic genera as partition functions -- references]]

[[!redirects Dirac-Ramond operators]]


direct category

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Category theory
+-- {: .hide}
[[!include category theory - contents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Idea

A *direct category* is a [[category]] in which the morphisms only go in "one direction."

A direct category can be thought of as a category of [[geometric shapes for higher structures]] which includes only the "inclusions of faces," not any "degeneracy" maps.  (The more general notion of [[Reedy category]] can also include degeneracies.)  The objects of a direct category admit no nontrivial automorphisms, but the notion can also be generalized to allow for such automorphisms.

Alternately, a direct category can be thought of as a combinatorial representation of a *single* geometric shape.  In this case, each object is a "face" and each morphism is the inclusion of a lower-dimensional face in a higher-dimensional one.  If $D$ is a direct category regarded as a category *of* geometric shapes, then each $d\in D$ can be represented by the [[slice category]] $D/d$, a direct category regarded as a single geometric shape.  (In general, however, $D/d$ may admit more automorphisms than $d$, so it may need to be equipped with a "labeling" or "orientation" to truly recapture the shape $d$.)

Finally, a direct category can also be thought of as a [[categorification]] of the notion of [[well-founded relation]] from [[posets]] to [[categories]].

## Definition

### Standard definition

A category $D$ is a **direct category** if the following *equivalent* conditions are satisfied.

* $D$ contains no infinite descending chains of nonidentity morphisms $\cdots \to\cdot \to\cdot\to\cdot$ (including cycles of length $\gt 0$).
* The relation $a\prec b$ on $ob(D)$ defined by "there exists a nonidentity morphism from $a$ to $b$" is [[well-founded relation|well-founded]].
* There exists a function $d\colon ob(D)\to Ord$, where $Ord$ is the class of [[ordinals]], such that every nonidentity morphism of $D$ raises the degree.
* There exists an identity-reflecting functor (that is, it maps a morphism to an identity if and only if the morphism is itself an identity) $d : D\to \mathbf{Ord}$, where $\mathbf{Ord}$ is the large poset of ordinals viewed as a category.
* $D$ is a [[Reedy category]] (in particular an _[[elegant Reedy category]]_) in which $D_-$ consists only of identity maps (or equivalently $D_+$ is all of $D$).

In particular, a [[poset]] is a direct category just when its strict order relation $\lt$ is well-founded.  Thus, direct categories can be seen as a categorification of well-founded relations.

If $D^{op}$ is a direct category, then we say that $D$ is an **inverse category**.

### Direct versus one-way categories

Of course, a direct category can have no nonidentity [[endomorphisms]], since any such endomorphism would be a cycle of length 1.  A category with this property is sometimes called a **one-way category**.

A direct category is also necessarily [[skeleton|skeletal]], since any nonidentity isomorphism and its inverse would form a cycle of length 2.  The notion of generalized direct category, below, relaxes this requirement.

Conversely, a skeletal one-way category can have no cycles of nonidentity morphisms of any finite length, since by the one-way property all morphisms in such a cycle would be isomorphisms, hence endomorphisms by skeletality, and hence identities by the one-way property.  Since any infinite chain in a [[finite category]] contains a cycle, we see that

* A finite category is a direct category if and only if it is one-way and skeletal.

Since this condition is self-dual, we also see that

* A finite category is direct if and only if it is inverse.

An infinite category can be one-way and skeletal without being direct or inverse, such as the [[poset]] $\mathbb{Z}$ with its usual ordering.  However, if it additionally has **finite fan-out**, meaning that for each object $x$ there are altogether only finitely many morphisms with [[source]] $x$ (and arbitrary target), then it must be an inverse category, for any infinite non-cyclic chain would induce infinitely many distinct morphisms out of any of its objects by composition.  In [[FOLDS]], skeletal one-way categories with finite fan-out are called **simple categories** and used as signatures; thus

* Any simple category (in the sense of FOLDS) is an inverse category.

However, a category can be inverse without having finite fan-out.  Let $S$ be an infinite set and consider the poset $S \cup \{\infty\}$, with $S$ having the discrete ordering (no nonidentity arrows) and $\infty$ being less than every element of $S$.  This does not have finite fan-out, since $\infty$ is the source of infinitely many distinct arrows, but it is inverse, since there are clearly no chains of nonidentity arrows of length $\gt 1$.

### Allowing automorphisms

Some categories of geometric shapes, such as the [[tree category]] $\Omega$ and the [[cycle category]] $\Lambda$, include automorphisms of their objects.  By analogy with the notion of [[generalized Reedy category]], we can define $D$ to be a **generalized direct category** by replacing "identity" with "isomorphism" in the above definition.  Thereby we obtain the following equivalent conditions for $D$ to be a generalized direct category.

* $D$ contains no infinite descending chains of noninvertible morphisms $\cdots \to\cdot \to\cdot\to\cdot$.
* The relation $a\prec b$ on $ob(D)$ defined by "there exists a noninvertible morphism from $a$ to $b$" is [[well-founded relation|well-founded]].
* There exists a function $d\colon ob(D)\to Ord$, where $Ord$ is the class of [[ordinals]], such that every noninvertible morphism of $D$ raises the degree.
* There exists an isomorphism-reflecting functor (that is, it maps a morphism to an isomorphism if and only if the morphism is itself an isomorphism) $d : D\to \mathbf{Ord}$, where $\mathbf{Ord}$ is the large poset of ordinals viewed as a category.
* $D$ is a [[generalized Reedy category]] in which $D_-$ consists only of isomorphisms (or equivalently $D_+$ is all of $D$).

Of course, $\Omega$ and $\Lambda$ are not generalized direct categories themselves, since they have degree-lowering degeneracies as well, but their full subcategories of coface maps are generalized-direct.  More generally:

* If $R$ is any generalized Reedy category, then $R_+$ is a generalized direct category.

### Direct categories versus finite simplicial sets
If a direct category has finitely many objects then its [[nerve]]  is a finite [[simplicial set]]. Conversely, if a finite simplicial set is the nerve of a category then the category is a direct category with finitely many objects.

## Properties

### Model structures

Every direct category (and every inverse category) is in particular a [[Reedy category]], in fact an _[[elegant Reedy category]]_. Therefore whenever $M$ is a [[model category]] there is a [[Reedy model structure]] on $M^D$.  In the case of direct and inverse categories, these model structures are even easier to describe, since either the latching or the matching objects are degenerate.

## Examples

### Inside the simplex category

The [[wide subcategory]] $\Delta_+$ of the [[simplex category]] on the injective maps is direct. That is, the category with objects being the totally ordered sets $\{0\}, \{0, 1\}, \{0, 1, 2\}...$, and the morphisms being injective order-preserving maps. Its [[presheaves]] are [[semi-simplicial objects]]/[[semi-simplicial sets]] as opposed to [[simplicial objects]]/[[simplicial sets]].

### The direct category of corollas

We now define a "[[universal property|universal]]" generalized direct category which contains "all" [[geometric shapes for higher structures]].  This is based on ([Borisov](#Borisov)).

A functor $f\colon D\to E$ between generalized direct categories is called a **dependency** if it is equivalent to the inclusion of a [[sieve]], or equivalently if it is a [[fully faithful functor]] and a [[discrete fibration]] in the generalized sense of Street.  If $D$ and $E$ are [[skeleton|skeletal]] (as they must be if they are *non-generalized* direct categories), then any dependency must be isomorphic to the inclusion of a sieve.  One also usually works only with skeletal generalized direct categories, although the definition does not require it.

If we regard direct categories as a categorification of well-founded relations, then dependencies are a categorification of injective [[simulation]]s, i.e. the inclusions of initial segments.

Define a **corolla** to be a generalized direct category with a [[weak limit|weakly]] [[terminal object]]; we call this object the **vertex** of the corolla.  If a generalized direct category is finite and skeletal, then it is a corolla if and only if it has a unique object which is not the source of any noninvertible morphism.  Corollas are the direct categories which it is most natural to regard as *single* geometric shapes; other direct categories are more like "pasting diagrams" of geometric shapes.

Let $Corolla$ be the category of [[small category|small]] skeletal corollas and dependencies.

+-- {: .num_theorem #UnivDirect}
###### Theorem
$Corolla$ is a (large) generalized direct category.
=--
+-- {: .proof}
###### Proof
Suppose that $\cdots \overset{d_2}{\to} C_2 \overset{d_1}{\to} C_1 \overset{d_0}{\to} C_0$ is a descending chain of noninvertible dependencies.  Any dependency whose target is a corolla and whose image contains the vertex must be an equivalence, and hence an isomorphism if the corollas are skeletal; thus none of the dependencies $C_{n+1} \to C_{n}$ can have the vertex in their image.  Let $c_n\in C_0$ be the image of the vertex $\star_n \in C_n$ under the composite $d_0\circ \dots \circ d_{n-1}$.  Then we have a noninvertible morphism $c_{n+1} \to c_n$ for each $n$, arising from some map $d_n(\star_{n+1}) \to \star_n$ which exists since $\star_n$ is weakly terminal.  This is a contradiction, since $C_0$ is a generalized direct category.
=--

Of course, the same is true for any subcategory of $Corolla$.  In particular, it is very natural to consider only the category $FinCorolla$ of *finite* corollas, which is moreover essentially small (though not finite).

We next observe that $Corolla$ is the "universal" generalized direct category in a certain sense.  Let $D$ be any generalized direct category; then the slice category $D/d$ is a corolla for any $d\in D$.  Moreover, if $f\colon d\to d'$ is a morphism in $D$ which is [[monomorphism|monic]], then the "composition" functor $\Sigma_f\colon D/d \to D/d'$ is a dependency.  Thus, if every morphism in $D$ is monic, we have a functor $ext_D\colon D \to Corolla$ with $ext_D(d)= D/d$ and $ext_D(f)=\Sigma_f$.

Now if $f,g\colon d\to d'$ are parallel morphisms and $\Sigma_f=\Sigma_g$, then in particular $f = \Sigma_f(id_d) = \Sigma_g(id_d) = g$; thus $ext_D$ is [[faithful functor|faithful]].  Therefore, if $D$ is a direct category in which all morphisms are monic, and in which $D/d \cong D/d'$ implies $d=d'$ (which includes many examples), then $D$ is equivalent to a [[subcategory]] of $Corolla$.

This subcategory of $Corolla$ is usually *not* full, however.  In particular, for $d\in D$ the corolla $D/d$ will generally admit more automorphisms than $d$ has in $D$.  For instance, if $D$ is a non-generalized direct category, then $d$ has no nontrivial automorphisms, whereas $D/d$ generally will.  In particular, if the objects of $D$ have any sort of "orientation" or "labeling," then this information is forgotten by the functor $ext_D$.

## Related concepts

* [[test category]]

* [[filtered category]], [[sifted category]]

## References
 {#References}

* [[William Dwyer]], [[Daniel Kan]], §4.1 in: *Function complexes for diagrams of simplicial sets*, Indagationes Mathematicae (Proceedings) **86** 2 (1983) 139-147 [doi:10.1016/1385-7258(83)90051-3, [pdf](https://core.ac.uk/download/pdf/82652265.pdf)]

* {#Borisov} [[Dennis Borisov]], _Comparing definitions of weak higher categories, I_ ([arXiv:0909.2534](http://arxiv.org/abs/0909.2534))

* {#Barwick} [[Clark Barwick]], Def. 1.1 in: _On Reedy Model Categories_ ([arXiv:0708.2832](https://arxiv.org/abs/0708.2832))

[[!redirects direct category]]
[[!redirects direct categories]]
[[!redirects inverse category]]
[[!redirects inverse categories]]
[[!redirects one-way category]]
[[!redirects one-way categories]]
[[!redirects category with finite fan-out]]
[[!redirects categories with finite fan-out]]
[[!redirects categories with finite fan-outs]]
[[!redirects simple category]]
[[!redirects simple categories]]


direct image

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Topos Theory
+-- {: .hide}
[[!include topos theory - contents]]
=--
#### Category Theory
+-- {: .hide}
[[!include category theory - contents]]
=--
=--
=--

This page is about **direct images of sheaves** and related subjects.  For the set-theoretic operation, see [[image]].

***

#Contents#
* table of contents
{:toc}

## Idea

The [[right adjoint]] part $f_*$ of any [[geometric morphism]]

$$
  (f^* \dashv f_*) \;\; : E_1  \leftrightarrows E_2
$$

of [[topos]]es is called a **direct image**.

More generally, pairs of [[adjoint functor]]s between the [[categories of sheaves]] appear in various other setups apart from [[geometric morphism]]s of [[topoi]], for instance on [[abelian categories]] of [[quasicoherent sheaves]], bounded [[derived categories]] of [[coherent sheaves]] and the term _direct image_ is used for the right adjoint part of these, too.

Specifically for [[Grothendieck topos]]es:
a [[morphism of sites|morphism of sites]] $f : X \to Y$ induces a [[geometric morphism]] of [[Grothendieck topos]]es

$$
  (p^* \dashv p_*)
  \;\;\;
  :
  \;\;\;
  Sh(X) \stackrel{}{\stackrel{\overset{p_*}{\to}}{\overset{p^*}{\leftarrow}}}
  Sh(Y)
$$

between the [[category of sheaves|categories of sheaves]] on the sites, with

* $p_*$ the **direct image**

* and $p^*$ its [[left adjoint]]: the [[inverse image]].

## Definition

Given a [[site|morphism of sites]] $f : X \to Y$ coming from a [[functor]] $f^t : S_Y \to S_X$, the **direct image** operation on [[presheaf|presheaves]] is the functor

$$
  f_* : PSh(X) \to PSh(Y)
$$
$$
  f_* F : S_Y^{op} \stackrel{f^t}{\to} S_X^{op} \stackrel{F}{\to} Set
  \,.
$$

The restriction of this operation to [[sheaf|sheaves]], which respects sheaves, is the **direct image of sheaves**

$$
  f_* : Sh(X) \to Sh(Y)
  \,.
$$

## Examples

### Global sections

For $X$ a [[site]] with a [[terminal object]], let the morphism of [[site]]s be the canonical morphism $p : X \to {*}$.

* The direct image $p_*$ is the [[global section]]s functor;

* the inverse image $p^*$ is the [[constant sheaf]] functor;

* the left adjoint to $p^*$  is $\Pi_0$, the functor of geometric connected components (see [[homotopy group of an ∞-stack]]).

### Restriction and extension of sheaves

See

* [[restriction and extension of sheaves]]

for the moment.

### Direct image with compact supports
 {#DirectImageWithCompactSupport}

Let $f:X\to Y$ be a morphism of [[locally compact topological spaces]]. Then there exist a unique [[subfunctor]] $f_!: Sh(X)\to Sh(Y)$ of the direct image functor $f_*$ such that for any [[abelian sheaf]] $F$ over $X$ the sections of $f_!(F)$ over $U^{open}\subset X$ are those sections $s\in f_*(U)= \Gamma(f^{-1}(U),F)$ for which the restriction $supp(s)|f : supp(s)\hookrightarrow U$ is a [[proper map]].

This is called the [[direct image with compact support]].

It follows that $f_!$ is left exact.

Let $p:X\to {*}$ be the map into the one point space. Then for any $F\in Sh(X)$ the [[abelian sheaf]] $p_!F$ is the [[abelian group]] consisting of sections $s\in \Gamma(X,F)$ such that $supp(s)$ is compact. One writes $\Gamma_c(X,F) \coloneqq p_! F$ and calls this group a group of sections of $F$ with compact support. If $y\in Y$, then the fiber $(f_! F)_y$ is isomorphic to $\Gamma_c(f^{-1}y,F|_{f^{-1}(y)})$.

### Derived direct image

+-- {: .num_prop}
###### Proposition

Let $f^{-1} \colon Y \to X$ be a [[morphism of sites]].
Then the $q$th [[derived functor]] $R^q f_\ast$ of the induced [[direct image]] functor sends $\mathcal{F} \in Ab(Sh(X_{et}))$ to the [[sheafification]] of the [[presheaf]]

$$
  U_Y
  \mapsto
  H^q(f^{-1}(U_Y), \mathcal{F})
  \,,
$$

where on the right we have the degree $q$ [[abelian sheaf cohomology]] [[cohomology group|group]] with [[coefficients]] in the given $\mathcal{F}$.

=--

(e.g. [Tamme, I (3.7.1), II (1.3.4)](#Tamme), [Milne, 12.1](#Milne)).

+-- {: .proof}
###### Proof

We have a [[commuting diagram]]

$$
  \array{
    Ab(PSh(X)) &\stackrel{(-)\circ f^{-1}}{\longrightarrow}& Ab(PSh(Y))
    \\
    \uparrow^{\mathrlap{inc}} && \downarrow^{L}
    \\
    Ab(Sh(X)) &\stackrel{f_\ast}{\longrightarrow}& Ab(Sh(Y))
  }
  \,,
$$

where the right vertical morphism is [[sheafification]]. Because $(-) \circ f^{-1}$ and $L$ are both [[exact functors]] it follows that for $I^\bullet \to \mathcal{F}$ an [[injective resolution]] that

$$
  \begin{aligned}
     R^p f_\ast(\mathcal{F})
     & :\simeq
     H^p( f_\ast I)
     \\
     & = H^p(L I^\bullet(f^{-1}(-)))
     \\
     & = L (H^p(I^\bullet)(f^{-1}(-)))
  \end{aligned}
$$

=--

## Related concepts

* [[derived direct image]]

## References

e.g.

* [[Günter Tamme]], section II 1 of _[[Introduction to Étale Cohomology]]_
 {#Tamme}

* [[James Milne]], section 7 of _[[Lectures on Étale Cohomology]]_
 {#Milne}

[[!redirects direct images]]
[[!redirects direct image functor]]