Perry-Schwarz action

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#### Differential geometry
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#Contents#
* table of contents
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## Idea

The _Perry-Schwarz Lagrangian_ is [[Lagrangian density]]/[[action functional]] for the [[self-dual higher gauge field]] in 6d and/or the [[M5-brane]] [[Green-Schwarz sigma model]], after [[KK-compactification]] to 5 [[worldvolume]] [[dimensions]].

The construction is closely related to the following basic fact relating [[self-dual higher gauge theory|self-dual]] [[differential 3-forms]] on 6d [[Minkowski spacetime]] and [[D=5 Maxwell theory]]:

[[!include relation between 5d Maxwell theory and self-dual 3-forms in 6d -- section]]

## Details

### For the free self-dual field and trivial target space metric
 {#ForTheFreeSelfDualField}

We review the definitions from [Perry-Schwarz 96, Section 2 "The Free Theory"](#PerrySchwarz96) (following [Henneaux-Teitelboim 88](#HenneauxTeitelboim88)), for the [[worldvolume]] [[Lagrangian density]] of just the [[free field|free]] [[self-dual higher gauge field]] on a [[circle principal bundle]]-[[worldvolume]] for would-be [[target space]] being [[Minkowski spacetime]].

In doing so, we translate to [[coordinate chart|coordinate]]-invariant [[Cartan calculus]]-formalism and generalized to [[KK-compactification]] on possibly non-[[trivial bundle|trivial]] [[circle principal bundle]]:

#### Worldvolume and self-duality

Let

$$
  (\Sigma^6, g)
$$

be a [[pseudo-Riemannian manifold]] of [[dimension]] 6 and
of [[signature of a quadratic form|signature]] $(-,+,+,+,+,+)$, to be called the _[[worldvolume]]_.

In this dimension and with this signature, the [[Hodge star operator]]
squares to $+1$. This allows to consider for a [[differential 3-form]]
$$
  H \;\in\; \Omega^3\big(\Sigma^6\big)
$$

the condition that it be self-dual ([PS 96 (2)](#PerrySchwarz96))

\[
  \label{SelfDualityIn6d}
  H
  \;=\;
  \star
  H
  \,.
\]

We will assume in the following that $H$ is [[exact differential form]], hence that there exists a [[differential 2-form]]

$$
  B \in \Omega^2\big( \Sigma^6 \big)
$$

such that ([PS 96 (4)](#PerrySchwarz96))

$$
  H = d B
  \,.
$$

#### $S^1$-compactification

Consider then on $\Sigma^6$ the structure of an
$S^1 = U(1)$-[[circle principal bundle|principal bundle]]

\[
  \label{FibrationWorldvolume}
  \,
\]
\begin{xymatrix}
    S^1 \ar[r]
    &
    \Sigma^6
    \ar[d]
    \\
    & \Sigma^5
\end{xymatrix}

Write

\[
  \label{FiberVectorField}
  v^5 \in \Gamma( T \Sigma^6 )
\]

for the [[vector field]] which reflects the infinitesimal [[circle group]]-[[action]] on (eq:FibrationWorldvolume). We will write

$$
  \mathcal{L}_{v^5}
  \;=\;
  \big[d, \iota_{v^5} \big]
  \;\colon\;
  \Omega^\bullet\big( \Sigma^6 \big)
  \longrightarrow
  \Omega^\bullet\big( \Sigma^6  \big)
$$

for the [[Lie derivative]] of [[differential forms]] along $v^5$, and make use of [[Cartan's magic formula]] expressing it as an [[anti-commutator]], as shown.

Next consider an [[Ehresmann connection]] on the $S^1$-bundle (eq:FibrationWorldvolume), hence a [[differential 1-form]]

$$
  \theta^5
  \;\in\;
  \Omega^1\big( \Sigma^6 \big)
$$

such that

\[
  \label{EhresmannConditions}
  \iota_{v^5} \theta^5 = 1
  \phantom{AA}
  \text{and}
  \phantom{AA}
  \mathcal{L}_{v^5} \theta^5 = 0
\]

So in particular

$$
  \theta^5 \wedge \iota_{v^5}
  \;:\;
  \Omega^\bullet\big( \Sigma^6\big)
  \longrightarrow
  \Omega^\bullet\big( \Sigma^6\big)
$$

is a [[projection operator]]:

$$
  \theta^5 \wedge \iota_{v^5}
  \circ
  \theta^5 \wedge \iota_{v^5}
  \;=\;
  \theta^5 \wedge \iota_{v^5}
$$

The complementary projection is that onto [[horizontal differential forms]]

$$
  (-)^{\mathrm{hor}}
  :=
  \big(\mathrm{id} - \theta^5 \iota_{v^5})
  \;:\;
  \Omega^\bullet\big( \Sigma^6\big)
  \longrightarrow
  \Omega^\bullet\big( \Sigma^6\big)
$$

We require $v^5$ (eq:FiberVectorField) to be a [[spacelike]] [[isometry]]. This means that

\[
  \label{HodgeStarCommutingWithIsometryContraction}
  \star \circ \iota_{v^5}
  =
  -
  \theta^5 \wedge \circ \star
  \;:\;
  \Omega^3\big( \Sigma^6\big)
  \longrightarrow
  \Omega^4\big( \Sigma^6 \big)
\]

#### Self-duality after $S^1$-compactification

Set ([PS 96 (5)](#PerrySchwarz96))

\[
  \label{DefOfCalF}
  \mathcal{F}
  \;\coloneqq\;
  \iota_{v^5} H
\]

and ([PS 96 (6)](#PerrySchwarz96))

\[
  \label{DefOfTildeH}
  \tilde H
  \;\coloneqq\;
  \iota_{v^5} \star H
\]

With this notation the self-duality condition (eq:SelfDualityIn6d)
is equivalently ([PS 96 (9)](#PerrySchwarz96), see (eq:EquivalentIncarnationsOfSelfDuality) below):

\[
  \label{SelfDualityIntermsOfcalF}
  \mathcal{F}
  \;=\;
  \tilde H
\]

To make this fully explicit, notice that we have the following chain of logical equivalences:

\[
  \label{EquivalentIncarnationsOfSelfDuality}
  \begin{aligned}
    \big(
      H = \star H
    \big)
    & \Leftrightarrow
    \left(
      \array{
         &
         \phantom{\text{and}\;}
         \iota_{v_5} H  = \iota_{v^5} \star H
         \\
         &
         \text{and}\;
         \theta^5 \wedge H = \theta^5 \wedge \star H
      }
    \right)
    \\
    & \Leftrightarrow
    \big(
      \iota_{v^5} H = \iota_{v^5} \star H
    \big)
    \\
    &\Leftrightarrow
    \big(
      \mathcal{F}
      \;=\;
      \widetilde H
    \big)
  \end{aligned}
\]

Here the first step is decomposition of the self-duality equation into components, the second step follows by (eq:HodgeStarCommutingWithIsometryContraction) and the third step invokes the definitions (eq:DefOfCalF) and (eq:DefOfTildeH) and the fourth step the equality (eq:DecompositionOfCalF).

#### The gauge field

Define the [[vector potential]] ([PS 96 above (4)](#PerrySchwarz96))

\[
  \label{AField}
  A
  \;\coloneqq\;
  -
  \iota_{v^5} B
\]

With this we have

$$
  B
  \;=\;
  A \wedge \theta^5
  +
  B^{\mathrm{hor}}
  \,.
$$

Set also ([PS 96 above (4)](#PerrySchwarz96))

$$
  F
  \;\coloneqq\;
  \big( d A \big)^{\mathrm{hor}}
$$

then ([PS 96 (5)](#PerrySchwarz96))

\[
  \label{DecompositionOfCalF}
  \begin{aligned}
    \mathcal{F}
    &\coloneqq
    \iota_{v^5} H
    \\
    & =
    \iota_{v^5} d B
    \\
    & =
    - d \iota_{v^5} B + [\iota_{v^5}, d] B
    \\
    & =
    d A + \mathcal{L}_5 B
    \\
    & =
    F
    + \theta^5 \wedge \iota_{v^5} d A
    + \mathcal{L}_5 B^{\mathrm{hor}}
    +
    \underset{
      = -\theta^5 \wedge \mathcal{L}_{v^5} A
    }{
      \underbrace{
        \mathcal{L}_5 \theta^5 \wedge \iota_{v^5} B
      }
    }
    \\
    &
    =
    F
    +
    \mathcal{L}_{v^5} B^{\mathrm{hor}}
  \end{aligned}
\]

where in the last step under the brace we used (eq:EhresmannConditions) and (eq:AField).

Hence in terms of $F$ and $B^{\mathrm{hor}}$ the self-duality condition
(eq:SelfDualityIn6d), (eq:SelfDualityIntermsOfcalF)
is equivalently expressed as on the right of the following

\[
  \label{SelfDualityDecomposition}
  \big(
    H = \star H
  \big)
  \;\Leftrightarrow\;
  \big(
    \widetilde H
    \;=\;
    F + \mathcal{L}_{v^5} B^{\mathrm{hor}}
  \big)
\]

#### Weak self-duality and PS-equations of motion

Notice that

$$
  \begin{aligned}
    \theta^5 \wedge d
      \big( (d A)^{hor} \big)
    & =
    \theta^5 \wedge d
       \big( d A - \theta^5 \wedge \iota_{v_5} d A\big)
    \\
    & =
    \theta^5 \wedge (d \theta^5) \wedge \iota_{v_5} d A
  \end{aligned}
  \,.
$$

Hence assume now hat the Ehresmann connection is [[flat connection|flat]],
hence $d \theta^5 = 0$.

Then the self-duality condition in the form (eq:SelfDualityDecomposition)

$$
  \widetilde H
  -
  \mathcal{L}_{v^5} B^{\mathrm{hor}}
  \;=\;
  (d A)^{\mathrm{hor}}
$$

implies, after applying $\theta^5 \wedge d$ to both sides, the second-order equation ([PS 96 (16)](#PerrySchwarz96))

\[
  \label{TheEquationOfMotion}
  (H = \star H)
  \;\;\;\Rightarrow\;\;\;
  \theta^5
  \wedge
  d
  \big(
    \widetilde H
    -
    \mathcal{L}_{v^5} B^{\mathrm{hor}}
  \big)
  \;=\;
  0
\]

This equation by itself is hence a weakened form of the self-duality condition, a kind of "self-duality up to horizontally closed terms".

The proposal of [Perry-Schwarz 96, Sec. 2](#PerrySchwarz96) is to take this as the relevant [[equation of motion]] for the theory on $S^1$.

#### Lagrangian density

Therefore one is looking now for a [[Lagrangian density]] whose [[Euler-Lagrange equations]] are (eq:TheEquationOfMotion):

The _Perry-Schwarz-Lagrangian_ is ([PS 96 (17)](#PerrySchwarz96))

\[
  \label{CartanCalculusPerrySchwarzLagrangian}
  L
  \;\coloneqq\;
  -
  \tfrac{1}{2}
  \big(
    \tilde H
    -
    \mathcal{L}_{v^5} B^{\mathrm{hor}}
  \big)
  \wedge
  \star
  \tilde H
\]

With (eq:HodgeStarCommutingWithIsometryContraction) the Lagrangian  (eq:CartanCalculusPerrySchwarzLagrangian) becomes

\[
  \label{LagrangianUsingIsometry}
  \begin{aligned}
  L
  & =
  -
  \tfrac{1}{2}
  \big(
  \tilde H
  -
  \mathcal{L}_{v^5} B^{\mathrm{hor}}
  \big)
  \wedge
  H
  \wedge \theta^5
  \\
  & =
  -
  \tfrac{1}{2}
  \big(
  \iota_{v^5} \star H
  -
  \mathcal{L}_{v^5} B^{\mathrm{hor}}
  \big)
  \wedge
  H
  \wedge \theta^5
  \end{aligned}
\]

where in the second line we inserted the definition  (eq:DefOfTildeH).

Notice that (eq:LagrangianUsingIsometry) is the quadratic part of the following form-valued [[bilinear form]] on 2-form fields:

$$
  (B, B^\prime)
  \;\mapsto\;
  -
  \tfrac{1}{2}
  \big(
  \iota_{v^5} \star (d B)
  -
  \mathcal{L}_{v^5} B^{\mathrm{hor}}
  \big)
  \wedge
  (d B^\prime)
  \wedge \theta^5
$$

Moreover, this bilinear form is _symmetric_ up to a total derivative. For the first summand this is manifest from its incarnation in (eq:CartanCalculusPerrySchwarzLagrangian), since the Hodge pairing is symmetric, and for the second term this follows by "local integration by parts".

As a consequence, the [[Euler-Lagrange equations]] of the Perry-Schwarz Lagrangian density (eq:LagrangianUsingIsometry) may be computed from twice the variation of just the second factor

$$
  \begin{aligned}
    \delta L_{\mathrm{sd}}
    & =
    2
    \Big(
      -
      \tfrac{1}{2}
      \big(
        \iota_{v^5} \star H
        -
        \mathcal{L}_{v^5} B^{\mathrm{hor}}
      \big)
      \wedge
      d(\delta B)
      \wedge
      \theta^5
    \Big)
    \\
    & =
    \Big(
    d
    \big(
      \iota_{v^5} \star H
      -
      \mathcal{L}_{v^5} B^{\mathrm{hor}}
    \big)
    \Big)
    \wedge
    (\delta B)
    \wedge \theta^5
    +
    d(\cdots)
  \end{aligned}
$$

to indeed be (eq:TheEquationOfMotion):

\[
  \label{EquationsOfMotion}
  \theta^5 \wedge
  d
  \big(
    \iota_{v^5} \star H
    -
    \mathcal{L}_{v^5} B^{\mathrm{hor}}
  \big)
  \;=\;
  0
  \,.
\]

Notice that if we do use the self-duality condition (eq:SelfDualityIn6d)
on the Perry-Schwarz Lagrangian (eq:LagrangianUsingIsometry) it becomes

\[
  \label{NonCovariantLagrangianInFlatSpacetime}
  L_{\mathrm{sd}}
  \;=\;
  -
  \tfrac{1}{2}
  \big(
  \iota_{v^5} H
  -
  \mathcal{L}_{v^5} B^{\mathrm{hor}}
  \big)
  \wedge
  H
  \wedge \theta^5
  \phantom{AAAAA}
  \text{if} \;\; H = \star H
\]

#### Example: Reduction to 5d Maxwell theory

Consider the special case that

$$
  \mathcal{L}_{v^5} B = 0
  \,,
$$

which corresponds to keeping only the 0-mode under [[KK-compactification]] along the circle fiber.

Then (eq:DecompositionOfCalF) becomes

$$
  \mathcal{F} = F
$$

and so the self-duality condition (eq:SelfDualityDecomposition) now becomes

$$
  \iota_{v^5} \star H \;=\; F
  \,.
$$

which means that

$$
  H = F \wedge \theta^5 + \star_5 F
$$

> (check relative sign)

Since $d H = d \circ d B = 0$, this implies

$$
  \begin{aligned}
    \big(
      d H = 0
    \big)
    & \Leftrightarrow
    \Big(
      d
      \big(
        F \wedge \theta^5
        +
        \star_5 F
      \big)
      = 0
    \Big)
    \\
    & \Leftrightarrow
    \left(
    \left\{
      \array{
        d_5 F & = 0
        \\
        d_5 \star_5 F & = 0
      }
    \right.
    \right)
  \end{aligned}
$$

These are of course [[Maxwell's equations]] on $\Sigma^5$.

([PS 96 above (16)](#PerrySchwarz96))

### For the full interacting M5-brane sigma model
 {#TheFullInteractingModel}

The full interacting PS Lagrangian  ([PS 96 (63)](#PerrySchwarz96)) has more terms..

(...)

([Schwarz 97](#Schwarz97), [APPS 97](#APPS97))

(...)

## Related concepts

* [[D=5 super Yang-Mills theory]]

## References

The Perry-Schwarz action is due to

* {#PerrySchwarz96} [[Malcolm Perry]], [[John Schwarz]], _Interacting Chiral Gauge Fields in Six Dimensions and Born-Infeld Theory_, Nucl. Phys. B489 (1997) 47-64 ([arXiv:hep-th/9611065](http://arxiv.org/abs/hep-th/9611065))

* {#Schwarz97} [[John Schwarz]], _Coupling a Self-Dual Tensor to Gravity in Six Dimensions_, Phys. Lett. B395:191-195, 1997 ([cds:317663](http://cds.cern.ch/record/317663), doi:10.1016/S0370-2693(97)00094-4)

* {#APPS97} [[Mina Aganagic]], Jaemo Park, Costin Popescu, [[John Schwarz]], _World-Volume Action of the M Theory Five-Brane_, Nucl.Phys. B496 (1997) 191-214 ([arXiv:hep-th/9701166](http://arxiv.org/abs/hep-th/9701166))

A similar construction but with compactification along the [[timelike]] direction is due to

* {#HenneauxTeitelboim88} [[Marc Henneaux]], [[Claudio Teitelboim]], _Dynamics of chiral (self-dual) $p$-forms_, Physics Letters B Volume 206, Issue 4, 2 June 1988, Pages 650-654 (doi:10.1016/0370-2693(88)90712-5)

The [[double dimensional reduction]] to the [[Green-Schwarz sigma-model]] of the [[D4-brane]]:

* [APPS 97, Section 6](#APPS97)

* {#APPS97b} [[Mina Aganagic]], Jaemo Park, Costin Popescu, [[John Schwarz]], Section 6 of _Dual D-Brane Actions_, Nucl. Phys. B496 (1997) 215-230 ([arXiv:hep-th/9702133](https://arxiv.org/abs/hep-th/9702133))

The covariant version via a [[scalar field|scalar]] [[auxiliary field]] is due to

* [[Paolo Pasti]], [[Dmitri Sorokin]], [[Mario Tonin]], _On Lorentz Invariant Actions for Chiral P-Forms_, Phys.Rev. D55 (1997) 6292-6298 ([arXiv:hep-th/9611100](https://arxiv.org/abs/hep-th/9611100))

* {#PST97} [[Paolo Pasti]], [[Dmitri Sorokin]], [[Mario Tonin]], _Covariant Action for a D=11 Five-Brane with the Chiral Field_, Phys. Lett. B398 (1997) 41 ([arXiv:hep-th/9701037](https://arxiv.org/abs/hep-th/9701037))

* {#BLNPST97} [[Igor Bandos]], [[Kurt Lechner]], [[Alexei Nurmagambetov]], [[Paolo Pasti]], [[Dmitri Sorokin]], [[Mario Tonin]], _Covariant Action for the Super-Five-Brane of M-Theory_, Phys. Rev. Lett. 78 (1997) 4332-4334 ([arXiv:hep-th/9701149](http://arxiv.org/abs/hep-th/9701149))

Speculations about non-abelian generalizations (for several coincident M5-branes):

* [[Chong-Sun Chu]], _A proposal for the worldvolume action of multiple M5-branes_, 2013 ([pdf](http://hep.phy.ntnu.edu.tw/old_version/talks/2013/2013-05-09Chong-SunChu.pdf))

The above text follows

* {#FSS19c} [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], Section 2 of: _[[schreiber:Super-exceptional embedding construction of the M5-brane|Super-exceptional geometry: origin of heterotic M-theory and super-exceptional embedding construction of M5]]_ ([arXiv:1908.00042](https://arxiv.org/abs/1908.00042))

[[!redirects Perry-Schwarz actions]]

[[!redirects Perry-Schwarz action functional]]
[[!redirects Perry-Schwarz action functionals]]

[[!redirects Perry-Schwarz Lagrangian]]
[[!redirects Perry-Schwarz Lagrangians]]

[[!redirects Perry-Schwarz Lagrangian density]]
[[!redirects Perry-Schwarz Lagrangian densities]]


Persi Diaconis

American mathematician, working in statistics and probability theory, and magician.

Standford University [webpage](http://statweb.stanford.edu/~cgates/PERSI/).

## Selected writings

On [[Cayley graph spectra]]:

* {#DiaconisShahshahani81} [[Persi Diaconis]], [[Mehrdad Shahshahani]], *Generating a random permutation with random transpositions*, Z. Wahrscheinlichkeitstheorie verw Gebiete 57, 159–179 (1981) ([doi:10.1007/BF00535487](https://doi.org/10.1007/BF00535487))

On [[representation theory]] and [[geometric group theory]] in [[probability theory]] and [[statistics]] ([[Cayley distance]], [[Kendall distance]], ...):

* {#Diaconis88} [[Persi Diaconis]], *Group Representations  in Probability  and  Statistics*, IMS Lecture Notes Monogr. Ser., 11: 198pp. (1988) ([jstor:i397389](https://www.jstor.org/stable/i397389), [ISBN: 0940600145](https://projecteuclid.org/ebooks/institute-of-mathematical-statistics-lecture-notes-monograph-series/Group-representations-in-probability-and-statistics/toc/10.1214/lnms/1215467407), [pdf](https://jdc.math.uwo.ca/M9140a-2012-summer/Diaconis.pdf))

* [[Persi Diaconis]], [[Phil Hanlon]], Section 4 of: *Eigen Analysis for Some Examples of the Metropolis Algorithm*, in Donald Richards (ed.) *Hypergeometric Functions on Domains of Positivity, Jack Polynomials, and Applications*, Contemporary Mathematics Vol. 138, AMS 1992 ([doi:10.1090/conm/138](http://dx.doi.org/10.1090/conm/138), [EFS NSF 392](https://statistics.stanford.edu/research/eigen-analysis-some-examples-metropolis-algorithm), [pdf](https://statweb.stanford.edu/~cgates/PERSI/papers/eigen-metalg.pdf))

## Related pages

* [[Bayesian reasoning]]

category: people


persistence diagram

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Homological algebra
+--{: .hide}
[[!include homological algebra - contents]]
=--
#### Representation theory
+-- {: .hide}
[[!include representation theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In [[persistent homology]] a *persistence diagram* or *barcode* is a way to encode the [[isomorphism class]] of a [[persistence module]] in terms of a [[multiset]] of [[pairs]] of [[numbers]] $a \leq b$ which stand for "features" reflected in the module which "appear" at stage $a$, "persist" until stage $b$ and then "disappear".

Concretely, [[Gabriel's theorem]] for [[ADE classification|A-type]] [[quivers]] (and analogously its generalizations to infinite linear diagrams and/or infinite-dimensional modules) says that every ([[zigzag persistence|zig-zag]]) [[persistence module]] (over some [[ground field]] $\mathbb{K}$), such as

\begin{tikzcd}
  V_1
  \ar[r, "{f_1}"]
  &
  V_2
  \ar[from=r, "{f_2}"{swap}]
  &
  V_3
  \ar[r, "{f_3}"]
  &
  \cdots
  \ar[r, "f_{n-1}"]
  &
  V_n
\end{tikzcd}

is [[generalized the|the]] [[direct sum]] (in the [[category]] of [[quiver representations]]) of *interval* modules $I_{a,b}$ of the form:

\begin{tikzcd}[column sep = small, row sep=small]
  1
  \ar[r,-]
  &
  \cdots
  \ar[r,-]
  &
  (a-1)
  \ar[r,-]
  &
  a
  \ar[r,-]
  &
  \cdots
  \ar[r,-]
  &
  b
  \ar[r,-]
  &
  (b+1)
  \ar[r,-]
  &
  \cdots
  \ar[r,-]
  &
  n
  \\
  \mathbb{K}^0
  \ar[r, -, "{0}"]
  &
  \cdots
  \ar[r, -, "{0}"]
  &
  0
  \ar[r,-, "{0}"]
  &
  \mathbb{K}^1
  \ar[r,-, "{\mathrm{id}}"]
  &
  \cdots
  \ar[r,-, "{\mathrm{id}}"]
  &
  \mathbb{K}^1
  \ar[r,-,"{0}"]
  &
  \mathbb{K}^0
  \ar[r,-,"{0}"]
  &
  \cdots
  \ar[r,-,"{0}"]
  &
  \mathbb{K}^0
  \mathrlap{\,.}
\end{tikzcd}

Hence the [[multiset]] of [[pairs]] $(a,b)$ labeling its inverval [[direct sum|summands]] completely characterizes the given persistence module, up to [[isomorphism]].

Regarding these pairs as [[intervals]], the collection they form is naturally envisioned as a collection of "bars" and then known as the *barcode* of the persitence module.

Regarding these pairs instead as points in the [[plane]], $(a,b) \in \mathbb{R}^2$, the [[multiset|multi]]-[[subset]] which they form is called the *persistence diagram* of the persistence module.

\begin{remark}
  While for a given [[quiver]]-shape, persistence diagrams/barcodes carry the same information as the [[isomorphism classes]] of the [[persistence modules]] of this shape, it makes sense to compare persistence diagrams arising from quivers of different shape. For example, the *[[diamond principle]]* gives conditions under which [[persistence modules]] whose arrow directions differs in two consecutive positions still have bijective persistence diagrams.
\end{remark}

## Properties

* [[stability of persistence diagrams]]

## References

### General

See also the references at *[[persistent homology]]*.

Introduction and survey:

* [[Robert Ghrist]], _Barcodes: The Persistent Topology of Data_, Bull. Amer. Math. Soc. 45 (2008), 61-75 ([doi:10.1090/S0273-0979-07-01191-3](https://doi.org/10.1090/S0273-0979-07-01191-3), [pdf](https://www.math.upenn.edu/~ghrist/preprints/barcodes.pdf))

* [[Steve Y. Oudot]], *Persistence theory: from quiver representations to data analysis*, Mathematical Surveys and Monographs **209** AMS (2015) $[$[pdf](https://geometrica.saclay.inria.fr/team/Steve.Oudot/books/o-pt-fqrtda-15/surv-209.pdf), [ISBN:978-1-4704-3443-4](https://bookstore.ams.org/surv-209/)$]$

* [[Dan Burghelea]], _"Barcodes" for continuous maps and a brief introduction to Alternative Morse Theory_, [arXiv:2305.19828](https://arxiv.org/abs/2305.19828)

### Kernel methods and topological machine learning

On [[kernel methods]] applicable to [[persistence diagrams]]/[[barcodes]] for making [[topological data analysis]] amenable to "[[topological machine learning|topological]]" [[machine learning]]:

* {#RHBK15} Jan Reininghaus, [[Stefan Huber]], Ulrich Bauer, Roland Kwitt, *A Stable Multi-Scale Kernel for Topological Machine Learning*, NIPS'15: Proceedings of the 28th International Conference on Neural Information Processing System, **2** (2015 3070–3078  $[$[arXiv:1412.6821](https://arxiv.org/abs/1412.6821)$]$

* {#KHNL15} Roland Kwitt, [[Stefan Huber]], Marc Niethammer, Weili Lin, [[Ulrich Bauer]], *Statistical Topological Data Analysis -- A Kernel Perspective*, in: *Advances in Neural Information Processing Systems*  (NIPS 2015) $[$[ISBN:9781510825024](https://proceedings.neurips.cc/paper/2015), [doi:10.5555/2969442.2969582](https://dl.acm.org/doi/10.5555/2969442.2969582)$]$

* {#RSL} [[Bastian Rieck]], Filip Sadlo, Heike Leitte, *Topological Machine Learning with Persistence Indicator Functions*,  In: *Topological Methods in Data Analysis and Visualization V* TopoInVis (2017) 87-101 Mathematics and Visualization. Springer, $[$[arXiv:1907.13496](https://arxiv.org/abs/1907.13496), [doi:10.1007/978-3-030-43036-8_6](https://doi.org/10.1007/978-3-030-43036-8_6)$]$

* Raphael Reinauer, Matteo Caorsi, Nicolas Berkouk, *Persformer: A Transformer Architecture for Topological Machine Learning* $[$[arXiv:2112.15210](https://arxiv.org/abs/2112.15210)$]$

[[!redirects persistence diagrams]]

[[!redirects barcode]]
[[!redirects barcodes]]

[[!redirects persistence barcode]]
[[!redirects persistence barcodes]]


persistence object

# Contents
* table of contents
{: toc}

## Idea

A _persistence object_ of a [[category]] $C$ is a [[functor]] from a [[poset]], often a [[Cartesian product]] of [[linear orders]], to $C$.
This is a [[concept with an attitude]]: One calls such functors "persistence objects" when one is interested in determining their [[persistence diagrams]] or other measures of "persistence" as used in [[topological data analysis]].

The main example in this context arises when $C$ is a [[category of vector spaces]] or more generally a [[category of modules]], in which case one speaks of *[[persistence modules]]* as used in *[[persistent homology]]*. Alternatively, $C$ could be a [[category of groups]], such as [[homotopy groups]], or even of full [[homotopy types]], which is the case of interest in *[[persistent homotopy]]*.

When the indexing poset is the poset of [[real numbers]] or, more generally, a product of copies of the poset of real numbers, the collection of persistent objects of a fixed category $C$ admits a distance called the _interleaving distance_, which, informally, measures how isomorphic any two persistence objects are.

The key property which one will typically demand of a good theory of persistence objects is a notion of [[persistence diagrams]] (measuring "how persistent" a given persistence object is) which is
*[[stability of persistence diagrams|stable]]* with respect to interleaving distance.

## Definitions

Let $P$ be a partially ordered set, [seen as a category](partial+order#AsACategoryWithExtraProperties), and let $C$ be a category.
A **$P$-persistent object** of $C$ is a functor $P \to C$.
The category of $P$-persistent objects of $C$ is the [[functor category]] $C^P$.

Let $(\mathbb{R}, \leq)$ be the poset of real numbers with its standard order, and let $C$ be a fixed category.
Let $X : \mathbb{R} \to C$ be a persistent object of $C$, and let $\epsilon \geq 0 \in \mathbb{R}$.
The **$\epsilon$-shift** of $X$ is the persistent object $X[\epsilon] : \mathbb{R} \to C$ such that, for all $r \in \mathbb{R}$ we have $X[\epsilon](r) = X(r+\epsilon)$, and for all $r \leq s \in \mathbb{R}$, the structure morphism $X[\epsilon](r) \to X[\epsilon](s)$ is equal to the structure morphism $X(r+\epsilon) \to X(s+\epsilon)$.
The $\epsilon$-shift construction gives a functor $(-)[\epsilon] : C^\mathbb{R} \to C^\mathbb{R}$.
Note that the structure morphisms of $X$ give a natural transformation $\eta^X_\epsilon : X \to X[\epsilon]$.

An **$\epsilon$-interleaving** between $X$ and $Y$ consists of natural transformations $f : X \to Y[\epsilon]$ and $g : Y \to X[\epsilon]$ such that $g[\epsilon] \circ f = \eta^X_{2\epsilon}$ and $f[\epsilon] \circ g = \eta^Y_{2\epsilon}$.

As an example, note that a $0$-interleaving is precisely an isomorphism in the category $C^\mathbb{R}$.

The **interleaving distance** between persistent objects $X, Y \in C^{\mathbb{R}}$ is
\[
    d_I(X,Y) = inf\; \big(
        \big\{\epsilon \geq 0 \;:\;
            X \;\text{and}\; Y\;\text{are}\; \epsilon\text{-interleaved}\;\big\}
        \cup \{ \infty \} \big).
\]

## Examples

Let $(\mathbb{R}, \leq)$ denote the poset of real numbers with its standard order, and let $Vect_k$ denote the category of vector spaces over a fixed field $k$.
The $\mathbb{R}$-persistent objects of $Vect_k$ are known as [[persistence module|_persistence modules_]], and are a central object of study in [[topological data analysis]].

Let $n \geq 2$ be a natural number.
Persistent objects of the form $\mathbb{R}^n \to Vect_k$ are known as _multiparameter persistence modules_.

Let $\mathbb{R}_{\geq 0}$ denote the poset of non-negative real numbers with its standard order, and let $Set$ denote the category of sets.
The $\mathbb{R}_{\geq 0}$-persistent objects of $Set$ are known as _persistent sets_.

## Related concepts

* [[persistence module]]

## References
 {#References}

The terminology "persistence object" is used for instance in

* [[Donald Pinckney]], Section 3.1 *[Persistence objects](https://donaldpinckney.com/machine%20learning/2019/05/02/tda.html#persistence-objects)* in *[Topological Data Analysis and Persistent Homology](https://donaldpinckney.com/machine%20learning/2019/05/02/tda.html)* (2019)

General [[category theory|category theoretic]] discussion of persistence objects (not using that terminology, though):

* {#BubenikScott14} [[Peter Bubenik]], [[Jonathan Scott]] _Categorification of Persistent Homology_, Discrete & Computational Geometry **51** (2014) 600-627 [[doi:10.1007/s00454-014-9573-x](https://doi.org/10.1007/s00454-014-9573-x)]

* {#CagliariFerriPozzi01} Francesca Cagliari, Massimo Ferri & Paola Pozzi. _Size Functions from a Categorical Viewpoint_, Acta Applicandae Mathematica, 2001.

Original references on multiparameter persistent homotopy groups and multiparameter persistence modules:

* {#FrosiniMulazzani99} P. Frosini, M. Mulazzani. _Size homotopy groups for computation of natural size distances_, Bulletin of the Belgian Mathematical Society Simon Stevin, 1999.

* {#CarlssonZomorodian09} G. Carlsson, A. Zomorodian. _The theory of multidimensional persistence_, Discrete and Computational Geometry, 2009.

Survey on multiparameter persistent homology:

* {#BotnanLesnick22} Magnus Bakke Botnan, Michael Lesnick. _An Introduction to Multiparameter Persistence_. ([arXiv:2203.14289](https://arxiv.org/abs/2203.14289))

[[!redirects persistence objects]]

[[!redirects persistent object]]


persistent cohomotopy

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Homotopy theory
+--{: .hide}
[[!include homotopy - contents]]
=--
#### Spheres
+--{: .hide}
[[!include spheres -- contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea
 {#Idea}

### General
 {#IdeaGeneral}

*Persistent cohomotopy* (not yet an established term) would be the study of [[cohomotopy]] in situations where the [[domain]] [[topological space|space]] and/or its [[continuous function|map]] to an [[n-sphere]] may depend on a parameter, to yield not just a single [[cohomotopy set]], but a [[filtered object|filtered]] set (a filtered [[group]] when the [[dimension of a cell complex|dimension]] of the domain space is large enough). One concrete implementation of this general notion is considered in [Franek & Krčál 2017](#FranekKrcal17), see [below](#Definition).

Generally, one may understand persistent cohomotopy both as the [[duality|dual]] of *[[persistent homotopy]]* as well as a [[non-abelian cohomology]]-version of the more widely considered notion of [[persistent homology]]:

| | coarse |  intermediate | fine  |
|--|--|--|--|
| **[[homology]]** | [[ordinary homology]] | [[generalized homology]] | [[homotopy groups|homotopy]] |
| **[[cohomology]]** | [[ordinary cohomology]] | [[Whitehead-generalized cohomology|generalized cohomology]] | [[cohomotopy]] |
|  |  |  |  |
| **persistent homology** | [[persistent homology|persistent ordinary homology]] | persistent generalized homology | [[persistent homotopy]] |
| **persistent cohomology** | persistent ordinary cohomology | persistent generalized cohomology | [[persistent cohomotopy]] |

### Detecting data meeting targets
 {#DetectingDataMeetingTargets}

While [[persistent homology]], as a tool in [[topological data analysis]], is traditionally meant to discover [[persistence diagram|persistent]] [[cycles]] in a given data set, it is typically unclear what a persistent cycle actually *means* for the practical interpretation of the data -- certainly this information is not provided by the mathematics.

\begin{imagefromfile}
    "file_name": "IndicatorFunctionOnData-220523.jpg",
    "float": "right",
    "width": 420,
    "unit": "px",
    "margin": {
        "top": -40,
        "bottom": 0,
        "right": 0,
        "left": 10
    }
\end{imagefromfile}

In contrast, [[persistent cohomotopy]] in [[TDA]] is the effective answer to a concrete and common question in [[data analysis]]:

Given a large-[[dimension of a CW-complex|dimensional]] [[topological space|space]] of data, and a small [[natural number|number]] $n$ of ([[real number|real]]) indicator values assigned to each data point with given precision $1/r$, *does _any_ data meet a prescribed target indication precisely?*

A fundamental theorem of persistent Cohomotopy ([Franek, Krčál & Wagner 2018](persistent+cohomotopy#FranekKrcalWagner18), [Franek & Krčál 2017, p. 5](persistent+cohomotopy#FranekKrcal17), see Thm. \ref{GuaranteeThatSolutionsExist} below) shows that (1.) the answer to this question is detected by a certain [[Cohomotopy]]-class and (2.) in a fair range of dimensions, this Cohomotopy class is provably [[computability|computable]], hence the above question is effectively [[decidability|decidable]]. $\phantom{----}$ (figure from [[schreiber:New Foundations for TDA -- Cohomotopy|SS22]])

Alternatively, with tools from [[persistent homology theory]] an answer to this question is given by the method of *[[well groups]]* -- but (1.) it is known that well groups are in general too coarse to provide a complete answer and (2.) despite effort it remains unknown if well groups are actually [[computability|computable]] in relevant cases, see [Franek & Krčál 2016](persistent+cohomotopy#FranekKrcal16).

In this concrete sense (and generally by the [above](#IdeaGeneral) discussion), persistent cohomotopy may be understood as an enhancement or refinement of ([[well groups]] in) [[persistent homology]]:

> (graphics from [[schreiber:New Foundations for TDA -- Cohomotopy|SS 22]])

### Use of cobordism theory
 {#UseOfCobordismTheory}

If the [[topological space]] $X$ of data may be assumed to be a [[smooth manifold]] (indeed, in typical examples $X$ is itself a large-dimensional [[Cartesian space]]) then persistent cohomotopy may be understood dually via [[Pontryagin's theorem]] as characterizing [[iso-hypersurfaces]] of data (close to a given target indicator) by [[framed cobordism|framed]] *[[cobordism theory]]* ([Franek & Krčál 2017, p. 8-9](persistent+cohomotopy#FranekKrcal17)). The full implications of this relation for topological data analysis remain to be explored.

## Definition
 {#Definition}

We spell out the definition considered in [Franek & Krčál 2017](#FranekKrcal17).

For

* $X$ a [[topological space]],

* $n \in \mathbb{N}$ a [[natural number]],

* $r \in \mathbb{R}_{\gt 0}$ a [[positive number|positive]] [[real number]],

* $f \,\colon\, X \xrightarrow{\;\;\;\;\;\;} \mathbb{R}^n$ a [[continuous map]],

such that with

* $A_r \,\coloneqq\, f^{-1}\big( \mathbb{R}^n \setminus B_r \big) \;\xhookrightarrow{i_r}\; X\;$

  (the [[preimage]] under $f$ of the [[complement]] of the [[open ball]] of [[radius]] $r$ around the origin in the [[Cartesian space]])

we have that

* $(X,A_r)$ is a [[CW-pair]] of [[dimension of a cell complex|dimension]] $dim(X) \leq 2n - 3$,

consider the following induced [[exact sequence]] of [[cohomotopy]]-groups (where square brackets $[X,Y]$ denote the [[hom-set]] $Ho(X,Y)$ in the [[classical homotopy category]], hence [[homotopy classes]] of [[continuous functions]] $X \to Y$):

\begin{tikzcd}[row sep=small]
  \pi^{n-1}
  \big(
    X
  \big)
  \ar[r, "{ \iota_r^\ast}"]
  \ar[d, phantom, "{\coloneqq}"{rotate=-90}]
  &
  \pi^{n-1}
  \big(
    A
  \big)
  \ar[d, phantom, "{\coloneqq}"{rotate=-90}]
  \ar[r, "{ \delta_r }"]
  &
  \pi^{n}
  \big(
    X
  \big)
  \ar[d, phantom, "{\coloneqq}"{rotate=-90}]
  \\
  \big[
    X
    ,\,
    S^{n-1}
  \big]
  \ar[r, "{ [\iota_r. S^{n-1}] }"]
  &
  \big[
    A
    ,\,
    S^{n-1}
  \big]
  \ar[r]
  &
  \big[
    X
    ,\,
    S^n
  \big]
  \mathrlap{\,.}
\end{tikzcd}

Now define what we may call **the $n$-cohomotopy of $X$ at resolution $1/r$ relative to $f$**, to be (this is [FK17 (2)](#FranekKrcal17)):

\begin{equation}
  \label{CohomotopyAtSomeResolution}
  \pi^n_{(f,r)}
  \;\coloneqq\;
  image(\delta_r)
  \;\simeq\;
  \Big\{
    [g/A_r]
    \,\in\,
    \pi^n(X/A_r)
    \,\big\vert\,
    (X, A_r) \xrightarrow{ \exists \, g } (\mathbb{R}^n,\, \mathbb{R}^n \setminus B_r)
  \Big\}
  \,,
\end{equation}

hence that subgroup of the $n$-cohomotopy group of the [[quotient topological space]] $X/A_r$ whose elements may be represented by [[continuous maps]] defined on all of $X$.

Notice that the function $f$ itself canonically represents an element

\[
  \label{FunctionAsClassInThePersistenceCohomotopyRelativeToIt}
  [f]_r
    \;\in\;
  \pi^n_{(f,r)}(X)
\]

which provides a base point, making this a [[pointed object|pointed]] group.

Moreover, for $r_1 \leq r_2$ we have the evident inclusion

$$
  A_{r_2} \xhookrightarrow{\; i_{r_1, r_2} \;}  A_{r_1}
$$

and thus a canonical comparison [[homomorphisms]] of pointed groups:

\begin{tikzcd}[row sep=small]
  \pi^n_{(f,r_1)}(X)
  \ar[d, hook]
  \ar[rr]
  &&
  \pi^n_{(f,r_2)}(X)
  \ar[d, hook]
  \\
  \big[
    X/A_{r_1}
    ,\,
    S^n
  \big]
  \ar[rr, "{ \big[ X/{i_{r_1,r_2}}, S^n\big] }"]
  &&
  \big[
    X/A_{r_2}
    ,\,
    S^n
  \big]
  \,.
\end{tikzcd}

Therefore, regarding the [[positive number|positive]] [[real numbers]] [[ordering|ordered]] by $\leq$ as a [[poset]] and thus as a [[category]], the resulting [[direction|directed]] [[diagram]] of [[pointed objects|pointed]] [[groups]]

\[
  \label{TheCohomotopyPersistenceModule}
  \pi^n_{(f,\bullet)}
  (X)
  \;\colon\;
  (\mathbb{R}, \leq)
  \xrightarrow{\;\;\;}
  Grp^{\mathbb{Z}/}
\]

is the **cohomotopy persistence module** ([FK17, p. 5](#FranekKrcal17), rhyming on *[[persistence module]]* as used in [[persistent homology theory]]) of $X$ relative to $f$.

\linebreak

## Properties

For $r \in \mathbb{R}_{\gt 0}$, say that a continuous function

$$
  g\;\colon\; X \xrightarrow{\;\;} \mathbb{R}^n
$$

is an *$r$-deformation* of the given $f$ if its values are pointwise within a radius $r$ of those of $f$, i.e. if

\[
  \label{rDeformation}
  \vert g-f\vert_\infty
  \;\coloneqq\;
  \underset{x \in X}{max}
  \vert g(x) - f(x)\vert
  \;\lt\;
  r
  \,.
\]

\begin{theorem}\label{GuaranteeThatSolutionsExist}
**([Franek & Krčál 2017](#FranekKrcal17), following [2016, p. 5](#FranekKrcal16))**
\linebreak
Under the above assumptions, the following are equivalent:

* there exists an $r$-deformation (eq:rDeformation) of $f$ with *no* zeros on $X \setminus A_r$;

* the Cohomotopy class $[f]_r \in \pi_n^{(f,r)}(X)$ (eq:FunctionAsClassInThePersistenceCohomotopyRelativeToIt) is trivial.

Moreover, both questions are effectively [[computability|computable]]/[[decidability|decidable]] in the given dimension range.
\end{theorem}

The proof of the computability statement in Thm. \ref{GuaranteeThatSolutionsExist} uses a general result about computability of [[cohomotopy]]-sets from [CKMSVW14](computational+topology#CKMSVW14).

\linebreak

## Examples
 {#Examples}

Consider the simplest instructive example, where $X \subset \mathbb{R}$ is an [[interval]] (closed or open, such as the [[real line]] itself) and $n = 1$.

Notice that $X$ is [[contractible topological space|contractible]], so that its plain [[cohomotopy]] in [[positive number|positive]] degree is necessarily trivial, $\pi^{n \geq 1}(X) = \ast$.

Now consider a [[continuous function]]

$$
  f
  \;\colon\;
  X
  \xrightarrow{\;\;}
  \mathbb{R}^1
$$

to the [[real numbers]], and consider the question whether any continuous map that might differ from $f$ by up to $\pm r$ ever takes the value $0 \in \mathbb{R}$.

In this simple case one can decide this immediately by appeal to the [[intermediate value theorem]]: Any such function $r$-close to $f$ will have to pass through zero at least once if the values of $f$ ever pass between $(-\infty, -r)$ and $(r, \infty)$. But precisely this is also detected by the 1-cohomotopy class of $f$  at resolution $1/r$ (relative to itself) in the sense of (eq:CohomotopyAtSomeResolution), as illustrated in the following figure:

\begin{tikzpicture}

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%%%%%%%%%%%%%%%

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%%%%%%%%%%%%%%%

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> (graphics adapted from [[schreiber:New Foundations for TDA -- Cohomotopy|SS 22]])

Here the [[curve]] indicates the [[graph of a function|graph]] of $f$ and "gray" stands for the region $A_r \,\subset\, X$, while "red" stands for $\mathbb{R}^1 \setminus B_r$.

One sees on the left that if the cohomotopy class of $f$ at resolution $1/r$ is non-trivial, then there is at least one zero.

{#IntuitivelyClear} In the situation on the right it is intuitively clear that a deformation of $f$ with magnitude smaller than $r$ may move the curve/[[graph of a function|graph]] away from the horizontal 0-line. In terms of [[homotopy theory]] this is the statement that in this case the [[quotient]]-map  $X/gray \to \mathbb{R}^1/red$ of $f$ admits a [[homotopy]] to the [[constant function]], which shows that its [[homotopy class]] (here: a [[cohomotopy]]-class) is trivial:

> (graphics from [[schreiber:New Foundations for TDA -- Cohomotopy|SS22]])

\linebreak

## References

* {#FranekKrcal16} [[Peter Franek]], [[Marek Krčál]], _On Computability and Triviality of Well Groups_, Discrete Comput Geom **56** (2016) 126 ([arXiv:1501.03641](https://arxiv.org/abs/1501.03641), [doi:10.1007/s00454-016-9794-2](https://doi.org/10.1007/s00454-016-9794-2))

* {#FranekKrcal17} [[Peter Franek]], [[Marek Krčál]], _Persistence of Zero Sets_, Homology, Homotopy and Applications, **19** 2 (2017)  ([arXiv:1507.04310](https://arxiv.org/abs/1507.04310), [doi:10.4310/HHA.2017.v19.n2.a16](http://dx.doi.org/10.4310/HHA.2017.v19.n2.a16))

* {#FranekKrcalWagner18} [[Peter Franek]], [[Marek Krčál]], [[Hubert Wagner]], _Solving equations and optimization problems with uncertainty_, J Appl. and Comput. Topology **1** (2018) 297 ([arxiv:1607.06344](https://arxiv.org/abs/1607.06344), [doi:10.1007/s41468-017-0009-6](https://doi.org/10.1007/s41468-017-0009-6))

Review:

* [[Peter Franek]], [[Marek Krčál]], _Cohomotopy groups capture robust Properties of Zero Sets via Homotopy Theory_, talk at [ACAT meeting 2015](https://www2.ist.ac.at/acat) ([pdf slides](https://www2.ist.ac.at/fileadmin/user_upload/events_pages/acat/ACAT2015_Marek_Krcal.pdf))

* {#SS22} [[Urs Schreiber]] on joint work with [[Hisham Sati]]: *[[schreiber:New Foundations for TDA -- Cohomotopy]]*, (May 2022)

[[!redirects persistent cohomotopy theory]]

[[!redirects persistent Cohomotopy]]
[[!redirects persistent Cohomotopy theory]]


persistent homology

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Homological algebra
+--{: .hide}
[[!include homological algebra - contents]]
=--
#### Representation theory
+-- {: .hide}
[[!include representation theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{: toc}

## Idea
 {#Idea}

*Persistent homology* is the study of [[sequences]] of [[linear maps]] between [[vector spaces]]

$$
  \cdots
   \xrightarrow{\;}
  V_i
    \xrightarrow{\phi_i}
  V_{i + 1}
    \xrightarrow{\phi_{i + 1}}
  V_{i + 2}
    \xrightarrow{\phi_{i + 2}}
  \cdots
  \,,
$$

under the aspect of which elements ([[vectors]]) $v_i \in V_i$ "persist" (ie. remain non-[[zero]]) under which number of iterations of these linear maps. One then refers to these sequences as *[[persistence modules]]* (hence a [[concept with an attitude]]).

In the archetypical applications of interest, these vector spaces are [[ordinary homology|ordinary]] [[homology groups]] $V_i \,=\, H_n(X_i)$ (over a given [[ground field]]) of [[topological spaces]] $X_i$ which themselves are stages

$$
  \cdots \hookrightarrow X_i \xhookrightarrow{\iota_i} X_{i+ 1} \xhookrightarrow{\iota_{i + 1}} X_{i + 2} \xhookrightarrow{} \cdots \xhookrightarrow{x} X
$$

of a [[filtered topological space]], and one is interested in deducing "relevant" properties of this filtration by finding those [[homology]] [[cycles]] which stand out as persisting over a large range of steps.

This question, in turn, has been motivated from problems in [[topological data analysis]], where sequences of spaces $X_i$ arise as coarse-grained versions, at varying stages of resolution $i$ (say as measured in some ambient [[metric space]]) of a given [[discrete space|discrete set]] of data points. In this example, the more the homology cycles *persist* as the resolution $i$ is varied, the more one is willing to assume that they signal relevant structure that is hidden in the data, while cycles that do not persist for long would be interpreted as irrelevant noise effects.

Of course, these simple basic ideas may be (and still are being) refined and generalized in a number of ways.

A particularly important generalization turns out to be that from [[linear order|directed 1-dimensional]] sequences of linear maps to arbitrary [[zigzags]] of these, then called *[[zigzag persistence modules]]*. Again, this is naturally motivated in the case of homology groups of a [[filtered topological space]], where one may form the sequence of [[cospans]]

$$
  \cdots \leftarrow X_i \to X_i \cup X_{i + 1} \leftarrow X_{i + 1} \to X_{i + 1} \cup X_{i + 2} \leftarrow \cdots
$$

of inclusions into [[unions]] of consecutive filter stages.

Seen in this generality,  the notion of finite-length [[zigzag persistence modules]] happens to coincide with that of [[ADE classification|A-type]] [[quiver representations]], a fruitful coincidence that serves to bring well-developed tools of [[quiver]]-[[representation theory]] to bear on persistent homology theory.

In particular, the founding result of [[quiver representation theory]] -- namely  [[Gabriel's theorem]] -- serves, in hindsight, to establish the formal notion of persistence itself: The theorem implies (see [there](Gabriel's+theorem#ATypeQuivers)) that every ([[zigzag persistence|zigzag]]) [[persistence module]] is the [[direct sum]] of *interval modules* $I_{a,b}$, namely those which are [[zero]] except over the [[interval]] $i \in [a,b]$, where they are given by the [[identity]] on the [[ground field]] (the latter regarded as the [[dimension of a vector space|1-dimensional]] [[vector space]] over itself).

Hence the canonical [[linear basis]]-elements of these interval modules $I_{a,b}$ are exactly the persistent cycles which *persist from $a$ to $b$*, so that [[Gabriel's theorem]] guarantees that any (zigzag) persistence module may be completely decomposed into (the [[linear span]] of) these elements. Numerous generalizations of this theorem exist (taking it out of the context of [[quiver]]-theory) and show that this state of affairs remains true notably when one allows the indices $i$ to be taken from sets with [[infinite set|infinite]] [[cardinality]].

In other words, the [[isomorphism class]] of any (zigzag) persistence module is equivalently encoded in a [[multiset]] of intervals. These multisets are known as *[[barcodes]]* (due to the evident graphical representation of a multiset of intervals) or as *[[persistence diagrams]]* (the latter usually when $[a,b]$ is regarded as a point $(a,b) \in \mathbb{R}^2$ in the [[plane]]). These [[persistence diagrams]] are meant to be the [[invariants]] of interest in persistent homology theory.

Besides the foundational theorem that guarantees the existence of persistence diagrams, the fundamental theorem of persistent homology has come to be the *[[stability of persistence diagrams|stability theorem]]*: This says that [[persistence diagrams]] are indeed *useful invariants*, namely in that they remain "stable" under small variations (think: noise, measurement errors) of input data such as the above topological filter stages $X_i$.

For example, as one changes a little the [[metric]] with which to produce the topological space $X_i$ of coarse grained data at resolution stage $i$, the homology groups $H(X_i)$ may change dramatically, but the [[stability of persistence diagrams|stability theorem]] ensures that effect on the [[persistence diagram]] remains small.

It is in this way that persistent homology may be used, in particular, to produce robust invariants of noisy data, which has made it the archetypical tool in [[topological data analysis]].

\linebreak

## Properties
 {#Properties}

* [[stability of persistence diagrams]]

  ([Cohen-Steiner, Edelsbrunner & Harer 2007](#CohenSteinerEdelsbrunnerHarer07))

* [[diamond principle]]

  (Carlsson et al.)

## Software

One can compute intervals for homological features algorithmically over field coefficients and software packages are available for this purpose. See for instance [Perseus](http://www.sas.upenn.edu/~vnanda/perseus). The principal algorithm is based on bringing the filtered complex to its **canonical form** by upper-triangular matrices from ([Barannikov1994, §2.1] (https://www.researchgate.net/profile/Serguei_Barannikov/publication/267672645_The_Framed_Morse_complex_and_its_invariants/links/5970e947458515fa8de6e724/The-Framed-Morse-complex-and-its-invariants.pdf))

## Related concepts

* [[persistence module]]

* [[persistence diagram]]

* [[well group]]

* [[topological data analysis]]

* [[persistent homotopy theory]]

* [[computational topology]]

* [[quiver representation]], [[Gabriel's theorem]]

* [[persistence object]]

## References

### General

Introduction and survey:

* [[Robert Ghrist]], _Barcodes: The Persistent Topology of Data_, Bull. Amer. Math. Soc. 45 (2008), 61-75 ([doi:10.1090/S0273-0979-07-01191-3](https://doi.org/10.1090/S0273-0979-07-01191-3), [pdf](https://www.math.upenn.edu/~ghrist/preprints/barcodes.pdf))

* [[Herbert Edelsbrunner]], [[John Harer]], *Persistent homology -- a survey*, in: *Surveys on Discrete and Computational Geometry: Twenty Years Later*, Contemporary Mathematics *453* (2008) $[$[doi:10.1090/conm/453](http://dx.doi.org/10.1090/conm/453)$]$

* [[Gunnar Carlsson]], *Topology and data*, Bull. Amer. Math. Soc. 46 (2009), no. 2, 255-308 $[$[doi:10.1090/S0273-0979-09-01249-X](https://doi.org/10.1090/S0273-0979-09-01249-X)$]$

* [[Herbert Edelsbrunner]], [[Dmitriy Morozov]], *Persistent homology: theory and practice*, in: *European Congress of Mathematics Kraków, 2–7 July, 2012* EMS $[$[doi:10.4171/120-1/3](https://www.ems-ph.org/books/show_abstract.php?proj_nr=170&vol=1&rank=3), [pdf](http://mrzv.org/publications/persistent-homology-theory-practice/ecm)$]$

* [[Donald Pinckney]], *[Topological Data Analysis and Persistent Homology](https://donaldpinckney.com/machine%20learning/2019/05/02/tda.html)* (2019)

* [[Stefan Huber]], *Persistent Homology in Data Science* In: *Data Science – Analytics and Applications*, Springer (2021) $[$[doi:10.1007/978-3-658-32182-6_13](https://doi.org/10.1007/978-3-658-32182-6_13), [pdf](https://www.sthu.org/research/publications/files/Hub20.pdf)$]$

Review with emphasis of [[zigzag persistence]] with relation to [[quiver representation theory]]:

* [[Steve Y. Oudot]], *Persistence Theory: From Quiver Representations to Data Analysis*, Mathematical Surveys and Monographs **209** AMS (2015) $[$[pdf](https://geometrica.saclay.inria.fr/team/Steve.Oudot/books/o-pt-fqrtda-15/surv-209.pdf), [ISBN:978-1-4704-3443-4](https://bookstore.ams.org/surv-209/)$]$

* [[Gunnar Carlsson]], *Persistent Homology and Applied Homotopy Theory*, in: [[Handbook of Homotopy Theory]], CRC Press (2019) $[$[arXiv:2004.00738](https://arxiv.org/abs/2004.00738), [doi:10.1201/9781351251624](https://doi.org/10.1201/9781351251624)$]$

See also

* [[Tim Porter]], _Observing Information: Applied Computational Topology_ 2008 ([[ACT-OIT.pdf|Slides:file]])

* Bei Wang, _Topological Data Analysis_, Lecture 2010 ([[WangTopologicalDataAnalysis.pdf:file]])

* Wikipedia, _[Persistent homology](https://en.wikipedia.org/wiki/Persistent_homology)_

The concept of persistent homology originates around:

* [[Herbert Edelsbrunner]], D. Letscher, and A. Zomorodian, *Topological persistence and simplification*, Discrete & Computational Geometry, **28** (2002) 511–533 $[$[doi:10.1007/s00454-002-2885-2](https://doi.org/10.1007/s00454-002-2885-2)$]$

The generalization to "[[zigzag persistence]]" with relation to [[ADE-classification|A-type]] [[quiver representation]]-theory is due to:

* {#CarlssonDeSilva10} [[Gunnar Carlsson]], [[Vin de Silva]], *Zigzag Persistence*, Found Comput Math **10** (2010) 367–405 $[$[arXiv:0812.0197](https://arxiv.org/abs/0812.0197), [doi:10.1007/s10208-010-9066-0](https://doi.org/10.1007/s10208-010-9066-0)$]$

and specifically for [[level sets]]:

* [[Gunnar Carlsson]], [[Vin de Silva]], [[Dmitriy Morozov]], *Zigzag persistent homology and real-valued functions*, in: *SCG '09: Proceedings of the twenty-fifth annual symposium on Computational geometry (2009) 247–256 $[$[doi:10.1145/1542362.1542408](https://doi.org/10.1145/1542362.1542408)$]$

* [[Gunnar Carlsson]], [[Vin de Silva]], [[Sara Kališnik]], [[Dmitriy Morozov]], *Parametrized Homology via Zigzag Persistence*, Algebr. Geom. Topol. **19** (2019) 657-700 $[$[arXiv:1604.03596](https://arxiv.org/abs/1604.03596), [doi:10.2140/agt.2019.19.657](https://doi.org/10.2140/agt.2019.19.657)$]$

The [[stability of persistence diagrams|stability result]] is due to:

* {#CohenSteinerEdelsbrunnerHarer07} [[David Cohen-Steiner]], [[Herbert Edelsbrunner]], [[John Harer]], *Stability of Persistence Diagrams*, Discrete & Computational Geometry **37** (2007) 103–120 $[$[doi:10.1007/s00454-006-1276-5](https://doi.org/10.1007/s00454-006-1276-5)$]$

Bar-codes were introduced under the name of *canonical forms invariants of filtered complexes* in

*  Serguei Barannikov, _Framed Morse complex and its invariants_, [pdf](https://www.researchgate.net/profile/Serguei_Barannikov/publication/267672645_The_Framed_Morse_complex_and_its_invariants/links/5970e947458515fa8de6e724/The-Framed-Morse-complex-and-its-invariants.pdf) Advances in Soviet Mathematics __21__ 93–115 (1994)

See also

* A. Zomorodian, [[Gunnar Carlsson]], *Computing persistent homology*, Discrete Comput. Geom. __33__, 249–274 (2005) ([doi:10.1007/s00454-004-1146-y](https://doi.org/10.1007/s00454-004-1146-y))

* [[Gunnar Carlsson]], V. de Silva, _Zigzag persistence_ ([arXiv:0812.0197](http://arxiv.org/abs/0812.0197))

* Robert J. Adler, Omer Bobrowski, Matthew S. Borman, Eliran Subag, Shmuel Weinberger, _Persistent homology for random fields and complexes_ Institute of Mathematical Statistics Collections __6__:124–143, 2010 ([arxiv/1003.1001](http://arxiv.org/abs/1003.1001))

* [[Paweł Dłotko]], [[Hubert Wagner]], _Computing homology and persistent homology using iterated Morse decomposition_ $[$[arxiv/1210.1429](http://arxiv.org/abs/1210.1429)$]$

* [[Robert MacPherson]], Benjamin Schweinhart, _Measuring shape with topology_, J. Math. Phys. __53__, 073516 (2012) ([doi:10.1063/1.4737391](http://dx.doi.org/10.1063/1.4737391))

* Robert J. Adler, Omer Bobrowski, Shmuel Weinberger, _Crackle: the persistent homology of noise_, [arxiv/1301.1466](http://arxiv.org/abs/1301.1466)

*  D. Le Peutrec, N. Nier, C. Viterbo, _Precise Arrhenius Law for p-forms: The Witten Laplacian and Morse–Barannikov Complex_, Annales Henri Poincaré __14__ (3): 567–610 (2013) [doi](https://10.1007/s00023-012-0193-9)

* Ulrich Bauer, Michael Kerber, Jan Reininghaus, _Clear and compress: computing persistent homology in chunks_, [arxiv/1303.0477](http://arxiv.org/abs/1303.0477)

* [[Sara Kališnik]], _Persistent homology and duality_, 2013 ([pdf](http://www.matknjiz.si/doktorati/2013/Kalisnik-14521-4.pdf), [[KalisnikPersistent.pdf:file]])

* Francisco Belchí Guillamón, Aniceto Murillo Mas, _A-infinity persistence_, [arxiv/1403.2395](http://arxiv.org/abs/1403.2395)

* João Pita Costa, Mikael Vejdemo Johansson, Primož Škraba, _Variable sets over an algebra of lifetimes: a contribution of lattice theory to the study of computational topology_, [arxiv/1409.8613](http://arxiv.org/abs/1409.8613)

* Genki Kusano, Kenji Fukumizu, Yasuaki Hiraoka, _Persistence weighted Gaussian kernel for topological data analysis_, [arxiv/1601.01741](http://arxiv.org/abs/1601.01741)

* Heather A. Harrington, Nina Otter, Hal Schenck, [[Ulrike Tillmann]], _Stratifying multiparameter persistent homology_, [arxiv/1708.07390](https://arxiv.org/abs/1708.07390)

* {#BerkoukGinotOudot19} [[Nicolas Berkouk]], [[Grégory Ginot]], [[Steve Oudot]], *Level-sets persistence and sheaf theory* $[$[arXiv:1907.09759](https://arxiv.org/abs/1907.09759)$]$

* [[Wojciech Chachólski]], Alessandro De Gregorio, Nicola Quercioli, Francesca Tombari: *Symmetries of data sets and functoriality of persistent homology*, Theory and Applications of Categories **39** 23 (2023) 667-686 [[tac:39/23](http://www.tac.mta.ca/tac/volumes/39/23/39-23abs.html)]

The following paper uses persistent homology to single out features relevant for training [[neural networks]]:

* Jean-Baptiste Bardin, Gard Spreemann, [[Kathryn Hess]], _Topological exploration of artificial neuronal network dynamics_, [arxiv/1810.01747](https://arxiv.org/abs/1810.01747)

Application to [[topological data analysis]] in [[cosmological structure formation]]:

* Matteo Biagetti, [[Alex Cole]], [[Gary Shiu]], _The Persistence of Large Scale Structures I: Primordial non-Gaussianity_ ([arXiv:2009.04819](https://arxiv.org/abs/2009.04819))

Application of [[topological data analysis]] ([[persistent homology]]) to analysis of [[phase transitions]]:

* [[Alex Cole]], Gregory J. Loges, [[Gary Shiu]], _Quantitative and Interpretable Order Parameters for Phase Transitions from Persistent Homology_ ([arXiv:2009.14231](https://arxiv.org/abs/2009.14231))

[[!include cohomotopy in topological data analysis -- references]]

### Further variants

See also

* [[Robert MacPherson]], Amit Patel, _Persistent Local Systems_ ([arXiv:1805.02539](https://arxiv.org/abs/1805.02539))

category: topology, applications

[[!redirects persistent homology]]
[[!redirects persistent homology theory]]

[[!redirects persistence module]]
[[!redirects persistence modules]]


persistent homotopy

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{:toc}

## Idea
 {#Idea}

*Persistent homotopy* studies [[homotopy types]] (of [[topological spaces]]) as a parameter varies, hence [[filtered object in an (infinity,1)-category|filtered]] [[homotopy types]] (of [[filtered topological spaces]]), with focus on which elements of [[homotopy groups]] at given stage *persist* how far through the filtering to later stages.
A key application is to [[Vietoris-Rips complexes]] of discrete subsets in a [[metric space]].

Notably in [[topological data analysis]] (TDA) these VR complexes arise as "point clouds" of datapoints, and the corresponding persistent homotopy is thought to detect relevant structure hidden in such data. As such, persistent homotopy refines the traditional use of [[persistent homology]] in TDA.

In general, persistent homotopy theory is to [[persistent homology]] as  [[homotopy theory]] is to [[homology theory]]: [[homotopy]] is a finer invariant than [[homology]],  the former sees the full [[homotopy type]] of a [[topological space]], the latter at most the underlying [[stable homotopy type]].

In other words, [[homology]] involves a kind of linearization or [[abelianization]] which loses information that is retained in the homotopy type (see the [Hurewicz theorem](Hurewicz+theorem#HurewiczTheorem)). Therefore persistent homotopy is in general a finer invariant of [[filtered topological spaces]] than [[persistent homology]].  In fact, traditional [[persistent homology]] considers only *[[ordinary homology]]* which is the coarsest of all [[generalized homology]] invariants. Hence in between the coarse invariant of [[persistent homology]] and the fine invariants of persistent homotopy will be intermediate invariants that would deserve to be called *persistent generalized homology* -- but these have not yet found much attention, certainly not in the context of [[topological data analysis]].

However, besides [[homology]] there is, [[duality|dually]], also *[[cohomology]]*, whose analogous [[homotopy theory|homotopy theoretic]] refinement is ([[non-abelian cohomology|non-abelian cohomology theories]], but in particular:) *[[cohomotopy|co-homotopy]]*. The generalization of [[cohomotopy]] to the context of persistence lends itself to the analysis of persistence of [[level sets]] of [[continuous functions]]: see at *[[persistent cohomotopy]]* (and see the references [below](#CohomotopyInTopologicalDataAnalysisReferences)).

| | coarse |  intermediate | fine  |
|--|--|--|--|
| **[[homology]]** | [[ordinary homology]] | [[generalized homology]] | [[homotopy groups|homotopy]] |
| **[[cohomology]]** | [[ordinary cohomology]] | [[Whitehead-generalized cohomology|generalized cohomology]] | [[cohomotopy]] |
|  |  |  |  |
| **persistent homology** | [[persistent homology|persistent ordinary homology]] | persistent generalized homology | [[persistent homotopy]] |
| **persistent cohomology** | persistent ordinary cohomology | persistent generalized cohomology | [[persistent cohomotopy]] |

## Related concepts

* [[Vietoris-Rips complex]]

* [[homotopy theory]]

* [[persistence object]]

* [[parameterized homotopy theory]]

* [[persistent homology]]

* [[filtered topological space]]

## References

### General

Original articles with focus on establishing the homotopy-version of the [[stability of persistence diagrams|stability theorem]] and the persistent version of [[Whitehead theorem|Whitehead's theorem]]:

* {#BlumbergLesnick17} [[Andrew J. Blumberg]], [[Michael Lesnick]], *Universality of the Homotopy Interleaving Distance* $[$[arXiv:1705.01690](https://arxiv.org/abs/1705.01690)$]$

* {#Jardine19} [[J. F. Jardine]], *Data and homotopy types* $[$[arXiv:1908.06323](https://arxiv.org/abs/1908.06323)$]$

* {#LanariScoccola22} Edoardo Lanari, [[Luis Scoccola]], _Rectification of interleavings and a persistent Whitehead theorem_, Algebraic & Geometric Topology (to appear), $[$[arXiv:2010.05378] (https://arxiv.org/abs/2010.05378)$]$

Review:

* [[J. F. Jardine]], *Persistent homotopy theory* (2020) $[$[pdf](https://www.uwo.ca/math/faculty/jardine/lectures/persist-OSU-beamer.pdf), [[Jardine-PersistentHomotopy.pdf:file]]$]$

Further discussion:

* [[Facundo Mémoli]], [[Ling Zhou]], *Persistent Homotopy Groups of Metric Spaces* $[$[arXiv:1912.12399](https://arxiv.org/abs/1912.12399), [talk](Center+for+Quantum+and+Topological+Systems#ZhouAtHomotopicalPerspectivesOnTDAJune2022)$]$

Discussion with focus on the [[van Kampen theorem]], [[excision]] and the [[Hurewicz theorem]] in persistent homotopy:

* Mehmet Ali Batan, Mehmetcik Pamuk, Hanife Varli, *Persistent Homotopy* $[$[arXiv:1909.08865](https://arxiv.org/abs/1909.08865)$]$

[[!include cohomotopy in topological data analysis -- references]]

[[!redirects persistent homotopy theory]]

[[!redirects persistent homotopy type]]
[[!redirects persistent homotopy types]]


Pertti Lounesto

* [webpage](https://users.aalto.fi/~ppuska/mirror/Lounesto/)

## related $n$Lab entries

* [[Clifford algebra]]

category: people


perturbation theory

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#Contents#
* table of contents
{:toc}

## Idea

### General

_Perturbation theory_ is a general method of finding (or even defining) the solution of equations of mathematical physics by expanding them with respect to a small parameter in the vicinity of known, defined or well-understood solution (for which the small parameter is $0$). It is used in the study of PDEs involving operators depending on small parameter, in classical and celestical mechanics, in quantum mechanics, and in the statistical and [[perturbative quantum field theory]].

One of the varieties of perturbation theory provides a method to make sense of and handle the [[path integral]] involved in the [[quantization]] of [[classical field theory]] to [[quantum field theory]].

It is based on the observation that the [[quantization]] of free [[classical field theory|classical field theories]], whose [[action functional]] contains only the kinetic term, is well understood; therefore, the quantization of a functional consisting of a kinetic term and polynomial interaction terms may be expanded like a [[Taylor series]] in the interaction terms, thus yielding what looks like a series of [[correlator]]s in a free field theory.  If the **coupling constant** -- the parameter in front of the interaction terms -- is small enough, one says one is in the _weakly coupled regime_ of the theory and expects this perturbation series to approximate the desired answer. Usually, even for that to work the [[action functional]] first has to be subjected to [[renormalization]].

### More details

Suppose we're working with a quantum system that's nearly a [[quantum harmonic oscillator]], but not quite; that is, the quadratic potential $V_0 = \frac{1}{2}k x^2 - \frac{1}{2}$ is only a good local approximation to the real potential $V_0 + \lambda V$.  Then we can write the [[Hamiltonian]] as $H = H_0 + \lambda V,$ where $V$ is a function of the [[position]] $x$ and the [[momentum]] $p$ (or equivalently, of $z = \frac{p+i x}{\sqrt{2}}$ and $\frac{d}{dz}$) and $\lambda$ is small.

Now we solve Schrödinger's equation perturbatively.  We know that
\[\psi(t) = e^{-itH} \psi(0),\]
and we assume that
\[e^{-itH}\psi(t) \approx e^{-itH_0} \psi(t)\]
so that it makes sense to solve it perturbatively.  Define
\[\psi_1(t) = e^{itH_0} e^{-itH}\psi(t)\]
and
\[V_1(t) = e^{itH_0} \lambda V e^{-itH_0}.\]
After a little work, we find that
\[\frac{d}{dt}\psi_1(t) = -i V_1(t) \psi_1(t),\]
and integrating, we get
\[\psi_1(t) = -i\int_0^t V_1(t_0) \psi_1(t_0) dt_0 + \psi(0).\]
We feed this equation back into itself recursively to get
\[\array{ \psi_1(t) & = & -i \int_0^t V_1(t_0) \left[-i\int_0^{t_0} V_1(t_1) \psi_1(t_1) dt_1 + \psi(0) \right] dt_0 + \psi(0) \\ & = & \left[\psi(0)\right] + \left[\int_0^t i^{-1} V_1(t_0)\psi(0) dt_0\right] + \left[\int_0^t\int_0^{t_0} i^{-2} V_1(t_0)V_1(t_1) \psi_1(t_1) dt_1 dt_0\right] \\ & = & \sum_{n=0}^{\infty} \int_{t \ge t_0 \ge \ldots \ge t_{n-1} \ge 0} i^{-n} V_1(t_0)\cdots V_1(t_{n-1}) \psi(0) dt_{n-1}\cdots dt_0 \\ & = & \sum_{n=0}^{\infty} (-\lambda i)^n \int_{t \ge t_0 \ge \ldots \ge t_{n-1} \ge 0} e^{-i(t-t_0)H_0} V e^{-i(t_0-t_1)H_0} V \cdots V e^{-i(t_{n-1}-0)H_0} \psi(0) dt_{n-1}\cdots dt_0. }\]
So here we have a sum of a bunch of terms; the $n$th term involves $n$ interactions with the potential interspersed with evolving freely between the interactions, and we integrate over all possible times at which those interactions could occur.

Here's an example [[Feynman diagram]] for this simple system, representing the fourth term in the sum above:

The lines represent evolving under the free Hamiltonian $H_0$, while the dots are interactions with the potential $V$.

As an example, let's consider $V = (z + \frac{d}{dz})$ and choose $\lambda = \frac{1}{\sqrt{2}}$ so that $\lambda V = p.$  When $V$ acts on a state $\psi = z^n,$ we get $V \psi = z^{n+1} + nz^{n-1}.$  So at each interaction, the system either gains a photon or changes phase and loses a photon.

## Properties

### Divergence/Convergence
 {#DivergenceConvergence}

Despite what one might naively expect, the [[perturbation series]] of natural [[quantum field theories]] have a vanishing [[radius of convergence]], they are [[asymptotic series]].

Roughly this can be understood as follows: since the pertrubation is in the [[coupling constant]] about vanishing coupling, a non-zero [[radius of convergence]] would imply that the theory is finite also for _negative_ coupling (where "things fly apart"), which will not happen in realistic theories.

More in detail, [[theory (physics)|theories]] with [[non-perturbative effects]] such as [[instantons]] [[field (physics)|field]] configurations (such as [[Yang-Mills theory]], hence [[QCD]], [[QED]]), [[branes]] (such as [[string theory]]), etc., are expected to have a  [[path integral]] which as a [[function]] of the [[coupling constant]] $g$ schematically looks like

$$
  Z(g)
  =
  \sum_n a_n g^n
  +
  e^{-A/g}
  \sum_n a_n^{(1)} g^n
  +
  \mathcal{O}(e^{-2A/g})
  \,,
$$

where the first sum is the [[perturbation series]] itself and where the terms with a prefactor of the form $\exp(-A/g)$ are the contributions of the [[instantons]] ($A$ is the contribution of the [[instanton]] [[action functional]]). Since all the [[derivatives]] of the function $g \mapsto e^{-1/g}$ vanish at [[coupling constant]] $g = 0$, the [[Taylor series]] of this part of the path integral does not appear in [[perturbation series]], even though it is present. Therefore this is called a _[[non-perturbative effect]]_.

See the [references below](#ReferencesDivergenceConvergence) for details. The mathematics behind this is called _[[resurgence theory]]_.

(See also at _[string theory FAQ -- Isn't it fatal that the string perturbation series does not converge?](string+theory+FAQ#NonConvergenceOfPerturbationSeries)_.)

## Related concepts

* [[perturbative quantum field theory]]

  * [[causal perturbation theory]], [[perturbative algebraic quantum field theory]]

  * [[renormalization]]

  * [[Feynman diagram]]

  * [[scattering amplitude]], [[string scattering amplitude]]

  * [[vacuum]], [[tachyon]]

* [[non-perturbative quantum field theory]]

  * [[non-perturbative effect]]

  * [[soliton]], [[instanton]]

  * [[Borel resummation]], [[resurgence theory]]

* [string theory FAQ -- Isn't it fatal that the string perturbation series does not converge?](string%20theory%20FAQ#NonConvergenceOfPerturbationSeries)

## References

### General

A solid mathematical formulation of perturbation theory has been given in

* K. Hepp.: _Théorie de la Renormalisation_ Lect. Notes in Phys. Springer (1969)

* O. Steinmann, _Perturbation expansion in axiomatic field theory_ Lect. Notes in Phys. 11, Springer (1971)

### On Divergence/Convergence and Non-perturbative effects
 {#ReferencesDivergenceConvergence}

The argument that the perturbation series of realistic [[quantum field theories]] such as [[QED]] necessarily diverges goes back to

* {#Dyson52} [[Freeman Dyson]], _Divergence of perturbation theory in quantum electrodynamics_, Phys. Rev. 85, 631, 1952 ([spire](http://inspirehep.net/record/29799?ln=en))

  **Abstract**: _An argument is presented which leads tentatively to the conclusion that all the power-series expansions currently in use in quantum electrodynamics are divergent after the renormalization of mass and charge. The divergence in no way restricts the accuracy of practical calculations that can be made with the theory, but raises important questions of principle concerning the nature of the physical concepts upon which the theory is built._

and is made more precise in

* {#Lipatov77} [[Lev Lipatov]], _Divergence of the Perturbation Theory Series and the Quasiclassical Theory_, Sov.Phys.JETP 45 (1977) 216–223 ([pdf](http://jetp.ac.ru/cgi-bin/dn/e_045_02_0216.pdf))

recalled for instance in

* {#Suslov05} [[Igor Suslov]], section 1 of _Divergent perturbation series_, Zh.Eksp.Teor.Fiz. 127 (2005) 1350; J.Exp.Theor.Phys. 100 (2005) 1188 ([arXiv:hep-ph/0510142](https://arxiv.org/abs/hep-ph/0510142))

* Justin Bond, last section of _Perturbative QFT is Asymptotic; is Divergent; is Problematic in Principle_ ([pdf](https://mcgreevy.physics.ucsd.edu/s13/final-papers/2013S-215C-Bond-Justin.pdf))

* {#BakulevShirkov10} Alexander P. Bakulev, [[Dmitry Shirkov]], section 1.1 of _Inevitability and Importance of Non-Perturbative Elements in Quantum Field Theory_, Proceedings of the 6th Mathematical Physics Meeting, Sept. 14--23, 2010, Belgrade, Serbia (ISBN 978-86-82441-30-4), pp. 27--54 ([arXiv:1102.2380](https://arxiv.org/abs/1102.2380))

* {#HollandsWald14} [[Stefan Hollands]], [[Robert Wald]], section 4.1 of _Quantum fields in curved spacetime_, Physics Reports Volume 574, 16 April 2015, Pages 1-35 ([arXiv:1401.2026](https://arxiv.org/abs/1401.2026))

* Marco Serone, from 2:46 on in _A look at $\phi^4_2$ using perturbation theory_ ([recording](https://www.youtube.com/watch?v=J4nxvY1rOhI))

Exposition also in:

* [[Jakob Schwichtenberg]], *[divergence-perturbation-series-qft](http://jakobschwichtenberg.com/divergence-perturbation-series-qft)*

For the example of [[phi^4 theory]] this non-convergence of the perturbation series is discussed in

* {#Helling} Robert Helling, p. 4 of _Solving classical field equations_ ([pdf](http://homepages.physik.uni-muenchen.de/~helling/classical_fields.pdf))

For the example of [[phi^4 theory]] this non-convergence of the perturbation series is discussed in

* {#Helling} Robert Helling, p. 4 of _Solving classical field equations_ ([pdf](http://homepages.physik.uni-muenchen.de/~helling/classical_fields.pdf))

A general introduction on divergence of perturbation theory, asymptotic series and [[non-perturbative effects]] is for instance on the first pages of

* [[Marcos Marino]], _Lectures on non-perturbative effects in large N gauge theories, matrix models and strings_ ([arXiv:1206.6272](http://arxiv.org/abs/1206.6272))

See also

* Carl M. Bender, Carlo Heissenberg, _Convergent and Divergent Series in Physics_ ([arXiv:1703.05164](https://arxiv.org/abs/1703.05164))

Further discussion is for instance in

* Physics.Stack exchange discussion, _[Is QCD free from all divergences?](http://physics.stackexchange.com/questions/30105/is-qcd-free-from-all-divergences)_

### In AQFT
 {#ReferencesInAQFT}

Perturbation theory in the spirit of [[AQFT]], namely in [[locally covariant perturbative quantum field theory]] is discussed in the following articles.

The observation that in [[perturbation theory]] the [[renormalization|Stückelberg-Bogoliubov-Epstein-Glaser]] local [[S-matrix|S-matrices]] yield a [[local net of observables]] was first made in

* V. Il'in, D. Slavnov, _Observable algebras in the S-matrix approach_ Theor. Math. Phys. **36** , 32 (1978) 578-585

which was however mostly ignored and forgotten. It is taken up again in

* [[Romeo Brunetti]], [[Klaus Fredenhagen]], _Microlocal Analysis and Interacting Quantum Field Theories: Renormalization on Physical Backgrounds_  Commun. Math. Phys. 208:623-661 (2000) ([arXiv:math-ph/9903028](http://arxiv.org/abs/math-ph/9903028))

(a quick survey is in section 8, details are in section 2).

Further developments along these lines are then

* [[Michael Dütsch]], [[Klaus Fredenhagen]], _Algebraic quantum field theory, perturbation theory, and the loop expansion_,
Commun. Math. Phys. 219:5-30 (2001) ([arXiv:hep-th/0001129](http://xxx.uni-augsburg.de/abs/hep-th/0001129))

* [[Michael Dütsch]], [[Klaus Fredenhagen]], _Perturbative algebraic quantum field theory and deformation quantization_,
Proceedings of the Conference on Mathematical Physics in Mathematics and Physics, Siena June 20-25 (2000) ([arXiv:hep-th/0101079](http://xxx.uni-augsburg.de/abs/hep-th/0101079))

(relation to [[deformation quantization]])

* [[Romeo Brunetti]], [[Michael Dütsch]], [[Klaus Fredenhagen]], _Perturbative Algebraic Quantum Field Theory and the Renormalization Groups_, Adv. Theor. Math. Physics 13 (2009), 1541-1599 ([arXiv:0901.2038](http://arxiv.org/abs/0901.2038))

(relation to [[renormalization]])

* [[Michael Dütsch]], [[Klaus Fredenhagen]], _A local (perturbative) construction of observables in gauge theories: the example of qed_, Commun. Math. Phys. 203 (1999), no.1, 71-105,  ([arXiv:hep-th/9807078](http://xxx.uni-augsburg.de/abs/hep-th/9807078)).

(relation to [[gauge theory]] and [[QED]])

Reviews includes

* [[Klaus Fredenhagen]], [[Katarzyna Rejzner]], _Perturbative algebraic quantum field theory_ ([arXiv:1208.1428](http://arxiv.org/abs/1208.1428))

* [[Klaus Fredenhagen]], [[Katarzyna Rejzner]], _Perturbative Construction of Models of Algebraic Quantum Field Theory_ ([arXiv:1503.07814](http://arxiv.org/abs/1503.07814))

and a textbook acount is in

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

Further developments in perturbation theory in [[AQFT on curved spacetimes]] icludes

* {#KhavkineMoretti16} [[Igor Khavkine]], [[Valter Moretti]], _Analytic Dependence is an Unnecessary Requirement in Renormalization of Locally Covariant QFT_, Communications in Mathematical Physics, March 2016 ([arXiv:1411.1302](http://arxiv.org/abs/1411.1302), [publisher](http://link.springer.com/article/10.1007%2Fs00220-016-2618-7))

### In BV-BRST formalism

Perturbative quantization in [[BV-BRST formalism]] is nicely systematically discussed in section 5 of

* [[Kevin Costello]], [[Owen Gwilliam]], _Factorization algebras in quantum field theory_ ([pdf](http://www.math.northwestern.edu/~costello/factorization.pdf))

in the broad context of _[[factorization algebras]]_ (see there for further references). In particular the relation to Feynman diagrams is discussed in

* [[Owen Gwilliam]], [[Theo Johnson-Freyd]], _How to derive Feynman diagrams for finite-dimensional integrals directly from the BV formalism_ (2011) ([arXiv:1202.1554](http://arxiv.org/abs/1202.1554))

[[!redirects perturbation theories]]

[[!redirects perturbation series]]

[[!redirects perturbation]]
[[!redirects perturbations]]


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=--
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This page collects material related to the textbook

* [[Katarzyna Rejzner]],

  _Perturbative Algebraic Quantum Field Theory_,

  Mathematical Physics Studies, Springer 2016

  ([web](https://link.springer.com/book/10.1007%2F978-3-319-25901-7))

on [[perturbative quantum field theory]] in its mathematically precise formulation known as _[[perturbative algebraic quantum field theory]]_, based on [[causal perturbation theory]].

## Related textbooks

* [[Günter Scharf]], _[[Finite Quantum Electrodynamics -- The Causal Approach]]_, Springer 1995

* [[Günter Scharf]], _[[Quantum Gauge Theories -- A True Ghost Story]]_, Wiley 2001

* [[Urs Schreiber]], _[[geometry of physics -- perturbative quantum field theory|Perturbative quantum field theory]]_, 2017

* [[Michael Dütsch]], _[[From classical field theory to perturbative quantum field theory]]_, 2018

category: reference