Green-Schwarz mechanism

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#Contents#
* table of contents
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## Idea

A **Green-Schwarz mechanism** (named after [[Michael Green]] and [[John Schwarz]]) is a modification of an [[action functional]] of a [[quantum field theory]] involving higher [[gauge field]]s that makes a [[quantum anomaly]] of the original action functional disappear.

More in detail:

* An [[action functional]] in [[path integral|path integral quantization]] is said to be [[quantum anomaly|anomalous]] if it is only locally identified with a function on the configuration space of fields, but is globally instead a section of a line [[bundle]] (usually equipped with [[connection on a bundle|connection]]).

* Given two anomalous action functionals in this sense, it may happen that while the two corresponding line bundles on configuration space are each nontrivial, their [[tensor product]] becomes trivializable. In this case one can consider the non-anomalous action function given by the sum of the two anomalous action functionals. This is what is called **anomaly cancellation** of one piece of an action functional against another.

* The two main sources of examples for action functionals that are anomalous are

  * the standard [[fermionic action functional]] (see there for details) for chiral fermions

    * this is a section of the [[Pfaffian line bundle]] for the
      [[Dirac operator]]

  * the action functional for [[differential cohomology|differential cocycles]] (higher connection, higher [[gauge theory]]) in the presence of [[electric charge|electric]] and [[magnetic charge]]s:

    * this is a [[section]] of the [[transgression]] of the
      [[line bundle]] arising as the [[cup product]] of
      the [[electric charge|electric]] and
      [[magnetic charge|magnetic differential cocycles]].

A **Green--Schwarz mechanism** is the addition of an action functional for higher differential cocycles with [[magnetic charge]]s such that their [[quantum anomaly]] cancels a given [[Pfaffian line bundle]]: so it is a choice of by itself ill-defined action functional for higher gauge theory that cancels the ill-definedness of an action functional for chiral fermions.

In the more strict and original sense of the word, _the_ Green--Schwarz mechanism is the application of this procedure in the theory called [[heterotic string theory|heterotic supergravity]]: there it so happens that the Pfaffian line bundle of the fermionic action has as [[Chern class]] the [[transgression]] of a degree-12 class in [[ordinary differential cohomology]] that factorizes as $I_8 \wedge I_4$. Since heterotic supergravity contains a higher gauge field that couples to [[string theory|strings]], this is precisely of the form $J_{electric} \wedge J_{magnetic}$ that the anomaly for the corresponding higher gauge theory in the presence of magnetic charges gives rise to. So the original Green--Schwarz anomaly cancellation mechanism consist of modifying the "naive" action functional for heterotic supergravity by adding the contribution that corresponds to adding a magnetic current of the form

$$
  j_B \,\coloneqq\, I_4
  \,.
$$

## The higher magnetic charge anomaly

### The higher abelian Yang-Mills action functional...
  {#YMactionfunctional}

Consider on some [[spacetime]] $X$ a [[gauge field]] $[\nabla] \in H_{diff}^{n+1}(X)$ modeled in [[ordinary differential cohomology]] in degree $n+1$: a [[circle n-bundle with connection]]. For instance

* in degree $n=1$ this is an [[electromagnetic field]];

* in degree $n=2$ this is a [[Kalb-Ramond field]];

* in degree $n = 3$ a [[supergravity C-field]].

Canonically associated to the [[gauge field]] is its [[field strength]]: the [[curvature]] [[differential form]]

$$
  F_\nabla \in \Omega^{n+1}_{cl}(X)
  \,.
$$

The abelian [[Yang-Mills action functional]] for our gauge field (the [[action functional]] of higher order [[electromagnetism]]) is the [[function]]

$$
  \exp\big(
    \mathrm{i} S_{YM}(-)
  \big)
  \;\colon\;
  H_{diff}^{n+1}(X) \to \mathbb{C}
$$

that sends the field $\nabla$

$$
  \exp\big( \mathrm{i} S_{YM}(-)\big)
  \;\colon\;
  [\nabla]
   \mapsto
  \exp\big(
    - \mathrm{i}
    \textstyle{\int}_X F_\nabla \wedge \star F_\nabla
  \big)
$$

to the [[exponential]] of the [[integral]] over the [[spacetime]] $X$ of the  [[differential form]] obtained as the [[wedge product]] of the [[curvature form]] with its image under the [[Hodge star operator]] correspondonding to the [[pseudo-Riemannian manifold|pseudo-Riemannian metric]] on $X$.

The fact that this [[map]] defined in terms of [[cocycles]] is a well defined function on [[cohomology]] means that the [[action functional]] is [[gauge transformation|gauge invariant]]. At this point this is just the trivial statement that under a [[gauge transformation]]

$$
  \nabla \stackrel{g}{\to} \nabla'
$$

the [[curvature]] is invariant (due to the [[infinity-group|higher]] [[gauge group]] being [[abelian infinity-group|abelian]]): $F_\nabla = F_{\nabla'}$.

### ... with electric charge ...
  {#WithElectricCharge}

The [above](#YMactionfunctional) [[action functional]] describes the dynamics of the [[gauge field]] all by itself, with no interactions with other fields or with [[relativistic particle|fundamental particles]]/[[brane|fundamental branes]].

A distribution of  $n$-[[electric charge]] on $X$ is modeled itself by a [[cocycle]] $\hat j_E$ in [[ordinary differential cohomology]] in degree $dim X - n$

$$
  [\hat j_E]
    \;\in\;
  H_{diff}^{dim X - n}(X)
  \,,
$$

so that the [[curvature]] $j_E$ of $\hat j_E$ is the [[electric current]] form. The [[action functional]] that encodes the [[Lorentz force]] of the [[gauge field]] exerted on this electric charge distribution is locally on [[coordinate charts]] $U \subset X$ given by the [[integral]] $\int_X A_U \wedge j_E$, where $A_U$ is the local [[connection on an infinity-bundle|connection]] $n$-form of the gauge field $\nabla$.

Globally, this contribution is given by the push-forward

$$
  2 \pi \mathrm{i}
  \textstyle{\int}_X (-)
  \;\colon\;
  H_{flat}^{dim X + 1}(X) \to H_{flat}^1(*)
  \;=\;
  U(1)
  \,.
$$

of the [[cup product]] $\hat j_E \cdot \nabla$ in [[ordinary differential cohomology]].

In total then, the [[action functional]] of higher abelian [[Yang-Mills theory]] in the presence of [[electric charge]] is the [[function]]

$$
  \exp\big(
    \mathrm{i}
    S_{YM}(-)
      +
    \mathrm{i}
    S_{el}(-)
  \big)
    \;\;\colon\;\;
  H^{n+1}_{diff}(X)
   \times
  H^{dim X - n}_{diff}(X)
   \longrightarrow
  \mathbb{C}
$$

given by

$$
  \big(
    [\nabla], [\hat j_E]
  \big)
     \;\mapsto\;
  \exp\big(
    \mathrm{i}
    \textstyle{\int}_X F_\nabla \wedge \star F_\nabla
  \big)
  \,
  \exp\big(
    2 \pi \mathrm{i} \int_X \hat j_E \cdot \nabla
  \big)
  \,.
$$

[cf. [Freed 2002 (2.17)](#Freed00)]

### ... and with magnetic charge.

We now consider one more additional term in the [[action functional]], one that describes moreover the interaction of our [[gauge field]] with a distribution of $n$-[[magnetic charge]] on $X$, in addition to the interaction with the distribution of [[electric charge]] described
[above](#WithElectricCharge).

The [[magnetic charge]] distribution itself is also modeled as a  [[cocycle]] $\hat j_B$ in [[ordinary differential cohomology]]. As opposed to the [[electric charge]] it is however not part of the dynamics but of the kinematics of the system: it does not manifestly show up in the integral expression for the [[action functional]], but does modify the nature of the configuration space that this action functional is defined on.

Namely the kinematic higher [[Maxwell equations]] is a condition of the form

$$
  \mathrm{d} F_\nabla
    \;=\;
  j_B
  \,,
$$

where $j_B$ is the [[magnetic charge]] distribution. If $F_\nabla$ is the [[curvature]] of a [[circle n-bundle with connection]], then necessarily $d F_\nabla = 0$. Therefore the system of higher electromagnetism in the presence of magnetic charge cannot be modeled any more by [[cocycles]] in [[ordinary differential cohomology]].

One finds instead that one has to model $\nabla$ not as a [[circle n-bundle with connection]], but as an $n$-[[twisted bundle]] with connection, where the [[twisted cohomology|twist]] is $\hat j_B$.

We shall write $C^{n+1}_{diff}(X)_{\hat j_B}$ for the [[∞-groupoid]] of twisted cocycles for this fixed twist. The crucial point is now the following:

the [above](#WithElectricCharge) expression

$$
  \exp\big(
    \mathrm{i}
       S_{el}(\nabla, \hat j_E)
  \big)
    \;\colon\;
  \exp\big(
    2 \pi \mathrm{i}
    \textstyle{\int}_X \hat j_E \cdot \nabla
  \big)
$$

for the electric coupling can still be given sense, even with $\nabla \in C^{n+1}_{diff}(X)_{\hat j_B}$, but it no longer has the interpretation of a [[circle group]]-valued [[function]]. Rather, it has now the interpretation of a [[section]] of a [[line bundle]]

\begin{tikzcd}[sep=30pt]
  &
  \mathrm{Anom}_{\hat j_B}
  \ar[d, ->>]
  \\
  \mathrm{Conf}
  \ar[r, equals]
  \ar[
    ur,
    dashed,
    "{
      \exp(\mathrm{i} S_{el})
    }"
  ]
  &
  \mathrm{Conf}
\end{tikzcd}

on configuration space. The [[characteristic class]] of this line bundle -- its  [[Chern class]] -- is hence the _magnetic anomaly_ in higher gauge theory.

[cf. [Freed 2002 (2.29)](#Freed00)]

In the next section we formalize properly the notion of this line bundle on configuration space.

### The anomaly line bundle

In order to formalize this we have to refine the formalization of the structure of the configuration space. So far we had regarded the [[set]] $H^{dim X - n}_{diff}(X) \times H_{diff}^{n+1}(X)$ of gauge equivalence classes of field configurations. This is the set of connected components of the full [[cocycle]] [[∞-groupoid]]

$$
  C^{dim X - n}_{diff}(X) \times C_{diff}^{n+1}(X)
  \in
  \infty Grpd
$$

whose

* [[objects]] are field configurations on $X$;

* [[morphisms]] are [[gauge transformations]];

* [[2-morphisms]] are gauge transformations of gauge transformation,

* and so on.

Moreover, this cocycle [[∞-groupoid]] is not just a [[discrete ∞-groupoid]] but it naturally has _smooth structure_ making it a [[smooth ∞-groupoid]]: an [[∞-stack]] over the [[category]] [[SmoothMfd]]. We shall write

$$
  Conf
  \;\coloneqq\;
  \Big[
    X
    ,\,
    \big(
      \mathbf{B}^{n}U(1)
        \times
      \mathbf{B}^{dim X - n-1}U(1)
    \big)_{conn}
  \Big]
  \;\in\;
  SmoothGrpd_{\infty}
$$

for this smooth $\infty$-groupoid of configurations of the physical system -- defined as the [[internal hom]] in terms of the [closed monoidal structure on the (∞,1)-topos](infinity1-topos#ClosedMonoidalStructure)
[[Smooth∞Grpd]] of $X \in SmoothMfd \hookrightarrow Smooth\infty Grpd$ into the target object of the higher [[gauge theory]], (this object is discussed in detail here; it is presented under the [[Dold-Kan correspondence]] by the [[Deligne complex]] of sheaves on [[CartSp]]).

This smooth structure is characterized by saying that for any $U \in$ [[SmoothMfd]] the $U$-parameterized smooth families of field configurations, gauge transformations, etc. form the [[∞-groupoid]]

$$
  Conf(U)
   \simeq
  C^{dim X - n}_{diff}(U \times X) \times C_{diff}^{n+1}(U \times X )
$$

of gauge fields on the [[product]] of [[spacetime]] $X$ with the parameter space $U$. (See for instance [[Lie integration]] and [[connection on an ∞-bundle]] for details on how differential forms on $U \times X$ encode $U$-families of forms on $X$).

This way the configuration space of higher electromagnetism in the presence of electric and magnetic charge is naturally incarnated as an object in the [[cohesive (∞,1)-topos]] of [[smooth ∞-groupoid]]s, and accordingly all the differential geometric structures in cohesive $(\infty,1)$-topos are available. In particular we may speak of [[circle n-bundle with connection|line bundles with connection]] on $Conf$, given for instance by morphisms

$$
  Anom_{\hat j_B}
  \;\colon\;
  Conf \to \mathbf{B} U(1)_{conn}
  \,.
$$

in [[Smooth∞Grpd]].

We say that:

* the underlying class in [[ordinary cohomology]]

  $$
    [Anom_{\hat j_B}] \in H^2(Conf, \mathbb{Z})
  $$

  is **the anomaly** of the system of higher electromagnetism coupled to electric and magnetic charge;

* its [[curvature]] 2-form

  $$
    Curv_{Anom_{\hat j_B}}
    \;\colon\;
    Conf \to \Omega^2_{flat}
  $$

  is the **differential anomaly**.

One finds that this curvature 2-form is given by the [[fiber integration]] of the wedge product of the [[electric current]] $(n+1)$-form with the [[magnetic charge]] $dim X - n$-form over $X$:

$$
  Curv_{Anom_{\hat j_B}}
   \,=\,
  \textstyle{\int}_X
  \,
  j_E \wedge j_B
  \,.
$$

This means that for every parameter space $U \in $ [[SmoothMfd]] and every morphism $\phi : U \to Conf$ -- which corresponds by the nature of the [[∞-stack]] $Conf$ to a field configuration $(\nabla, \hat j_E) \in C^{n+1}_{diff}(U \times X) \times C^{dim X - n}_{diff}(U \times X)$ -- the pullback of this differential form to $U$ yields the ordinary differential form $\int_X j_E \wedge j_B$ which is the image of $(\nabla, \hat j_E)$ under the [[fiber integration]] map

$$
  \textstyle{\int}_X(-)
  \;\colon\;
  \Omega^\bullet(U \times X) \to \Omega^\bullet(U)
  \,.
$$

### The Green-Schwarz mechanism

We can now state the **Green-Schwarz mechanism** itself.

Let $\hat Conf \in $ [[Smooth∞Grpd]] be the configuration space of a physical system that contains among its fields higher abelian gauge theory with [[electric charge]] with configuration space $Conf$

$$
  \hat Conf \,=\, Conf_{rest} \times Conf
$$

and equipped with an [[action functional]]

$$
  \exp\big(
    \mathrm{i} S_{rest}(-) + \mathrm{i} S_{el}(-)
  \big)
  \;\colon\;
  \hat Conf \to Anom_{rest}
$$

that is a [[section]] of an anomaly line bundle $Anom_{rest}$

$$
  \array{
    && Anom_{rest}
    \\
     & {}^{\mathllap{\exp(S_{tot})}}\nearrow & \downarrow
    \\
    Conf_{rest} &=& Conf_{rest}
  }
$$

such that the [[curvature]] 2-form of $Anom_{tot}$ happens to be of the form

$$
  Curv{Anom_{ref}}
  \,=\,
  \textstyle{\int}_X I_{n+2} \wedge j_{el}
  \,,
$$

for some $I_{n+2} \in \Omega^{n+2}_{cl}(X)$ (understood, as explained [above](https://ncatlab.org/nlab/show/Green-Schwarz+mechanism#the_anomaly_line_bundle), as forms in $\Omega^{n+2}_{cl}(X\times U)$) and
$j_{el} \in \Omega^{dim X - n}(X)$.

Then the **Green-Schwarz mechanism** is the map that changes this physical system by adding magnetic charge to it, given by a cocycle  $\hat j_B$ with

$$
  [\hat j_B]
  \,=\,
  - [Anom_{rest}]
$$

$$
  j_B
  \,=\,
  - I_{n+2}
  \,.
$$

This means, by the above, that the new [[action functional]] is now a section

$$
  \exp\big(
    \mathrm{i} S_{rest}(-) + \mathrm{i} S_{el}(-)
  \big)
  \,\colon\,
  Conf_{rest} \times Conf
    \to
  Anom_{rest} \otimes Anom_{\hat j_B}
$$

of the [[tensor product]] of the two anomaly line bundles.
The [[first Chern class]] of the tensor product is the sum of the two 1st Chern-classes, hence by definition of $j_B$ they cancel, so that $Anom_{rest} \otimes Anom_{\hat j_B}$ is trivializatable as a line bundle with connection.

A choice of such trivialization identifies the section then with an ordinary function

$$
  \exp\big(
    \mathrm{i} S_{rest}(-) + \mathrm{i} S_{el}(-)\big)
  \;\colon\;
  Conf_{rest} \times Conf \to U(1)
  \,.
$$

This is the anomaly-free action functional after the Green-Schwarz mechanism has been applied.

## Examples

### Heterotic supergravity

The original work of Green-Schwarz concerned anomaly cancellation in the effective [[supergravity]] theory on a $dim X = 10$-dimensional target [[spacetime]] in [[heterotic string theory]].

The configurations of this theory are given by

* a [[Spin]]-[[principal bundle]] [[connection on a bundle|with connection]] $\hat \omega$ -- the [[spin connection]]
  (the [[graviton]]);

* a [[unitary group]]-[[principal bundle]] [[connection on a bundle|with connection]] $\hat A$ (the [[Yang-Mills field]]);

* a [[circle n-bundle with connection|circle 2-bundle with connection]] $\hat B$ -- the [[Kalb-Ramond field]].

* [[section]]s of the [[associated bundle|associated]] [[spinor bundle]]s: the fermions $\psi$ ([[gravitino]], [[gaugino]], [[dilatino]]).

The [[path integral]] over the fermionic part of the action

$$
  \exp(i S_{ferm}(-)) : (\omega, A, B, \psi) \mapsto   \exp(i \int_X \bar \psi D_{\omega,A} \psi)
$$

is an anomalous action functional on the configuration space of the remaining bosonic fields $(\hat A, \hat \omega)$, a [[section]] of a [[Pfaffian line bundle]], whose curvature form turns out to be

$$
  curv_{Pfaff} = \int_X I_4 \wedge I_8
$$

with

$$
  I_4 = \frac{1}{2} p_1(F_\omega) - ch_2(F_A)
$$

the difference between the (image in [[de Rham cohomology]] of the) first fractional [[Pontryagin class]] of the $Spin$-principal bundle and second [[Chern class]] of the [[unitary group]]-[[principal bundle]]

and

$$
  I_8 = \frac{1}{48} p_2(F_\omega) - ch_4(F_A) +
  ...
$$

where the ellipses indicate decomposable [[curvature characteristic form]]s.

Therefore in this case the Green-Schwarz mechanism consists of

1. adding to the system _fivebrane [[magnetic charge]]_ $j_B \in \Omega^{2+2}$ given by $I_4$.

   This means that the Kalb-Ramond field $\hat B$ becomes a twisted field whose [[field strength]] $H$ is no longer closed, but satisfies the kinematical [[Maxwell equation]]

   $$
     d H = I_4
     \,.
   $$

1. adding to the system _string [[electric charge]]_ $j_E \in \Omega^{10 - 2}(X)$ .

   This means that to the [[action functional]] is added the factor

  $$
    \exp(i \int_X \hat B \cdot \hat I_8 )
  $$

  which is locally on $U \hookrightarrow X$ given in the exponent by the integral

  $$
    \int_U B_U \wedge I_8
     \,.
  $$

The nature of the field configuration obtained this way -- spin connection with twist of th Kalb-Ramond field by the Pontryagin class -- may be understood conciesely as constituting a [[differential string structure|twisted differential string structure]] on $X$. See there for more details.

### Axions and the strong CP problem in heterotic supergravity

The Green-Schwarz anomaly cancellation mechanism naturally makes the twisted [[B-field]] in [[heterotic string theory]] behave like the [[axion]] with the correct potential to serve as the [[theta angle]] and serve as the solution to the [[strong CP problem]] ([Svrcek-Witten 06](#SvrcekWitten06)):

> An obvious question about the axion hypothesis is how natural it really is. Why introduce a global PQ "symmetry" if it is not actually a symmetry? What is the sense in constraining a theory so that the classical Lagrangian possesses a certain symmetry if the symmetry is actually anomalous? It could be argued that the best evidence that PQ "symmetries" are natural comes from string theory, which produces them without any contrivance. ... the string compactifications always generate PQ symmetries, often spontaneously broken at the string scale and producing axions, but sometimes unbroken.([Svrcek-Witten 06, pages 3-4](#SvrcekWitten06))

In [[heterotic string theory]] [[KK-compactification|KK-compactified]] to 4d, the 4d [[B-field]], dualized, serves as the axion field, called the "model independent axion" ([Svrcek-Witten 06, starting p. 15](#SvrcekWitten06)).

This works as follows: By the [[Green-Schwarz anomaly cancellation]] mechanism, then [[B-field]] in [[heterotic string theory]] is a twisted 2-form field, whose [[field strength]] $H$ locally has in addition to the exact differential $d B$ also a fundamental 3-form contribution, so that

$$
  H = d B + C
$$

(locally). Moreover, the differential $d H$ is constrained to be the Pontryagin 4-form of the gauge potential $\nabla$ (minus that of the  [[Riemann curvature]], but let's suppress this notationally for the present purpose):

$$
  d H = tr \left(F_\nabla \wedge F_\nabla\right)
  \,.
$$

Now suppose [[KK-compactification]] to 4d has been taken care of, then this constraint may be implemented in the [[equations of motion]] by adding it to the [[action functional]], multiplied with a [[Lagrange multiplier]] :

$$
  S
    =
  \underset{
      \text{kinetic action} \atop \text{for B-field}
   }{
        \underbrace{\int_X H \wedge \star H}
   }
    +
   \underset{
      \text{Green-Schwarz constraint}
   }{
    \underbrace{
      \int_X a \left( d H - tr(F_\nabla \wedge F_\nabla) \right)
    }
   }
  \,.
$$

Now by the usual argument, one says that instead of varying by $a$ and thus implementing the [[Green-Schwarz anomaly cancellation]] constraint, it is equivalent to fist vary with respect to the other fields, and then insert the resulting equations in terms of $a$ into the action functional.

Now since we are dealing with a twisted [[B-field]], with free 3-form component $C$, we actually vary with respect to $C$. This yields the [[Euler-Lagrange equation]] [[equation of motion|of motion]]

$$
  d a = \star H
  \,,
$$

hence the usual relation or [[electro-magnetic duality]], expressing what used to be the [[Lagrange multiplier]] now as the magentic dual [[field (physics)|field]] to the twisted [[B-field]].

Plugging this algebraic [[equation of motion]] back into the above [[action functional]] for $H$ gives

$$
  \tilde S
    =
  \underset{\text{kinetic action} \atop \text{for axion field}}{\underbrace{\int_X d a \wedge \star d a}}
  +
  \underset{\text{axionic} \atop \text{interaction}}{\underbrace{\int_X a \, tr(F_\nabla \wedge F_\nabla)}}
  \,.
$$

## Related concepts

* [[Freed-Witten anomaly]]

## References

### General

The original articles are

* [[Michael Green]], [[John Schwarz]], _Anomaly Cancellation in Supersymmetric $D=10$ Gauge Theory and Superstring Theory_, Phys. Lett. B 149 (1984) 117-122 ([spire:15583](https://inspirehep.net/literature/15583), doi:10.1016/0370-2693(84)91565-X)

* {#CHSW85} [[Philip Candelas]], [[Gary Horowitz]], [[Andrew Strominger]], [[Edward Witten]], p. 49 of: _Vacuum configurations for superstrings_, Nuclear Physics B Volume 258, 1985, Pages 46-74 (doi:10.1016/0550-3213(85)90602-9)

Rederivation using the [[elliptic genus]]/[[Witten genus]] (then: "character-valued partition function"):

* [[A. N. Schellekens]], [[Nicholas P. Warner]], *Anomalies and modular invariance in string theory*, Physics Letters B 177 (3-4), 317-323, 1986 (doi:10.1016/0370-2693(86)90760-4)

* [[A. N. Schellekens]], [[Nicholas P. Warner]], *Anomalies, characters and strings*, Nuclear Physics B Volume 287, 1987, Pages 317-361 (doi:10.1016/0550-3213(87)90108-8)

* [[Wolfgang Lerche]], [[Bengt Nilsson]], [[A. N. Schellekens]], *Heterotic string-loop calculation of the anomaly cancelling term*, Nuclear Physics B Volume 289, 1987, Pages 609-627 (doi:10.1016/0550-3213(87)90397-X)

* [[Wolfgang Lerche]], [[Bengt Nilsson]], [[A. N. Schellekens]],  [[Nicholas P. Warner]], *Anomaly cancelling terms from the elliptic genus*, Nuclear Physics B Volume 299, Issue 1, 28 March 1988, Pages 91-116 (doi:10.1016/0550-3213(88)90468-3)

Review:

* [[Luis Alvarez-Gaumé]], [[Miguel A. Vázquez-Mozo]], *Anomalies and the Green-Schwarz Mechanism*, in: *[[Handbook of Quantum Gravity]]*, Springer (2023) [[arXiv:2211.06467](https://arxiv.org/abs/2211.06467)]

Review (with an eye towards [[KK-compactification]] to 6d, see also at [[D=6 N=(1,0) SCFT]]):

* [[Michael Green]], [[John Schwarz]], [[Pete West]], Section 2 of: _Anomaly-free chiral theories in six dimensions_, Nuclear Physics B Volume 254, 1985, Pages 327-348 (doi:10.1016/0550-3213(85)90222-6)

* [[Augusto Sagnotti]], _A Note on the Green-Schwarz Mechanism in Open-String Theories_, Phys. Lett. B **294** (1992) 196-203 [[arXiv:hep-th/9210127](https://arxiv.org/abs/hep-th/9210127)]

A clear and precise account of what the relevant anomalies are and what the Green-Schwarz mechanism is to cancel them is given in (see also the relevant bits at _[[eta invariant]]_)

* {#Witten99} [[Edward Witten]], section 2.2 of: _World-Sheet Corrections Via D-Instantons_, JHEP **2000** 02:030 (2000) [[arXiv:9907041](http://arxiv.org/abs/hep-th/9907041), [doi:10.1088/1126-6708/2000/02/030](https://doi.org/10.1088/1126-6708/2000/02/030)]

Review, broader context and further discussion is given in

* {#Freed00} [[Daniel Freed]], _[[Dirac charge quantization and generalized differential cohomology]]_, Surveys in Differential Geometry **7** (2002) 129-194  [[arXiv:hep-th/0011220](http://arxiv.org/abs/hep-th/0011220), [doi:10.4310/SDG.2002.v7.n1.a6](https://dx.doi.org/10.4310/SDG.2002.v7.n1.a6)]

An account of historical developments is in section 7 of

* [[John Schwarz]], _The Early Years of String Theory: A Personal Perspective_ in _[The birth of string theory](books+about+string+theory#TheBirthOfStringTheory)_ ([arXiv:0708.1917](http://arxiv.org/abs/0708.1917))

The full formula for the differential form data including the fermionic contributions is in

* {#BBDFLPPRRTZ} L. Bonora, M. Bregola, R. D'Auria, P. Fré K. Lechner, P. Pasti, I. Pesando, M. Raciti, F. Riva, M. Tonin and D. Zanon, _Some remarks on the supersymmetrization of the Lorentz Chern-Simons form in $D = 10$ $N= 1$ supergravity theories_, Physics Letters B 277 (1992) ([[BonoraSuperGS.pdf:file]])

and references given there.

Discussion relating to [[axions]]:

* {#SvrcekWitten06} Peter Svrcek, [[Edward Witten]], _Axions In String Theory_, JHEP 0606:051,2006 ([arXiv:hep-th/0605206](http://arxiv.org/abs/hep-th/0605206))

[[!include higher gauge theory of the Green-Schwarz mechanism -- references]]

[[!redirects Green-Schwarz mechanisms]]

[[!redirects Green Schwarz mechanism]]
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Greenberg scheme

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## Idea

In the context of [[motivic integration]], the term _Greenberg scheme_ refers to one of multiple generalizations of [[arc spaces]] (here referred to as "arc schemes"). Greenberg schemes are built on [[schemes]] over a _complete_ [[discrete valuation ring]] (DVR). In the locally Noetherian setting, if $R$ is a complete DVR then  Greenberg schemes of [[formal schemes]] generalize this prior notion of Greenberg schemes. Greenberg schemes associated to such formal schemes are the setting for motivic integration.

## Definitions

Let $R$ be a complete DVR, let $\mathfrak{m}$ be its unique [[maximal ideal]], and let $k=R/\mathfrak{m}$ be its residue field.

### The ring schemes $\mathscr{R}_n$

For every non-negative integer $n\ge0$, the functor $\mathscr{R}_n\colon\mathbf{Alg}_k\to\mathbf{Ring}$ from the category of $k$-algebras to the category of rings given by
$$
A\mapsto A\otimes_k R_n = A\otimes_k R/\mathfrak{m}^{n+1}
$$
is a [[representable functor]] if $R$ is equicharacteristic (that is, $R$ and $k$ have the same characteristic). Denote by $\mathscr{R}_\infty$ the limit of these functors, which is again representable. As schemes, these are identified with $\mathbb{A}_k^n$ and $\mathbb{A}_k^\infty$, respectively, but the ring scheme structures endowed on $\mathscr{R}_n$ and $\mathscr{R}_\infty$ are not the same as the usual ones on affine space.

In the mixed characteristic case, let $W(-)$ be the functor which assigns a $k$-algebra to its [[ring of Witt vectors]]. The functor
$$
\mathscr{R}_n\colon A\mapsto W(A)\otimes_{W(k)} R_n
$$
is not always representable, but its fpqc-sheafification (its [[sheafification]] over the [[fpqc site]]) _is_ representable; in this case, we replace the functor $\mathscr{R}_n$ with its fpqc-sheafification.

### Functorial description of $\mathrm{Gr}_n$

Given a $k$-scheme $(Y,\mathscr{O}_Y)$, define the [[locally ringed space]] $h_n(Y)=(Y,\mathscr{R}_n(\mathscr{O}_Y))$ whose structure sheaf is valued in $R_n$-algebras, given by
$$
U\mapsto\mathrm{Hom}_k(U,\mathscr{R}_n)
$$
which has the structure of an $R_n$-algebra given by the ring scheme structure on $\mathscr{R}_n$. $h_n(-)$ is isomorphic to the functor $(-)\times_{k} R_n$, and thus $h_n(Y)$ is an $R_n$-scheme.

With this construction, the _Greenberg functor_ $\mathrm{Gr}_n$ can be defined abstractly as the right adjoint to the functor $h_n$; that is, to every $R$-scheme $X$, let
$$
\mathrm{Gr}_n(X)(-)\coloneqq
\mathrm{Hom}_{\mathbf{Sch}_R}(h_n(-), X).
$$
This functor is representable; write the corresponding scheme as $\mathrm{Gr}_n(X)$.

The Greenberg schemes $\mathrm{Gr}_n(X)$ form an inverse system, where the morphisms $\mathrm{Gr}_m(X)\to\mathrm{Gr}_n(X)$ for $m\ge n$ come from the obvious natural transformations $h_n\Rightarrow h_m$. The limit of these functors
$$
\mathrm{Gr}_\infty(X)(-)\coloneqq
\lim_n\,\mathrm{Hom}_{\mathbf{Sch}_R}(h_n(-),X)
$$
is also represented by a $k$-scheme.

### Greenberg schemes of formal schemes

To any locally Noetherian $R$-adic formal scheme $\mathfrak{X}$, one can define the _Greenberg scheme_ $\mathrm{Gr}_n(\mathfrak{X})$ by simply applying the already constructed Greenberg functor to $\mathfrak{X}_n=\mathfrak{X}\times_{R} R_n$. Similarly, the limit of these functors
$$
\mathrm{Gr}_\infty(\mathfrak{X})\coloneqq
\lim_n\,\mathrm{Gr}_n(\mathfrak{X})
$$
again is representable, hence defines a $k$-scheme, called the _Greenberg scheme_ of $\mathfrak{X}$.

This construction generalizes Greenberg schemes associated to locally Noetherian $R$-schemes, and therefore generalizes, in particular, the construction of arc schemes.

## Related concepts

* [[arc space]]

* [[motivic integration]]

* [[p-adic integration]]

* [[motivic cohomology]]

## References

{#CNS18} Antoine Chambert-Loir, Johannes Nicaise, Julien Sebag, _Motivic integration_, Progress in Mathematics 325 Birkhaeuser 2018 ([doi:10.1007/978-1-4939-7887-8](https://doi.org/10.1007/978-1-4939-7887-8))


Greg Egan

[Greg Egan](http://www.gregegan.net) is an Australian science fiction author, with several articles in the journal _Classical and Quantum Gravity_ to his name. His writing is very 'hard', in the sense that it contains lots of scientific and mathematical detail. Egan contributes images to [[John Baez]]'s blog [Visual Insight](http://blogs.ams.org/visualinsight/).

Along with [[J. Daniel Christensen]], he presented an algorithm, written in C++, to understand the asymptotic behavior of the [[10j symbol]], a [[spin network]] that arises in the [[partition function]] for the [[Barrett-Crane model]] of [[4d Riemannian quantum gravity]]. This partition function is the summation over all spin labelings of the faces in the dual [[2-skeleton]] of a fixed [[triangulation]] of spacetime, applied to an expression which contains the product of a [[10j symbol]] for each [[dual vertex]].

There are famously [no pictures of him on the web](http://gregegan.customer.netspace.net.au/images/GregEgan.htm).

category: people

##References

* J. Daniel Christensen and Greg Egan, _An  Efficient  Algorithm  for the  Riemannian 10j Symbols_ [arxiv:gr-qc/0110045v3](http://arxiv.org/abs/gr-qc/0110045v3)

* Baez, John C., J. Daniel Christensen, and Greg Egan. _Asymptotics of 10j Symbols_ Classical and Quantum Gravity 19, no. 24 (December 21, 2002): 6489–6513. [https://doi.org/10.1088/0264-9381/19/24/315.](https://arxiv.org/abs/gr-qc/0208010)


Greg Friedman

* [webpage](http://faculty.tcu.edu/gfriedman/)

## Selected writings

On [[simplicial sets]]:

* {#Friedman08} [[Greg Friedman]], _An elementary illustrated introduction to simplicial sets_, Rocky Mountain J. Math. 42(2): 353-423 (2012) ([arXiv:0809.4221](http://arxiv.org/abs/0809.4221), [doi:10.1216/RMJ-2012-42-2-353](https://projecteuclid.org/journals/rocky-mountain-journal-of-mathematics/volume-42/issue-2/Survey-Article-An-elementary-illustrated-introduction-to-simplicial-sets/10.1216/RMJ-2012-42-2-353.full))

On [[intersection homology]]:

* [[Greg Friedman]], *Singular Intersection Homology*,  Cambridge University Press 2020 ([doi:10.1017/9781316584446](https://doi.org/10.1017/9781316584446), [pdf](http://faculty.tcu.edu/gfriedman/IHbook.pdf))

## Related $n$Lab entries

* [[product of simplices]]

category: people


Greg Kuperberg

* [personal page](https://www.math.ucdavis.edu/~greg/)

* [institute page](https://cs.ucdavis.edu/directory/greg-kuperberg)

## Selected writings

On [[involutory Hopf algebras]] and [[3-manifolds]]:

* {#Kuperberg91} [[Greg Kuperberg]], *Involutory Hopf algebras and 3-manifold invariants*, Internat. J. Math. 2 (1991), no. 1, 41--66 ([arXiv:math/9201301](https://arxiv.org/abs/math/9201301))

On [[quantum probability]], [[quantum mechanics]] and [[quantum computing]]:

* {#Kuperberg05} [[Greg Kuperberg]], _A concise introduction to quantum probability, quantum mechanics, and quantum computation_ (2005) [[pdf](http://www.math.ucdavis.edu/~greg/intro-2005.pdf), [[Kuperberg-ConciseQuantum.pdf:file]]]

On [[von Neumann algebras]], [[noncommutative measure theory]] and [[quantum relations]]:

* {#WeaverKuperberg11} [[Nik Weaver]], [[Greg Kuperberg]], *A von Neumann Algebra Approach to Quantum Metrics/Quantum Relations*, Memoirs of the AMS **215** 1010 (2011) [[ams:memo-215-1010](https://bookstore.ams.org/memo-215-1010)]

## Related entries

* [[quantum probability]]

* [[quantum computation]]

category: people


Greg Stevenson

* [webpage](http://www.math.uni-bielefeld.de/~gstevens/)

## Writings

* _Tensor actions and locally complete intersection_ PhD thesis 2011 ([pdf](http://www.math.uni-bielefeld.de/~gstevens/Stevenson_thesis.pdf))

 on [[spectra of tensor triangulated categories]] and specifically on [[actions]] of [[tensor triangulated categories]].

category: people


GregB

I help manage [Physics Forums](https://www.physicsforums.com/), a science community of scientists, students and hobbyists.


Gregg Jaeger

* [personal page](https://www.bu.edu/cgs/profile/gregg-jaeger/)

## Selected writings

On the "[[potentiality]]" of [[quantum states]] in [[quantum mechanics]], according to [[Werner Heisenberg]]:

* [[Gregg Jaeger]], *Quantum potentiality revisited*, Phil. Trans. R. Soc. A **375** 20160390 (2017) [[doi:10.1098/rsta.2016.0390](https://doi.org/10.1098/rsta.2016.0390)]

## Related entries

* [[philosophy of physics]]

category: people


Gregg Zuckerman

__Gregg Jay Zuckerman__ is a mathematician at Yale University.

* Wikipedia: [Gregg Zuckerman](http://en.wikipedia.org/wiki/Gregg_Zuckerman), [Zuckerman functor](http://en.wikipedia.org/wiki/Zuckerman_functor)

* [website](http://directory.math.yale.edu/public_html/People/gjz2.html)

## Selected writings

On the [[variational bicomplex]]:

* {#Zuckerman87} [[Gregg J. Zuckerman]], *Action principles and global geometry*, in: [[Shing-Tung Yau]] (ed.) *Mathematical Aspects of String Theory*, World Scientific (1987) 259-284 [[[ZuckermanVariation.pdf:file]], [doi:10.1142/0383](https://doi.org/10.1142/0383)]

* [[Bong H. Lian]], [[Gregg J. Zuckerman]], _New perspectives on the BRST-algebraic structure of string theory_, Commun. Math. Phys. **154** (1993) 613-646 [[doi:10.1007/BF02102111](https://doi.org/10.1007/BF02102111)]

On [[representation theory]] and [[derived functors]]:

* _Construction of representations via derived functors_, unpublished lecture series, Institute for Advanced Study 1978.

On [[modular forms]] in [[string theory]]:

* _Modular forms, strings, and ghosts_, Proc. Sympos. Pure Math, 1989

* _Semi-infinite cohomology and string theory_, PNAS

## related $n$Lab entries

* [[cohomological induction]]

* [[geometric representation theory]]

* [[covariant phase space]]

* [[variational calculus]]

* [[semi-infinite cohomology]]

* [[TQFT]]

category: people

[[!redirects Gregg J. Zuckerman]]


Gregor Masbaum

* [webpage](http://webusers.imj-prg.fr/~gregor.masbaum/)

category: people


Gregor Schaumann

[website](http://www.algeo.math.uni-erlangen.de/staff/schaumann-gregor)

category: people


Gregory Adkins

* [webpage](https://www.fandm.edu/greg-adkins)

## Selected writings

On [[baryons]] as [[Skyrmions]]:

* {#AdkinsNappiWitten83} [[Gregory Adkins]], [[Chiara Nappi]], [[Edward Witten]], _Static Properties of Nucleons in the Skyrme Model_, Nucl. Phys. B228 (1983) 552 ([spire:190174](http://inspirehep.net/record/190174), doi:10.1016/0550-3213(83)90559-X)

Inclusion of [[vector mesons]] into the [[Skyrmion model]]:

Original proposal for inclusion of the [[ω-meson]]:

* {#AdkinsNappi84} [[Gregory Adkins]], [[Chiara Nappi]], _Stabilization of Chiral Solitons via Vector Mesons_, Phys. Lett. 137B (1984) 251-256 ([spire:194727](http://inspirehep.net/record/194727), doi:10.1016/0370-2693(84)90239-9)

Original proposal for inclusion of the [[ρ-meson]]:

* {#Adkins86} [[Gregory Adkins]], _Rho mesons in the Skyrme model_, Phys. Rev. D 33, 193 (1986) ([spire:16895](http://inspirehep.net/record/16895),  [doi:10.1103/PhysRevD.33.193](https://doi.org/10.1103/PhysRevD.33.193))

category: people


Gregory Arone

* [webpage](http://pi.math.virginia.edu/~zga2m/)

## Selected writings

On [[Goodwillie calculus]] and [[chromatic homotopy theory]]:

* {#AroneMahowald98} [[Greg Arone]], [[Mark Mahowald]], _The Goodwillie tower of the identity functor and the unstable periodic homotopy of spheres_, Inventiones mathematicae, February 1999, Volume 135, Issue 3, pp 743-788 ([pdf](http://hopf.math.purdue.edu/Arone-Mahowald/ArMahowald.pdf), [doi:10.1007/s002220050300](https://doi.org/10.1007/s002220050300))

* {#AroneChing19} [[Gregory Arone]], [[Michael Ching]], *Goodwillie Calculus*, in: *[[Handbook of Homotopy Theory]]* Taylor & Francis (2019) [[arXiv:1902.00803](https://arxiv.org/abs/1902.00803),[doi:10.1201/9781351251624](https://doi.org/10.1201/9781351251624)]

## Related entries

* [[Goodwillie calculus]]

* [[Goodwillie-Taylor tower]]

* [[Goodwillie derivatives of the identity functor]]

category: people

[[!redirects Greg Arone]]


Gregory Chaitin

* [Wikipedia entry](http://en.wikipedia.org/wiki/Gregory_Chaitin)

## related $n$Lab entries

* [[complexity theory]]

* [[Chaitin's incompleteness theorem]]

category: people


Gregory Dungan

* [Institute page](https://www.fmarion.edu/directory/dr-dungan-gregory/)

* [Mathematics Genealogy page](https://www.genealogy.math.ndsu.nodak.edu/id.php?id=194873)

## Selected writings

On [[model categories]] and [[model structures on chain complexes]]:

* {#Dungan10} [[Gregory Dungan]], _Review of model categories_, 2010 ([pdf](http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.696.123&rep=rep1&type=pdf), [[DunganModelCategories.pdf:file]])

category: people


Gregory Karpilovsky

## Selected writings

On [[non-abelian cohomology|non-abelian]] [[group cohomology]], [[graded algebras]], [[crossed homomorphisms]], ...:

* {#Karpilovsky87} [[Gregory Karpilovsky]], *The Algebraic Structure of Crossed Products*, Mathematics Studies **142**, North Holland 1987 ([ISBN:9780080872537](https://www.elsevier.com/books/the-algebraic-structure-of-crossed-products/karpilovsky/978-0-444-70239-5))

category: people


Gregory L. Naber

* [InSpire page](https://inspirehep.net/authors/1893381)

## Selected writings

On [[mathematical physics]]: [[topology]] ([[differential geometry|differential]]) [[geometry]] and [[gauge fields]]:

* [[Gregory L. Naber]], *Topology, Geometry and Gauge fields -- Foundations*,  Texts in Applied Mathematics **25** (2011) [[doi:10.1007/978-1-4419-7254-5](https://doi.org/10.1007/978-1-4419-7254-5)]

* [[Gregory L. Naber]], *Topology, Geometry and Gauge fields -- Interactions*, Applied Mathematical Sciences **141** (2011) [[doi:10.1007/978-1-4419-7895-0](https://doi.org/10.1007/978-1-4419-7895-0)]

category: people

[[!redirects Gregory Naber]]


Gregory Landweber

* [webpage](http://math.bard.edu/greg/)

## related $n$Lab entries

* [[adinkra]]

category: people

[[!redirects Greg Landweber]]


Gregory Moore

* [website](http://www.physics.rutgers.edu/~gmoore)
* wikipedia: [Greg Moore (physicist)](https://en.wikipedia.org/wiki/Greg_Moore_(physicist))

## Selected writings

* _[[The Impact of D-Branes on Mathematics]]_ (2014)

* _[[Physical Mathematics and the Future]]_ (2014)

* _PiTP lectures on BPS states and wall-crossing in
d = 4, N = 2 theories_ [pdf](http://www.physics.rutgers.edu/grad/695/PiTP-LectureNotes.pdf)

Early argument that the [[RR-field]] [[flux]] [[differential form|density]]-expressions for [[D-brane charge]] are of the form of [[Chern characters]] on [[topological K-theory]], leading to the [[K-theory classification of D-brane charge]]:

* {#GreenHarveyMoore97} [[Michael Green]], [[Jeffrey A. Harvey]], [[Gregory Moore]], *I-brane inflow and anomalous couplings on D-branes*, Class. Quant. Grav. **14** (1997) 47-52 [[arXiv:hep-th/9605033](https://arxiv.org/abs/hep-th/9605033), [doi:10.1088/0264-9381/14/1/008](https://doi.org/10.1088/0264-9381/14/1/008)]

On [[anyons]] satisfying [[non-abelian group|non-abelian]] [[braid group statistics]] in the [[fractional quantum Hall effect]]:

* [[Gregory Moore]], [[Nicholas Read]], _Nonabelions in the fractional quantum Hall effect_, Nucl. Phys. 360B (1991)362 ([pdf](http://www.physics.rutgers.edu/~gmoore/MooreReadNonabelions.pdf), doi:10.1016/0550-3213(91)90407-O)

On [[boundary conditions]] ([[BCFT]]/[[D-branes]]) for the [[gauged WZW model]] via [[parafermions]]:

* [[Juan Maldacena]], [[Gregory Moore]], [[Nathan Seiberg]], *Geometrical interpretation of D-branes in gauged WZW models*, JHEP 0107:046 (2001) $[$[arXiv:hep-th/0105038](https://arxiv.org/abs/hep-th/0105038), [doi:10.1088/1126-6708/2001/07/046](https://doi.org/10.1088/1126-6708/2001/07/046)$]$

On the [[K-theory classification of D-brane charge]]:

* [[Greg Moore]], _K-Theory from a physical perspective_, in: [[Ulrike Tillmann]] (ed.) _Topology, Geometry and Quantum Field Theory_, Proceedings of the 2002 Oxford Symposium in Honour of the 60th Birthday of Graeme Segal, Cambridge University Press (2004)  ([arXiv:hep-th/0304018](http://arxiv.org/abs/hep-th/0304018), [doi:10.1017/CBO9780511526398.011](https://doi.org/10.1017/CBO9780511526398.011))

On [[AdS3-CFT2]] for [[D1/D5 brane bound states]] and [[black hole entropy]] [[black holes in string theory|in string theory]]:

* [[Robbert Dijkgraaf]], [[Juan Maldacena]], [[Gregory Moore]], [[Erik Verlinde]], _A Black Hole Farey Tail_ ([arXiv:hep-th/0005003](https://arxiv.org/abs/hep-th/0005003), [spire:526744](http://inspirehep.net/record/526744))

On [[Lorentzian orbifolds]] as [[target spacetimes]] in [[string theory]]:

* Hong Liu, [[Gregory Moore]], [[Nathan Seiberg]], _Strings in a time-dependent orbifold_, JHEP 0206:045, 2002 ([arXiv:hep-th/0204168](https://arxiv.org/abs/hep-th/0204168))

* Hong Liu, [[Gregory Moore]], [[Nathan Seiberg]], _Strings in time-dependent orbifolds_, JHEP 0210:031, 2002 ([arXiv:hep-th/0206182](https://arxiv.org/abs/hep-th/0206182))

On [[quantization]] of the [[electromagnetic field]] in view of [[Dirac charge quantization]] and [[higher U(1)-gauge theory]]:

* [[Daniel S. Freed]], [[Gregory W. Moore]], [[Graeme Segal]], p. 7 of: *The Uncertainty of Fluxes*, Commun. Math. Phys. 271:247-274, 2007 ([arXiv:hep-th/0605198](https://arxiv.org/abs/hep-th/0605198), [doi:10.1007/s00220-006-0181-3](https://doi.org/10.1007/s00220-006-0181-3))

* [[Daniel Freed]], [[Gregory Moore]], [[Graeme Segal]], _Heisenberg Groups and Noncommutative Fluxes_, Annals Phys. 322:236-285 (2007) ([arXiv:hep-th/0605200](http://arxiv.org/abs/hep-th/0605200))

On potential [[experiments]] detecting [[uncertainty of fluxes]] in [[quantum electrodynamics|quantum]] [[electromagnetism]]:

* [[Alexei Kitaev]], [[Gregory W. Moore]], [[Kevin Walker]], *Noncommuting Flux Sectors in a Tabletop Experiment* [[arXiv:0706.3410](https://arxiv.org/abs/0706.3410)]

On relating the [[Monster vertex operator algebra]] and its cousins to [[quantum error correction]]:

* [[Jeffrey Harvey]], [[Gregory Moore]], _Moonshine, Superconformal Symmetry, and Quantum Error Correction_ ([arXiv:2003.13700](https://arxiv.org/abs/2003.13700))

On the [[10-fold way]]:

* {#GeikoMoore21} Roman Geiko, [[Gregory W. Moore]]: _Dyson’s classification and real division superalgebras_, Journal of High Energy Physics **2021** 4 (2021) 299 [[doi: 10.1007/jhep04(2021)299](http://dx.doi.org/10.1007/jhep04%282021%29299)]

## Related $n$Lab entries

* [[Kalb-Ramond field]]

* [[RR-field]]

* [[FQFT]]

* [[orientifold]]

* [[Donaldson-Thomas invariant]]

category: people

[[!redirects Gregory W. Moore]]
[[!redirects Greg Moore]]


Gregory Taroyan

* [arXiv page](https://arxiv.org/search/math?searchtype=author&query=Taroyan%2C+G)

## Selected writings

On [[derived stacks]] and [[derived manifolds]]:

* [[Gregory Taroyan]], *Equivalent models of derived stacks* [[arXiv:2303.12699](https://arxiv.org/abs/2303.12699)]

category: people


Gribov ambiguity

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## Idea

The __Gribov ambiguity__ is a common problem in [[gauge fixing]] procedure, namely the unique and continuous choice of a representative of each gauge orbit is often impossible globally. In particular most common gauge fixing procedures have this problem. Gribov introduced what is now called __Gribov regions__ and their boundaries, __Gribov horizons__.

## References

* V.N. Gribov, _Quantization of Nonabelian Gauge Theories_, Nucl.Phys. B139 (1978) 1.
* wikipedia [Gribov ambiguity](http://en.wikipedia.org/wiki/Gribov_ambiguity)
* D. Zwanziger, _Action from the Gribov horizon_, Nucl. Phys. B321 (1989) 591.
* D. Zwanziger, _Local and renormalizable action from the Gribov horizon_,
Nucl. Phys. B323 (1989) 513.
* N. Vandersickel, D. Zwanziger, _The Gribov problem and QCD dynamics_,
[arXiv:hep-th/1202.1491](http://arxiv.org/abs/1202.1491)
[[!redirects Gribov horizon]]
[[!redirects Gribov region]]


Grigori Perelman

* [Wikipedia entry](http://en.wikipedia.org/wiki/Grigori_Perelman)

category: people


Grigori Vilkovisky

## related $n$Lab entries

* [[Batalin-Vilkovisky formalism]]

category: people


Grigorios Giotopoulos

Member of [[CQTS]] at NYU Abu Dhabi.

* [institute page](https://nyuad.nyu.edu/en/research/faculty-labs-and-projects/cqts/grigorios-giotopoulos.html) at NYU Abu Dhabi

* [inspire page](https://inspirehep.net/authors/1844380)

## Selected writings

On the [[first-order formulation of gravity]] via [[L-infinity algebras|$L_\infty$-algebras]]:

* Marija Dimitrijević Ćirić, [[Grigorios Giotopoulos]], Voja Radovanović, [[Richard J. Szabo]], *$L_\infty$-Algebras of Einstein-Cartan-Palatini Gravity*, J. Math. Phys., **61** 112502 (2020) [[arXiv:2003.06173](https://arxiv.org/abs/2003.06173), [doi:10.1063/5.0011344](https://doi.org/10.1063/5.0011344)]

On [[smooth sets]] as a [[convenient category of spaces|convenient category]] for [[variational calculus]] in [[Lagrangian quantum field theory|Lagrangian]] [[classical field theory]]:

* [[Grigorios Giotopoulos]], [[Hisham Sati]], *Field Theory via Higher Geometry I: [[schreiber:Smooth Sets of Fields]]* [[arXiv:2312.16301](https://arxiv.org/abs/2312.16301)]

Expositions:

* [[Grigorios Giotopoulos]],  *Classical field theory in the topos of smooth sets*, [talk at](Center+for+Quantum+and+Topological+Systems#GiotopoulosOct2023) [[CQTS]] (Oct 2023) [[[Giotopoulos-FieldTheoryInSmoothSets.pdf:file]], video:[YT](https://youtu.be/7Bw9CJct8QY)]

* [[Grigorios Giotopoulos]], *Towards Non-Perturbative Lagrangian Field Theory via the Topos of Smooth Sets*, [talk at](M-Theory+and+Mathematics#Giotopoulos2024) *[[M-Theory and Mathematics]]*, [[CQTS]] (Jan 2024)

category: people


Grigory Garkusha

__Grigory Garkusha__ is a mathematician at Swansea, formerly in Sankt Petersburg. His interests include algebraic geometry, algebraic K-theory, [[reconstruction theorem]]s for schemes, [[derivator]]s, Quillen model categories etc.

* [web](http://www.swansea.ac.uk/staff/academic/science/mathematics/garkushagrigory), [publications](http://www.swansea.ac.uk/staff/academic/science/mathematics/garkushagrigory/researchpublications,137383,en.php)

category: people
[[!redirects G. Garkusha]]


Gromoll-Meyer sphere

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## Idea

What is called the _Gromoll-Meyer sphere_ ([Gromoll-Meyer 74](#GromollMeyer74)) is an example of an [[exotic 7-sphere]]. It arises as a [[double coset space]]/[[biquotient]] of the [[quaternionic unitary group]] [[Sp(2)]] $\simeq$ [[Spin(5)]] by two copies of [[Sp(1)]] $\simeq$ [[SU(2)]].

## Definition

Consider the [[quaternion unitary group]] [[Sp(2)]] $\simeq$ [[Spin(5)]] and its two [[subgroup]] inclusion of [[Sp(1)]] $\simeq$ [[SU(2)]] as

$$
  Sp(1)
  \;\simeq\;
  \left\{
    \left(
    \array{
      q & 0
      \\
      0 & q
    }
    \right)
    \;\vert\;
    q \in Sp(1)
  \right\}
  \phantom{AA}
   \text{and}
   \phantom{AA}
  Sp(1)
  \;\simeq\;
  \left\{
    \left(
    \array{
      q & 0
      \\
      0 & 1
    }
    \right)
    \;\vert\;
    q \in Sp(1)
  \right\}
$$

Then the [[double coset space]]/[[biquotient]]  $Sp(1)\backslash Sp(2) /Sp(1)$ with respect to the left and right multiplication [[action]], respectively, of these two subgroups is, with respect to its canonically induced geometric structures:

1. [[homeomorphism|homeomorphic]] to the [[topological manifold|topological]] [[7-sphere]];

1. [[diffeomorphism|diffeomorphic]] to an [[exotic 7-sphere]].

As such, it is called the _Gromoll-Meyer sphere_, due to [Gromoll-Meyer 74](#GromollMeyer74)

$$
  Sp(1)\backslash Sp(2) /Sp(1)
  \;\simeq\;
  S^7_{exotic}
$$

## Properties

* The Gromoll-Meyer sphere is the only [[exotic 7-sphere]] that can be modeled by a [[biquotient]] of a [[compact Lie group]] ([KZ02, Corollary C](#KZ02)).

* The Gromoll-Meyer sphere is a [[3-sphere]]-[[bundle]] over the [[4-sphere]].

* It is a generator of the [[group]] of [[diffeomorphism]] [[equivalence class|classes]] of [[orientation|oriented]] [[homotopy spheres]] in [[dimension]] 7, which is of [[order of a group|order]] 28.

## Related concepts

[[!include coset space structure on n-spheres -- table]]

## References

The construction is due to

* {#GromollMeyer74} [[Detlef Gromoll]], Wolfgang Meyer, _An Exotic Sphere With Nonnegative Sectional Curvature_, Annals of Mathematics Second Series, Vol. 100, No. 2 (Sep., 1974), pp. 401-406 ([jstor:1971078](https://www.jstor.org/stable/1971078))

Review includes

* {#JoachimWraith} [[Michael Joachim]], D. J. Wraith, pages 7-8 of _Exotic spheres and curvature_ ([pdf](https://ivv5hpp.uni-muenster.de/u/joachim/exo.pdf))

Generalization of the construction to a large class of [[exotic 7-spheres]]:

* C. Duran, T. Puettmann, A. Rigas, _An infinite family of Gromoll-Meyer spheres_. Arch. Math. (2010) 95: 269 ([arXiv:math/0610349](https://arxiv.org/abs/math/0610349), [doi:10.1007/s00013-010-0161-x](https://doi.org/10.1007/s00013-010-0161-x))

See also

* Jost-Hinrich Eschenburg, Martin Kerin, _Almost positive curvature on the Gromoll-Meyer sphere_,  Proc. Amer. Math. Soc ([arXiv:0711.2987](https://arxiv.org/abs/0711.2987))

* Carlos Durán, Thomas Püttmann, _A minimal Brieskorn 5-sphere in the Gromoll-Meyer sphere and its applications_, Michigan Math. J. Volume 56, Issue 2 (2008), 419-451 ([euclid](https://projecteuclid.org/euclid.mmj/1224783521))

* Llohann D. Sperança, _Pulling back the Gromoll-Meyer construction and models of exotic spheres_, Proceedings of the American Mathematical Society 144.7 (2016): 3181-3196 ([arXiv:1010.6039](https://arxiv.org/abs/1010.6039))

* Llohann D. Sperança, _Explicit Constructions over the Exotic 8-sphere_ ([pdf](https://www.ime.unicamp.br/~rigas/sigma8EncontroTopol.pdf), [[SperancaExoticSpheres.pdf:file]])

For a proof that the Gromoll-Meyer sphere is the only exotic sphere that is a [[biquotient]] of a [[compact Lie group]] see

* {#KZ02} Vitali Kapovitch, Wolfgang Ziller, _Biquotients with singly generated rational cohomology_, ([arXiv:math/0210231](https://arxiv.org/abs/math/0210231))

[[!redirects Gromoll-Meyer spheres]]


Gromov's non-squeezing theorem

Gromov's non-squeezing theorem is an analogue of [[Heisenberg uncertainty relations]] in [[symplectic geometry]] proved in seminal work of Gromov,

* [[Mikhail Gromov]], _Pseudo holomorphic curves in symplectic manifolds_, Invent. Math. 82 (1985) 307--347

Some generalizations and consequences are studied in terms of symplectic capacities. A survey:

* Maurice de Gosson, Franz Luef, _Symplectic capacities and the geometry of uncertainty: The irruption of symplectic topology in classical and quantum mechanics_, Physics Reports __484__:5, (2009) 131--179 [doi](https://doi.org/10.1016/j.physrep.2009.08.001)

* A. Sukhov, A. Tumanov, _Gromov's non-squeezing theorem and Beltrami type equation_, Commun. Part. Diff. Eq. __39__:10 (2014) [doi](https://doi.org/10.1080/03605302.2014.908392)

#### Non-squeezing theorems in [[contact geometry]]

* Sheng-Fu Chiu, _Non-squeezing property of contact balls_, [doi](https://doi.org/10.1215/00127094-3715517) [arXiv:1405.1178](https://arxiv.org/abs/1405.1178)

> In this paper we solve a contact non-squeezing conjecture proposed by Eliashberg, Kim and Polterovich.
Let BR be the open ball of radius $R$ in $\mathbb{R}^{2n}$ and let $\mathbb{R}^{2n}\times S^1$ be the prequantization space equipped with the standard contact structure. Following [[Dmitry Tamarkin|Tamarkin]]’s idea, we apply [[microlocal analysis|microlocal]] category methods to prove that if $R$ and $r$ satisfy $1\leq \pi r^2 \lt \pi R^2$, then it is impossible to squeeze the contact ball $B_R\times S^1$ into $B_r\times S^1$ via compactly supported contact isotopies.

* Peter Albers, Will J. Merry, _Orderability, contact non-squeezing, and Rabinowitz Floer homology_, [arxiv:1302.6576](https://arxiv.org/abs/1302.6576)

category: geometry


Gromov-Witten invariants

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

#Contents#
* table of contents
{:toc}

## Idea

_Gromov-Witten theory_ studies [[symplectic manifolds]] via maps of [[Riemann surfaces]] into it. In contrast to the [[fundamental group]], where one studies maps from the circle into the space, there is no composition law for Riemann surfaces. Instead, one considers [[pseudoholomorphic curves]] in a fixed homology class and with fixed boundary conditions such that only a finite number of such maps exists. These numbers are symplectic invariants. Their computation is difficult and uses enumerative and analytic techniques from [[Floer homology]].

Gromov-Witten invariants are used to deform the cup product of cohomology classes (using only maps from the Riemann sphere into the manifold), leading to [[quantum cohomology]].

Gromov-Witten theory originates in physics from the [[A-model]]. A powerful tool to compute Gromov-Witten invariants is [[mirror symmetry]].

The precise mathematical definition uses the notion of the moduli space of [[stable maps]].

## Definition

...

## Relation to TQFT

Gromov-Witten invariants may be understood (and have originally been found) as arising from a particular [[TQFT]], or actually a [[TCFT]], called the [[A-model]].

For a useful exposition of this see ([Tolland](#Tolland)).

## Related concepts

* [[stable map]]

* [[orbifold cohomology]]

* [[quantum sheaf cohomology]]

* [[Donaldson theory]]

* [[A-model]]

* [[GW/DT correspondence]]

* [[enumerative geometry]]

## References

### Expositions

Introductory notes:

* Simon Rose, _Introduction to Gromov-Witten theory_ ([arXiv:1407.1260](http://arxiv.org/abs/1407.1260))

* {#Bertram02} [[Albrecht Bertram]], *Stable Maps and Gromov-Witten Invariants*, School and Conference on Intersection Theory and Moduli Trieste, 9-27 September 2002 ([pdf](http://users.ictp.it/~pub_off/lectures/lns019/Bertram/Bertram.pdf), [[Bertram_StableMaps.pdf:file]])

* {#Katz06} [[Sheldon Katz]], *Enumerative Geometry and String Theory*, Student Mathematical Library **32**, AMS 2006 ([ISBN:978-1-4704-2143-4](https://bookstore.ams.org/stml-32), [spire:739788](https://inspirehep.net/literature/739788))

Seminar notes:

* [[basic ideas of moduli stacks of curves and Gromov-Witten theory]]

And this introductory bit on the [[moduli stack]] of [[elliptic curve]]s:

* [[A Survey of Elliptic Cohomology - elliptic curves]].

An exposition of GW theory as a [[TCFT]] is at

* {#Tolland} [[AJ Tolland]], _Gromov-Witten Invariants and Topological Field Theory_ ([blog](http://sbseminar.wordpress.com/2008/12/11/gromov-witten-invariants-and-topological-field-theory))

The origin of Gromov-Witten theory in and relation to [[string theory]] and other [[physics]] motivation is recalled and surveyed in

* Daniel Grunberg, _Gromov-Witten Theory and Threshold Corrections_ ([arXiv:hep-th/0605087](http://arxiv.org/abs/hep-th/0605087))

### Via geometric quantization

Discussion in the context of [[geometric quantization]] is in

* {#CladerPriddisShoemaker13} Emily Clader, Nathan Priddis, Mark Shoemaker, _Geometric Quantization with Applications to Gromov-Witten Theory_ ([arXiv:1309.1150](http://arxiv.org/abs/1309.1150))

### As a TCFT

* [[Kevin Costello]], _The Gromov-Witten potential associated to a TCFT_ ([arXiv:0509264](http://arxiv.org/abs/math/0509264))

See also the references at [[A-model]].

### General

A discussion by quantization of quadratic Hamiltonians is in

* [[Alexander Givental]], _Gromov-Witten invariants and quantization of quadratic Hamiltonians_ ([pdf](http://math.berkeley.edu/~giventh/papers/gwi.pdf))

* [[Maxim Kontsevich]], [[Yuri Manin]], _Gromov-Witten classes, quantum cohomology, and enumerative geometry_, Comm. Math. Phys. 164 (1994), no. 3, 525--562 ([euclid](http://projecteuclid.org/euclid.cmp/1104270948)).

* [[Yuri Manin]], _Frobenius manifolds, quantum cohomology and moduli spaces_, Amer. Math. Soc., Providence, RI, 1999,

* W. Fulton, R. Pandharipande, _Notes on stable maps and quantum cohomology_, in: Algebraic Geometry, Santa Cruz 1995 ed. Kollar, Lazersfeld, Morrison. Proc. Symp. Pure Math. 62, 45–96 (1997)

* J Robbin, D A Salamon, _A construction of the Deligne-Mumford orbifold_,  J. Eur. Math. Soc. (JEMS)  8  (2006),  no. 4, 611--699 ([arxiv](http://arxiv.org/abs/math/0407090); [pdf at JEMS](http://www.ems-ph.org/journals/show_pdf.php?issn=1435-9855&vol=8&iss=4&rank=3)); corrigendum  J. Eur. Math. Soc. (JEMS)  9  (2007),  no. 4, 901--905 ([pdf at JEMS](http://www.ems-ph.org/journals/show_pdf.php?issn=1435-9855&vol=9&iss=4&rank=11)).

* J Robbin, Y Ruan, D A Salamon, _The moduli space of
regular stable maps_,  Math. Z.  259  (2008),  no. 3, 525--574 ([doi](http://dx.doi.org/10.1007/s00209-007-0237-x)).

* Martin A. Guest, _From quantum cohomology to integrable systems_, Oxford Graduate Texts in Mathematics, 15. Oxford University Press, Oxford, 2008. xxx+305 pp.

* Joachim Kock, Israel Vainsencher, _An invitation to quantum cohomology. Kontsevich's formula for rational plane curves_, Progress in Mathematics, 249. Birkhäuser Boston, Inc., Boston, MA, 2007. xiv+159 pp.

* Dusa McDuff, Dietmar Salamon, _Introduction to symplectic topology_, 2 ed. Oxford Mathematical Monographs 1998. x+486 pp.

* Sheldon Katz, _Enumerative geometry and string theory_, Student Math. Library __32__. IAS/Park City AMS & IAS 2006. xiv+206 pp.

* Eleny-Nicoleta Ionel, Thomas H. Parker, _Relative Gromov-Witten invariants_, Ann. of Math. (2) 157 (2003), no. 1, 45--96 ([doi](http://dx.doi.org/10.4007/annals.2003.157.45)).

* [[Edward Frenkel]], [[Constantin Teleman]],
  [[AJ Tolland]], _Gromov-Witten Gauge Theory I_ ([arXiv:0904.4834](http://arxiv.org/abs/0904.4834))

* [[Constantin Teleman]], _The structure of 2D semi-simple field theories_ ([arXiv:0712.0160](http://arxiv.org/abs/0712.0160))

* Oliver Fabert, _Floer theory, Frobenius manifolds and integrable systems_, ([arxiv/1206.1564](http://arxiv.org/abs/1206.1564))

A generalization is discussed in

* [[Edward Frenkel]], A. Losev, [[Nikita Nekrasov]], _Instantons beyond topological theory I_ ([arXiv:hep-th/0610149](http://arxiv.org/abs/hep-th/0610149))

* [[Edward Frenkel]], A. Losev, [[Nikita Nekrasov]], _Instantons beyond topological theory II_ ([arXiv:hep-th/0610149](http://arxiv.org/abs/0803.3302))

Expositions and summaries of this are in

* [[Edward Frenkel]], A. Losev, [[Nikita Nekrasov]], _Notes on instantons in topological field theory and beyond_ ([arXiv:hep-th/0702137](http://arxiv.org/abs/hep-th/0702137))

* [[Jacques Distler]], _Localized_ (2006) ([blog](Localized))

### In higher differential geometry / on orbifolds

GW theory of [[orbifolds]] (hence in [[higher differential geometry]]) has been introduced in

* [[Weimin Chen]], [[Yongbin Ruan]], _Orbifold Gromov-Witten Theory_, in _Orbifolds in mathematics and physics_ (Madison, WI, 2001), 25–85, Contemp. Math., 310, Amer. Math. Soc., Providence, RI, 2002 ([arXiv:math/0103156](http://arxiv.org/abs/math/0103156))

A review with further pointers is in

* Dan Abramovich, _Lectures on Gromov-Witten invariants of orbifolds_ ([arXiv:math/0512372](http://arxiv.org/abs/math/0512372))

### In terms of motives

That the [[path integral as a pull-push transform|pull-push quantization]] of Gromov-Witten theory is naturally understood as a "[[motivic quantization]]" in terms of [[Chow motives]] of [[Deligne-Mumford stacks]] was suggested in

* [[Kai Behrend]], [[Yuri Manin]], _Stacks of Stable Maps and Gromov-Witten Invariants_ ([arXiv:alg-geom/9506023](http://arxiv.org/abs/alg-geom/9506023))
 {#BehrendManin95}

Further investigation of these stacky Chow motives then appears in

* [[Bertrand Toën]], _On motives for Deligne-Mumford stacks_, International Mathematics Research Notices 2000, 17 (2000) 909-928 ([arXiv:math/0006160](http://arxiv.org/abs/math/0006160), [web](http://hal.archives-ouvertes.fr/hal-00773027), [pdf](http://hal.archives-ouvertes.fr/docs/00/77/30/27/PDF/motdm.pdf))
 {#Toen00}

* Utsav Choudhury, _Motives of Deligne-Mumford Stacks_ ([arXiv:1109.5288](http://arxiv.org/abs/1109.5288))

On its relation to [[tropical geometry]]:

* [[Grigory Mikhalkin]]. *Enumerative Tropical Algebraic Geometry in $\mathbb{R}^2$*. Journal of the American Mathematical Society, Vol. 18, No. 2 (Apr., 2005), pp. 313-377. ([doi](https://doi.org/10.1090/S0894-0347-05-00477-7))

* Emil Albrychiewicz, Kai-Isaak Ellers, Andrés Franco Valiente, [[Petr Hořava]]. *Tropological Sigma Models*. (2023). ([arXiv:2311.00745](https://arxiv.org/abs/2311.00745))

[[!redirects Gromov-Witten invariant]]
[[!redirects Gromov-Witten invariants]]
[[!redirects Gromov–Witten invariant]]
[[!redirects Gromov–Witten invariants]]
[[!redirects Gromov--Witten invariant]]
[[!redirects Gromov--Witten invariants]]

[[!redirects Gromov-Witten theory]]
[[!redirects Gromov–Witten theory]]
[[!redirects Gromov--Witten theory]]

[[!redirects Gromov-Ruan-Witten invariant]]
[[!redirects Gromov-Ruan-Witten invariants]]


Gross-Neveu model

## Idea

A toy [[model (in theoretical physics)|model]] for [[quantum chromodynamics]].

## References

See also

* Wikipedia, _[Gross-Neveu model](https://en.m.wikipedia.org/wiki/Gross%E2%80%93Neveu_model)_

Via [[intersecting D-brane models]]/[[AdS/QCD]]:

* E. Antonyan, [[Jeffrey Harvey]], [[David Kutasov]], _The Gross-Neveu Model from String Theory_, Nucl. Phys. B776:93-117, 2007 ([arxiv:hep-th/0608149](https://arxiv.org/abs/hep-th/0608149))

Relation to [[coadjoint orbits]]:

* [[Dmitri Bykov]], *Sigma models as Gross-Neveu models. II* [[arXiv:2310.15394](https://arxiv.org/abs/2310.15394)]

Relation to [[flag manifold sigma-models]]:

* [[Dmitri Bykov]], *Quantum flag manifold σ-models and Hermitian Ricci flow*, Commun. Math. Phys. **401** 1-32 (2023) [[arXiv:2006.14124](https://arxiv.org/abs/2006.14124), [doi:10.1007/s00220-022-04532-5](https://doi.org/10.1007/s00220-022-04532-5)]

[[!redirects Gross-Neveu models]]


Grothendieck > history

< [[Grothendieck]]
[[!redirects Grothendieck -- history]]


Grothendieck category

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Additive and abelian categories
+--{: .hide}
[[!include additive and abelian categories - contents]]
=--
#### Homological algebra
+--{: .hide}
[[!include homological algebra - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

_Grothendieck categories_ are those [[abelian category|abelian categories]] $\mathcal{A}$

* such that for [[presheaf|presheaves]] on a [[site]] with values in $\mathcal{A}$ there is an existence theorem for the  [[sheafification]] functor;

* such that all [[complexes]] in $\mathcal{A}$ (with respect to a [[category with translation|translation]]) are [[quasi-isomorphism|quasi-isomorphic]] to [[injective object|homotopically injective]] complexes (so that [[derived functor on a derived category|derived functor]] can be computed on homotopically injective replacements).

More abstractly, Grothendieck categories are precisely [[Ab-enriched]] [[Grothendieck topos]]es. This follows from the [[Gabriel-Popescu theorem]] together with the theory of [[enriched sheaves]].

## Definition

In terms of the AB$n$ hierarchy discussed at [[additive and abelian categories]] we have

A **Grothendieck category** is an [[additive and abelian categories|AB5-category]] which has a [[separator|generator]].

This means that a **Grothendieck category** is an [[abelian category]]

* that admits a [[separator|generator]];

* that admits small [[colimits]];

* such that small [[filtered category|filtered]] [[colimits]] are _[[exact functor|exact]]_ in the following sense:

  * for $I$ a [[direction|directed set]] and $0 \to A_i \to B_i \to C_i \to 0$ an [[exact sequence]] for each $i \in I$, then $0 \to colim_i A_i \to colim_i B_i \to colim_i C_i \to 0$ is also an [[exact sequence]].

Dually a *co-Grothendieck category* is an AB5$^*$ category
with a [[cogenerator]]. The category of abelian groups is not a co-Grothendieck category. Any abelian category which is simultaneously Grothendieck and co-Grothendieck has just a single object (see Freyd's book, p.116).

## Properties

A Grothendieck category $C$ satisfies the following properties.

* it admits small [[limits]];

* if a functor $F : C^{op} \to Set$ commutes with small [[limits]], the $F$ is [[representable functor|representable]];

* if a functor $F : C^{op} \to Set$  commutes with small [[colimits]], then $F$ has a [[adjoint functor|right adjoint]].

* The [[Gabriel-Popescu theorem]] exhibits any [[Grothendieck abelian category]] as a [[reflective subcategory]] of a [[category of modules]] over a [[ring]].

* Any [[Grothendieck abelian category]] is [[locally presentable]].

  ([Beke (2000), Prop. 3.10](#Beke00), see also [Krause (2015), Corollary 5.2](#Krause) and references therein.)

* More generally, any [[cocomplete]] [[abelian category]] with a [[set]] of [[generators]] in which $\kappa$-[[filtered colimits]] commute with [[finite limits]] for some [[regular cardinal]] $\kappa$ is a [[locally presentable category]].  See [Positselski-Rosicky](#PositselskiRosicky), Theorem 2.2.

* If $C$ is equipped with [[category with translation|translation]] $T : C \to C$, then for every [[complex]] $X \in Cplx(C)$ there exists a [[quasi-isomorphism]] of [[complex]]es $X \to I$ such that $I$ is [[injective object|homotopically injective]].

* it satisfies Pierre Gabriel's sup property: every small family of subobjects of a given object $X$ has a [[supremum]] which is a subobject of $X$;

* it admits an [[injective object|injective]] [[cogenerator]] (see [Kashiwara-Schapira](#KS), Theorem 9.6.3).

Much of the localization theory of rings generalizes to general Grothendieck categories.

## Examples

\begin{example}\label{CateoriesOfModules}
**($R$[[Mod]] is Grothendieck abelian)**
\linebreak
For $R$ a [[commutative ring]], its [[category of modules]] $R$[[Mod]] is a Grothendieck category.
(See e.g [Kiersz 06, prop. 4](#Kiersz) for the proof that filtered colimits here are exact.)
This statement remains true [[internalization|internal]] to any [[Grothendieck topos]] [[Johnstone (1977), Thm. 8.11 (iii)](#Johnstone1977)].
\end{example}

\begin{example}\label{CategoriesOfVectorSpaces}
**([[Vect]] is Grothendieck abelian)**
\linebreak
Taking $R$ in Exp. \ref{CateoriesOfModules} to be a [[field]] it follows that [[categories of vector spaces]] are Gorthendieck abelian.
\end{example}

\begin{example}\label{ChainComplexesInGrothAbCat}
**([[category of chain complexes|$Ch_\bullet(\mathcal{A}_{Gr})$]] is Grothendieck abelian)**
\linebreak
For $\mathcal{A}$ a Grothendieck category, the (unbounded) [[category of chain complexes]] $Ch_\bullet(\mathcal{A})$ in $\mathcal{A}$ is again a Grothendieck category (e.g. [Hovey (1999), p. 3](#Hovey99)).

With Exp. \ref{CateoriesOfModules} it follows that the usual categories $Ch_\bullet(R Mod)$ of [[chain complexes]] of [[modules]] over a [[ring]] are all Grothendieck abelian.
\end{example}

\begin{example}
**([[category of ind-objects|$Ind(\mathcal{A})$]] is Grothendieck abelian)**
\linebreak
For $\mathcal{A}$ a [[small category|small]] [[abelian category]], the category $Ind(\mathcal{A})$ of [[ind-objects]] in $C$ is a Grothendieck category. \end{example}

## Related concepts

* [[sheaf of abelian groups]]

* [[abelian sheaf cohomology]]

## References

A dedicated survey is

* Grigory Garkusha, _Grothendieck categories_, Algebra i Analiz 2001, 55 pp. [pdf](https://www.researchgate.net/profile/Grigory_Garkusha/publication/2142579_Grothendieck_Categories/links/570e36d408aec783ddd1ba89.pdf)

Grothendieck categories are mentioned at the end of section 8.3 in

* {#KS} [[Masaki Kashiwara]], [[Pierre Schapira]], _[[Categories and Sheaves]]_

The relation to complexes is in section 14.1.

The proof that filtered colimits in $R Mod$ are exact is spelled out for instance in

* {#Kiersz} Andy Kiersz, _Colimits and homological algebra_, 2006 ([pdf](http://www.math.uchicago.edu/~may/VIGRE/VIGRE2006/PAPERS/Kiersz.pdf))

Proof that $R$[[Mod]] internal to any Grothendieck topos is Grothendieck abelian:

* {#Johnstone1977} [[Peter Johnstone]]: Theorem 8.11 in: *[[Topos Theory]]*, Academic Press (1977) Dover reprint (2014)

See also:

* [[Peter Freyd]], _Abelian categories_, Harper (1966O)

* Nicolae Popescu, *An introduction to Abelian categories with applications to rings and modules*_, Academic Press 1973

The fact that all Grothendieck categories are [[locally presentable]]:

* {#Beke00} [[Tibor Beke]], Prop. 3.10 in: *Sheafifiable homotopy model categories*, Math. Proc. Cambridge Philosophical Society **129** 3 (2000) 447-475 [[arXiv:math/0102087](https://arxiv.org/abs/math/0102087), [doi:10.1017/S0305004100004722](https://doi.org/10.1017/S0305004100004722)]

* {#Krause15} [[Henning Krause]], Corollary 5.2 in: _Deriving Auslander's formula_, Documenta Math. **20** (2015) 669-688 [[arXiv:1409.7051](https://arxiv.org/abs/1409.7051)]

A generalization to $\kappa$-Grothendieck categories (defined using $\kappa$-filtered colimits) is proved in Theorem 2.2 of

* {#PositselskiRosicky} [[Leonid Positselski]], [[Jiří Rosický]], _Covers, envelopes, and cotorsion theories in locally presentable abelian categories and contramodule categories_, [arXiv:1512.08119](https://arxiv.org/abs/1512.08119).

The duality of Grothendieck categories with categories of modules over [[linearly compact ring]]s is discussed in

*  U. Oberst, _Duality theory for Grothendieck categories and linearly compact rings_, J. Algebra __15__ (1970), p. 473 --542, [pdf](https://core.ac.uk/download/pdf/82581773.pdf)

Discussion of [[model structures on chain complexes]] in Grothendieck abelian categories is in

* [[Denis-Charles Cisinski]], F. Déglise, _Local and stable homologial algebra in Grothendieck abelian categories_, Homology, Homotopy and Applications, vol. 11 (1) (2009)  ([pdf](http://www.intlpress.com/HHA/v11/n1/a11/v11n1a11.pdf))

Formalization of Grothendieck categories as [[univalent categories]] in [[homotopy type theory]]:
Formalization of [[abelian category|abelian]] [[univalent categories]] of [[ring]]-[[modules]], in [[homotopy type theory]] ([[univalent foundations of mathematics]]):

* [[Jarl G. Taxerås Flaten]], Section 3.1 in *Univalent categories of modules* [[arXiv:2207.03261](https://arxiv.org/abs/2207.03261)]

On Grothendieck abelian categories of chain complexes:

* {#Hovey99} [[Mark Hovey]], p. 3 of: *Model category structures on chain complexes of sheaves* (1999) [[K-theory:0366](https://faculty.math.illinois.edu/K-theory/0366), [pdf](https://faculty.math.illinois.edu/K-theory/0366/sheaves.pdf), [[Hovey-SheavesOfChainComplexesPreprint.pdf:file]]]

which was published as

* {#Hovey01} [[Mark Hovey]], _Model category structures on chain complexes of sheaves_, Trans. Amer. Math. Soc. **353** 6 (2001) [[ams:S0002-9947-01-02721-0](https://www.ams.org/journals/tran/2001-353-06/S0002-9947-01-02721-0), [jstor:221954](https://www.jstor.org/stable/221954)]

[[!redirects Grothendieck categories]]

[[!redirects Grothendieck abelian category]]
[[!redirects Grothendieck abelian categories]]


Grothendieck connection

#Contents#
* automatic table of contents goes here
{:toc}

## Idea

A _Grothendieck connection_ $\nabla$ is a way to encode the parallel transport of a _flat_ [[connection on a bundle|connection]] along [[infinitesimal object|infinitesimal]] paths.

In the language of [[schreiber:∞-Lie theory|∞-Lie theory]] it is a morphism from the [[infinitesimal path ∞-groupoid]] $\Pi_{inf}(X) \coloneqq \Im(X)$ of a suitable [[space]] $X$ (a [[scheme]] in [[algebraic geometry]] or, more generally, a space in [[synthetic differential geometry]]) to some [[Lie ∞-groupoid]] $A$:

$$
  \nabla : \Pi_{inf}(X) \to A
  \,.
$$

Here $\Pi_{inf}(X)$ is effectively the [[infinitesimal singular simplicial complex]] of $X$ and $A$ is modeled typically as a [[sheaf]] of [[∞-groupoid]]s as described at [[models for ∞-stack (∞,1)-toposes]].

This was originally considered by [[Alexander Grothendieck|Grothendieck]] for [[scheme]]s in the context of [[algebraic geometry]] for the case that $A$ is a 1-[[groupoid]] -- but the generalization to [[∞-groupoid]]s in [[synthetic differential geometry]] is immediate.

## Details

In his study of [[crystalline cohomology]], [[Alexander Grothendieck|Grothendieck]] has noticed that flat [[connection on a bundle|connection]]s correspond to the [[descent]] data for [[deRham complex|deRham]] descent; he called them the costratification of the $n$-th order if one considers the $n$-the infinitesimal neighborhoods.

Already in EGA, Grothendieck has introduced a notion of [[regular differential operator]] and of jet-spaces, which were later by Malgrange and Spencer transferred into the study of differential equations and analytic deformation theory.

De Rham descent for a relative [[scheme]] (= morphism of schemes) $f: X\to S$ is formulated in terms of resolutions or [[infinitesimal object|infinitesimal]] thickenings of the diagonal for the [[simplicial object|simplicial]] scheme

$$
  \cdots
  X
  \times_S X \times_S X
  \stackrel{\to}{\stackrel{\to}{\to}}
   X \times_S X \stackrel{\to}{\to}
   X
$$

corresponding to $f$. Usually one assumes that $X$ is a [[smooth scheme]] of finite type over $S$. By separatedness the diagonal $\Delta: X\hookrightarrow X\times_S X$ is a [[closed subscheme|closed immersion of schemes]]. Let then $\mathcal{I}$ be the corresponding defining ideal sheaf (locally it is generated by the elements of the form $t\otimes 1 - 1\otimes t$, cf. [[Kähler differential]]). The definining ideal $\mathcal{I}^{n+1}$ defines for the $n$-th infinitesimal neighborhood $X^{(n)}$ and the diagonal subscheme $X\cong \Delta(X)\hookrightarrow X\times_S X$; there is a series of inclusions

$$
X\hookrightarrow X^{(1)}\hookrightarrow X^{(2)}\hookrightarrow \ldots \hookrightarrow X^{(\infty)}=\hat{X}
$$

where $\hat{X}$ is the completion (the corresponding [[formal scheme]]). Let $\mathcal{F}\to Sch$ be a [[Grothendieck fibration]] over the category of schemes (or the subcategory of the category of schemes containing all schemes in our consideration) classified under the [[Grothendieck construction]] by some [[pseudofunctor]] $A : Sch^{op} \to Cat$. A typical examples would be the [[stack]] of [[quasicoherent sheaves]] of $\mathcal{O}$-modules.

Consider now the projections $d_0,d_1:X\times_S X\to X$.
Then

+-- {: .un_defn}
###### Definition (Grothendieck connection)

A **Grothendieck connection** on an object $\rho\in\mathcal{F}$ is a [[descent]] datum of the form $(\rho,\theta)$ where $\theta: d_0^*\rho\to d_1^*\rho$ is an isomorphism satisfying the cocycle (and the normalization) condition and such that the restriction (pullback along inclusion) on the diagonal is the identity.

=--

The constructions can be almost literally transferred to the [[synthetic differential geometry]] using [[infinitesimal object|infinitesimal neighborhoods]] in that setup.

More recently, some partial generalizations were found in the purely algebraic framework. One could take any [[coring]] (or even an additive [[comonad]]) with a [[grouplike element]] and define corresponding connections and descent data and generalize the correspondence found by Grothendieck (see [[semi-free dga]] and Menini-Stefan reference below).

## Literature and links

Related entries include [[deRham space]], [[crystal]], [[crystalline cohomology]]

* A. Grothendieck, _Crystals and the de Rham cohomology of schemes_, p. 306--358 of _Dix exposes sur la cohomologie des schemas_, North Holland 1968, [Dix exp. pdf](http://www.math.jussieu.fr/~leila/grothendieckcircle/DixExp.pdf)

* P. Berthelot,  A. Ogus, _Notes on crystalline cohomology_, Princeton Univ. Press 1978. vi+243, ISBN0-691-08218-9

* N. Katz, _Nilpotent connections and the monodromy theorem : applications of a result of Turrittin_, Publications Mathématiques de l'IHÉS, 39 (1970), p. 175-232  [numdam](http://www.numdam.org/item?id=PMIHES_1970__39__175_0)

* A. Grothendieck, (EGA IV.16) _Éléments de géométrie algébrique_ (rédigés avec la collaboration de Jean Dieudonné) : IV. Étude locale des schémas et des morphismes de schémas, Quatrième partie. Publications Mathématiques de l'IHÉS, 32 (1967), p. 5-361 [numdam](http://www.numdam.org/numdam-bin/fitem?id=PMIHES_1967__32__5_0)

* SGA III, vol. 1, Expose VIIa (P. Gabriel) ETUDE INFINITESIMALE DES SCHEMAS EN GROUPES, 409-473

* wikipedia: [Grothendieck connection](http://en.wikipedia.org/wiki/Grothendieck_connection)

* B. Osserman, _Connections, curvature and $p$-curvature_, [pdf](http://www.math.ucdavis.edu/~osserman/math/connections.pdf) (expositional preprint)

* A. Beilinson, I. N. Bernstein, _A proof of Jantzen conjecture_, Adv. in Soviet Math. 16, Part 1 (1993), 1-50. MR95a:22022

* C. Menini, D. Ştefan, _Descent theory and Amitsur cohomology of triples_, J. Algebra 266 (2003), no. 1, 261--304 (doi).

* T. Brzeziński, R. Wisbauer, _Corings and comodules_, London Math. Soc. Lec. Note Series 309, Cambridge 2003.

* L. Breen, Messing, _Combinatorial differential forms_, [arxiv:math/0005087](http://arxiv.org/abs/math/0005087)

* Notes in Gaitsgory's grad student seminar [pdf](http://www.math.harvard.edu/~gaitsgde/grad_2009/SeminarNotes/Nov17-19%28Crystals%29.pdf)