F-branes -- table

**from [[M-branes]] to F-branes: [[superstrings]], [[D-branes]] and [[NS5-branes]]**

| **[[M-theory]]** on $S^1_A \times S^1_B$-[[elliptic fibration]] | [[KK-compactification]] on $S^1_A$ | **[[type IIA string theory]]** | [[T-duality|T-dual]] [[KK-compactification]] on $S^1_B$ | **[[type IIB string theory]]** | geometrize the [[axio-dilaton]] | **[[F-theory]]** on [[elliptic fibration|elliptically fibered]]-[[K3]] fibration | [[duality between F-theory and heterotic string theory]] | **[[heterotic string theory]]** on [[elliptic fibration]] |
|--|--|--|--|--|--|--|--|--|
| [[M2-brane]] [[wrapped brane|wrapping]] $S_A^1$ | [[double dimensional reduction]] $\mapsto$ | [[type IIA superstring]] | $\mapsto$ | [[type IIB superstring]] | $\mapsto$ |" | $\mapsto$ | [[heterotic superstring]] |
| [[M2-brane]] [[wrapped brane|wrapping]] $S_B^1$ | $\mapsto$ | [[D2-brane]] | $\mapsto$ | [[D1-brane]] | $\mapsto$ |" | |  |
| [[M2-brane]] [[wrapped brane|wrapping]] $p$ times around $S_A^1$ and $q$ times around $S_B^1$ | $\mapsto$ | $p$ [[strings]] and $q$ [[D2-branes]] | $\mapsto$ | [[(p,q)-string]] | $\mapsto$ | " |  | |
| [[M5-brane]] [[wrapped brane|wrapping]] $S_A^1$ | [[double dimensional reduction]] $\mapsto$ | [[D4-brane]] | $\mapsto$ | [[D5-brane]] | $\mapsto$ | " |  |  |
| [[M5-brane]] [[wrapped brane|wrapping]] $S_B^1$ | $\mapsto$ | [[NS5-brane]] | $\mapsto$ | [[NS5-brane]] | $\mapsto$ | " | $\mapsto$ | [[NS5-brane]] |
| [[M5-brane]] [[wrapped brane|wrapping]] $p$ times around $S_A^1$ and $q$ times around $S_B^1$ | $\mapsto$ | $p$ [[D4-brane]] and $q$ [[NS5-branes]] | $\mapsto$ | [[(p,q)5-brane]] | $\mapsto$ |"  |  | |
| [[M5-brane]] [[wrapped brane|wrapping]] $S_A^1 \times S_B^1$ | $\mapsto$ |  | $\mapsto$ | [[D3-brane]] | $\mapsto$ | "|  |  |
| [[KK-monopole]]/[[ADE singularity|A-type ADE singularity]] (degeneration locus of $S^1_A$-circle fibration, [[Sen limit]] of $S^1_A \times S^1_B$ [[elliptic fibration]]) | $\mapsto$ | [[D6-brane]] | $\mapsto$ | [[D7-branes]] | $\mapsto$ | A-type [[nodal curve]] cycle degeneration locus of [[elliptic fibration]]  ([Sen 97, section 2](F-theory#Sen97b)) | | [[special unitary group|SU]]-[[M-theory lift of gauge enhancement on D6-branes|gauge enhancement]] |
| [[KK-monopole]] [[orientifold]]/[[ADE singularity|D-type ADE singularity]]  | $\mapsto$ | [[D6-brane]] with [[O6-planes]] | $\mapsto$ | [[D7-branes]] with [[O7-planes]] | $\mapsto$ | D-type [[nodal curve]] cycle degeneration locus of [[elliptic fibration]]  ([Sen 97, section 3](F-theory#Sen97b)) | | [[special orthogonal group|SO]]-[[M-theory lift of gauge enhancement on D6-branes|gauge enhancement]] |
|  exceptional [[ADE-singularity]] | $\mapsto$ |  |  $\mapsto$ |  | $\mapsto$ | [exceptional ADE-singularity of elliptic fibration](elliptic+fibration#ClassificationOfSingularFibers) | $\mapsto$ | [[E6]]-, [[E7]]-, [[E8]]-[[gauge enhancement]]  |

(e.g. [Johnson 97](F-theory#Johnson97), [Blumenhagen 10](F-theory#Blumenhagen10))


F-category

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# $\mathcal{F}$-categories
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## Idea

An **$\mathcal{F}$-category** is like a [[2-category]], but with two types of 1-morphism, one of which we think of as "stricter" than the other.  The stricter morphisms are called **tight** and the less strict ones are called **loose**.

## Definition

### Strict $\mathcal{F}$-categories

Let $\mathcal{F}$ denote the category whose [[objects]] are [[functors]] that are [[fully faithful functors|fully faithful]] and injective on objects, and whose [[morphisms]] are [[commutative squares]] (a [[full subcategory]] of the [[arrow category]] of [[Cat]]).  We call the objects of $\mathcal{F}$, for the nonce, _full embeddings_.  Then $\mathcal{F}$ is [[cartesian closed category|cartesian closed]], [[complete category|complete]] and [[cocomplete category|cocomplete]], hence a [[Benabou cosmos]].

A **strict $\mathcal{F}$-category** is a [[enriched category|category enriched over]] $\mathcal{F}$.  Therefore, between every two objects, an $\mathcal{F}$-category $K$ has an object $K(x,y)\in \mathcal{F}$, hence a full embedding $K(x,y)_\tau \to K(x,y)_\lambda$.  The objects of $K(x,y)_\tau$ are called **tight morphisms** $x\to y$, and the objects of $K(x,y)_\lambda$ are called **loose morphisms** $x\rightsquigarrow y$.

Since full embeddings are injective on objects, "being tight" is a [[stuff, structure, property|property]] of a loose morphism.  (This would still be true in the "up to unique isomorphism" sense even if we did not ask for injectivity on objects, but when dealing with strict things, it is easier to keep them as strict as possible.)  And since full embeddings are fully faithful, the 2-cells between two tight morphisms are the same whether we regard them as tight or as loose.

For any $\mathcal{F}$-category $K$, the objects, tight morphisms, and 2-cells form a strict 2-category $K_\tau$, and the objects, loose morphisms, and 2-cells form a strict 2-category $K_\lambda$.  There is an obvious strict 2-functor
$$K_\tau \to K_\lambda$$
which is the identity on objects, strictly [[faithful functor|faithful]] on 1-morphisms, and [[locally fully faithful 2-functor|locally fully faithful]].  Since $K$ can be recovered from this 2-functor, an equivalent definition of a strict $\mathcal{F}$-category is as a strict 2-functor with these properties.

Strict $\mathcal{F}$-categories are equivalent to [[double categories]] with [[companions]].

### Weak $\mathcal{F}$-categories

Probably the best "fully weak" version of $\mathcal{F}$-categories is obtained by redefining $\mathcal{F}$ to consist of fully faithful functors, with squares that commute up to specified isomorphism, and then by considering $\mathcal{F}$-[[enriched bicategories]] rather than enriched categories.  Such a thing would be equivalent to an identity-on-objects and locally-fully-faithful pseudofunctor between bicategories.

One could consider semi-strict versions as well, in which (for example) the tight morphisms form a strict 2-category.

## Examples

* Any [[proarrow equipment]] is an $\mathcal{F}$-category (perhaps weak, perhaps semi-strict).

* An example of a semi-strict $\mathcal{F}$-category is the [[localization]] 2-functor $Cat(S) \to Cat(S)[W^{-1}]$ for a class of [[weak equivalences]] $W$.

* For any ([[strict 2-monad|strict]]) [[2-monad]] $T$, there are strict $\mathcal{F}$-categories of $T$-algebras whose tight and loose morphisms are, respectively:

  1. strict and pseudo $T$-morphisms
  1. strict and [[lax morphism|lax]] $T$-morphisms
  1. strict and colax $T$-morphisms
  1. pseudo and lax $T$-morphisms
  1. pseudo and colax $T$-morphisms

  In fact, this can be generalized to any $\mathcal{F}$-monad on an $\mathcal{F}$-category.

* Any 2-category gives rise to two $\mathcal{F}$-categories:

  * In a **chordate** $\mathcal{F}$-category, all morphisms are tight.
  * In an **inchordate** $\mathcal{F}$-category, only identities are tight.

* The [[lax slice 2-category]] is an $\mathcal{F}$-category whose tight 2-category is the (pseudo) slice 2-category.  This $\mathcal{F}$-category allows a definition of [[fibration in a 2-category|fibrations]] using [[lax F-adjunctions]].

* $\mathcal{F}$ itself becomes an $\mathcal{F}$-category in the usual way.  Its tight morphisms are just the morphisms in the underlying ordinary category $\mathcal{F}$, while its loose morphisms are simply functors between the loose parts (the codomains of the full embeddings).

## $\mathcal{F}$-weighted limits

The general machinery of [[enriched category]] theory applied to $\mathcal{F}$ gives us a notion of [[weighted limit]].  Note first that an $\mathcal{F}$-enriched *diagram* in an $\mathcal{F}$-category is a diagram of morphisms in which some are required to be tight, and others are not (but could "accidentally" be tight).

In general, a weighted limit of such a diagram in a (strict) $\mathcal{F}$-category is a weighted (strict) [[2-limit]] in its 2-category of loose morphisms, with the property that certain specified projections from the limit object are tight and "jointly detect tightness", in the sense that a morphism into the limit is tight if and only if its composites with all of the specified projections are tight.  Details and examples can be found in ([LS](#LS)).

One of the most important things about $\mathcal{F}$-categories is that they allow us to define the classes of [[rigged limits]], which are the $\mathcal{F}$-weighted limits that are [[created limit|created]] by the forgetful functors from the various $\mathcal{F}$-categories of algebras and strict/pseudo/lax/colax morphisms over a 2-monad (or an $\mathcal{F}$-monad).

## Related pages

* [[rigged limit]]

* [[F-functor]]

* [[lax F-natural transformation]]

* [[lax F-adjunction]]

The [[1-category]]-theoretic version:

* [[M-categories]] (cf. [[relative categories]]$\;$[as enriched categories](relative+category#AsEnrichedCategories))

## References

$\mathcal{F}$-categories were introduced in the following, where they were called **enhanced 2-categories**:

* {#LS} [[Stephen Lack]] and [[Mike Shulman]], "Enhanced 2-categories and limits for lax morphisms", [arXiv](http://arxiv.org/abs/1104.2111).

[[!redirects F-category]]
[[!redirects F-categories]]
[[!redirects strict F-category]]
[[!redirects strict F-categories]]
[[!redirects enhanced 2-category]]
[[!redirects enhanced 2-categories]]


F-functor

# $\mathcal{F}$-functors

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## Idea

A [[functor]] between [[F-categories]].

[[!redirects F-functors]]


f-invariant

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#### Elliptic cohomology
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#Contents#
* table of contents
{:toc}

## Idea

The _$f$-invariant_ ([Laures 99](#Laures99)) is the third in a sequence of [[homotopy invariants]] of "stable maps", i.e. of [[morphisms]] in the [[stable homotopy category]] (in particular: of [[stable homotopy groups of spheres]]), being elements of [[Ext-groups]] between the [[homology groups]]/[[cohomology groups]] of the [[domain]] and [[codomain]] of the map, with respect to some suitable choice of [[Whitehead-generalized cohomology theory]] $E$.

The previous two invariants in the sequence are the _[[d-invariant]]_ and the _[[e-invariant]]_. All these are elements that appear on the second page of the $E$-[[Adams spectral sequence]].

In higher analogy to how the [[e-invariant]] exists if the [[d-invariant]] vanishes and then makes sense for $E = KU$ ([[complex topological K-theory]]), so the f-invariant exists when the [[e-invariant]] vanishes and then makes sense of $E$ an [[elliptic cohomology theory]].

Moreover, in higher analogy to how the [[e-invariant]], when expressed in terms of [[MU]]-[[bordism theory]], computes the [[Todd genus]]/[[A-hat genus]] of a [[manifold with corners]] (see [there](Adams+e-invariant#AsACobordismInvariantOfManifoldsWithBoundary)), so the f-invariant computes an [[elliptic genus]] of a [[manifold with corners]] ([Laures 00](#Laures00)).

(...)

Let $X \overset{\phi}{\longrightarrow} Y$ be morphism in the [[stable homotopy category]] out of a [[finite spectrum]] $X$ (for instance the image under [[suspension]] $\Sigma^\infty$ of a morphism in the [[classical homotopy category]] of [[pointed homotopy types]] out of a [[finite CW-complex]]).

If $E$ be a [[multiplicative cohomology theory]] satisfying the flatness assumptions used in the [[Adams spectral sequence]], and such that the [[e-invariant]] of $\phi$ in $E$ vanishes. Then the _$f$-invariant_ of $\phi$ ([Laures 99](#Laures99)) is a certain element in the second [[Ext-group]] $Ext^2_{E_\bullet(E)}\big( E_\bullet(X), E_\bullet(Y) \big)$.

(...)

## Related concepts

* [[Adams spectral sequence]]

* [[d-invariant]], [[e-invariant]]

* [[manifold with corners]]

* [[MUFr]]

## References

The concept is due to

* {#Laures99} Gerd Laures, _The topological $q$-expansion principle_, Topology Volume 38, Issue 2, March 1999, Pages 387-425 (doi:10.1016/S0040-9383(98)00019-6)

* {#Laures00} Gerd Laures, _On cobordism of manifolds with corners_, Trans. Amer. Math. Soc. 352 (2000) ([doi:10.1090/S0002-9947-00-02676-3](https://doi.org/10.1090/S0002-9947-00-02676-3))

  > (see [Genauer 12](manifold+with+boundary#Genauer12) for more on the bordism classes of [[manifolds with corners]])

Further discussion in:

* Hanno von Bodecker, _On the geometry of the f-invariant_ ([arXiv:0808.0428](https://arxiv.org/abs/0808.0428))

* [[Mark Behrens]], Gerd Laures, _$\beta$-family congruences and the f-invariant_ ([arXiv:0809.1125](https://arxiv.org/abs/0809.1125))

* [[Ulrich Bunke]], [[Niko Naumann]], _The f-invariant and index theory_, Manuscripta Math. 132, 365–397 (2010) ([arXiv:0808.0257](https://arxiv.org/abs/0808.0257), [https://doi.org/10.1007/s00229-010-0351-7](https://doi.org/10.1007/s00229-010-0351-7))

* Gerd Laures, _Toda brackets and congruences of modular forms_ ([arXiv:1102.3783](https://arxiv.org/abs/1102.3783), [euclid:agt/1513715257](https://projecteuclid.org/euclid.agt/1513715257))

See also:

* [[Lennart Meier]], [MO comment](https://mathoverflow.net/a/179639/381), 2014


F-norm

# F-norms
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{: toc}

## Idea

An F-norm is a non-homogeneous variant of a [[norm]]: a translation-invariant [[metric]] on a [[vector space]] that satisfies properties in between being a [[G-norm]] (on the underlying [[abelian group]] of the vector space) and being a norm.  As with norms, there is a semi- variant.

## Definitions

Let $K$ be a [[topological field]] (typically the [[real numbers]] or the [[complex numbers]], but conceivably only a [[topological ring]], or at least a [[commutative ring|commutative]] one); we will call the elements of $K$ _scalars_.  Let $V$ be a [[vector space]] (or [[module over a ring|module]]) over $K$; we will call the elements of $V$ _vectors_.  Let $\|{-}\|$ be a [[function]] from (the underlying set of) $V$ to the set of [[real numbers]].

+-- {: .num_defn #Gseminorm}
###### Definition

If

1. ${\|0_V\|} = 0$ (or even just ${\|0\|} \leq 0$),
2. ${\|{-x}\|} = {\|x\|}$ (or even just ${\|{-x}\|} \leq {\|x\|}$) for each vector $x$, and
3. ${\|x + y\|} \leq {\|x\|} + {\|y\|}$ for each vector $x$ and vector $y$ (the [[triangle inequality]]),

then ${\|{-}\|}$ is a __[[G-seminorm]]__.
=--

This is enough to prove that ${\|x\|} \geq 0$ for each $x$ in $V$, making $(x,y) \mapsto {\|y - x\|}$ (precisely) a translation-invariant [[pseudometric]] on $V$.

Note that addition $(x,y) \mapsto x + y\colon V \times V \to V$ is a [[short map]] under this pseudometric and so certainly [[continuous map|continuous]].

+-- {: .num_defn #Fseminorm}
###### Definition

If

1. $\|{-}\|$ is a G-seminorm and
2. scalar multiplication $(a,x) \mapsto a x\colon K \times V \to V$ is continuous (relative to the topology on $K$ and the pseudometric on $V$),

then $\|{-}\|$ is an __F-seminorm__.
=--

If the topology on $K$ is given by an [[absolute value]] $|{-}|$, then we can go further:

+-- {: .num_defn #seminorm}
###### Definition

If

1. ${\|x + y\|} \leq {\|x\|} + {\|y\|}$ for each vector $x$ and vector $y$ and
2. ${\|a x\|} = {|a|} {\|x\|}$ for each scalar $a$ and vector $x$,

then $\|{-}\|$ is a __[[seminorm]]__.
=--

Every seminorm is automatically an F-seminorm.

No longer assuming anything further about $K$, there are some subsidiary definitions:

+-- {: .num_defn #Fnorm}
###### Definition

If

1. $\|{-}\|$ is an F-seminorm and
2. $x = 0_V$ whenever $x$ is a vector and ${\|x\|} = 0$,

then $\|{-}\|$ is an __F-norm__.
=--

Thus an F-norm is precisely an F-seminorm whose induced pseudometric is a [[metric]].  (Compare the relationship between [[G-norms]] and [[norms]] with G-seminorms in \ref{Gseminorm} and seminorms in \ref{seminorm} above.)

+-- {: .num_defn #Fspace}
###### Definition

If

1. $\|{-}\|$ is an F-norm and
2. $x$ converges under the metric on $V$ whenever $x$ is a [[net]] of vectors and $\lim_{i,j} {\|x_j - x_i\|} = 0$ in $\mathbb{R}$,

then $(V,{\|{-}\|})$ is an __F-space__.
=--

In other words, an F-space is a vector space equipped with an F-norm whose induced metric is [[complete metric|complete]] (or equivalently such that the topology on $V$ is [[complete topological group|complete]]).

+-- {: .num_defn #Frechetspace}
###### Definition

If

1. $(V,{\|{-}\|})$ is an F-space and
2. $(V,{\|{-}\|})$ is [[locally convex space|locally convex]] as a topological vector space,

then $(V,{\|{-}\|})$ is a __[[Fréchet space]]__.
=--

Finally, $F Sp$ is the [[category]] whose [[objects]] are F-spaces and whose morphisms are [[short linear maps]]; that said, often people really study the [[essential image]] of that category within the category of [[topological vector spaces]], or [[equivalence of categories|equivalently]] the category whose objects are F-spaces and whose morphisms are [[continuous map|continuous]] linear maps.  (This is especially so with Fréchet spaces, which have a common alternative definition that makes no reference to a canonical metric.)

## Examples

The usual examples of F-spaces that are not Fréchet spaces are the [[Lebesgue spaces]] $l^p$ for $p \lt 0 \lt 1$.  These use a modified $p$-[[p-norm|norm]] in which
$$ {\|x\|_p} = \sum_i {|x_i|^p} $$
(so without the $p$th root) to ensure that the [[triangle inequality]] (\ref{Gseminorm}.3) holds.

## Properties

The uniqueness theorem for [[complete space|complete]] [[norms]] in [[dream mathematics]] applies also to F-norms: assuming [[excluded middle]], [[dependent choice]], and the (classically false) [[Borel property]], two complete F-norms on a given [[vector space]] over the [[real numbers]] must be [[topological equivalence|topologically equivalent]].  See [[norm#dreamUnique]].

category: analysis

[[!redirects F-space]]
[[!redirects F-spaces]]
[[!redirects F space]]
[[!redirects F spaces]]
[[!redirects Fspace]]
[[!redirects Fspaces]]

[[!redirects F-norm]]
[[!redirects F-norms]]
[[!redirects F norm]]
[[!redirects F norms]]
[[!redirects Fnorm]]
[[!redirects Fnorms]]

[[!redirects F-seminorm]]
[[!redirects F-seminorms]]
[[!redirects F seminorm]]
[[!redirects F seminorms]]
[[!redirects Fseminorm]]
[[!redirects Fseminorms]]


F-star-algebra

#Contents#
* table of contents
{:toc}

## Idea

There are several natural generalizations of the notion of [[C-star-algebra]] whic allow the [[linear operators]] involved to be not necessarily [[bounded operators]]. (See [Bagarello](#Bagarello) for a list of literature.) Some variant of these are called _F-star-algebras_, short for _Fréchet-[[star algebra]]_.

## References

* F. Bagarello, _Applications of Topological *-Algebras of Unbounded Operators_ ([arXiv:math/9803133](http://arxiv.org/abs/math/9803133))
 {#Bagarello}

[[!redirects F-star-algebras]]
[[!redirects F*-algebra]]
[[!redirects F*-algebras]]


F-theory

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#### Elliptic cohomology
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#Contents#
* table of contents
{:toc}

## Idea
 {#Idea}

_F-theory_ is a toolbox for describing [[type IIB string theory]] backgrounds -- _including_ [[non-perturbative effects]] induced from the presence of [[D7-branes]] and [[(p,q)-strings]] -- in terms of [[complex numbers|complex]] [[elliptic fibrations]] whose fiber modulus $\tau$ encodes the [[axio-dilaton]] (the [[string coupling constant]] and the degree-0 [[RR-field]]) transforming under the $SL(2, \mathbb{Z})$ [[S-duality]]/[[U-duality]] [[group]]. See also at _[[duality in string theory]]_.

More technically, F-theory is what results when [[KK-compactification|KK-compactifying]] [[M-theory]] on an [[elliptic fibration]] (which yields [[type IIA superstring theory]] compactified on a [[circle]]-[[fiber bundle]]) followed by [[T-duality]] with respect to one of the two cycles of the elliptic fiber. The result is (uncompactified) [[type IIB superstring theory]] with [[axio-dilaton]] given by the moduli of the original elliptic fibration, see [below](#From11dSupergravity).

Or rather, this is [[type IIB string theory]] with some [[non-perturbative effects]] included, reducing to [[perturbative string theory]] in the [[Sen limit]]. With a full description of [[M-theory]] available also F-theory should be a full non-perturbative description of type IIB string theory, but absent that it is some kind of approximation. For instance while the [[modular group|modular]] [[structure group]] of the [[elliptic fibration]] in principle encodes (necessarily non-perturbative) [[S-duality]] effects, it is presently not actually known in full detail how this affects the full theory, notably the proper charge quantization law of the 3-form fluxes, see at _[S-duality -- Cohomological nature of the fields under S-duality](S-duality#CohomologicalNatureOfTypeIIFieldsUnderSDuality)_  for more on that.

## Properties

### Relation to M-theory
 {#From11dSupergravity}

The [[duality between M-theory and F-theory]]:

The following line of argument shows why first compactifying M-theory on a torus $S_1^A \times S_1^B$ to get type IIA on a circle and then T-dualizing that circle to get type IIB indeed only depends on the shape $\frac{R_A}{R_B}$ of the torus, but not on its other geometry.

By the [[dualities in string theory]], 10-dimensional [[type II string theory]] is supposed to be obtained from the [[UV-completion]] of [[11-dimensional supergravity]] by first [[Kaluza-Klein mechanism|dimensionally reducing]] over a circle $S^1_A$ -- to obtain [[type IIA supergravity]] -- and then applying [[T-duality]] along another circle $S^1_B$ to obtain [[type IIB supergravity]].

To obtain type IIB sugra in noncompact 10 dimensions this way, also $S^1_B$ is to be compactified (since [[T-duality]] sends the radius $r_A$ of $S^1_A$ to the inverse radius $r_B = \ell_s^2 / R_A$ of $S^1_B$).
Therefore type IIB sugra in $d = 10$ is obtained from 11d sugra compactified on the [[torus]] $S^1_A \times S^1_B$. More generally, this torus may be taken to be an [[elliptic curve]] and this may vary over the 9d base space as an [[elliptic fibration]].

Applying T-duality to one of the compact direction yields a 10-dimensional theory which may now be thought of as encoded by a 12-dimensional elliptic fibration. This 12d elliptic fibration encoding a 10d type II supergravity [[vacuum]] is the input data that F-theory is concerned with.

A schematic depiction of this story is the following:

|  |  |  |
|--|--|--|
| [[11d supergravity|M-theory]] in $d = 11$ | | F-theory in $d = 12$ |
| $\downarrow$ [[Kaluza-Klein mechanism|KK-reduction]] along [[elliptic fibration]] | | $\downarrow$ [[axio-dilaton]] is modulus of [[elliptic fibration]] |
| [[type IIA string theory|IIA string theory]] in $d = 9$ | $\leftarrow$[[T-duality]]$\rightarrow$ | [[type IIB string theory|IIB string theory]] in $d = 10$ |

In the simple case where the elliptic fiber is indeed just $S^1_A \times S^1_B$, the [[imaginary part]] of its complex modulus is

$$
  Im(\tau) = \frac{R_A}{R_B}
  \,.
$$

By following through the above diagram, one finds how this determines the [[coupling constant]] in the [[type II string theory]]:

First, the [[KK-compactification]] of [[M-theory]] on $S^1_A$ yields a type IIA [[string coupling]]

$$
  g_{IIA} = \frac{R_A}{\ell_s}
  \,.
$$

Then the T-duality operation along $S^1_B$ divides this by $R_B$:

$$
  \begin{aligned}
    g_{IIB} & = g_{IIA} \frac{\ell_s}{R_B}
    \\
    & = \frac{R_A}{R_B}
    \\
    & = Im(\tau)
   \end{aligned}
   \,.
$$

#### Relation to M-theory on $G_2$-manifolds

In order to get minimal [[N=1 d=4 supergravity]] after [[KK-compactification]], one needs [[M-theory on G2-manifolds]] and [[F-theory on CY4-manifolds]].

Discussion of the relation between then [[G2-manifold]] fibers for [[M-theory on G2-manifolds]] and the corresponding [[Calabi-Yau manifold|Calabi-Yau 4-manifold]] fibers in F-theory includes ([Gukov-Yau-Zaslow 02](#GukovYauZaslow02), [Belhaj 02](#Belhaj02)).

#### Relation to heterotic M-theory on ADE-singularities

Relation to [[heterotic M-theory on ADE-singularities]]:

* [[Monika Marquart]], [[Daniel Waldram]], _F-theory duals of M-theory on $S^1/\mathbb{Z}_2 \times T^4 / \mathbb{Z}_N$_ ([arXiv:hep-th/0204228](https://arxiv.org/abs/hep-th/0204228))

* Christoph Lüdeling, [[Fabian Ruehle]], _F-theory duals of singular heterotic K3 models_, Phys. Rev. D 91, 026010 (2015) ([arXiv:1405.2928](https://arxiv.org/abs/1405.2928))

### Relation to orientifold type II backgrounds
 {#RelationToOrientifolds}

The general [[vacuum]] of [[type II superstring theory]] (including [[type I superstring theory]]) is an [[orientifold]].

The [[target space]] data of an [[orientifold]] is a $\mathbb{Z}_2$-[[principal bundle]]/[[local system]], possibly singular (hence possibly on a [[smooth stack]]). On the other hand, the non-singular part of the [[elliptic fibration]] that defines the F-theory is a $SL_2(\mathbb{Z})$-[[local system]] (being the "homological invariant" of the [[elliptic fibration]]).

An argument due to ([Sen 96](#Sen96), [Sen 97a](#Sen97a)) says that the F-theory data does induce the [[orientifold]] data along the [[subgroup]] inclusion $\mathbb{Z}_2 \hookrightarrow SL_2(\mathbb{Z})$. See at _[[Sen limit]]_.

The degeneration locus of the elliptic fibration -- where the [[discriminant]] $\Delta$ vanishes and its fibers are the [[nodal curve]] -- is interpreted as that of [[D7-branes]] and [[O-planes|O7-planes]] ([Sen 97a, (3)](#Sen97a), [Blumenhagen 10, (11)](#Blumenhagen10)), exhibiting [[gauge enhancement]] [Sen 97b](#Sen97b) see [below](#SingularLocusAndD7Branes).

Reasoning like this might suggest that in generalization to how type II [[orientifolds]] involve $\mathbb{Z}_2$-[[equivariant]] [[K-theory]] (namely [[KR-theory]]), so F-theory should involve $SL_2(\mathbb{Z})$-equivariant [[elliptic cohomology]]. This was conjectured in ([Kriz-Sati 05, p. 3, p.17, 18](#KrizSati05)). For more on this see at _[[modular equivariant elliptic cohomology]]_.

### Relation to heterotic string theory

The [[duality between F-theory and heterotic string theory]]:A subclass of K3 manifolds elliptically fibered.

F-theory on an [[elliptic fibration|elliptically fibered]] [[K3]] is supposed to be equivalent to [[heterotic string theory]] [[KK-compactification|compactified]] on a 2-[[torus]]. An early argument for this is due to ([Sen 96](#Sen96)).

More generally, heterotic string theory on an elliptically fibered Calabi-Yau $Z \to B$ of complex dimension $(n-1)$ is supposed to be equivalent $F$-theory on an $n$-dimensional $X\to B$ with elliptic K3-fibers.

A detailed discussion of the [[equivalence]] of the respective [[moduli spaces]] is originally due to ([Friedman-Morgan-Witten 97](#FriedmanMorganWitten97)). A review of this is in ([Donagi 98](#Donagi98)).

### Singular locus of the elliptic fibration and D7-branes
 {#SingularLocusAndD7Branes}

In passing from [[M-theory]] to [[type IIA string theory]], the locus of any [[Kaluza-Klein monopole]] in 11d becomes the locus of [[D6-branes]] in 10d. The locus of the [[Kaluza-Klein monopole]] in turn (as discussed there) is the locus where the $S^1_A$-circle fibration degenerates. Hence in F-theory this is the locus where the fiber of the $S^1_A \times S^1_B$-[[elliptic fibration]] degenerates to the [[nodal curve]]. Since the [[T-duality|T-dual]] of [[D6-branes]] are [[D7-branes]], it follows that [[D7-branes]] in F-theory "are" the singular locus of the elliptic fibration.

Now an [[elliptically fibered complex K3-surface]]

$$
  \array{
    T &\longrightarrow& K3
    \\
    && \downarrow
    \\
    && \mathbb{C}\mathbb{P}^1
  }
$$

may be parameterized via the [[Weierstrass elliptic function]] as the solution locus of the equation

$$
  y^2 = x^3 + f(z) x + g(z)
$$

for $x,y,z \in \mathbb{C}\mathbb{P}^1$, with $f$ a [[polynomial]] of degree 8 and $g$ of degree twelve. The [[j-invariant]] of the complex [[elliptic curve]] which this parameterizes for given $z$ is

$$
  j(\tau(z)) = \frac{4 (24 f)^3}{27 g^2 + 4 f^3}
  \,.
$$

The [[poles]] $j\to \infty$ of the [[j-invariant]] correspond to the [[nodal curve]], and hence it is at these poles that the [[D7-branes]] are located.

\begin{imagefromfile}
        "file_name": "K3CobordismBetween24ThreeSpheres.jpg",
        "web": "nlab",
        "width": 260,
        "unit": "px",
        "float": "right",
        "margin": {
            "top": -40,
            "right": 0,
            "bottom": 20,
            "left": 20,
            "unit": "px"
        },
        "alt": "homotopy pasting diagram exhibiting the homotopy Whitehead integral",
        "caption": "from [SS21](https://ncatlab.org/schreiber/show/Equivariant+Cohomotopy+and+Oriented+Cohomology+Theory)"
\end{imagefromfile}

Since the order of the poles is 24 (the polynomial degree of the [[discriminant]] $\Delta = 27 g^2 + 4 f^3$, see at _[[elliptically fibered K3-surface]]  -- [singular points](elliptic+fibration+of+a+K3-surface#SingularPoints)_) there are necessarily _24 D7-branes_ ([Sen 96, page 5](#Sen96), [Sen 97b](#Sen97b), see also [Morrison 04, sections 8 and 17](#Morrison04), [Denef 08, around (3.41)](#Denef08), [Douglas-Park-Schnell 14](duality+between+M%2FF-theory+and+heterotic+string+theory#DouglasParkSchnell14)).

Under [[T-duality]] this translates to 24 [[D6-branes]] in [[type IIA string theory]] on [[K3]] ([Vafa 96, Footnote 2 on p. 6](#Vafa96)).

Notice that the _net charge_ of these 24 D7-branes is supposed to vanish, due to [[S-duality]] effects (e.g. [Denef 08, below (3.41)](#Denef08)).

For analogous discussion of 24 NS5-branes in [[heterotic string theory]] on [[K3]] see [Schwarz 97, around p. 50](duality+in+string+theory#Schwarz97).

For more see at _[[24 branes transverse to K3]]_.

### F-brane scan
 {#Fbranescan}

[[!include F-branes -- table]]

### S-duality operation on $(p,q)$-branes

The F-theory picture gives a geometric interpretation of the [[S-duality]] expected in [[type II string theory]], by which all branes carry _two_ integer charges $(p,q)$ acted on by $SL(2,\mathbb{Z})$. For instance the fundamental string ([[F1-brane]]) and the [[D1-brane]] combine to the $(p,q)$-string, and similarly the [[NS5-brane]] and the [[D5-brane]] combine to a $(p,q)$-5-brane.

Namely in the F-theory picture this comes from [[wrapped brane|wrapping]] the [[M2-brane]] and the [[M5-brane]], respectively, on either of the two cycles of the elliptic fibration (and the [[T-duality|T-dualizing]]).

(e.g. [Johnson 97, p. 4](#Johnson97))

### Model building and phenomenology
 {#ModelBuildingAndPhenomenology}

For F-theory a fairly advanced [[model (physics)|model]] building and [[string phenomenology]] has been developed. A detailed review is in ([Denef 08](#Denef08)).

Via the relation between [[supersymmetry and Calabi-Yau manifolds]] there is particular interest in F-theory compactified on [[Calabi-Yau variety|Calabi-Yau spaces]] of ([[complex manifold|complex]]) [[dimension]] 4. For more on this see at _[[F/M-theory on elliptically fibered Calabi-Yau 4-folds]]_.

A large number of realizations of the exact field content of the [[standard model of particle physics]] (or rather the [[MSSM]]) is claimed to be realized in in F-theory in [Cvetic-Halverson-Lin-Liu-Tian 19](#CveticHalversonLinLiuTian19).

## Related concepts

* [[F'-theory]]

* [[D=12 supergravity]], [[D=14 supersymmetry]], [[bosonic M-theory]]

[[!include F-theory compactifications -- table]]

* [[F-theory on Spin(7)-manifolds]]

## References
 {#References}

### General

Discussion of [[T-duality]] in the strong coupling limit is due to

* {#Schwarz96} [[John Schwarz]], _M Theory Extensions of T Duality_ ([arXiv:hep-th/9601077](https://arxiv.org/abs/hep-th/9601077))

* {#Russo97} [[Jorge Russo]], _T-duality in M-theory and supermembranes_, Phys.Lett. B400 (1997) 37-42 ([arXiv:hep-th/9701188](https://arxiv.org/abs/hep-th/9701188))

That [[S-duality]] of [[type II string theory]] may be interpreted in terms of [[conformal transformations]] on the [[fiber]] for [[M-theory]] compactified on a [[torus]] was originally observed in

* [[John Schwarz]], _An $SL(2,\mathbb{Z})$-Multiplet of Type IIB Superstrings_, Phys.Lett.B 360:13-18, 1995; ERRATUM-ibid.B364:252, 1995 ([arXiv:hep-th/9508143](https://arxiv.org/abs/hep-th/9508143))

* [[Paul Aspinwall]], _Some Relationships Between Dualities in String Theory_, Nucl.Phys.Proc.Suppl. 46 (1996) 30-38 ([arXiv:hep-th/9508154](https://arxiv.org/abs/hep-th/9508154))

The original article on F-theory as such is

* {#Vafa96} [[Cumrun Vafa]], _Evidence for F-theory_, Nucl. Phys. B469:403-418, 1996, ([arXiv:hep-th/9602022](http://arxiv.org/abs/hep-th/9602022))

An early survey of its relation to [[M-theory]] with [[M5-branes]] is in

* {#Johnson97} [[Clifford Johnson]], _From M-theory to F-theory, with Branes_, Nucl.Phys. B507 (1997) 227-244 ([arXiv:hep-th/9706155](http://arxiv.org/abs/hep-th/9706155))

A more recent survey is

* {#Blumenhagen10} [[Ralph Blumenhagen]], _Basics of F-theory from the Type IIB Perspective_ ([arXiv:1002.2836](http://arxiv.org/abs/1002.2836))

Lecture notes include

* {#Morrison04} [[David Morrison]], _TASI Lectures on Compactification and Duality_ ([arXiv:hep-th/0411120](http://arxiv.org/abs/hep-th/0411120))

* [[Timo Weigand]], _Lectures on F-theory compactifications and model building_ Class. Quantum Grav. 27 214004 ([arXiv:1009.3497](http://arxiv.org/abs/1009.3497))

* {#Weigand18} [[Timo Weigand]], _TASI Lectures on F-theory_ ([arXiv:1806.01854](https://arxiv.org/abs/1806.01854))

Textbook accounts include

* [[Katrin Becker]], [[Melanie Becker]], [[John Schwarz]], section 8.4 of: _String Theory and M-Theory: A Modern Introduction_, Cambridge University Press 2007 ([doi:10.1017/CBO9780511816086](https://doi.org/10.1017/CBO9780511816086))

Further survey includes

* [[Timo Weigand]], _F-theory: Progress and Prospects_, 2014 ([pdf](https://www.theorie.physik.uni-muenchen.de/activities/workshops/archive_workshops_conferences/string_pheno_ringberg/slides_frontiers/weigand.pdf))

* {#Vafa15} [[Cumrun Vafa]], _Reflections on F-theory_, 2015 ([pdf](http://wwwth.mpp.mpg.de/conf/f-theory15/talks/Vafa.pdf))

Related conferences include

* _[Physics and Geometry of F-theory 2014](http://www.match.uni-heidelberg.de/GPF/)_

* _[Physics and Geometry of F‑theory 2015](http://f-theory15.mpp.mpg.de)_

### Relation to orientifolds

F-theory lifts of [[orientifold]] backgrounds were first identified in

* {#Sen96} [[Ashoke Sen]], _F-theory and Orientifolds_, Nucl. Phys. B475:562-578,1996 ([arXiv:hep-th/9605150](http://arxiv.org/abs/hep-th/9605150))

* {#Sen97a} [[Ashoke Sen]], _Orientifold Limit of F-theory Vacua_, Nucl. Phys. Proc. Suppl. 68 (1998) 92 (Nucl. Phys. Proc. Suppl. 67 (1998) 81) ([arXiv:hep-th/9702165](http://arxiv.org/abs/hep-th/9702165))

and the corresponding [[gauge enhancement]] in

* {#Sen97b} [[Ashoke Sen]], _A Note on Enhanced Gauge Symmetries in M- and String Theory_, JHEP 9709:001,1997 ([arXiv:hep-th/9707123](http://arxiv.org/abs/hep-th/9707123))

with more details including

* Zurab Kakushadze, [[Gary Shiu]], S.-H. Henry Tye, _Type IIB Orientifolds, F-theory, Type I Strings on Orbifolds and Type I - Heterotic Duality_, Nucl.Phys. B533 (1998) 25-87 ([arXiv:hep-th/9804092](http://arxiv.org/abs/hep-th/9804092))

This is further expanded on in

* [[Hisham Sati]], _The Elliptic curves in gauge theory, string theory, and cohomology_, JHEP 0603 (2006) 096 ([arXiv:hep-th/0511087](http://arxiv.org/abs/hep-th/0511087))

### Relation to elliptic cohomology

A series of articles arguing for a relation between the [[elliptic fibration]] of F-theory and [[elliptic cohomology]] (see also at [[modular equivariant elliptic cohomology]])

* {#KrizSati05} [[Igor Kriz]], [[Hisham Sati]], _Type II string theory and modularity_, 	JHEP 0508 (2005) 038 ([arXiv:hep-th/0501060](http://arxiv.org/abs/hep-th/0501060))

### Relation to heterotic string string theory

* {#FriedmanMorganWitten97} Robert Friedman, [[John Morgan]], [[Edward Witten]], _Vector Bundles And F Theory_ ([arXiv:hep-th/9701162](http://arxiv.org/abs/hep-th/9701162))

* {#Donagi98} [[Ron Donagi]], _ICMP lecture on heterotic/F-theory duality_ ([arXiv:hep-th/9802093](http://arxiv.org/abs/hep-th/9802093))

### Relation to M-theory on $G_2$-manifolds

* {#GukovYauZaslow02} [[Sergei Gukov]], [[Shing-Tung Yau]], [[Eric Zaslow]], _Duality and Fibrations on $G_2$ Manifolds_ ([arXiv:hep-th/0203217](http://arxiv.org/abs/hep-th/0203217))

* {#Belhaj02} Adil Belhaj, _F-theory Duals of M-theory on $G_2$ Manifolds from Mirror Symmetry_ ([arXiv:hep-th/0207208](http://arxiv.org/abs/hep-th/0207208))

* [[Mariana Graña]], C. S. Shahbazi, [[Marco Zambon]], _$Spin(7)$-manifolds in compactifications to four dimensions_, JHEP11(2014)046 ([arXiv:1405.3698](http://arxiv.org/abs/1405.3698))

### Relation to the 6d superconformal theory

Realization to the [[6d (2,0)-supersymmetric QFT]] is discussed in

* {#HeckmannMorrisonVafa13} Jonathan Heckman, [[David Morrison]], [[Cumrun Vafa]], _On the Classification of 6D SCFTs and Generalized ADE Orbifolds_ ([arXiv:1312.5746](http://arxiv.org/abs/1312.5746))

### Phenomenology and model building
 {#ReferencesPhenomnenology}

A large body of literature is concerned with particle physics [[string phenomenology]] modeled in the context of F-theory, in particular [[GUTs]]:

* {#Denef08} [[Frederik Denef]], _Les Houches Lectures on Constructing String Vacua_, in: _[[String theory and the real world]]_ ([arXiv:0803.1194](http://arxiv.org/abs/0803.1194), [spire:780946](https://inspirehep.net/literature/780946))

* [[Chris Beasley]], [[Jonathan Heckman]], [[Cumrun Vafa]], _GUTs and Exceptional Branes in F-theory - I_, JHEP 0901:058,2009 ([arXiv:0802.3391](http://arxiv.org/abs/0802.3391))

* [[Chris Beasley]], [[Jonathan Heckman]], [[Cumrun Vafa]], _GUTs and Exceptional Branes in F-theory - I_ ([arxiv:0802.3391](http://arxiv.org/abs/0802.3391)), _II: Experimental Predictions_ ([arxiv:0806.0102](http://arxiv.org/abs/0806.0102))

* {#Wijhnholt14} [[Martin Wijnholt]], _String compactification_, [PITP 2014](https://pitp2014.ias.edu) lecture notes ([[WijnholtCompactification14.pdf:file]], [slides for lecture 1](https://static.ias.edu/pitp/2014/sites/pitp2014.ias.edu/files/PITP2014_P1_wijnholt.pdf), [slides for lecture 2](https://static.ias.edu/pitp/2014/sites/pitp2014.ias.edu/files/PITP2014_P2_wijnholt.pdf), [slides for lecture 3](https://static.ias.edu/pitp/2014/sites/pitp2014.ias.edu/files/PITP2014_P3_wijnholt.pdf))

* {#Zoccarato14} Gianluca Zoccarato, _Yukawa couplings at the point
of $E_8$ in F-theory_, 2014 ([pdf](http://stringpheno2014.ictp.it/parallels/tuesday/F-theory(B)/zoccarato.pdf))

Discussion of the _exact_ [[gauge group]] of the [[standard model of particle physics]], $G = \big( SU(3) \times SU(2) \times U(1)\big)/\mathbb{Z}_6 $ including its $\mathbb{Z}_6$-[[quotient group|quotient]] (see [there](standard+model+of+particle+physics#GaugeGroup)) and the exact [[fermion]] field content, realized in F-theory is in

* Denis Klevers, Damian Kaloni Mayorga Pena, Paul-Konstantin Oehlmann, Hernan Piragua, Jonas Reuter, _F-Theory on all Toric Hypersurface Fibrations and its Higgs Branches_, JHEP01(2015)142 ([arXiv:1408.4808](https://arxiv.org/abs/1408.4808))

* {#CveticLin17} [[Mirjam Cvetic]], Ling Lin, section 3.3 of _The global gauge group structure of F-theory compactifications with $U(1)$s_ ([arXiv:1706.08521](https://arxiv.org/abs/1706.08521))

Based on this large number of realizations of the exact field content of the [[standard model of particle physics]] (or rather [[MSSM]]) in [[F-theory]] is claimed to be realized in

* {#CveticHalversonLinLiuTian19} [[Mirjam Cvetic]], [[James Halverson]], Ling Lin, Muyang Liu, Jiahua Tian, _A Quadrillion Standard Models from F-theory_ ([arXiv:1903.00009](https://arxiv.org/abs/1903.00009))

* {#TaylorTurner19} [[Washington Taylor]], Andrew P. Turner, _Generic construction of the Standard Model gauge group and matter representations in F-theory_ ([arXiv:1906.11092](https://arxiv.org/abs/1906.11092))

Review:

* [[Fernando Marchesano]], [[Bert Schellekens]], [[Timo Weigand]], *D-brane and F-theory Model Building*, in *[[Handbook of Quantum Gravity]]*, Springer (2023) [[arXiv:2212.07443](https://arxiv.org/abs/2212.07443)]

Realization of [[E7|$E_7$]]-[[GUT]] models in F-theory:

* Shing Yan Li, [[Washington Taylor]], *Towards natural and realistic $E_7$ GUTs in F-theory* [[arXiv:2401.00040](https://arxiv.org/abs/2401.00040)]

#### Cosmological constant

An argument for [[non-perturbative effect|non-perturbative]] non-[[supersymmetry|supersymmetric]] 4d [[string phenomenology]] with fundamentally vanishing [[cosmological constant]], based on 3d [[M-theory on 8-manifolds]]
decompactified at strong coupling to 4d via [[duality between M-theory and type IIA string theory]] (recall the [[super 2-brane in 4d]]):

* {#Witten00} [[Edward Witten]], _The Cosmological Constant From The Viewpoint Of String Theory_, lecture at [DM2000](http://inspirehep.net/record/972507) ([arXiv:hep-ph/0002297](https://arxiv.org/abs/hep-ph/0002297))

  (see p. 7)

* [[Edward Witten]], _Strong coupling and the cosmological constant_, Mod. Phys. Lett. A 10:2153-2156, 1995 ([arXiv:hep-th/9506101](https://arxiv.org/abs/hep-th/9506101))

* [[Edward Witten]], Section 3 of _Some Comments On String Dynamics_, talk at [Strings95](https://cds.cern.ch/record/305869) ([arXiv:hep-th/9507121](http://arxiv.org/abs/hep-th/9507121))

The realization of this scenario in [[F-theory]]:

* [[Cumrun Vafa]], Section 4.3 of: _Evidence for F-Theory_, Nucl. Phys. B469:403-418, 1996 ([arxiv:hep-th/9602022](https://arxiv.org/abs/hep-th/9602022))

* [[Jonathan Heckman]], Craig Lawrie, Ling Lin, Gianluca Zoccarato, _F-theory and Dark Energy_, Fortschritte der Physik  ([arXiv:1811.01959](https://arxiv.org/abs/1811.01959), [doi:10.1002/prop.201900057]( https://doi.org/10.1002/prop.201900057))

* [[Jonathan Heckman]], Craig Lawrie, Ling Lin, Jeremy Sakstein, Gianluca Zoccarato, _Pixelated Dark Energy_ ([arXiv:1901.10489](https://arxiv.org/abs/1901.10489))

#### Flavour anomalies

Realization in [[F-theory]] of [[GUT]]-models with [[Z'-bosons]] and/or [leptoquarks]] addressing the [[flavour anomalies]] and the [(g-2) anomalies](anomalous+magnetic+moment#Anomalies):

* Miguel Crispim Romao, Stephen F. King, George K. Leontaris, _Non-universal $Z'$ from Fluxed GUTs_, Physics Letters B Volume 782, 10 July 2018, Pages 353-361 ([arXiv:1710.02349](https://arxiv.org/abs/1710.02349))

* A. Karozas, G. K. Leontaris, I. Tavellaris, N. D. Vlachos, _On the LHC signatures of $SU(5) \times U(1)'$ F-theory motivated models_ ([arXiv:2007.05936](https://arxiv.org/abs/2007.05936))

### 4-Form flux and instantons
 {#FluxAndInstantons}

The image of the [[supergravity C-field]] from [[11-dimensional supergravity]] to F-theory yields the _$G_4$-[[flux compactification|flux]]_.

* Andres Collinucci, Raffaele Savelli, _On Flux Quantization in F-Theory_ (2010) ([arXiv:1011.6388](http://arxiv.org/abs/1011.6388))

* Sven Krause, Christoph Mayrhofer, [[Timo Weigand]], _$G_4$ flux, chiral matter and singularity resolution in F-theory compactifications_ ([arXiv:1109.3454](http://arxiv.org/abs/1109.3454))

* {#GKP12} [[Thomas Grimm]], Denis Klevers, Maximilian Poretschkin, _Fluxes and Warping for Gauge Couplings in F-theory_ ([arXiv:1202.0285](http://arxiv.org/abs/1202.0285))

* {#KMW12} Sven Krause, Christoph Mayrhofer, [[Timo Weigand]], _Gauge Fluxes in F-theory and Type IIB Orientifolds_ (2012) ([arXiv:1202.3138](http://arxiv.org/abs/1202.3138))

and with [[M5-brane]] [[instanton]] contributions:

* Max Kerstan, [[Timo Weigand]], _Fluxed M5-instantons in F-theory_ ([arXiv:1205.4720](http://arxiv.org/abs/1205.4720))

Reviewed in

* {#Weigand12} [[Timo Weigand]], _Fluxes and M5-instantons in F-theory_, 2012 ([pdf slides](http://people.physik.hu-berlin.de/~ahoop/weigand.pdf))

For more on this see also at _[[F/M-theory on elliptically fibered Calabi-Yau 4-folds]]_.


F-theory compactifications -- table

**[[F-theory]] [[KK-compactification|KK-compactified]] on [[elliptic fibration|elliptically fibered]] [[complex analytic space|complex analytic]] [[fiber bundle|fiber]] $\Sigma$**

| $dim_{\mathbb{C}}(\Sigma)$ | 1 | 2 | 3 | 4 | 5 |
| ---------------------------|---|---|---|---|--|
|  | [[F-theory]] | [[F/M-theory on elliptically fibered Calabi-Yau 2-folds|F-theory on CY2]] | [[F/M-theory on elliptically fibered Calabi-Yau 3-folds|F-theory on CY3]] | [[F/M-theory on elliptically fibered Calabi-Yau 4-folds|F-theory on CY4]] | [[F/M-theory on elliptically fibered Calabi-Yau 5-folds|F-theory on CY5]] |


F-theory on Spin(7)-manifolds

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

#Contents#
* table of contents
{:toc}

## Idea

The [[KK-compactification]] of [[F-theory]] on [[Spin(7)-manifolds]] to 4d. Differs subtly from [[F-theory on CY4]].

Related by [[T-duality]] to 3d [[M-theory on Spin(7)-manifolds]]

## Properties

### (Non-)Supersymmetry and (vanishing) cosmological constant

What has been called _[[Witten's dark fantasy]]_ in [Heckmann-Lawrie-Lin-Zoccarato 19, Section 8](#HeckmannLawrieLinZoccarato19) is an argument, going back to [Witten 95a](#Witten95a), [Witten 95b, Sec. 3](#Witten95b), [Witten 00, p. 7](#Witten00) for the existence of [[non-perturbative effect|non-perturbative]] non-[[supersymmetry|supersymmetric]] 4d [[string vacua]]/[[string phenomenology]] with fundamentally _vanishing_ [[cosmological constant]] ("[[dark energy]]").

The original idea was formulated in terms of 3d [[M-theory on 8-manifolds]] decompactified at strong coupling to 4d via [[duality between M-theory and type IIA string theory]] (recall the [[super 2-brane in 4d]]).

Based on the observation of [Vafa 96, Section 4.3](#Vafa96) that the argument should have a natural realization in 4d [[F-theory on Spin(7)-manifolds]] ([[T-duality|T-dual]] to the previous perspective), a detailed construction was finally laid out in [Bonetti-Grimm-Pugh 13](#BonettiGrimmPugh13), [Heckmann-Lawrie-Lin-Zoccarato1 18](#HeckmannLawrieLinZoccarato118), [Heckman-Lawrie-Lin-Sakstein-Zoccarato 19](#HeckmanLawrieLinSaksteinZoccarato19).

The key technical point is the claim that a careful analysis of [[D=4 N=1 supergravity]] obtained after [[KK-compactification]] of  [[F-theory on Spin(7)-manifolds]] [[T-duality|T-dual to]] [[M-theory on Spin(7)-manifolds]] reveals a "1/2 supersymmetry" where

1. the [[vacuum state]] is [[supersymmetry|supersymmetric]] and hence has vanishing [[cosmological constant]];

1. but no [[finite number|finite]]-[[energy]]-excitation of the vacuum appears supersymmetrically,

   hence [[fermions]] and [[bosons]] in the model do _not_ appear in supersymmetric spectra.

([Vafa 96, Sec. 4.3](#Vafa96) [BGP 13](#BonettiGrimmPugh13), [HLLZ 18, Sec. 4](#HeckmannLawrieLinZoccarato118))

### Relation to J-twisted Cohomotopy
 {#RelationToJTwistedCohomotopy}

On a [[spin-manifold]] [[8-manifold|of dimension 8]] a choice of topological [[Spin(7)]]-[[G-structure|structure]] is equivalently a choice of [[cocycle]] in [[J-homomorphism|J-]][[twisted Cohomotopy cohomology theory]]. This follows ([[schreiber:Twisted Cohomotopy implies M-theory anomaly cancellation|FSS 19, 3.4]]) from

1. the standard [[coset space]]-[[structures]] on the [[7-sphere]] (see [here](7-sphere#CosetSpaceRealization))

1. the fact that [[coset spaces]] $G/H$ are the [[homotopy fibers]] of the maps $B H \to B G$ of the corresponding [[classifying spaces]] (see [here](coset#ForInfinityGroups))

\begin{imagefromfile}
  "file_name": "ExceptionalSpheres.jpg",
  "width": 730
\end{imagefromfile}

## Related concepts

[[!include F-theory compactifications -- table]]

## References

### General

The concept goes back to

* {#Vafa96} [[Cumrun Vafa]], Section 4.3 of: _Evidence for F-Theory_, Nucl. Phys. B469:403-418, 1996 ([arxiv:hep-th/9602022](https://arxiv.org/abs/hep-th/9602022))

See also:

* [[Mariana Graña]], C. S. Shahbazi, [[Marco Zambon]], _$Spin(7)$-manifolds in compactifications to four dimensions_, JHEP11(2014)046 ([arXiv:1405.3698](http://arxiv.org/abs/1405.3698))

* [[Andreas Braun]], [[Sakura Schafer-Nameki]], _Spin(7)-Manifolds as Generalized Connected Sums and 3d $N=1$ Theories_, JHEP06(2018)103 ([arXiv:1803.10755](https://arxiv.org/abs/1803.10755))

  (generalization of [[compact twisted connected sum G2-manifolds]])

Relating [[M-theory on Spin(7)-manifolds]] with [[F-theory on Spin(7)-manifolds]] via [[Higgs bundles]]:

* {#CHRTZ20} [[Mirjam Cvetic]], [[Jonathan Heckman]], Thomas B. Rochais, Ethan Torres, [[Gianluca Zoccarato]], _Geometric Unification of Higgs Bundle Vacua_ ([arXiv:2003.13682](https://arxiv.org/abs/2003.13682))

### Witten's Dark Fantasy

Already [Vafa 96, Section 4.3](#Vafa96) mentions the relation to [[Witten's Dark Fantasy]], then developed in

* {#BonettiGrimmPugh13} [[Federico Bonetti]], [[Thomas Grimm]], [[Tom Pugh]], _Non-Supersymmetric F-Theory Compactifications on $Spin(7)$ Manifolds_, JHEP 01 (2014) 112 ([arXiv:1307.5858](https://arxiv.org/abs/1307.5858))

* {#BonettiGrimmPugh13b} [[Federico Bonetti]], [[Thomas Grimm]], [[Eran Palti]], [[Tom Pugh]], _F-Theory on $Spin(7)$ Manifolds: Weak-Coupling Limit_, JHEP 02 (2014) 076 ([arXiv:1309.2287](https://arxiv.org/abs/1309.2287))

* {#HeckmannLawrieLinZoccarato118} [[Jonathan Heckman]], Craig Lawrie, Ling Lin, [[Gianluca Zoccarato]], _F-theory and Dark Energy_, Fortschritte der Physik  ([arXiv:1811.01959](https://arxiv.org/abs/1811.01959), [doi:10.1002/prop.201900057]( https://doi.org/10.1002/prop.201900057))

* {#HeckmanLawrieLinSaksteinZoccarato19} [[Jonathan Heckman]], Craig Lawrie, Ling Lin, Jeremy Sakstein, [[Gianluca Zoccarato]], _Pixelated Dark Energy_ ([arXiv:1901.10489](https://arxiv.org/abs/1901.10489))

See also

* [[Tom Pugh]], _M-theory on $Spin(7)$-manifold duals and their F-theory duals_ ([pdf](/nlab/files/PughMTheoryOnSpin7.pdf))

* [[Federico Bonetti]], [[Thomas Grimm]], [[Eran Palti]], [[Tom Pugh]], _F-Theory on Spin(7) Manifolds: Weak-Coupling Limit_, J. High Energ. Phys. (2014) 2014: 76 ([arXiv:1309.2287](https://arxiv.org/abs/1309.2287))


F. Conduché > history

see at *[[François Conduché]]*


F. Marmolejo > history

see at *[[Francisco Marmolejo]]*


F/M-theory on elliptically fibered Calabi-Yau 3-folds

## Related concepts

[[!include F-theory compactifications -- table]]

[[!include KK-compactifications of M-theory -- table]]

## References

* [[David Morrison]], [[Washington Taylor]], _Classifying bases for 6D F-theory models_ ([arXiv:1201.1943](http://arxiv.org/abs/1201.1943))

* [[David Morrison]], [[Washington Taylor]], _Toric bases for 6D F-theory models_ ([arXiv:1204.0283](http://arxiv.org/abs/arXiv:1204.0283))

* Gabriella Martini, [[Washington Taylor]], _6D F-theory models and elliptically fibered Calabi-Yau threefolds over semi-toric base surfaces_ ([arXiv:1404.6300](http://arxiv.org/abs/1404.6300))


F/M-theory on elliptically fibered Calabi-Yau 4-folds

+-- {: .rightHandSide}
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###Context###
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#### Cohomology
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[[!include cohomology - contents]]
=--
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#Contents#
* table of contents
{:toc}

## Idea

### General

The [[KK-compactification|KK]] [[flux compactification]] of [[M-theory]]/[[F-theory]] on an [[elliptic fibration|elliptically fibered]] [[Calabi-Yau variety]] of ([[complex manifold|complex]]) [[dimension]] 4 (hence real dimension 8, see also at _[[M-theory on 8-manifolds]]_) yields [[type IIB superstring theory]] (with some [[S-duality]]-related [[non-perturbative effects]] included) [[KK-compactification|KK-compactified]] on Calabi-Yau 3-folds.

This is subtly different from [[F-theory on Spin(7)-manifolds]], see at _[[Witten's Dark Fantasy]]_.

Realization of [[moduli stabilization]] of those flux compactifications is by considering higher dimensional gauge fields in the compact space, the curvature of those fields parametrizes the mass term.

### Motivation from string model building/phenomenology

By the discussion at _[[supersymmetry and Calabi-Yau manifolds]]_ such compactifications of [[type IIB superstring theory]] yield the [[UV-completion]] of an [[effective field theory]] in 4-dimensions which exhibits $N =1 $ [[supersymmetry]].

[[!include N=1 susy compactifications -- table]]

For this reason F/M-theory compactifications on elliptically fibered Calabi-Yau 4-folds are a major contender in [[string phenomenology]] [[model (physics)|model]] building. See at _[string phenomenology -- Models in type II / F-theory](string+phenomenology#ModelsInTypeIIWithIntersectingBranes)_ for more on this.

### Motivation from mathematics

At the same time these types of [[F-theory]] [[KK-compactifications]] are of interest also for their pure [[mathematics]] and as such received a fair amount of attention notably in [[algebraic geometry]].

For instance the [[duality between F-theory and heterotic string theory]] translates into a class of rich equivalences of [[moduli spaces]] of bundles on different [[algebraic varieties]] which is rather non-obvious without the [[string theory]]-perspective.

From the abstract of ([Donagi 98](#Donagi98)).

> The [[heterotic string theory|heterotic string]] [[KK-compactification|compactified]] on an $(n-1)$-[[dimension|dimensional]] [[elliptic fibration|elliptically fibered]] [[Calabi-Yau variety|Calabi-Yau]] $Z \to B$ is conjectured to be [[duality in string theory|dual]] to [[F-theory]] [[KK-compactification|compactified]] on an $n$-dimensional [[Calabi-Yau variety|Calabi-Yau]] $X \to B$, fibered over the same base with [[elliptic fibration|elliptic]] [[K3]] fibers. In particular, the [[moduli]] of the two theories should be [[isomorphism|isomorphic]]. The cases most relevant to the physics are $n=2$, $3$, $4$, i.e. the [[KK-compactification|compactification]] is to [[dimensions]] $d=8$, $6$ or $4$ respectively. Mathematically, the richest picture seems to emerge for $n=3$, where the [[moduli space]] involves an analytically [[integrable system]] whose fibers admit rather different descriptions in the two theories.

Moreover, the description of the [[supergravity C-field]] (in the physics literature: the "4-form flux", see at _[[flux compactification]]_), being (up to various twists and subtleties, see there) a [[cocycle]] in degree-4 [[ordinary differential cohomology]] on the [[Calabi-Yau variety]]  involves the [[moduli spaces of (higher) line bundles]] on the $CY_4$ which under decomposition of the C-field into various [[cup product in ordinary differential cohomology|cup-product]] factors associated with the various [[divisor (algebraic geometry)|divisor]] of the $CY_4$ involves [[intermediate Jacobians]] and [[Artin-Mazur formal groups]] in degrees 0,1,2 with rich theory.

## Related concepts

[[!include F-theory compactifications -- table]]

* [[F-theory on Spin(7)-manifolds]]

[[!include KK-compactifications of M-theory -- table]]

## References

Reviews of the general setup include

* [[Thomas Grimm]], Raffaele Savelli, sections 3.1, 3.2 of _Gravitational Instantons and Fluxes from M/F-theory on Calabi-Yau fourfolds_ ([arXiv:1109.3191](http://arxiv.org/abs/1109.3191))

Discussion of the [[duality between F-theory and heterotic string theory]] for compactifications on elliptically fibered Calabi-Yau manifolds includes

* {#Donagi98} [[Ron Donagi]], _ICMP lecture on heterotic/F-theory duality_ ([arXiv:hep-th/9802093](http://arxiv.org/abs/hep-th/9802093))

* {#DonagiMarkman95} [[Ron Donagi]], [[Eyal Markman]], _Spectral curves, algebraically completely integrable Hamiltonian systems, and moduli of bundles_ ([arXiv:alg-geom/9507017](http://arxiv.org/abs/alg-geom/9507017))

Discussion of relation to [[M-theory on G2-manifolds]]/[[M-theory on Spin(8)-manifolds]] includes

* [[Mariana Graña]], C. S. Shahbazi, [[Marco Zambon]], _$Spin(7)$-manifolds in compactifications to four dimensions_, JHEP11(2014)046 ([arXiv:1405.3698](http://arxiv.org/abs/1405.3698))

For more on this see at _[[M-theory on Spin(7)-manifolds]]_

[[!redirects F-theory on elliptically fibered Calabi-Yau 4-folds]]

[[!redirects F/M-theory on CY4]]
[[!redirects F-theory on CY4]]

[[!redirects F-theory on CY4-manifolds]]

[[!redirects F-theory on CY4-folds]]


F/M-theory on elliptically fibered Calabi-Yau 5-folds

+-- {: .rightHandSide}
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###Context###
#### String theory
+-- {: .hide}
[[!include string theory - contents]]
=--
#### Cohomology
+--{: .hide}
[[!include cohomology - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

### General

The [[KK-compactification|KK]] [[flux compactification]] of [[M-theory]]/[[F-theory]] on an [[elliptic fibration|elliptically fibered]] [[Calabi-Yau variety]] of ([[complex manifold|complex]]) [[dimension]] 5 (hence real dimension 10) yields, on the [[F-theory]] side, [[type IIB superstring theory]] (with some [[S-duality]]-related [[non-perturbative effects]] included) [[KK-compactification|KK-compactified]] on Calabi-Yau 4-folds, and hence $D=2$, $\mathcal{N}=(2,0)$ QFTs such as [[D=2 N=(2,0) SCFT]], [[T-duality|T-dual]], on the [[M-theory]]-side, to [[supersymmetric quantum mechanics]] (see also: [[BFSS matrix model]] and [E10](E10#AsUDualityOf1dMTheory)).

## Related concepts

[[!include F-theory compactifications -- table]]

\linebreak

[[!include KK-compactifications of M-theory -- table]]

## References

For [[M-theory]] and [[supersymmetric quantum mechanics]]:

* [[Alexander Haupt]], [[Andre Lukas]], [[Kellogg Stelle]], _M-theory on Calabi-Yau Five-Folds_, JHEP 0905:069, 2009 ([arXiv:0810.2685](https://arxiv.org/abs/0810.2685))

For [[F-theory]] and [[D=2 N=(2,0) SCFT]]:

* [[Sakura Schäfer-Nameki]], [[Timo Weigand]], _F-theory and 2d (0,2) Theories_, J. High Energ. Phys. 2016, 59 (2016) ([arXiv:1601.02015](https://arxiv.org/abs/1601.02015))

Survey:

* [[Sakura Schäfer-Nameki]], _F-theory and $2d$ $(0,2)$ Theories_, 2016 (talk slides [[SchaeferNamekiFTheoryAnd2DQFT.pdf:file]])

[[!redirects M-theory on Calabi-Yau 5-folds]]


F1-brane

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=--

#Contents#
* table of contents
{:toc}

## Idea

In [[type IIB string theory]] [[S-duality]] mixes the [[D1-brane]] with the [[superstring]] itself to form [[(p,q)-strings]]. To reflect how the string hence here appears on the same footing as a [[D-brane]], one also calls it the _F1-brane_ (for "fundamental string").

Similarly S-duality mixes the [[NS5-brane]] and the [[D5-brane]] and hence the former is also sometimes (rarely though) called the "F5".

## As a black brane

[[!include black branes in supergravity -- table]]

## Related concepts

[[!include table of branes]]

[[!redirects F1-branes]]

[[!redirects F1-brane]]
[[!redirects F1-branes]]

[[!redirects F-string]]
[[!redirects F-strings]]

[[!redirects F1-string]]
[[!redirects F1-strings]]


f1-meson

+-- {: .rightHandSide}
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### Context
#### Fields and quanta
+-- {: .hide}
[[!include fields and quanta - table]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The _$f_1$-meson_ is the [[light meson|light]] [[meson]] which (in the [[Wigner classification]]) is the [[Lorentz group|Lorentz]] [[spin group]] [[pseudo-vector representation]] and [[isospin]] [[scalar]].

This is the [[chiral partner]] of the [[omega-meson]].

\begin{imagefromfile}
    "file_name": "LightAndChiralPartnerMesonFieldsPinIII.jpg",
    "width": 600,
    "unit": "px",
    "margin": {
        "top": 0,
        "right": 10,
        "bottom": 0,
        "left": 20
    }
\end{imagefromfile}

## References

* Su Houng Lee, _Observing chiral partners in nuclear  medium_ ([[LeeChiralPartners.pdf:file]])

* M. Kirchbach, D. O. Riska, _The Coupling of the $f_1(1285)$ meson to the isoscalar axial current of the nucleon_, Nucl. Phys.A 594 (1995) 419-424 ([arXiv:nucl-th/9502018](https://arxiv.org/abs/nucl-th/9502018))

* [[Robert Pisarski]], p. 1 of: _Where does the Rho Go? Chirally Symmetric Vector Mesons in the Quark-Gluon Plasma_, Phys. Rev. D52 (1995) 3773-3776 ([arXiv:hep-ph/9503328](https://arxiv.org/abs/hep-ph/9503328))

* A. E. Dorokhova, N. I. Kochelevba. A. P. Martynenkoc. F. A. Martynenkoc. A. E. Radzhabovb, _The contribution of axial-vector mesons to hyperfine structure of muonic hydrogen_, Physics Letters B Volume 776, 10 January 2018, Pages 105-110 Physics Letters B ([doi:10.1016/j.physletb.2017.11.027](https://doi.org/10.1016/j.physletb.2017.11.027))

[[!redirects f1-mesons]]


F4

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#### Exceptional structures
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[[!include group theory - contents]]
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#### Lie theory
+--{: .hide}
[[!include infinity-Lie theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

One of the [[exceptional Lie groups]].

## Definition

+-- {: .num_prop #JordanAutomorphisms}
###### Definition/Proposition
**([[Jordan algebra]] [[automorphism group]] of [[octonion|octonionic]] [[Albert algebra]] is [[F4]])**

The [[group]] of [[automorphism]] with respect to the [[Jordan algebra]] structure $\circ$ on the [[octonion|octonionic]] [[Albert algebra]] is the [[exceptional Lie group]] $F_4$:

$$
  Aut\left(
    Mat_{3\times 3}^{herm}(\mathbb{O}), \circ
  \right)
  \;\simeq\;
  F_4
  \,.
$$

=--

(e.g. [Yokota 09, section 2.2](#Yokota09))

## Related concepts

* [[exceptional geometry]], [[exceptional generalized geometry]]

* [[G2]], **F4**,

  [[E6]], [[E7]],  [[E8]], [[E9]], [[E10]], [[E11]], $\cdots$

## References

### General

* [[José Figueroa-O'Farrill]], _A geometric construction of the exceptional Lie algebras F4 and E8_ ([arXiv:0706.2829](http://arxiv.org/abs/0706.2829))

* {#Yokota09} Ichiro Yokota, _Exceptional Lie groups_ ([arXiv:0902.0431](https://arxiv.org/abs/0902.0431))

Cohomological properties:

* Y. Cchoi, S. Yoon, _Homology of the triple loop space of the exceptional Lie group $F_4$_, J. Korean Math. Soc. 35 (1998), No. 1, pp. 149–164 ([pdf](http://www.mathnet.or.kr/mathnet/kms_tex/32004.pdf))

### In string theory

That the group $F_4$ controls the massless degrees of freedom of [[11-dimensional supergravity]] was observed and explored in

* T. Pengpan, [[Pierre Ramond]], _M(ysterious) Patterns in SO(9)_, Phys. Rept. 315:137-152,1999 ([arXiv:hep-th/9808190](http://arxiv.org/abs/hep-th/9808190))

* [[Pierre Ramond]], _Boson-Fermion Confusion: The String Path To Supersymmetry_, Nucl.Phys.Proc.Suppl. 101 (2001) 45-53 ([arXiv:hep-th/0102012](http://arxiv.org/abs/hep-th/0102012))

* [[Pierre Ramond]], _Algebraic Dreams_ ([arXiv:hep-th/0112261](http://arxiv.org/abs/hep-th/0112261))

* [[Hisham Sati]], _$\mathbb{O}P^2$-bundles in M-theory_, Commun. Num. Theor. Phys 3:495-530, 2009 ([arXiv:0807.4899](http://arxiv.org/abs/0807.4899))


Faa di Bruno formula

__Faà di Bruno formula__ is a remarkable combinatorial formula for higher derivatives of a composition of functions. There are various modern approaches to the related mathematics, using Joyal's theory of [[species]], operads, graphs/trees, [[combinatorial Hopf algebra]]s and so on.

### Literature

* [[Imma Gálvez-Carrillo]], [[Joachim Kock]], [[Andrew Tonks]], Groupoids and Faa di Bruno formulae for Green functions in bialgebras of trees [arxiv/1207.6404](http://arxiv.org/abs/1207.6404) (cf. also Kock's combinatorics [page](http://mat.uab.es/~kock/comb-pQFT.html))

> We prove a Faà di Bruno formula for the Green function in the bialgebra of P-trees, for any polynomial endofunctor P. The formula appears as relative homotopy cardinality of an equivalence of groupoids. For suitable choices of P, the result implies also formulae for Green functions in bialgebras of graphs.

* [[Doron Zeilberger]], _Toward a combinatorial proof of the Jacobian conjecture?_ in  _Combinatoire énumérative_ (Montreal, Que., 1985/Quebec, Que., 1985), 370–380, Lecture Notes in Math. __1234__, Springer 1986. [MR89c:05009](http://www.ams.org/mathscinet-getitem?mr=927775)
* Eliahu Levy, _Why do partitions occur in Faa di Bruno's chain rule for higher derivatives?_, [math.GM/0602183](http://arxiv.org/abs/math/0602183).
* E. Di Nardo, G. Guarino, D. Senato, _A new algorithm for computing the multivariate Faà di Bruno's formula_, [arxiv/1012.6008](http://arxiv.org/abs/1012.6008)
* Miguel A. Mendez, _Combinatorial differential operators in: Faà di
Bruno formula, enumeration of ballot paths, enriched rooted trees and
increasing rooted trees_, [arXiv:1610.03602](https://arxiv.org/abs/1610.03602)

In works of T. J. Robinson the formula is treated in the context of vertex algebras, calculus with [[formal power series]] and in logarithmic calculus, as well as in a connection to the [[umbral calculus]]:

* Thomas J. Robinson, _New perspectives on exponentiated derivations, the formal Taylor theorem, and Faà di Bruno's formula_, Proc.Conf.Vert.Op.Alg., Cont.Math. __497__ (2009) 185-198 [arxiv/0903.3391](http://arxiv.org/abs/0903.3391); _Formal calculus and umbral calculus_, Electronic Journal of Combinatorics, 17(1) (2010) R95 [arxiv/0912.0961](http://arxiv.org/abs/0912.0961)

### Faà di Bruno Hopf algebra

* Christian Brouder, Alessandra Frabetti, Christian Krattenthaler, _Non-commutative Hopf algebra of formal diffeomorphisms_, Adv. Math. __200__:2 (2006) 479-524 [pdf](http://www.sciencedirect.com/science/article/pii/S0001870805000265/pdf?md5=cb2367d0f3f8b07bf9799d61be34a595&pid=1-s2.0-S0001870805000265-main.pdf)

* Kurusch Ebrahimi-Fard, Frederic Patras, _Exponential renormalization_, Annales Henri Poincare 11:943-971,2010, [arxiv/1003.1679](http://arxiv.org/abs/1003.1679) [doi](http://dx.doi.org/10.1007/s00023-010-0050-7)

> Using Dyson's identity for Green's functions as well as the link between the Faà di Bruno Hopf algebra and the Hopf algebras of Feynman graphs, its relation to the composition of formal power series is analyzed.

* Hector Figueroa, Jose M. Gracia-Bondia, Joseph C. Varilly, _Faà di Bruno Hopf algebras_, article at Springer [[eom]], [math.CO/0508337](http://arxiv.org/abs/math/0508337)

* Jean-Paul Bultel, _Combinatorial properties of the noncommutative Faà di Bruno algebra_, J. of Algebraic Combinatorics 38:243–273 (2013) [MR3081645](http://www.ams.org/mathscinet-getitem?mr=3081645)

> We give a new combinatorial interpretation of the noncommutative [[Lagrange inversion]] formula, more precisely, of the formula of Brouder–Frabetti–Krattenthaler for the antipode of the noncommutative Faà di Bruno algebra.

[[!redirects Faà di Bruno Hopf algebra]]
[[!redirects Faà di Bruno formula]]
[[!redirects Faà di Bruno algebra]]


Fabian Hebestreit

* [webpage](http://www.math.uni-bonn.de/people/fhebestr/?language=en)

## Selected writings

On [[twisted K-theory]]:

* {#HebestreitSagave19} [[Fabian Hebestreit]], [[Steffen Sagave]], *Homotopical and operator algebraic twisted K-theory*, Mathematische Annalen **378** (2020) 1021-1059 ([arXiv:1904.01872](https://arxiv.org/abs/1904.01872), [doi:10.1007/s00208-020-02066-6](https://doi.org/10.1007/s00208-020-02066-6))

On the [[(∞,1)-Grothendieck construction]] via [[straightening and unstraightening]]:

* [[Fabian Hebestreit]], [[Gijs Heuts]], [[Jaco Ruit]], _A short proof of the straightening theorem_ ([arXiv:2111.00069](https://arxiv.org/abs/2111.00069)).

category: people


Fabian Meier

* [webpage](http://www.math.uni-bonn.de/people/fmeier/)

category: people


Fabian Ruehle

* [webpage](https://www2.physics.ox.ac.uk/contacts/people/ruehle)

## Selected writings

On [[heterotic line bundles]] seen in [[F-theory]] under [[duality between M/F-theory and heterotic string theory]]:

* [[Andreas Braun]], Callum Brodie, [[Andre Lukas]], [[Fabian Ruehle]], _NS5-Branes and Line Bundles in Heterotic/F-Theory Duality_, Phys. Rev. D 98, 126004 (2018) ([arXiv:1803.06190](https://arxiv.org/abs/1803.06190))

On [[machine learning]] the detection of [[unknots]]:

* [[Sergei Gukov]], [[James Halverson]], [[Fabian Ruehle]], [[Piotr Sułkowski]], *Learning to Unknot*, Mach. Learn.: Sci. Technol. **2** (2022) 025035 [[arXiv:2010.16263](https://arxiv.org/abs/2010.16263), [doi:10.1088/2632-2153/abe91f](https://iopscience.iop.org/article/10.1088/2632-2153/abe91f)]

category: people


Fabien Morel

**Fabien Morel** is a French [[algebraic geometry|algebraic geometer]] (currently a professor at LMU Munich). His work includes the construction

*   of $\mathbb{A}^1$-[[A1-homotopy theory|homotopy theory]] (two variants, one jointly with [[Vladimir Voevodsky]]),

* of [[algebraic cobordism]] with [[Marc Levine]].

and the study of [[profinite homotopy type|profinite homotopy]].

* [personal page](http://www.mathematik.uni-muenchen.de/~morel/)

## Selected writings

*  _Ensembles profinis simpliciaux et interprétation géométrique du foncteur $T$,_ Bull. Soc. Math. France 124 (1996), no. 2, 347--373.

* _Quelques remarques sur la cohomologie modulo p continue des pro-p-espaces et les résultats de J. Lannes concernant les espaces fonctionnels $Hom(BV,X)$_, Ann. Sci. École Norm. Sup. (4) 26 (1993), no. 3, 309--360.

On [[motivic homotopy theory]]:

* _An introduction to $\mathbb{A}^1$ homotopy theory_, ICTP Trieste July 2002 ([directory](http://publications.ictp.it/lns/vol15/vol15toc.html), [pdf](http://www.ictp.it/%7Epub_off/lectures/lns015/Morel/Morel.pdf), [ps](http://www.ictp.it/%7Epub_off/lectures/lns015/Morel/Morel.ps)).

* (with [[Vladimir Voevodsky]]), _$\mathbb{A}^1$-homotopy theory of schemes_ K-theory, 0305 ([web](http://www.math.uiuc.edu/K-theory/0305/) [pdf](http://www.math.uiuc.edu/K-theory/0305/nowmovo.pdf))

and [[h-cobordism]]:

* {#AsokMorel} [[Aravind Asok]], [[Fabien Morel]], _Smooth varieties up to $\mathbb{A}^1$-homotopy and algebraic h-cobordisms_, Adv. Math. **227** (5) (2011) 1990-2058 [[arXiv:0810.0324](http://arxiv.org/abs/0810.0324), [doi:10.1016/j.aim.2011.04.009](https://doi.org/10.1016/j.aim.2011.04.009)]

category: people
[[!redirects F. Morel]]


Fabienne Prosmans

* [webpage](http://www.anmath.ulg.ac.be/fp/)

category: people


Fabio Ferrari Ruffino

* [website](https://www.dm.ufscar.br/profs/ferrariruffino/)

* [old institute page](https://www.math.sissa.it/users/fabio-ferrari-ruffino)

## Selected writings

On [[bundle gerbes]] [[connections on bundle gerbes|with connection]] and [[Čech cohomology|Čech]]-[[Deligne cohomology]] for the description of [[B-fields]] in [[string theory]]:

* [[Loriano Bonora]], [[Fabio Ferrari Ruffino]], [[Raffaele Savelli]], *Classifying A-field and B-field configurations in the presence of D-branes*, JHEP 0812:078 (2008) [[arXiv:0810.4291](https://arxiv.org/abs/0810.4291), [doi:10.1088/1126-6708/2008/12/078](https://doi.org/10.1088/1126-6708/2008/12/078)]

* [[Fabio Ferrari Ruffino]], *Classifying A-field and B-field configurations in the presence of D-branes - Part II: Stacks of D-branes*, Nuclear Physics, Section B **858** (2012) 377-404 [[arXiv:1104.2798](https://arxiv.org/abs/1104.2798), [doi:10.1016/j.nuclphysb.2012.01.013](https://doi.org/10.1016/j.nuclphysb.2012.01.013)]

On [[twisted differential K-theory]] and its application to the [[K-theory classification of D-brane charge]]:

* [[Fabio Ferrari Ruffino]], [[Juan Carlos Rocha Barriga]], _Twisted differential K-characters and D-branes_, Nuclear Physics B Volume 960, November 2020, 115169 ([arXiv:2009.04223](https://arxiv.org/abs/2009.04223). [doi:10.1016/j.nuclphysb.2020.115169](https://doi.org/10.1016/j.nuclphysb.2020.115169))

category: people

[[!redirects Fabio Ruffino]]


Fabio Gadducci

* [personal page](https://pages.di.unipi.it/gadducci/)

* [GoogleScholar page](https://scholar.google.com/citations?user=ugYkdyYAAAAJ)

## Selected writings

On [[term]] [[rewriting]] and introducing the notion of *S-monoidal categories* (later called *[[gs-monoidal categories]]*) for this purpose:

* [[Fabio Gadducci]], _On the algebraic approach to concurrent term rewriting_, PhD thesis (1996) [[pdf](https://citeseerx.ist.psu.edu/document?repid=rep1&type=pdf&doi=1277313dd2c4fce640780b7472221156579e392f), [[Gadducci-AlgebraicRewriting.pdf:file]]]

* [[Andrea Corradini]], [[Fabio Gadducci]], _An algebraic presentation of term graphs, via gs-monoidal categories_, Applied Categorical Structures **7** (1999) 299-331 [[doi:10.1023/A:1008647417502](https://doi.org/10.1023/A:1008647417502)]

category: people


Fabio Pasquali

* [SemanticScholar page](https://www.semanticscholar.org/author/Fabio-Pasquali/2576030)

## Selected writings

On [[fibered category|fibrations]] of [[enriched groupoids]] as [[categorical semantics]] for [[identity types]] in [[homotopy type theory]]:

* [[Jacopo Emmenegger]], [[Fabio Pasquali]], [[Giuseppe Rosolini]], §3 in: *Elementary fibrations of enriched groupoids*, Mathematical Structures in Computer Science **31** 9 (2021) 958-978 [[doi:10.1017/S096012952100030X](https://doi.org/10.1017/S096012952100030X)]

category: people


Fabio Perroni

* [Institute page](https://perroni.dmg.units.it/)

* [Mathematics Genealogy page](https://www.genealogy.math.ndsu.nodak.edu/id.php?id=144531)

## Selected writings

On the [[Chen-Ruan cohomology|Chen-Ruan]] [[orbifold cohomology]] of [[ADE-singularities]]:

* [[Fabio Perroni]], *Orbifold Cohomology of ADE-singularities* ([arXiv:0510528](https://arxiv.org/abs/math/0510528))

* [[Fabio Perroni]], *Chen-Ruan cohomology of ADE-singularities*, International Journal of Mathematics, Vol. 18, No. 9 (2007) 1009-1059 ([arXiv:math/0605207](https://arxiv.org/abs/math/0605207), [doi:10.1142/S0129167X07004436](https://doi.org/10.1142/S0129167X07004436))

category: people


Fabio Riccioni

* [webpage](http://www.mth.kcl.ac.uk/staff/f_riccioni.html)

## Selected writings

On [[D=6 supergravity]]:

* [[Sergio Ferrara]], [[Fabio Riccioni]], [[Augusto Sagnotti]], _Tensor and Vector Multiplets in Six-Dimensional Supergravity_, Nucl. Phys. B519 (1998) 115-140 ([arXiv:hep-th/9711059](https://arxiv.org/abs/hep-th/9711059))

* [[Fabio Riccioni]], [[Augusto Sagnotti]], _Some Properties of Tensor Multiplets in Six-Dimensional Supergravity_, Nucl. Phys. Proc. Suppl. 67 (1998) 68-73 ([arXiv:hep-th/9711077](https://arxiv.org/abs/hep-th/9711077))

* [[Fabio Riccioni]], _All Couplings of Minimal Six-dimensional Supergravity_, Nucl. Phys. B605 (2001) 245-265 ([arXiv:hep-th/0101074](https://arxiv.org/abs/hep-th/0101074))

On [[black branes]]/[[solitonic branes|solitonic]] [[exotic branes]] and their [[double dimensional reduction]]:

* [[Eric A. Bergshoeff]], [[Fabio Riccioni]], *Solitonic branes and wrapping rules*, Phys. Part. Nuclei **43** (2012) 557–561 [[doi:10.1134/S106377961205005X](https://doi.org/10.1134/S106377961205005X)]

On [[defect branes]]:

* {#BershoeffOrtinRiccioni11} [[Eric Bergshoeff]], [[Tomas Ortin]], [[Fabio Riccioni]], _Defect Branes_, Nuclear Physics B **856** 2 (2012) 210-227 [[arXiv:1109.4484](https://arxiv.org/abs/1109.4484), [doi:10.1016/j.nuclphysb.2011.10.037](https://doi.org/10.1016/j.nuclphysb.2011.10.037)]

category: people


Fabio Ruffino

* [webpage](http://www.dm.ufscar.br/profs/ferrariruffino/)

## related $n$Lab entries

* [[D-brane charge]]

* [[Atiyah-Hirzebruch spectral sequence]]

category: people


Fabio Zanasi

* [website](http://www.zanasi.com/fabio/#/main.html)

### Selected writings

On [[string diagrams]] in [[computer science]]:

* [[Robin Piedeleu]], [[Fabio Zanasi]], _An Introduction to String Diagrams for Computer Scientists_ ([arXiv:2305.08768](https://arxiv.org/abs/2305.08768))

category: people


Fabrizio Gabbiani

* [GoogleScholar page](https://scholar.google.com/citations?user=njGtKXMAAAAJ&hl=en)

## Selected writings

On [[anyons]] and [[braid group statistics]] via [[algebraic quantum field theory]]:

* [[Jürg Fröhlich]], [[Fabrizio Gabbiani]], *Braid statistics in local quantum theory*, Reviews in Mathematical Physics, **2** 03 (1990) 251-353 $[$[doi:10.1142/S0129055X90000107](https://doi.org/10.1142/S0129055X90000107)$]$

and making explicit the role of the [[configuration space of points]]:

* {#FroehlichGabbianiMarchetti90} [[Jürg Fröhlich]], [[Fabrizio Gabbiani]], [[Pieralberto Marchetti]], *Braid statistics in three-dimensional local quantum field theory*, in: H.C. Lee (ed.) *Physics, Geometry and Topology* NATO ASI Series, **238** Springer (1990)  $[$[doi:10.1007/978-1-4615-3802-8_2](https://doi.org/10.1007/978-1-4615-3802-8_2), [[FroehlichGabbianiMarchetti-BraidStatistics.pdf:file]]$]$

category: people


FAC

* [[Jean-Pierre Serre]], _Faisceaux algébriques cohérents_, Ann. of Math. (2) 61, (1955). 197–278, [MR0068874](http://www.ams.org/mathscinet-getitem?mr=0068874), [doi](http://dx.doi.org/10.2307/1969915)

* English translation by Piotr Achinger and Łukasz Krupa: [pdf](http://achinger.impan.pl/fac/fac.pdf)

is considered one of the most important algebraic geometry articles of 20th century; its sheaf-theoretic setup has certainly influenced a lot Grothendieck in his creation of the theory of schemes.

category: reference

[[!redirects FAC]]
[[!redirects Faisceaux Algébriques Cohérents]]
[[!redirects Faisceaux algébriques cohérents]]


face

#Contents#
* table of contents
{:toc}

## Idea

A 2-[[cell]] in a [[cell complex]] or [[simplicial complex]] etc., such as a [[polyhedron]], is often called a _face_ (also _facet_ or _plaquette_, for instance in [[lattice gauge theory]]).

More generally one calls cells in the [[boundary]] of a [[cell complex]] faces, even if they are higher dimensional (for instance in the definition of [[simplices]] and [[simplicial sets]]).

## Related concepts

* [[edge]], [[vertex]]

* [[Euler characteristic of planar graphs]]

## References

See also

* Wikipedia, _Face (geometry)_

[[!redirects faces]]

[[!redirects facet]]
[[!redirects facets]]

[[!redirects plaquette]]
[[!redirects plaquettes]]


factor

The word _factor_ can have various meanings

* factor in a [[product]];

* factor in [[composition]] of [[morphisms]] in a [[category]]

* [[von Neumann algebra factor]]

* etc.

[[!redirects factors]]


factorial

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Arithmetic
+--{: .hide}
[[!include arithmetic geometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

For $k \in \mathbb{N}$ a [[natural number]], its _factorial_ $k! \in \mathbb{N}$ is the number obtained by [[multiplication|multiplying]] all positive natural numbers less than or equal to $k$:

$$
  k!
  \;\coloneqq\;
  1 \cdot 2 \cdot 3 \cdot 4 \cdot \cdots \cdot (k-1) \cdot k
  \,.
$$

In [[combinatorics]], the definition usually extends to $k = 0$ by setting $0! = 1$. This may be justified by defining $k!$ to be the number of [[permutations]] of a set with $k$ elements.

## Related concepts

* [[double factorial]]

* [[binomial coefficient]], [[multinomial coefficient]]

* [[Gamma function]]

* [[Euler beta function]]

* [[exponential series]]

## References

See also

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

[[!redirects factorials]]
[[!redirects factorial function]]


factorization algebra

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Higher algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
#### Quantum field theory
+--{: .hide}
[[!include functorial quantum field theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A _factorization algebra_ is an [[algebra over an operad]] where the [[operad]] in question is like the [[little disk operad]], but with each disk embedded into a given [[manifold]] $X$.

This way a factorization algebra is an assignment of a [[chain complex]] $V_D$ to each [[ball]] $D \subset X$ embedded in $X$, and for each collection of non-intersecting embedded balls $D_1 , \cdots, D_n \subset D \subset X$ sitting inside a bigger embedded ball $D$ in $X$ a morphism

$$
  V_{D_1} \otimes V_{D_2} \otimes \cdots \otimes V_{D_n} \to V_{D}
$$

such that composition of such operations is suitably respected.

In Euclidean ([[Wick rotation|Wick rotated]]) [[field theory]] factorization algebras serve to axiomatize the [[operator product expansion]].

## Definition

### Prefactorization algebra

+-- {: .num_defn}
###### Definition

For $X$ a [[topological space]] write $Fact_X$ be the colored [[operad]] in [[Set]] whose

* [[object]]s are the [[connected]] [[open subset]]s of $X$;

* the [[hom-set]] $Fact_X(\{U_i\}_i, V)$ is the singleton precisely if the
  $U_i$ are all in $V$ and are pairwise disjoint and is the [[empty set]] otherwise.

This specifies composition uniquely.

=--

+-- {: .num_defn #PrefactorizationAlgebra}
###### Definition

For $(C, \otimes)$ a [[symmetric monoidal category|symmetric monoidal]] [[abelian category]] let $End(C)$ be its [[endomorphism operad]]. A **prefactorization algebra** on $X$ with values in $C$ is an [[algebra over an operad]] over $Fact_X$ in $C$, hence a morphism of operads

$$
  \mathcal{F} : Fact_X \to End(C)
  \,.
$$

=--

These definitions appear in "[Factorization algebras in quantum field theory](https://web.archive.org/web/20150909210342/http://math.northwestern.edu/~costello/factorization.pdf)" by Costello and Gwilliam.

### Factorization algebras

+-- {: .num_defn #FactorizingCover}
###### Definition

For $X$ a [[topological space]] and $U \subset X$ an [[open subset]], a [[open cover]] $\{U_i \hookrightarrow U\}_{i \in I}$ is called a **factorizing cover** if for every [[finite set]] of points $\{x_1, \cdots, x_k\} \subset U$ there is a finite subset $\{U_{i_j}\}_{j \in J \subset I} $ of pairwise disjoint open subsets such that each point is contained in their union.

=--

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

Every [[Hausdorff space]] admits a factorizing cover.

=--

+-- {: .num_defn}
###### Notation

For a factorizing cover $\{U_i \to U\}_{i \in I}$ write $P I$ for the set of finite subsets $\alpha \subset I$ such that for $j,j' \in \alpha$ we have $U_j \cap U_{j'} = \emptyset$.

Given a [prefactorization algebra](#PrefactorizationAlgebra) $\mathcal{F}$ and $\alpha \in P I$ write

$$
  \mathcal{F}(\alpha) := \otimes_{j \in \alpha} F(U_j)
$$

and for $\alpha_1, \cdots, \alpha_k \in P I$ write

$$
  \mathcal{F}(\alpha_1, \cdots, \alpha_k) =
    \bigotimes_{(j_1, \cdots, j_k) \in \alpha_1 \times \cdots \times \alpha_k}
    \mathcal{F}(U_{j_1} \cap \cdots \cap U_{j_k})
  \,.
$$

For each $1 \leq i \leq k$ there is a canonical morphism

$$
  p_i : \mathcal{F}(\alpha_1,\cdots, \alpha_k)
  \to
   \mathcal{F}(\alpha_1, \cdots, \alpha_{i-1}, \alpha_{i+1}, \cdots, \alpha_k)
  \,.
$$

=--

+-- {: .num_defn}
###### Definition

A prefactorization algebra $\mathcal{F} : Fact_X \to End(C)$ is called a **factorization algebra** if for every open subset $U \subset X$ and every [factorizing cover](#FactorizingCover) $\{U_i \to U\}_{i \in I}$ the sequence

$$
  \bigoplus_{\alpha_1, \alpha_2 \in P I}
   \mathcal{F}(\alpha_1, \alpha_2)
  \stackrel{p_1 - p_2}{\to}
    \bigoplus_{\beta \in P I}
   \mathcal{F}(\beta)
   \to
   \mathcal{F}(U)
   \to 0
$$

is an [[exact sequence]].

=--

These definitions appear [here](http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.382.8322&rep=rep1&type=pdf).

See also at _[[cosheaf]]_.

### Homotopy factorization algebras

Let now $(C,\otimes)$ specifically be a [[category of chain complexes]].

+-- {: .num_defn}
###### Definition

A  [prefactorization algebra]{#PrefactorizationAlgebra} $\mathcal{F} : Fact_X \to End(X)$ is a **homotopy factorization algebra** if for all factorizing covers $\{U_i \to U \subset X\}_{i \in I}$ the canonical morpshim

$$
  \bigoplus_{k \geq 0} \bigoplus_{\alpha_1, \cdots, \alpha_k \in P I}
   \mathcal{F}(\alpha_1, \cdots, \alpha_k)[k-1]
   \to
  \mathcal{F}(U)
$$

is a [[quasi-isomorphism]], where the [[differential]] on the left is defined by (...).

=--

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

This is the analogue of a [[descent]] condition for [[simplicial presheaves]].

=--

These definitions appear [here](http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.382.8322&rep=rep1&type=pdf).

## Examples

* [[beta-gamma system]]

## Related concepts

Factorization algebras have some similarity with

* [[Hochschild homology]];

* [[blob homology]], [[factorization homology]], [[topological chiral homology]];

* [[local net]]s in [[AQFT]].

* [[chiral algebra]], [[vertex operator algebra]]

* [[nonabelian Poincaré duality]]

[[!include Isbell duality - table]]

## References

The notion of factorization algebra may be regarded as a slight variation on the concept _[[chiral algebra]]_ originally introduced in

* [[Alexander Beilinson]], [[Vladimir Drinfeld]],  _[[Chiral Algebras]]_.

A definition formulated genuinely in  [[Higher Algebra]] appears in section 4.1 _Topological Chiral Homology_ of

* [[Jacob Lurie]], _[[On the Classification of Topological Field Theories]]_

This discusses how locally constant factorization algebras obtained from [[En-algebras]] induce [[extended quantum field theory|extended]] [[FQFTs]].

A fairly comprehensive account of factorization algebras as a formalization of [[perturbation theory|perturbative]] [[quantum field theory]] (see at _[[factorization algebra of observables]]_) is in

* [[Kevin Costello]], [[Owen Gwilliam]], _[[Factorization algebras in perturbative quantum field theory]]_

* [[Owen Gwilliam]], _Factorization algebras and free field theories_ PhD thesis ([[GwilliamThesis.pdf:file]])

* [[Owen Gwilliam]], [[Kasia Rejzner]], _Comparing nets and factorization algebras of observables: the free scalar field_, [arxiv:1711.06674](https://arxiv.org/abs/1711.06674)

and the beginnings of

* {#Costello11} [[Kevin Costello]], *Notes on supersymmetric and holomorphic field theories in dimension 2 and 4*, talk at *[Geometric and Algebraic Structures in Mathematics](https://www.math.stonybrook.edu/dennisfest/)*, Stony Brook (2011) [[arXiv:1111.4234](https://arxiv.org/abs/1111.4234)]

* [[Kevin Costello]], [[Claudia Scheimbauer]], _Lectures on mathematical aspects of (twisted) supersymmetric gauge theories_, pp. 57-88 in: Mathematical aspects of QFTs, D. Calaque, T. Strobl editors, Springer 2015

Lecture notes:

* [[Kevin Costello]] (with [[Owen Gwilliam]]), _Factorization algebras in perturbative quantum field theory_ in [[Oberwolfach Workshop, June 2009 -- Strings, Fields, Topology|Strings, Field, Topology]], Oberwolfach report No 28, 2009 ([pdf](http://www.mfo.de/programme/schedule/2009/24/OWR_2009_28.pdf#page=12))

  This can also be found mentioned in the talk notes of the [[Northwestern TFT Conference 2009]], see in particular

  * notes by  [[Christoph Wockel]], [[costello.pdf:file]]

  * notes by Evan Jenkins on the same talk: [Factorization algebras in perturbative quantum gravity](http://www.math.uchicago.edu/~ejenkins/notes/nwtft/25may-kc.pdf)

* [[Araminta Amabel]], *Notes on Factorization Algebras and TQFTs* [[arXiv:2307.01306](https://arxiv.org/abs/2307.01306)]

  > (with relation to [[functorial field theory]])

There seems to be a close relation between the description of [[quantum field theory]] by factorization algebras and the proposal presented in

* [[Stefan Hollands]], ([arXiv:0802.2198](http://arxiv.org/abs/0802.2198))

The relation of locally constant factorization algebras to higher order [[Hochschild homology]] is in

* [[Gregory Ginot]], [[Thomas Tradler]], [[Mahmoud Zeinalian]], _Derived higher Hochschild homology, topological chiral homology and factorization algebras_, [arxiv/1011.6483](http://arxiv.org/abs/1011.6483)

* [[Gregory Ginot]], _Notes on factorization algebras, factorization homology and applications_, [arxiv/1307.5213](https://arxiv.org/abs/1307.5213)

A comparison with FQFT for TFTs is presented in

* C. Scheimbauer, _A factorization view on states/observables in topological field theories_ [youtube](https://www.youtube.com/watch?v=nN-qtnbtpW4) 19 min, string-math 2017, Hamburg

An [[(infinity,1)-category theory|(infinity,1)-category theoretic ]] treatment of higher factorization algebras is in

* [[Dennis Gaitsgory]], [[John Francis]] _Chiral Koszul duality_ ([arXiv:1103.5803](http://arxiv.org/abs/1103.5803))
 {#GaitsgoryFrancis}

A construction of [[chiral differential operator]]s via quantization of $\beta\gamma$ system in [[BV formalism]] with an intermediate step using factorization algebras:

* Vassily Gorbounov, Owen Gwilliami, Brian Williams, _Chiral differential operators via Batalin-Vilkovisky quantization_, [pdf](http://people.mpim-bonn.mpg.de/gwilliam/cdo.pdf)
* Brian Williams, _The Virasoro vertex algebra and factorization algebras on Riemann surfaces_, Lett. Math. Phys. __107__:12, 2189–2237 (2017) [doi](https://dx.doi.org/10.1007/s11005-017-0982-7)

A version of bosonic string theory related to factorization algebras is presented in

* Owen Gwilliam, Brian Williams, _The holomorphic bosonic string?, [arxiv/1711.05823](https://arxiv.org/abs/1711.05823)

A higher-dimensional generalization of vertex algebras is suggested in the framework of factorization algebras in

* Emily Cliff, _Universal factorization spaces and algebras_, [arxiv/1608.08122](http://arxiv.org/abs/1608.08122)

>  We introduce categories of weak factorization algebras and factorization spaces, and prove that they are equivalent to the categories of ordinary factorization algebras and spaces, respectively. This allows us to define the pullback of a factorization algebra or space by an \'etale morphism of schemes, and hence to define the notion of a universal factorization space or algebra. This provides a generalization to higher dimensions and to non-linear settings of the notion of a vertex algebra.

Relation to [[homotopical algebraic quantum field theory]]:

* [[Donald Yau]], _Homotopical quantum field theory_ [[arxiv/1802.08101](https://arxiv.org/abs/1802.08101)]

* [[Marco Benini]], Giorgio Musante, [[Alexander Schenkel]], *Quantization of Lorentzian free BV theories: factorization algebra vs algebraic quantum field theory* [[arXiv:2212.02546](https://arxiv.org/abs/2212.02546)]

[[conformal net|Conformal nets]] exhibit factorization algebras

* [[Andre Henriques]], _Conformal nets are factorization algebras_, [arXiv:1611.05529](https://arxiv.org/abs/1611.05529)

[[!redirects factorization algebras]]


factorization algebra of observables

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Higher algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
#### Quantum field theory
+--{: .hide}
[[!include functorial quantum field theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

Given a [[local action functional]] on [[field (physics)|fields]] which are [[sections]] of a suitable [[field bundle]], [[perturbation theory]] and [[renormalization]] allow to assign to [[open subsets]] of [[spacetime]]/[[worldvolume]] the corresponding [[quantum BV-complexes]]/[[Beilinson-Drinfeld algebras]]. This assignment is canonically equipped with the structure of a [[factorization algebra]], the _factorization algebra of observables_ of the [[theory (physics)|theory]].

This formalization of [[algebraic quantum field theory]] is similar to, but a bit different from, the notion of _[[local net of observables]]_.

## Examples

* Over a Riemann surface: [[vertex operator algebra]] ([Gwilliam, section 6.3, section 6.5](#Gwilliam))

* (...)

## Related concepts

[[!include Isbell duality - table]]

* [[chiral algebra]], [[topological chiral homology]]

## References

A fairly comprehensive account of factorization algebras as a formalization of [[perturbation theory|perturbative]] [[quantum field theory]] (see at _[[factorization algebra of observables]]_) is in

* [[Kevin Costello]], [[Owen Gwilliam]], _[[Factorization algebras in perturbative quantum field theory]]_

* {#Gwilliam} [[Owen Gwilliam]], _Factorization algebras and free field theories_ PhD thesis ([[GwilliamThesis.pdf:file]])

and the beginning of

* [[Kevin Costello]], *Notes on supersymmetric and holomorphic field theories in dimension 2 and 4*, talk at *[Geometric and Algebraic Structures in Mathematics](https://www.math.stonybrook.edu/dennisfest/)*, Stony Brook (2011) [[arXiv:1111.4234](https://arxiv.org/abs/1111.4234)]

Lecture notes include

* [[Kevin Costello]] (with [[Owen Gwilliam]]), _Factorizaton algebras in perturbative quantum field theory_ in [[Oberwolfach Workshop, June 2009 -- Strings, Fields, Topology|Strings, Field, Topology]], Oberwolfach report No 28, 2009 ([pdf](http://www.mfo.de/programme/schedule/2009/24/OWR_2009_28.pdf#page=12))

  This can also be found mentioned in the talk notes of the [[Northwestern TFT Conference 2009]], see in particular

  * notes by  [[Christoph Wockel]], [[costello.pdf:file]]

  * notes by Evan Jenkins on the same talk: [Factorization algebras in perturbative quantum gravity](http://www.math.uchicago.edu/~ejenkins/notes/nwtft/25may-kc.pdf)

[[!redirects factorization algebras of observables]]

[[!redirects factorization algebra of local observables]]
[[!redirects factorization algebras of local observables]]


+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Quantum field theory
+--{: .hide}
[[!include functorial quantum field theory - contents]]
=--
#### Physics
+--{: .hide}
[[!include physicscontents]]
=--
=--
=--

This entry provides hyperlinks for the text

* [[Kevin Costello]], [[Owen Gwilliam]],

  _Factorization algebras in perturbative quantum field theory_

  ([Volume 1](https://people.math.umass.edu/~gwilliam/vol1may8.pdf), [Volume 2](https://people.math.umass.edu/~gwilliam/vol2may8.pdf))

on the description of [[perturbation theory|perturbative]] [[quantum field theory]] via [[factorization algebras of observables]] (see there for more details). Vol. 1 appeared in series New Mathematical Monographs, vol. 31, Cambridge University Press (2016).

The construction proceeds by a refined (spacetime-local) form of [[deformation quantization]] via [[BV-complexes]]. For [[renormalization]] of [[interaction]] terms it relies on methods in

* [[Kevin Costello]], _[[Renormalization and Effective Field Theory]]_.

Some of the notions in the text are inspired or influenced by

* [[Alexander Beilinson]], [[Vladimir Drinfeld]], _[[Chiral Algebras]]_

For further related publications see the references at _[[factorization algebra]]_.

category: reference


factorization category

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

# Factorization categories
* table of contents
{: toc}

## Idea

The _factorization category_ (also called the _interval category_) $Fact(f)$ of a [[morphism]] $f$ in a [[category]] $C$ is a way of organizing its binary factorizations $f = p\circ q$ into a category.

## Definition

The [[objects]] of $Fact(f)$ are factorizations

\[
\begin{matrix}
  X &&\stackrel{f}{\to}&& Y
       \\
       & {}_{p_1} \searrow && \nearrow_{p_2}
       \\
       && D
\end{matrix}
\]

so that $f = p_2 p_1$, and a [[morphism]] from $(p_1, D, p_2)$ to $(q_1, E, q_2)$ is a morphism $h \colon D \to E$ making everything in sight commute. There's an obvious projection [[functor]]
\[
  P_f \colon Fact(f) \to C
\]
which maps $(p_1, D, p_2)$ to $D$ and $h\colon (p_1, D, p_2) \to (q_1, E, q_2)$ to $h$.

### As iterated comma categories

In terms of [[slice categories]], a morphism $f: A \to B$ can be viewed as

1. an object in $C/B$
2. or an object in $A / C$

Now, taking over/under slices again yields only one new thing; it is easy to see that

* $(C/B)/f \cong C/A$, and
* $f / (A / C) \cong B / C$

the cool fact is that the two other options yield $Fact(f)$

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

$Fact(f) \cong f/(C/B) \cong (A / C)/f$, and the following [[diagram]] [[commuting diagram|commutes]]
\[
\array{
    (A / C)/f &\stackrel{\cong}{\to}&   Fact(f)  & \stackrel{\cong}{\leftarrow} & f/(C/B)
    \\
    \pi^A_f \downarrow && P_f \downarrow && \pi^B_f
    \\
    A / C & \underset{\pi_A}{\to} & C & \underset{\pi_B}{\leftarrow} & C/B
  }
\]
=--

+-- {: .query}
[[Eduardo Pareja-Tobes]]: This should follow from properties of comma objects; I could add  here the proof from Lawvere-Menni paper below, but I think it would be better to have more conceptual proof
=--

## Properties

### Characterization in terms of initial and terminal objects

There is a fairly simple characterization of the categories arising as factorization categories of some $f$ in a category $C$. First of all, note that $Fact(f)$ always has

* an [[initial object]] $f = f\circ id$
* a [[terminal object]] $f = id \circ f$

conversely, for any category $D$ with initial and terminal objects $0, 1$ denoting the unique morphism $! \colon 0 \to 1$ we have that
\[
\pi_! \colon Fact(!) \to D
\]
is an [[equivalence]]. We get then

> a category is equivalent to some $Fact(f)$ iff it has initial and terminal objects

### Factorization categories vs the category of factorizations

We can view $Fact(f)$ as a full [[reflective subcategory]] of the [[over-category]] $tw(C) / f$; here $f$ is viewed as an object of the [[category of factorizations]] $tw(C)$ of its ambient category $C$. There's a functor

\[
  U_f \colon Fact(f) \to tw(C) / f
\]

which on objects is

\[
  U_f(p_1, p_2) =
\begin{matrix}
  X              & \overset{1_X}{\leftarrow}   & X            \\
  p_1\downarrow  &                           & \downarrow f \\
  D              & \underset{p_2}{\to}       & Y
\end{matrix}

\]

and on arrows $U(h) = (h, id)$.

This functor has a _[[left adjoint|left]]_ [[adjoint functor|adjoint]]

\[
  F_f \colon tw(C)/f \to Fact(f)
\]

* $F_f$ on objects:
\[
  F_f\left(\,
\begin{matrix}
  A             & \overset{h}{\leftarrow}   & X            \\
  g \downarrow  &                           & \downarrow f \\
  D             & \underset{q}{\to}         & Y
\end{matrix}
\, \right) \, = \quad
\begin{matrix}
  X &&\stackrel{f}{\to}&& Y
  \\
  & {}_{gh} \searrow && \nearrow_{q}
  \\
  && D
\end{matrix}
\]

* $F_f$ on arrows: picks the morphism which goes between $D$ and $D'$.

It is immediate to check that $F_f \circ U_f = 1_{Fact(f)}$.

## References

* [[Bill Lawvere]], [[Matias Menni]], The Hopf algebra of Möbius
intervals, _Theory and Applications of Categories_, 24:10 (2010), 221-265. ([tac](http://www.tac.mta.ca/tac/volumes/24/10/24-10abs.html))
* B. Klin, [[Vladimiro Sassone]], P. Sobocinski, _Labels from reductions: Towards a general theory,_ Algebra and coalgebra in computer science: first international conference, CALCO 2005

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[[!redirects factorisation category]]
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[[!redirects factorisation categories]]
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[[!redirects interval categories]]

category:computer science