n-truss

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### Context
#### Higher category theory
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[[!include higher category theory - contents]]
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#Contents#
* table of contents
{:toc}

## Idea

$n$-Trusses are the [[fundamental categories]] of [[n-mesh|$n$-meshes]]. They are central to the combinatorial classification of [[manifold diagrams]] and play an important role in [[framed combinatorial topology]].

## Definition

Trusses in $n$-dimensions, or $n$-trusses for short, are defined inductively in the dimension $n$.

### 1-Trusses

\begin{defn}
A _1-truss_ $T \equiv (T,\leq, \preceq, \mathrm{dim})$ is a set $T$ with two [[partial orders]] (the 'face' order $\leq$ and the 'frame' order $\preceq$) as well as a 'dimension' map $\dim : (T,\leq) \to [1]^{\mathrm{op}}$ such that

1. $(T,\preceq) \cong [n] = (0 \to 1 \to ... \to n)$
2. either $i \lt i+1$ or $i + 1 \lt i$ for all $i \prec n$
3. $\dim$ is [[conservative]].

\end{defn}

\begin{defn} \label{defn_1-truss_maps} A _1-truss map_ $F : T \to S$ is a function of sets that preserves both face and frame order. Further,

* $F$ is called _regular_ if $\dim \circ F \Rightarrow \dim$,
* $F$ is called _singular_ if $\dim \circ F \Leftarrow \dim$,
* $F$ is called _dimension-preserving_ if $\dim \circ F = \dim$,

where $\Rightarrow$ and $\Leftarrow$ denote [[natural transformations]] of functors $(T,\leq) \to [1]^{\mathrm{op}} = (0 \leftarrow 1)$.
\end{defn}

\begin{notn} Given a truss $T$, denote by $T_{(i)}$ the subset of objects $x$ with $\dim(x) = i$.
\end{notn}

### 1-Truss bundles

To define bundles of 1-trusses, we first define what are the valid fiber transitions. We dub these '1-truss bordisms'.

\begin{rmk} Below, a _Boolean profunctor_ is an ordinary [[profunctor]] $H : C$ ⇸ $D$ whose values are either the initial set $\emptyset \equiv \bot$ or the terminal set $\ast \equiv \top$. If $C$ and $D$ are [[discrete category|discrete]], then such a profunctor $H$ is simply a relation of sets. In this case, we call the profunctor $H$ a _function_ if it is a functional relation or a _cofunction_ if the dual profunctor $H^{\mathrm{op}}$ is a function.
\end{rmk}

\begin{rmk} \label{rmk_poset_maps_yield_profunctors} For any map of [[posets]] $F : P \to Q$, the fiber $F^{-1}(x \to y)$ over an arrow $x \to y$ of $Q$ defines a Boolean profunctor $F^{-1}(x)$ ⇸ $F^{-1}(y)$ by mapping $(a,b)$ to $\top$ iff $a \to b$ is an arrow in $P$.
\end{rmk}

\begin{defn} Given 1-trusses $T$ and $S$, a _1-truss bordism_ $R : T$ ⇸ $S$ is a Boolean profunctor $T$ ⇸ $S$ satisfying the following:

1. $R$ restricts to a function $R_{(0)} : T_{(0)}$ ⇸ $S_{(0)}$ and a cofunction $R_{(1)} : T_{(1)}$ ⇸ $S_{(1)}$.
2. Whenever $R(t,s) = \top = R(t',s')$, then either $t \prec t'$ or $s' \prec s$ but not both.

\end{defn}

Importantly, 1-truss bordisms are morphisms of a category $\mathfrak{T}^1$ that embeds into the category of [[Prof|profunctors]] $\mathbf{Prof}$ (unlike general Boolean profunctors). See the discussion of 'labels' [below](#labels) for details.

\begin{defn} \label{defn_1-truss_bundle}  A _1-truss bundle_ over a 'base' poset $(P,\leq)$ is a poset map $q : (T,\leq) \to (P,\leq)$ in which, for each $x \in P$, the fiber $(T^x,\leq) = q^{-1}(x)$ is equipped with the additional structures $(\preceq,\dim)$ of a 1-truss, and, for each arrow $x \to y$ in $P$, the fiber $q^{-1}(x \to y)$ is a 1-truss bordisms $T^x$ ⇸ $T^y$ (cf. Rmk. \ref{rmk_poset_maps_yield_profunctors}).
\end{defn}

\begin{defn} A _1-truss bundle map_ $F : q_1 \to q_2$ of 1-truss bundles $q_i : T_i \to P_i$ is a map $F : T_1 \to T_2$ that factors through $q_i$ by a 'base' map $F_0 : P_1 \to P_2$, such that $F$ is fiberwise a 1-truss map (cf. Def. \ref{defn_1-truss_maps}).
We further say $F$ is regular resp. singular resp. dimension-preserving if it is fiberwise so.
\end{defn}

### n-Truss bundles and n-trusses

\begin{defn} An _$n$-truss bundle_ $T$ over a poset $P$ is a tower of 1-truss bundles (see Def. \ref{defn_1-truss_bundle})
\begin{center}
\begin{tikzcd}
	{T_n} & {T_{n-1}} & \cdots & {T_1} & {T_0 = P}
	\arrow["{q_n}", from=1-1, to=1-2]
	\arrow["{q_{n-1}}", from=1-2, to=1-3]
	\arrow["{q_2}", from=1-3, to=1-4]
	\arrow["{q_1}", from=1-4, to=1-5]
\end{tikzcd}
\end{center}
\end{defn}

{#n-truss_bundle_map}
\begin{defn} An _$n$-truss bundle map_ $F : T \to T'$ is a tower of 1-truss bundle maps $F_i : q_i \to q'_i$ where $F_{i-1}$ is the base map of $F_i$ and $F_n \equiv F : T_n \to T'_n$. The adjectives 'regular' resp. 'singular' resp. 'dimension-preserving' apply to $F$ if they apply to all $F_i$. If $T$ and $T'$ have the same base $P$, then $F$ is called _base-preserving_ if $F_0 = \mathrm{id}_P$.
\end{defn}

\begin{terminology} An $n$-truss bundle over the terminal poset $\ast$ is called an _$n$-truss_.
\end{terminology}

We can think of $n$-trusses as the 'fibers' of $n$-truss bundles: indeed, given an $n$-truss bundle $p$ over $P$, then any point $x \in P$ determines an $n$-truss by (iteratively) restricting $p$ over $\{x\}$.

{#labels}
## Labels and classifying categories

### Truss bundles with labels

\begin{defn} Given a [[category]] $C$, a _$C$-labeled $n$-truss bundle_ $T = (\underline T, \mathsf{lbl}_T)$ over $P$ consists of an 'underlying' $n$-truss bundle $\underline T = (T_n \to ... \to T_1 \to P)$ together with a 'labeling' functor $\mathsf{lbl}_T : T_n \to C$. In other words, $T$ is of the form
\begin{center}
\begin{tikzcd}
	C & {T_n} & {T_{n-1}} & \cdots & {T_1} & {T_0 = P}
	\arrow["{q_n}", from=1-2, to=1-3]
	\arrow["{q_{n-1}}", from=1-3, to=1-4]
	\arrow["{q_2}", from=1-4, to=1-5]
	\arrow["{q_1}", from=1-5, to=1-6]
	\arrow["{\mathsf{lbl}_C}"', from=1-2, to=1-1]
\end{tikzcd}
\end{center}
\end{defn}

\begin{rmk} If $C = \ast$ is the terminal category in the previous definition, then we recover ordinary $n$-truss bundles.
\end{rmk}

\begin{defn} A _labeled $n$-truss bundle map_ $F = (\underline F, \mathsf{lbl}_F) : T \to T'$ from a $C$-labeled $n$-truss bundle $T$ to a $C'$-labeled $n$-truss bundle $T'$ consists of an $n$-truss bundle map $\underline F : \underline T \to \underline T'$ and a functor $\mathsf{lbl}_F : C \to C'$ such that $\mathsf{lbl}_{T'} \circ \underline F \cong \mathsf{lbl}_F \circ \mathsf{lbl}_{T}$ commutes up to natural isomorphism. Adjectives 'regular', 'singular', 'dimension-preserving', 'base-preserving' apply if they apply to $\underline F$. Further, we say $F$ is _label-preserving_ if $\mathsf{lbl}_F = \mathrm{id}_C$.
\end{defn}

Labeled truss bundles are a central ingredient in truss theory as the next section will demonstrate.

### Truss bundle classifications

Since fiber transitions in 1-truss bundles are 1-truss bordisms, it comes as no surprise that the category of 1-truss bordisms classifies 1-truss bundles.

\begin{defn}
Given 1-truss bordisms $R : T$ ⇸ $S$ and $Q : S$ ⇸ $U$, their composite profunctor $R \circ Q$  (composed as ordinary profunctors) is again a 1-truss bordism. (In contrast, composites of general Boolean profunctors (composed as ordinary profunctors) in general need not themselves be Boolean.) This defines the _category $\mathfrak{T}^1$ of 1-trusses and their bordisms_.
\end{defn}

\begin{thm} $1$-truss bundles over a base poset $P$ up to dimension-preserving base-preserving isomorphism bijectively correspond to functors $P \to \mathfrak{T}^1$ up to natural isomorphism.
\end{thm}

\begin{proof} Follows from the definition of 1-truss bundles.
\end{proof}

(There are also more categorical phrasings of this theorem, as well as the subsequent theorems below, which replace [[bijection]] with categorical [[equivalence]]; see e.g. [Dorn-Douglas 2021, Ch. 2](#DornDouglas21).)

The theorem now generalizes to labelled 1-truss bundles as follows.

\begin{defn}
Given a category $C$, the _category $\mathfrak{T}^1(C)$ of $C$-labeled 1-trusses and their bordisms_ is defined as follows: objects of $\mathfrak{T}^1(C)$ are $C$-labeled 1-truss bundles over $[0]$; morphisms are $C$-labeled 1-truss bundles over $[1]$ (with domain and codomain given by restricting to fibers over  $0$ resp. $1$); two morphisms compose to a third iff there is a $C$-labeled bundle over $[2]$ that restricts over $(0 \to 1)$, $(1 \to 2)$, and $(0 \to 2)$ to the first, second, resp. third morphism.
The fact that this defines a category is shown in [Dorn-Douglas 2021, Sec. 2.3.1](#DornDouglas21).
\end{defn}

\begin{thm} $C$-labelled $1$-truss bundles over a base poset $P$ up to dimension-preserving base-preserving label-preserving isomorphism bijectively correspond to functors $P \to \mathfrak{T}^1(C)$ up to natural isomorphism.
\end{thm}

\begin{proof} Follows from the definition of $\mathfrak{T}^1(C)$.
\end{proof}

\begin{rmk} _(Recovering the unlabeled case)_. In particular, the preceding two definitions coincide $\mathfrak{T}^1(\ast) \cong \mathfrak{T}^1$ up to (essentially unique!) isomorphism of categories when $C = \ast$ is terminal.
\end{rmk}

\begin{rmk} _(Functoriality of labeled bordism)_.
Importantly, the construction of $\mathfrak{T}(C)$ is functorial in $C$. Indeed, given a functor $F : C \to D$, then $\mathfrak{T}^1(F) : \mathfrak{T}^1(C) \to \mathfrak{T}^1(D)$ acts on objects and morphisms of $\mathfrak{T}^1(C)$ by post-composing their labelings with $F$. This yields the _labeled bordism_ functor
$$
    \mathfrak{T}^1 : \mathbf{Cat} \to \mathbf{Cat}
$$
\end{rmk}

For a given category $C \in \mathbf{Cat}$ we can thus apply the labeled bordism functor $n$ times to it: the resulting category $\mathfrak{T}^n(C)$ classifies $C$-labeled $n$-truss bundless as follows.

\begin{thm} $C$-labelled $n$-truss bundles over a base poset $P$ up to dimension-preserving base-preserving label-preserving isomorphism bijectively correspond to functors $P \to \mathfrak{T}^n(C)$ up to natural isomorphism.
\end{thm}

\begin{proof} Inductively apply the previous theorem, starting with the highest 1-truss bundle and working your way downwards.
\end{proof}

\begin{rmk} _($n$-Truss bundles over categories)_. The theorem makes it evident that nothing would have stopped us from defining $n$-truss bundles over _categories_ $B$ (in place of just posets): indeed, such bundles may be thought of as functors $B \to \mathfrak{T}^n(\ast)$ (or $B \to \mathfrak{T}^n(C)$ in the labeled case).
\end{rmk}

[[Lukas Heidemann]] points out the following nice perspective on the labeled bordism functor.

\begin{rmk} _(Universal construction of the labeled bordism functor)_ Applying the [[Conduché functor#conduch_functors_and_2functors_to_prof|profunctorial Grothendieck construction]] to the (frame-order-forgetting) functor $\mathfrak{T}^1 \to \mathbf{Prof}$, yields an [[Conduché functor|exponentiable fibration]] $E\mathfrak{T}^1 \to \mathfrak{T}^1$. By general nonsense, the composition of the pullback $\mathbf{Cat}_{/ \mathfrak{T}^1} \to \mathbf{Cat}_{/ E\mathfrak{T}^1}$ and forgetful functor $\mathbf{Cat}_{/ E\mathfrak{T}^1} \to \mathbf{Cat}$ has a right adjoint $\mathbf{Cat} \to \mathbf{Cat}_{/ \mathfrak{T}^1}$; this adjoint is exactly the functor $C \mapsto \mathfrak{T}^1(C \to \ast)$. (Note: more generally, this construction applies to all [[normal]] [[pseudofunctors]] $H : D \to \mathbf{Prof}$, where it characterizes the constructions of 'vertical comma categories' $H_{/\!/ C}$ for such functors $H$ (see [Dorn-Douglas 20, Term. 2.3.18](#DornDouglas21)) as right adjoints.)
\end{rmk}

\begin{rmk} _(Labels in $\infty$-categories)_ The construction of $\mathfrak{T}^1(-)$ generalizes to an endofunctor on $\infty$-categories $\mathbf{Cat}_\infty$, which immediately leads to a notion of truss bundles labeled in $\infty$-categories.
\end{rmk}

Thus there is a 'spectrum' of base/label structures on which we can reasonably consider truss bundles, ranging from posets over to categories to $\infty$-categories. Most of the theory works the same across the spectrum. In this article, we work  with the simplest possible choice, i.e. with posets (initially as base structure, but later even as label structures for the purpose of defining 'stratifications').

### Normalization theorem

Another important part of truss theory is normalization.

{#reduction}
\begin{defn} Given $C$-labeled $n$-truss bundles $T$ and $T'$ over $P$, a _reduction_ $F : T \to T'$ is a labeled $n$-truss bundle map which is:

1. regular;
2. endpoint-dimension-preserving, meaning it is dimension-preserving on the _endpoints_ of all 1-truss fibers;
3. base-preserving;
4. label-preserving-on-the-nose, meaning that $\mathsf{lbl}_F = \mathrm{id}$ and $\mathsf{lbl}_{T'} \circ F = \mathsf{lbl}_{T}$ commutes strictly.

\end{defn}

(There's a dual version of the definition that replaces 'regular' by 'singular'; cf. the section on 'duality' [below](#duality).)

\begin{terminology} We sometimes write reductions as $F : T \underset{\mathsf{red}}{\longrightarrow} T'$, and say $T'$ is a _reduct_ of $T$.
\end{terminology}

\begin{terminology} A labeled truss whose only reduct is itself is called _normalized_ (or, said to be _in normal form_).
\end{terminology}

\begin{thm}(Reduction ends in normal forms). The category $\mathsf{Norm}(T)$ of reducts of $T$ and reduction between them has a unique terminal object $[[T]]$ (called the normal form of $T$).
\end{thm}

Various proofs have been given, the first in [Dorn 2018, 5.2.2](#Dorn18) (in the case of open trusses, see Def. \ref{defn_open_trusses}, in which case condition 2. is implied by condition 1. above; the proof generalizes to the case of general trusses, however.), and another (shorter) proof in [Heidemann-Reutter-Vicary 2022](#HRV22) (also for open trusses).

A geometric derivation and interpretation of the normalization theorem can be given in terms of 'the existence of coarsest subdividing framed regular cell complexes of stratification in framed $\mathbb{R}^n$', see [Dorn-Douglas 2021, Ch. 5](#DornDouglas21).

## Combinatorialization theorems

Trusses are the combinatorial analogues (namely, the [[fundamental category|fundamental categorical structures]]) of certain framed stratified topological structures. We describe how this works in two examples.

### Combinatorial meshes

Bare (unlabeled) trusses are combinatorial analogues of [[meshes]]. The relation is given by the following theorem.

\begin{thm} $n$-Meshes up to framed stratified homeomorphism bijectively correspond to $n$-trusses.
\end{thm}

**Proof sketch**. Given an [[n-mesh|$n$-mesh]] $M$
\begin{center}
\begin{tikzcd}
	{M_n} & {M_{n-1}} & \cdots & {M_1} & {\ast}
	\arrow["{p_n}", from=1-1, to=1-2]
	\arrow["{p_{n-1}}", from=1-2, to=1-3]
	\arrow["{p_2}", from=1-3, to=1-4]
	\arrow["{p_1}", from=1-4, to=1-5]
\end{tikzcd}
\end{center}
apply to it the [[stratified space#the_category_of_stratifications|entrance path poset functor]] $\mathsf{Entr}(-)$ to obtain the corresponding $n$-truss
\begin{center}
\begin{tikzcd}
	{\mathsf{Entr}(M_n)} & {\mathsf{Entr}(M_{n-1})} & \cdots & {\mathsf{Entr}(M_1)} & {\ast}
	\arrow["{\mathsf{Entr}(p_n)}", from=1-1, to=1-2]
	\arrow["{\mathsf{Entr}(p_{n-1})}", from=1-2, to=1-3]
	\arrow["{\mathsf{Entr}(p_2)}", from=1-3, to=1-4]
	\arrow["{\mathsf{Entr}(p_1)}", from=1-4, to=1-5]
\end{tikzcd}
\end{center}
(framings and strata dimensions of the mesh canonically induce the required truss structures $\preceq$ and $\dim$). $\square$

\begin{rmk} The theorem has various (more categorical) generalizations, which essentially capture versions of the equivalence
$$
    \mathcal{M}\mathit{esh}_n(B,g) \simeq \mathsf{Truss}_n(\mathcal{E}\mathit{ntr}(B,g))
$$
of the $\infty$-category of mesh bundles over a base stratification $(B,g)$ and the $\infty$-category of truss bundles over the entrance path $\infty$-category $\mathcal{E}\mathit{ntr}(B,g)$ of $g$ (recall, entrance path $\infty$-categories are the duals of [[stratified space#fundamental_categories|exit path $\infty$-categories]]). See [Dorn-Douglas 2021, Ch. 4](#DornDouglas21) for details.
\end{rmk}

{#comb_mdiag}
### Combinatorial manifold diagrams

Certain labeled trusses are the 'combinatorial' analogues of [[manifold diagrams]]. We first define the class of labeled trusses we are interested in. We need four ingredients:

1. Truss stratifications (special case of labelings)
2. Open trusses (special type of trusses)
3. Open truss neighborhoods and stratifed neighborhoods
4. Truss products (or, more specifically, 'cylinders')

\begin{rmk} (Stratifying posets). Recall a characteristic map $f : X \to \mathsf{Entr}(f)$ of a stratification $(X,f)$ maps points in a stratum $s$ to the corresponding poset element $s \in \mathsf{Entr}(f)$. Considering posets $P$ as spaces (by their [[specialization topology]], with the convention that downward closed subposets are open), we may equally consider stratified posets by characteristic maps $f : P \to \mathsf{Entr}(f)$. Such characteristic maps are exactly poset maps which are poset quotients with connected preimages (see [Dorn-Douglas 2021, App. B.1](#DornDouglas21)).
\end{rmk}

\begin{defn} (Stratified truss). A _stratified $n$-truss_ $T$ is a labeled $n$-truss $T$ whose labeling $\mathsf{lbl}_T$ is a characteristic map.
\end{defn}

{#open_trusses}
\begin{defn} \label{defn_open_trusses} (Open truss). A 1-truss is called _open_ if its endpoint have dimension 1. A (labeled) $n$-truss $T$ is called open if all 1-truss fibers in all 1-truss bundles that comprise $T$ are open.
\end{defn}

\begin{defn} (Open neigborhood). Given an open $n$-truss $T$ and an element $x \in T_n$, define the _neighborhood_ $T^{\leq x}$ of $x$ to be the unique open truss that comes with a dimension-preserving map $F : T^{\leq x} \hookrightarrow T$ such that $F : T^{\leq x}_n \hookrightarrow T_n$ is an inclusion of the downward closure of $x$ into the poset $(T_n,\leq)$.
\end{defn}

{#atomic_trusses_terminology}
\begin{terminology} (Atomic truss). Given an open $n$-truss $T$ with $x \in T_n$ such that $T^{\leq x} = T$, we say $T$ is an _atomic_ open $n$-truss with _cone point_ $x$ (in [Dorn-Douglas 2021, Ch. 2](#DornDouglas21), atomic trusses are called 'truss braces').
\end{terminology}

{#stratified_neighborhood}
\begin{defn} (Stratified open neighborhood). Given a stratified open $n$-truss $T$ and an element $x \in T_n$, define the _stratified neighborhood_ $T^{\leq x}$ to be the unique stratified trusses that stratified embeds in $T$ with underlying truss map being the (non-stratified) neighborhood inclusion of $x$.
\end{defn}

\begin{terminology} (Cone types). A stratified open $n$-truss $T$ is said to be a _combinatorial cone type_ if the underlying truss of $T$ is atomic with cone point $x$, and $\{x\} = \mathsf{lbl}^{-1}\circ \mathsf{lbl}(x)$ (in words: '$x$ is its own stratum').
\end{terminology}

{#cylinder}
\begin{defn} (Cylinders). Given a labeled $m$-truss $T$ defined the _$k$-cylinder_ $\mathbb{I}^k \times T$ of $T$ to be the labeled $(m+k)$ obtained from $T$ be adding $k$ trivial truss bundles $\ast \to \ast$ to its underlying truss.
\end{defn}

\begin{rmk} (Products). More generally, one can similarly define labeled $(k+m)$-trusses $U \times T$ as _products_ between unlabeled $k$-trusses $U$ and labeled $m$-trusses $T$.
\end{rmk}

Recall the definition of [[manifold diagrams]]: these are framed conical (compactly triangulable) stratifications. Putting the preceding notions together, we obtain a combinatorial version of framed conicality as follows.

\begin{defn} A _combinatorial manifold $n$-diagram_ $T$ is a stratifed open $n$-truss such that for all $x \in T$ we have
$$
    [[T^{\leq x}]] = \mathbb{I}^k \times C_x
$$
where $C_x$ is a combinatorial cone type.
\end{defn}

{#manifold_diagram_classification}
\begin{thm} Manifold diagrams, up to framed stratified homeomorphism, bijectively correspond to normalized combinatorial manifold $n$-diagrams.
\end{thm}

**Proof sketch**. Given a manifold $n$-diagram $(\mathbb{R}^n,f)$ its corresponding normalized combinatorial manifold $n$-diagram can be constructed by first refining $f$ by the unique coarsest $n$-mesh $M$, and then labeling the $n$-truss $\mathsf{Entr}(M)$ with the labeling $\mathsf{Entr}(M \to f)$. $\square$

(See [Dorn-Douglas 2022, Sec. 2](#DornDouglas22) for details.)

{#duality}
## Duality

There is a [[covariant functor|covariant]] _dualization functor_
$$
    \dagger : \mathsf{Truss}_n \to \mathsf{Truss}_n
$$
from the category of $n$-trusses and $n$-truss maps to itself: the functor is defined by dualizing orders $\leq \mapsto \leq^{\mathrm{op}}$ and fiberwise replacing dimension maps $\dim \mapsto \dim^{\mathrm{op}}$ (using that $[1]^{\mathrm{op op}} \cong [1]^{\mathrm{op}}$ uniquely).

Dualization of $n$-trusses is an [[involution]]. It maps:

1. Singular maps to regular maps and vice-versa
2. Open trusses to _closed_ trusses and vice-versa

Geometrically, dualization dualizes stratifications in the sense of [[Poincare duality]], which can for instance be used to translate between [[manifold diagrams]] and cellular [[pasting diagrams]]. This is discussed in e.g. in [Dorn-Douglas 2022](#DornDouglas22).

Dualization can also be applied to truss bordisms, where it yields a [[contravariant functor|contravariant]] _dualization_ functor:

$$
    \dagger : \mathfrak{T}^n \to (\mathfrak{T}^n)^{\mathrm{op}}
$$

(or, in the labeled case: $\dagger : \mathfrak{T}^n(C) \to (\mathfrak{T}^n(C^{\mathrm{op}}))^{\mathrm{op}}$.)

## Remarks

* Open trusses have also been called 'singular cubes' (e.g. in [Dorn 2018](#Dorn18)) and '[[zig-zags]]' (e.g. in [Reutter-Vicary 2019](#RV19) and [Heidemann-Reutter-Vicary 2022](#HRV22)).
* There is an unrelated notion of [[truss]] used in context of [[ring theory]] and [quantum algebra](https://en.wikipedia.org/wiki/Quantum_algebra).

## Related concepts

* [[n-mesh|n-mesh]]
* [[manifold diagram|manifold diagram]]
* [[manifold-diagrammatic n-category]]
* [[framed combinatorial topology]]

## References

* {#Dorn18} [[Christoph Dorn]], _Associative $n$-categories_, PhD thesis ([arXiv:1812.10586](https://arxiv.org/abs/1812.10586)).

* {#RV19} [[David Reutter]], [[Jamie Vicary]], _High-level methods for homotopy construction in associative $n$-categories_, LICS '19: Proceedings of the 34th Annual ACM/IEEE Symposium on Logic in Computer ScienceJune **62** (2019) 1–13 [[arXiv:1902.03831](https://arxiv.org/abs/1902.03831), [doi:10.1109/LICS52264.2021.9470575](https://doi.org/10.1109/LICS52264.2021.9470575)]

* {#DornDouglas21} [[Christoph Dorn]] and [[Christopher Douglas]], _Framed combinatorial topology_, 2021 ([arXiv](https://arxiv.org/abs/2112.14700), [latest](https://cxdorn.github.io/fct-book/))

* {#HRV22} [[Lukas Heidemann]], [[David Reutter]], [[Jamie Vicary]], *Zigzag normalisation for associative $n$-categories*, Proceedings of the Thirty-Seventh Annual ACM/IEEE Symposium on Logic in Computer Science (LICS 2022) [[arXiv:2205.08952](https://arxiv.org/abs/2205.08952), [doi:10.1145/3531130.3533352](https://dl.acm.org/doi/10.1145/3531130.3533352)]

* {#DornDouglas22} [[Christoph Dorn]] and [[Christopher Douglas]], _Manifold diagrams and tame tangles_, 2022 ([arXiv](https://arxiv.org/abs/2208.13758), [latest](https://cxdorn.github.io/manifold-diagram-paper/))

[[!redirects n-trusses]]
[[!redirects trusses]]


n-type > history

< [[n-type]]
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n-types cover

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### Context
#### Higher category theory
+-- {: .hide}
[[!include higher category theory - contents]]
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# $n$-types cover

* table of contents
{: toc}

## Idea

The property that **$n$-types cover** is a property of a [[higher category theory|higher category]], or an [[axiom]] in its corresponding [[internal logic]], which says roughly that every [[object]] is [[cover|covered]] by one that is [[truncated object|truncated]] at some level.

## In a higher category

A [[n-category|higher category]] is said to satisfy the (external) property that **$n$-types cover**, or to have **enough $n$-types**, if for every [[object]] $X$ there exists an $n$-[[truncated object]] $Y$ and an [[effective epimorphism in an (∞,1)-category|effective epimorphism]] $Y\to X$.  When $n=0$ one also says that **sets cover** or that there are **enough sets**.

Usually the category in question is some sort of [[topos]] or [[higher topos]], or at least a [[pretopos]] of an appropriate sort.  In this case, the property that $n$-types cover means that the [[subcategory]] of $n$-[[truncated objects]] is "generating" in an appropriate sense.

When $n=-1$, however, having enough $n$-types in this sense is not really a useful notion, because $(-1)$-truncated objects ([[subterminal objects]]) are not closed under [[coproducts]].  In this case a better condition is that all maps out of subterminal objects are jointly [[effective epimorphism|effective epimorphic]].

### Examples

If $n\gt 0$, then any [[n-localic (∞,1)-topos]] has enough $(n-1)$-types, since every object is surjected onto by a coproduct of [[representable functor|representables]] which are $n$-truncated.  The converse seems plausible as well.

A specific example of a higher topos in which sets do not cover is the [[slice (∞,1)-topos]] $\infty Gpd / \mathbf{B}^2 \mathbb{Z}$ of [[∞Grpd]] over the double [[delooping]] of the group of [[integers]], which is equivalently the topos of [[∞-actions]] of the [[∞-group]] ([[2-group]]) $\mathbf{B} \mathbb{Z}$.  From this second perspective, a 1-truncated object of this topos is a [[groupoid]] (1-groupoid) together with an [[automorphism]] of its [[identity]] [[functor]], i.e. an element of its [[center]], and a morphism of such is a functor that respects these central elements.  Such an object is 0-truncated if it is essentially discrete, in which case its center is also trivial; thus the 0-truncated objects are just ordinary sets.  However, since functors must respect the central elements, there can be no surjective map from a 0-truncated object to a 1-truncated one whose chosen central element is nontrivial.

## In homotopy type theory

In [[homotopy type theory]], the (internal) axiom of **$n$-types cover** or **enough $n$-types** says that for any type $X$ there [[merely]] exists an $n$-type $Y$ and a surjection (i.e. a [[n-epimorphism|(-1)-connected map]]) $Y\to X$.  In symbols:

$$ \prod_{(X:Type)} {\Vert \sum_{(Y:n Type)} \sum_{(f:Y\to X)} surj(f) \Vert }$$

where $\Vert-\Vert$ denotes the $(-1)$-truncation.  As before, when $n=0$ we say that **[[h-set|sets]] cover** or that there are **enough sets**.

### In models
 {#InModels}

+-- {: .num_theorem}
###### Theorem
If an $(\infty,1)$-topos satisfies the external property that $n$-types cover, then its internal type theory satisfies the internal axiom that $n$-types cover.
=--
+-- {: .proof}
###### Proof
By the [[Kripke-Joyal semantics]] of homotopy type theory, the conclusion requires that for any map $X\to \Gamma$ there is an effective epi $p:\Delta\to \Gamma$, an $n$-truncated map $Y\to \Delta$, and an effective epi $Y\to p^*X$ in the slice category over $\Delta$.  Assuming the hypothesis, we can obtain this by letting $\Delta$ be an (external) $n$-type cover of $\Gamma$ and $Y$ an (external) $n$-type cover of $p^* X$.
=--

The converse, however, is not true.  For the internal axiom, like all internal statements, is preserved by passage to slices (i.e. introduction of a nonempty context), but we saw above that the slice topos $\infty Gpd / \mathbf{B}^2 \mathbb{Z}$ does not have enough sets, even though $\infty Gpd$ does.

For an example of a topos in whose internal logic sets do not cover, let $C$ be the $(2,1)$-category with two objects $a$ and $b$, one morphism $a\to b$, no morphisms $b\to a$, and $C(a,a) = \mathbf{B} \mathbb{Z}$ and $C(b,b)=1$, and consider the presheaf $(\infty,1)$-topos over $C$.  A 1-truncated object therein consists of two groupoids $X_a$ and $X_b$, an element of the center of $X_a$, and a functor $X_b \to X_a$ which relates the chosen central element in $X_a$ with the identity in the center of $X_b$.

If we take $X_b$ to be empty, then $X_a$ is a groupoid with an arbitrary chosen central element, and we can choose it to be one which is nontrivial and hence admits no surjection from a 0-type.  Let $\Gamma = 1$ for this $X$.  Since the terminal object $1$ is a representable (it is $C(-,b)$), it is projective; thus given $\Delta$ and $Y$ as above, we could find a section of $p:\Delta\to\Gamma$ and pull back $Y$ along it to obtain a 0-type cover of $X$ itself, which does not exist.

### Relation to other axioms

The internal axiom that sets cover is related to some forms of the [[axiom of choice]].  Let us denote by

* $AC_0 = $ every surjection between sets merely has a section.
* $AC_\infty = $ every surjection with codomain a set merely has a section.

Then we have

+-- {: .num_theorem}
###### Theorem
$AC_\infty \Leftrightarrow (AC_0$ and sets cover).
=--
+-- {: .proof}
###### Proof
This is a theorem inside homotopy type theory, and likewise for its proof.  Clearly $AC_\infty \Rightarrow AC_0$.  Given $X$, we have a map $X\to {\Vert X \Vert_0}$ which is 0-connected, hence also surjective, and its codomain is a set; thus by $AC_\infty$ it merely has a section.  Any section of a 0-connected map is surjective; thus $X$ is merely covered by $\Vert X\Vert_0$.  (Note that in this case we have a stronger version of "sets cover" in which the first $\sum$ is outside the truncation.)

Conversely, suppose $AC_0$ and sets cover, and let $X\to Z$ be a surjection with codomain a set.  Then there merely exists a set $Y$ and a surjection $Y\to X$, and the composite surjection $Y\to X\to Z$ merely has a section by $AC_0$.  The composite of this section with $Y\to X$ is a section of $X\to Z$.
=--

## Related concepts

* [[cover]]

* [[atlas]]

category: foundational axiom

[[!redirects sets cover]]
[[!redirects enough n-types]]
[[!redirects enough sets]]


n-vector bundle

+-- {: .rightHandSide}
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###Context###
#### Bundles
+-- {: .hide}
[[!include bundles - contents]]
=--
#### Cohomology
+--{: .hide}
[[!include cohomology - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

An _$n$-vector bundle_ is an [[fiber ∞-bundle]] whose [[fibers]] are [[n-vector space]]s of sorts.

## Examples

* For $\mathbf{B}U(1)$ the [[circle n-group|circle 2-group]] and $2 Vect$ the category of [[2-vector space]]s (objects are [[categories]] equivalence to $A$ [[Mod]] for some [[associative algebra]] or algebroid $A$, morphisms are [[bimodule]]s) there is a canonical 1-dimensional [[∞-representation]]

  $$
    \rho : \mathbf{B} \mathbf{B} U(1) \to 2 Vect
  $$

  on the 1-dimensional 2-vector space $Vect_{\mathcal{C}}$.

  For $g : X \to \mathbf{B}^2 U(1)$ a [[cocycle]] for a [[circle n-bundle|circle 2-bundle]], the composite

  $$
    \rho(g) : X \to \mathbf{B}^2 U(1) \stackrel{\rho}{\to} 2 Vect
  $$

  is the corresponding classifying map for the "associated line 2-bundle".

  A [[section]] of $\rho(g)$ is a [[twisted bundle|twisted vector bundle]] with twist given by $\rho$.

## Related concepts

* [[principal bundle]] / [[torsor]] / [[groupoid principal bundle]] / [[associated bundle]]

* [[principal 2-bundle]] / [[gerbe]] / [[bundle gerbe]]

* [[principal 3-bundle]] / [[bundle 2-gerbe]]

* [[principal ∞-bundle]] / [[associated ∞-bundle]]

* [[vector bundle]]

* [[2-vector bundle]]

* [[(∞,1)-vector bundle]] / **$(\infty,n)$-vector bundle**

[[!redirects n-vector bundles]]

[[!redirects (∞,n)-vector bundle]]


N. D. Hari Dass

* [Institute page](https://www.tifrh.res.in/~dass/)

* [InSpire page](https://inspirehep.net/authors/1012399)

* [GoogleScholar page](https://scholar.google.com/citations?user=4XrIHxQAAAAJ&hl=en)

## Selected writings

On [[string theory]] models for [[flux tubes]] in [[QCD]] ([[Polyakov gauge-string duality]]):

* [[N. D. Hari Dass]], *Strings to Strings -- Yang-Mills Flux Tubes, QCD Strings and Effective String Theories*, Lecture Notes in Physics **1018**, Springer (2024) \[doi:10.1007/978-3-031-35358-1\]

  Ch. 23:

   *Effective String Theories (EST) of Yang-Mills Flux Tubes* \[dou:10.1007/978-3-031-35358-1_22, arXiv:2312.10629\]

category: people


N. Raghavendra

Nothing here


N=1 susy compactifications -- table

**[[KK-compactifications]] of higher dimensional [[supergravity]] with minimal ($N=1$) supersymmetry**:

| perspective | [[KK-compactification]] with $N=1$ [[supersymmetry]] |
|-------------|------------------|
| [[M-theory]] | [[M-theory on G2-manifolds]] |
| [[F-theory]] | [[F-theory on CY4-manifolds]] |
| [[heterotic string theory]] | [[heterotic string theory on CY3-manifolds]] |


Nabil Iqbal

* [webpage](https://www.dur.ac.uk/research/directory/staff/?mode=staff&id=15884)

## related $n$Lab entries

* [[monopole]]

category: people


nabla

# Nabla
* table of contents
{: toc}

## Idea

The operator __nabla__ (also called __Atled__ or __Del__, and written __$\nabla$__) is an operator introduced in [[quaternion]]ic analysis and now heavily used in elementary [[vector analysis]].  The [[gradient]], [[curl]], and [[divergence]] may all be defined in terms of it.

## Definition

Informally, nabla is defined to be a [[vector]] (to be thought of as a [[tangent vector]] on some [[manifold]], which is often taken to be the [[cartesian space]] $\mathbb{R}^3$) of [[differential operator]]s
$$ \nabla = \sum_{k = 1}^n e_k \frac{\partial}{\partial x^k} ,$$
and it may be applied to a [[scalar field]] or [[vector field]] in any way that it may be possible to multiply a scalar or vector (respectively) by a vector.  Classically, this gives three applications: the [[gradient]] of a scalar field (corresponding to scalar multiplication), the [[divergence]] of a vector field (corresponding to the [[inner product|dot product]]), and the [[curl|curl or rotation]] of a vector field (corresponding to the [[cross product]]).

Formally, all uses may be combined into a single definition based on [[integration]] on [[hypersurface]]s.  If $M$ is a smooth [[Riemannian manifold]], $V$ and $W$ are [[vector bundles]] over $M$, $\odot$ is a smooth [[bilinear operator]] from the [[tangent bundle]] and $V$ to $W$, and $T$ is a smooth [[section]] of $V$, then $\nabla \odot T$ is a smooth section of $W$ whose value at any point $x$ is given by the integral formula
$$
(\nabla \odot T)(x) = lim_{vol D\to 0} \frac{1}{vol D} \oint_{\partial D} \vec{n} \odot T \,d S,\,\,\,\,\stackrel{\circ}{D}\ni x,
$$
where $D$ runs over the regions with smooth boundary $\partial{D}$ containing the point $x$ and $\vec{n}$ is the unit vector of outer normal to the surface $S=\partial D$.

The formula does not depend (when everything in smooth) on the shape of boundaries taken in limiting process, so one can typically take a [[coordinate chart]] containing balls $D = D_r$ with decreasing radius $r \gt 0$ in this particular coordinate chart.  The formula also does not depend on [[orientation]]; $\partial{D}$ is naturally a [[pseudo-oriented submanifold|pseudo-oriented]] [[submanifold]] and so $\vec{n} \odot T \,d S$ (also written $d\vec{S} \odot T$) is a $W$-valued [[pseudoform|pseudo]]-$(n-1)$-[[differential form|form]], and we can integrate pseudoforms on pseudo-oriented submanifolds.  Perhaps most surprisingly, although it makes no sense in general to integrate $W$-valued (pseudo)forms (unless $W$ is a [[trivial bundle]]), we may simply perform the integration using any coordinate chart as if we were in the cartesian space $\mathbb{R}^n$ (where every vector bundle is trivial), and the limit is again independent of this choice.

This formula can also be extended to the case where $T$, or anything else up to $M$ itself, is only $C^k$ and not smooth, at the risk that $\nabla \odot T$ might not be defined or that the choice of coordinate chart might make a difference even in the limit.  In that case, we consider that $\nabla \odot T$ is defined only if the choice of coordinate chart in fact makes no difference.

## Historical and terminological notes

The nabla is an old instrument in the shape of inverted triangle, sort of an Assyrian harp.  [[William Hamilton]] introduced the operator $\nabla$ (sometimes called Hamilton's or Hamiltonian operator, but distinguish strictly from [[Hamiltonian]] in the mechanics sense) and [[James Maxwell]] used it to write [[Maxwell's equations|his equations]].

* History of nabla [nabla.txt](http://www.mat.univie.ac.at/~neum/contrib/nabla.txt)

The terms 'Atled' and 'Del' are based on the idea that the nabla is an upside-down capital Greek Delta.  Sometimes the [[partial derivative]] symbol $\partial$ is taken to be the lowercase equivalent, as it is a variant (but not upside-down) of the lowercase Delta.

## Related concepts

* **nabla**

* [[gradient]]

  * [[gradient flow]]

  * [[symplectic gradient]]

* [[Hamiltonian flow]]

* [[curl]]

* [[divergence]]

## References

* G.E. Shilov, _Mathematical analysis (functions of several real variables)_, part 1--2

[[!redirects nabla]]
[[!redirects Atled]]
[[!redirects Del]]


Nadia Ott

* [institute page](https://www.math.upenn.edu/people/nadia-ott)

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

* [MathGenealogy page](https://www.mathgenealogy.org/id.php?id=266602)

## Selected writings

On the [[supergeometry|super]]-[[moduli space]] of [[super Riemann surfaces]] (proving it is generically not "projected" in the presence of [[Ramond punctures]], in that it does not [[retraction|retract]] onto its bosonic [[body]]):

* [[Nadia Ott]], [[Alexander A. Voronov]], *The supermoduli space of genus zero SUSY curves with Ramond punctures* [[arXiv:1910.05655](https://arxiv.org/abs/1910.05655)]

* [[Nadia Ott]], *The Supermoduli Space of Genus Zero Susy Curves with Ramond Punctures*, University of Minnesota (2020) [[proquest:28094035](https://www.proquest.com/openview/7b7eaefa3b66fdc1eb16eab554d54153/1/advanced)]

* [[Ron Donagi]], [[Nadia Ott]], *Supermoduli Space with Ramond punctures is not projected* [[arXiv:2308.07957](https://arxiv.org/abs/2308.07957)]

category: people


Nadrian Seeman

* [personal webpahe](http://seemanlab4.chem.nyu.edu/)

* [Institute page](https://as.nyu.edu/content/nyu-as/as/faculty/nadrian-seeman.html)

* [Wikipedia entry](https://en.wikipedia.org/wiki/Nadrian_Seeman)

## Selected writings

On [[topology]] ([[knot]]-structure) of [[molecules]] in [[quantum chemistry]]:

* [[Nils Baas]], [[Nadrian Seeman]], _On the chemical synthesis of new topological structures_, J Math Chem (2012) 50: 220 ([doi:10.1007/s10910-011-9907-3](https://doi.org/10.1007/s10910-011-9907-3))

* [[Nils Baas]], [[Nadrian Seeman]], [[Andrew Stacey]], _Synthesising Topological Links_, J Math Chem. 2015 Jan; 53(1): 183–199 (doi:10.1007\%2Fs10910-014-0420-3)

category: people

[[!redirects Ned Seeman]]


Nafaa Chbili

* [institute page](https://www.uaeu.ac.ae/en/cos/profile.shtml?email=NafaaChbili@uaeu.ac.ae)

* [personal page](http://faculty.uaeu.ac.ae/nafaachbili/)

## Selected writings

On the [[Khovanov homology]] of quasi-alternating [[links]]:

* Khaled Qazaqzeh, [[Nafaa Chbili]], *On Khovanov Homology of Quasi-Alternating Links*,  Mediterr. J. Math. **19** 104 (2022) [[arXiv:2009.08624](https://arxiv.org/abs/2009.08624), [doi:10.1007/s00009-022-02006-5](https://doi.org/10.1007/s00009-022-02006-5)]

* [[Nafaa Chbili]], *Quasi-alternating links, Examples and obstructions*, talk at *[QFT and Cobordism](https://nyuad.nyu.edu/en/events/2023/march/quantum-field-theories-and-cobordisms.html)*, [[CQTS]] (Mar 2023) [[web](Center+for+Quantum+and+Topological+Systems#ChbiliMar2023)]

category: people


Nagata-Smirnov metrization theorem

## Idea

The Nagata-Smirnov [[metrization theorem]] gives a necessary and sufficient condition under which a space is [[metrisable topological space|metrisable]].

## Statement

[[!include Nagata-Smirnov metrization theorem - main theorem]]

## Related theorems

* [[metrization theorem]]
* [[Urysohn metrization theorem]]

## Related concepts

* [[topological space]]
* [[metrisable topological space]]
* [[Polish space]]

## References

* [Nagata–Smirnov metrization theorem](https://en.wikipedia.org/wiki/Nagata%E2%80%93Smirnov_metrization_theorem), Wikipedia


Nagata-Smirnov metrization theorem - main theorem

[[!redirects Nagata-Smirnov metrization theorem/main theorem]]
[[!redirects Nagata-Smirnov metrization theorem/main theorem]]
\begin{theorem}\label{mainTheorem}
A [[topological space]] $ X $ is [[metrisable topological space|metrisable]] if and only if it is [[regular space|regular]], [[Hausdorff space|Hausdorff]] and has a [[countably locally finite set of subsets|countably locally finite]] [[topological base|base]].
\end{theorem}


Nahm's equation

[[!redirects Nahm transform]]

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

#Contents#
* table of contents
{:toc}

## Idea

(...)

## Properties

[[!include Dp-D(p+2)-brane intersections in fuzzy funnels -- section]]

## Related concepts

* [[Yang-Mills monopole]]

  * [[fuzzy funnel]]

  * [[D1-D3-brane intersection]]

  * [[Dp-D(p+2)-brane intersection]]

* [[Yang-Mills instanton]]

* [[ADHM construction]]

* [[magnetic monopole]]

* [[caloron correspondence]]

* [[moduli space of monopoles]]

## References

### General

The original articles;

* [[Werner Nahm]], _The construction of all self-dual multi-monopoles by the ADHM method_, In: Craigie et al. (eds.), _Monopoles in quantum theory_, Singapore, World Scientific 1982 ([spire:178340](http://inspirehep.net/record/178340))

* [[Werner Nahm]], _Self-dual monopoles and calorons_, In: G. Denardo, G. Ghirardi and T. Weber (eds.), _Group theoretical methods in physics_, Lecture Notes in Physics 201. Berlin, Springer-Verlag 1984 ([doi:10.1007/BFb0016145](https://doi.org/10.1007/BFb0016145))

* [[Simon Donaldson]], _Nahm's Equations and the Classification of Monopoles_, Comm. Math. Phys., Volume 96, Number 3 (1984), 387-407, ([euclid:cmp.1103941858](https://projecteuclid.org/euclid.cmp/1103941858))

Further discussion:

* Rafe Mazzeo, [[Edward Witten]], _The Nahm Pole Boundary Condition_, In: _The influence of Solomon Lefschetz in geometry and topology_, Contemporary Mathematics 621 (2014): 171 ([arXiv:1311.3167](https://arxiv.org/abs/1311.3167), [doi:10.1090/conm/621](http://dx.doi.org/10.1090/conm/621))

Review:

* Marcos Jardim, _A survey on Nahm transform_, J Geom Phys 52 (2004) 313-327 ([arxiv:math/0309305](https://arxiv.org/abs/math/0309305))

* Reinier Storm, _The Yang-Mills Moduli Space and
The Nahm Transform_ ([dspace:1874/285043](https://dspace.library.uu.nl/handle/1874/285043))

See also

* Wikipedia, _[Nahm equations](https://en.m.wikipedia.org/wiki/Nahm_equations)_

### In terms of Coulomb branch singularities in SYM

In terms of [[Coulomb branch]] [[singularities]] on [[super Yang-Mills theories]]:

* [[Mathew Bullimore]], [[Tudor Dimofte]], [[Davide Gaiotto]], _The Coulomb Branch of 3d $\mathcal{N}=4$ Theories_, Commun. Math. Phys. (2017) 354: 671 ([arXiv:1503.04817](https://arxiv.org/abs/1503.04817))

### In term of Dp-D(p+2) brane intersections

On transversal [[Dp-D(p+2) brane intersections]] as [[Yang-Mills monopoles]] / [[fuzzy funnel]]-solutions to [[Nahm's equation]]:

For transversal [[D1-D3 brane intersections]]:

* {#Diaconescu97} Duiliu-Emanuel Diaconescu,  _D-branes, Monopoles and Nahm Equations_, Nucl. Phys. B503 (1997) 220-238 ([arxiv:hep-th/9608163](https://arxiv.org/abs/hep-th/9608163))

* [[Amihay Hanany]], [[Edward Witten]], _Type IIB Superstrings, BPS Monopoles, And Three-Dimensional Gauge Dynamics_, Nucl. Phys. B492:152-190, 1997 ([arxiv:hep-th/9611230](https://arxiv.org/abs/hep-th/9611230))

* Jessica K. Barrett, Peter Bowcock, _Using D-Strings to Describe Monopole Scattering_ ([arxiv:hep-th/0402163](https://arxiv.org/abs/hep-th/0402163))

* Jessica K. Barrett, Peter Bowcock, _Using D-Strings to Describe Monopole Scattering - Numerical Calculations_ ([arxiv:hep-th/0512211](https://arxiv.org/abs/hep-th/0512211))

For transversal [[D2-D4-brane bound states]] (with an eye towards [[AdS/QCD]]):

* {#GZZ09} Alexander Gorsky, Valentin Zakharov, Ariel Zhitnitsky, _On Classification of QCD defects via holography_, Phys. Rev. D79:106003, 2009 ([arxiv:0902.1842](https://arxiv.org/abs/0902.1842))

For transversal [[D3-D5 brane intersections]]:

* {#GaiottoWitten08} [[Davide Gaiotto]], [[Edward Witten]], _Supersymmetric Boundary Conditions in N=4 Super Yang-Mills Theory_, J Stat Phys (2009) 135: 789 ([arXiv:0804.2902](https://arxiv.org/abs/0804.2902))

For transversal [[D6-D8 brane intersections]] (with an eye towards [[AdS/QCD]]):

* {#HLPY08} Deog Ki Hong, Ki-Myeong Lee, Cheonsoo Park, Ho-Ung Yee, Section V of: _Holographic Monopole Catalysis of Baryon Decay_, JHEP 0808:018, 2008 ([https:arXiv:0804.1326](https://arxiv.org/abs/0804.1326))

and as transversal [[D6-D8-brane bound states]] on a [[half NS5-brane]] in [[type I' string theory]]:

* {#HananyZaffaroni99} [[Amihay Hanany]], [[Alberto Zaffaroni]], _Monopoles in String Theory_, JHEP 9912 (1999) 014 ([arxiv:hep-th/9911113](https://arxiv.org/abs/hep-th/9911113))

Making explicit the completion of the $\mathfrak{su}(2)_{\mathbb{C}} \simeq \mathfrak{sl}(2,\mathbb{C})$-representation to a $\mathfrak{gl}(2,\mathbb{C})$-representation by adjoining the gauge field component $A_y$ to the scalar fields $\vec X$:

* [Gaiotto-Witten 08, Section 3.1.1](#GaiottoWitten08)

* [[Sergey Cherkis]], _Instantons on Gravitons_, around (21) in: Commun. Math. Phys. 306:449-483, 2011 ([arXiv:1007.0044](https://arxiv.org/abs/1007.0044))

### Lift to M-theory

The lift of [[Dp-D(p+2)-brane bound states]] in [[string theory]] to [[M2-M5-brane bound states]]/[[E-strings]] in  [[M-theory]], under [[duality between M-theory and type IIA string theory]]+[[T-duality]], via generalization of [[Nahm's equation]] (this eventually motivated the [[BLG-model]]/[[ABJM model]]):

* [[Anirban Basu]], [[Jeffrey Harvey]], _The M2-M5 Brane System and a Generalized Nahm's Equation_, Nucl.Phys. B713 (2005) 136-150 ([arXiv:hep-th/0412310](https://arxiv.org/abs/hep-th/0412310))

* {#BaggerLambertMukhiPapageorgakis13} [[Jonathan Bagger]], [[Neil Lambert]], [[Sunil Mukhi]], [[Constantinos Papageorgakis]], Section 2.2.1 of _Multiple Membranes in M-theory_, Physics Reports, Volume 527, Issue 1, 1 June 2013, Pages 1-100 ([arXiv:1203.3546](http://arxiv.org/abs/1203.3546), [doi:10.1016/j.physrep.2013.01.006](https://doi.org/10.1016/j.physrep.2013.01.006))

[[!redirects Nahm transforms]]

[[!redirects Nahm's equation]]
[[!redirects Nahm's equations]]

[[!redirects Nahm equation]]
[[!redirects Nahm equations]]


Naichung Conan Leung

* [webpage](http://www.ims.cuhk.edu.hk/leung/)

## Selected writings

On the [[Peterson isomorphism]] identifying [[quantum cohomology rings]] of [[flag varieties]] with a [[Pontryagin ring]]:

* [[Naichung Conan Leung]], [[Changzheng Li]], *Gromov-Witten invariants for $G/B$ and Pontryagin product for $\Omega K$*, Transactions of the American Mathematical Society **364** 5 (2012) 2567-2599  [[web](https://www.ams.org/journals/tran/2012-364-05/S0002-9947-2012-05438-9/), [arXiv:0810.4859](https://arxiv.org/abs/0810.4859), [jstor:41524936](https://www.jstor.org/stable/41524936)]

On [[quantum cohomology rings]] of [[flag varieties]] ([[quantum Schubert calculus]]):

* [[Naichung Conan Leung]], [[Changzheng Li]], *An update of quantum cohomology of homogeneous varieties* [[arXiv:1407.5905](https://arxiv.org/abs/1407.5905)]

## Related entries

* [[special holonomy]]

* [[G2-structure]]

* [[normed division algebra]]

category: people


naive G-spectrum

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

#Contents#
* table of contents
{:toc}

## Idea

For $G$ a [[topological group]], a _naive $G$-spectrum_ is a [[spectrum object]] in the [[category]] of [[G-spaces]] ([e.g. Carlsson 92, def. 3.1](#Carlsson92)). The [[stable homotopy theory]] of spectra with $G$-action is part of the subject of [[equivariant stable homotopy theory]].

There one typically considers a richer concept of _[[G-spectra]]_. In that context a spectrum with $G$-action is a [[G-spectrum]] for the "trivial universe" and is called a _naive G-spectrum_. Naive G-spectra are obtained by [[stabilization|stabilizing]] [[G-spaces]] with respect to the sphere $S^1$, while (genuine) [[G-spectra]] are obtained by inverting all [[representation spheres]].

## Properties

### Relation to $G$-spectra

For $U$ a universe of $G$-[[representations]] in the sense of [[genuine G-spectra]], there is the inclusion

$$
  i \;\colon\; U^G \longrightarrow U
$$

of the [[fixed points]]. A [[genuine G-spectrum]] modeled on $U^G$ is a [[spectrum with G-action]] ("[[naive G-spectrum]]"). The induced [[adjunction]]

$$
  (i_\ast \dashv i^\ast) \;\colon\; G Spectra \longrightarrow Spectra(PSh_\infty(Orb_G))
$$

has [[unit]] and [[counit]] which are [[equivalences]] on the underlying bare [[spectra]].

(e.g. [Carlsson 92, p. 14](#Carlsson92), [Greenlees May, p. 16](#GreenleesMay))

### Homotopy fixed points and homotopy quotients

Write

$$
  (-)^G \;\colon\; Spectra(G Top) \longrightarrow Spectra
  \,.
$$

for the [[fixed point]] ([[invariants]]) [[functor]] and

$$
  (-)/Q \;\colon\; Spectra(G Top) \longrightarrow Spectra
$$

for the [[quotient]] ([[coinvariants]]) functor.

Write $E G$, as usual, for a [[contractible topological space|contractible]] [[topological G-space]] whose $G$-[[action]] is [[free action|free]]. Write $E G_+$ for this regarded as a pointed $G$-space with a point freely adjoined.

Then there are two functor

$$
  EG_+ \wedge (-) , [EG_+,-] \;\colon\;  Spectra(G Top) \to Spectra(G Top)
$$

given by forming [[smash product]] with $E G_+$ and forming the [[internal hom]] out of $E G_+$.

Then the [[homotopy fixed point]] functor is

$$
  (-)^{h G} \coloneqq [E G_+, -]^G \;\colon\; Spectra(G Top) \longrightarrow Spectra
$$

and the [[homotopy quotient]] functor is

$$
  (-)//G \coloneqq (-)_{h G} \coloneqq (E G_+ \wedge (-))/G
  \;\colon\; Spectra(G Top) \longrightarrow Spectra
$$

### Relation to equivariant cohomology
 {#RelationToEquivariantCohomology}

Spectra with $G$-action [[Brown representability theorem|represent]] $\mathbb{Z}$-graded [[equivariant cohomology]] on [[G-spaces]].

For $X$ a [[G-space]] and $E$ a spectrum with $G$-action, then the corresponding cohomology is

$$
  E^\bullet(X) \coloneqq \pi_\bullet(Hom(X,E)^G)
  \,,
$$

where on the right we have the [[homotopy fixed points]] of the [[mapping spectrum]], which inherits a [[conjugation action]] by $G$ from the $G$-action on $X$ and $E$.

More abstractly, in terms of the [[tangent cohesive (∞,1)-topos]] $T PSh_\infty(Orb)_{/\mathbf{B}G}$ of the [[slice (∞,1)-topos]] of [[orbispaces]] over $\mathbf{B}G$, this means that

$$
  E(X) = \underset{\mathbf{B}G}{\prod} [X,E]
$$

is the [[dependent product]] over $\mathbf{B}G$ of the intrinsic [[cohomology]] of the tangent slice topos. See at _[[∞-action]]_ for more on this.

Notice here [[Elmendorf's theorem]] which identifies [[G-spaces]] with [[(∞,1)-presheaves]] over the [[orbit category]] $Orb_G$. It is via this equivalence that spectra with $G$-action represent [[equivariant cohomology]] in the form of [[Bredon cohomology]].

Hence exhibiting a spectrum $E$ with $G$-action as a [[spectrum]]-valued [[presheaf]] on the [[orbit category]] means to assign to any [[coset space]] $G/H$ of $G$ the $H$-[[homotopy fixed points]] of $E$:

$$
  E//G \;\colon\;  G/H \mapsto E^{H}
  \,.
$$

## References

* {#Carlsson92} [[Gunnar Carlsson]], _A survey of equivariant stable homotopy theory_, Topology, Vol 31, No. 1, pp. 1-27, 1992 ([pdf](http://www.maths.ed.ac.uk/~aar/papers/carlsson1.pdf))

* {#GreenleesMay} [[John Greenlees]], [[Peter May]], _[[Equivariant stable homotopy theory]]_ ([pdf](http://www.math.uchicago.edu/~may/PAPERS/Newthird.pdf))

On [[model category]] structure on naive $G$-spectra:

* {#Schwede97} [[Stefan Schwede]], _Spectra in model categories and applications to the algebraic cotangent complex_, Journal of Pure and Applied Algebra 120 (1997) 77-104 ([pdf](http://www.math.uni-bonn.de/people/schwede/modelspec.pdf))

[[!redirects naive G-spectra]]


Najib Idrissi

* [webpage](https://idrissi.eu)

## References

On [[graph complex]]-models for the [[rational homotopy type]] of [[configuration spaces of points]]:

* [[Najib Idrissi]], _The Lambrechts-Stanley Model of Configuration Spaces_, Invent. Math, 2018 ([arXiv:1608.08054](https://arxiv.org/abs/1608.08054), [doi:10.1007/s00222-018-0842-9](https://dx.doi.org/10.1007/s00222-018-0842-9))

* {#CamposIdrissiLambrechtsWillwacher18} [[Ricardo Campos]], [[Najib Idrissi]], [[Pascal Lambrechts]], [[Thomas Willwacher]], _Configuration Spaces of Manifolds with Boundary_ ([arXiv:1802.00716](https://arxiv.org/abs/1802.00716))

* [[Ricardo Campos]], Julien Ducoulombier, [[Najib Idrissi]], [[Thomas Willwacher]], _A model for framed configuration spaces of points_ ([arXiv:1807.08319](https://arxiv.org/abs/1807.08319))

* {#Idrissi21} [[Najib Idrissi]], [_Real homotopy of configuration spaces_](https://idrissi.eu/research/peccot)

category: reference


Nakanishi-Lautrup field

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#### Fields and quanta
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#Contents#
* table of contents
{:toc}

## Idea

In [[BV-BRST formalism]], for [[gauge fixing]] [[Yang-Mills theory]] (to [[Lorenz gauge]] or similar) a [[contractible chain complex]] of [[auxiliary field|auxiliary]] [[field bundles]] is introduced for two [[Lie algebra]]-valued [[field (physics)|fields]], one in degree zero, called the _Nakanishi-Lautrup field_, usually denoted "$B$" and one in degree -1, called the _[[antighost field]]_, usually denoted $\overline{C}$. See at _[[quantization of Yang-Mills theory]]_.

Beware that there are also the [[antifields]] of the [[ghost fields]], which technically are hence "anti-ghostfields" as opposed to the Nakanishi-Lautrup "antighost-fields".  Whoever is responsible for this bad terminology should be blamed.

## Related concepts

* [[field (physics)]]

  * [[ghost]],

  * [[antifield]], **antighost**

## References

Named after [[Benny Lautrup]] and Noburo Nakanishi, who is sometimes misspelled as "Takanishi".

* Noburo Nakanishi, _Covariant quantization of the electromagnetic Field in the Landau Gauge_, Prog. Theor. Phys. 35, 1111 (1966)
* Benny Lautrup, _Canonical quantum electrodynamics in covariant gauges_, Kong. Dan. Vid. Sel. Mat. Fys. Med. 35, 29 (1967)

Review for the case of [[electromagnetism]] and with [[path integral]] terminology is in

* {#Henneaux90} [[Marc Henneaux]], section 9.1 of _Lectures on the Antifield-BRST formalism for gauge theories_, Nuclear Physics B (Proceedings Supplement) 18A (1990) 47-106 ([pdf](http://www.math.uni-hamburg.de/home/schweigert/ws07/henneaux2.pdf))

while discussion for general [[Yang-Mills theory]] in the context of [[causal perturbation theory]]/[[perturbative algebraic quantum field theory]] is in

* {#Rejzner16} [[Katarzyna Rejzner]], section 7.2 of _Perturbative Algebraic Quantum Field Theory_, Mathematical Physics Studies, Springer 2016 ([web](https://link.springer.com/book/10.1007%2F978-3-319-25901-7))

[[!redirects Nakanishi-Lautrup fields]]

[[!redirects Lautrup-Nakanishi field]]
[[!redirects Lautrup-Nakanishi fields]]

[[!redirects Takanishi-Lautrup field]]
[[!redirects Takanishi-Lautrup fields]]

[[!redirects Lautrup-Takanishi field]]
[[!redirects Lautrup-Takanishi fields]]


Nakayama's lemma

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## Idea

Nakayama's lemma is a simple but fundamental result of commutative algebra frequently used to lift information from the [[fiber]] of a [[sheaf]] over a point (as for example a [[coherent sheaf]] over a [[scheme]]) to give information on the [[stalk]] at that point.

## Statement

We give a general version of the Nakayama's lemma (Prop. \ref{GeneralNakayamaLemma}) and then two special cases that are often considered (Prop. \ref{FirstSpecialVersion} and Prop. \ref{SecondSpecialVersion}).

\begin{proposition}\label{GeneralNakayamaLemma}
**(Nakayama's lemma**)
\linebreak
Let $R$ be a [[commutative ring]] and $M$ be a [[finitely generated]] [[module|$R$-module]]. Then:

1. $\text{supp}(M) = \mathcal{V}(Ann(M))$.

2. If moreover $I\subseteq R$ is an ideal with $supp(M)\cap V(I)$ inhabited, then $M\otimes_{R} R/I$ is nonzero.

\end{proposition}
Recall here that $supp(M) = \{\mathfrak{p}\in Spec(R)\ \text{ s.t. }\ M\otimes_{R} R_{\mathfrak{p}}\ \text{ is nonzero}\}$, $Ann(M) = \{f \in R\ \text{ s.t. }\ f M = 0\}$, and $\mathcal{V}(I) = \{\mathfrak{p} \in Spec(R)\ \text{ s.t. }\ \mathfrak{p} \supseteq I\}$. The first claim is not usually considered part of Nakayama's lemma but will be useful here and can be proved with a closely related argument to that which we will use for the main result.)

The following proof uses relatively little pre-existing commutative algebraic theory (i.e., just some basic facts about $\mathcal{V}$, nonzeroness in short exact sequences, localization, tensoring, and localizing/tensoring cyclic modules)
\begin{proof}

The argument proceeds in the following four steps:

1. As $M$ is finitely generated, it admits a filtration with cyclic cokernels. To be explicit, if the $n$-tuple of elements $(m_{k}\in M)_{k=0,\dots,n-1}$ generates $M$, then it begets an $n$-tuple of short exact sequences
\[\left(\ 0 \to M_{k} \to M_{k+1} \to R/I_{k} \to 0\ \right)_{k=0,\dots,n-1},\]
where $M_{k}$ is the submodule of $M$ spanned by $(m_{k}\in M)_{k=0,\dots,k-1}$ and in particular $M_{0}=0$ and $M_{n}=M$. (Here the maps $M_{k} \to M_{k+1}$ are the evident inclusions; as the cokernel is generated by the image of $m_{k}$, it is in particular isomorphic to $R/I_{k}$ for some ideal $I_{k}\subseteq R$.) The basic idea is that the process of extension by which $M$ is built up from these $R/I_{k}$s interacts sufficiently nicely with the operations of localization/taking fibers that the properties of the latter determine those of the former.

2. For instance, it is clear that any $f \in Ann(M)$ must descend to $0$ on all the $R/I_{k}$s; on the other hand, for all $k=0,\dots,n-1$ we have that $I_{k}M_{k+1} \subseteq M_{k}$. We conclude that $\prod_{k=0}^{n-1} I_{k} \subseteq Ann(M) \subseteq \bigcap_{k=0}^{n-1} I_{k}$, whence $\mathcal{V}(Ann(M)) = \bigcup_{k=0}^{n-1} \mathcal{V}(I_{k})$.

3. Let us now show that $supp(M) = \bigcup_{k=0}^{n-1} \mathcal{V}(I_{k})$; this will complete the proof of our first claim. Recall that for any $\mathfrak{p}\in Spec(R)$ the functor $-\otimes_{R} R_{\mathfrak{p}}$ is exact; we therefore in any case obtain an $n$-tuple of short exact sequences
\[\left(\ 0 \to M_{k} \otimes_{R} R_{\mathfrak{p}} \to M_{k+1} \otimes_{R} R_{\mathfrak{p}} \to R/I_{k} \otimes_{R} R_{\mathfrak{p}} \to 0\ \right)_{k=0,\dots,n-1}.\]
If $\mathfrak{p} \notin \bigcup_{k=0,\dots,n-1} \mathcal{V}(I_{k})$, then each $R/I_{k} \otimes_{R} R_{\mathfrak{p}}$ vanishes, so the above SESs collapse to isomorphisms
\[\left(\ 0 \to M_{k} \otimes_{R} R_{\mathfrak{p}} \to M_{k+1} \otimes_{R} R_{\mathfrak{p}} \to 0\ \right)_{k=0,\dots,n-1}\]
and thus
\[M \otimes_{R} R_{\mathfrak{p}} \simeq M_{n} \otimes_{R} R_{\mathfrak{p}} \simeq M_{0} \otimes_{R} R_{\mathfrak{p}} \simeq 0.\]
Conversely, if $\mathfrak{p} \notin \bigcup_{k=0,\dots,n-1} \mathcal{V}(I_{k})$, then there is a maximal index $k_{max}$ such that $\mathfrak{p} \supseteq I_{k_{max}}$ and in particular an epimorphism
\[M_{k_{max}+1} \otimes_{R} R_{\mathfrak{p}} \to R/I_{k_{max}} \otimes_{R} R_{\mathfrak{p}} \to 0,\]
with $R/I_{k_{max}} \otimes_{R} R_{\mathfrak{p}}$ nonzero. But there are also, as before, isomorphisms
\[\left(\ 0 \to M_{k} \otimes_{R} R_{\mathfrak{p}} \to M_{k+1} \otimes_{R} R_{\mathfrak{p}} \to 0\ \right)_{k=k_{max}+1,\dots,n-1}\]
whence
\[M \otimes_{R} R_{\mathfrak{p}} \simeq M_{n} \otimes_{R} R_{\mathfrak{p}} \simeq M_{k_{max}+1} \otimes_{R} R_{\mathfrak{p}}\]
is epic onto a nonzero module and thus itself nonzero.

4. The proof of the second claim is in the same vein, requiring just a little bit more delicacy. Suppose that $\mathfrak{p}$ inhabits $\text{supp}(M)\cap V(I)$; by the above this means that $\mathfrak{p}\supseteq I$ and that there exists some index $k$ for which $\mathfrak{p}\supseteq I_{k}$. Again consider the maximal index $k_{max}$ for which $\mathfrak{p} \supseteq I_{k_{max}}$. As before, there is an epimorphism
\[M_{k_{max}+1} \otimes_{R} R_{\mathfrak{p}} \to R/I_{k_{max}} \otimes_{R} R_{\mathfrak{p}} \to 0\]
and isomorphisms
\[\left(\ 0 \to M_{k} \otimes_{R} R_{\mathfrak{p}} \to M_{k+1} \otimes_{R} R_{\mathfrak{p}} \to 0\ \right)_{k=k_{max}+1,\dots,n-1}\]
By the right-exactness of the functor $-\otimes_{R} R/\mathfrak{p}$, we obtain from the former an epimorphism
\[M_{k_{max}+1} \otimes_{R} R_{\mathfrak{p}} /\mathfrak{p} \to R/I_{k_{max}} \otimes_{R} R_{\mathfrak{p}}/\mathfrak{p} \to 0\]
(implicitly using that $(- \otimes_{R} R_{\mathfrak{p}})\otimes_{R} R/\mathfrak{p}\ \simeq\ - \otimes_{R} R_{\mathfrak{p}}/\mathfrak{p}$).
By its functoriality, there are likewise isomorphisms
\[\left(\ 0 \to M_{k} \otimes_{R} R_{\mathfrak{p}}/\mathfrak{p} \to M_{k+1} \otimes_{R} R_{\mathfrak{p}}/\mathfrak{p} \to 0\ \right)_{k=k_{max}+1,\dots,n-1}\]
Now,
\[R/I_{k_{max}} \otimes_{R} R_{\mathfrak{p}}/\mathfrak{p} \simeq R_{\mathfrak{p}}/\mathfrak{p}\]
is nonzero (here we use that $\mathfrak{p} \supseteq I_{k_{max}}$). It follows that
\[M \otimes_{R} R_{\mathfrak{p}}/\mathfrak{p} \simeq M_{n} \otimes_{R} R_{\mathfrak{p}}/\mathfrak{p} \simeq M_{k_{max}+1} \otimes_{R} R_{\mathfrak{p}}/\mathfrak{p}\]
is epic onto a nonzero module and thus itself nonzero.
But
\[M \otimes_{R} R_{\mathfrak{p}}/\mathfrak{p} \simeq (M \otimes_{R} R/I) \otimes_{R} R_{\mathfrak{p}}/\mathfrak{p}\]
(here we use that $\mathfrak{p} \supseteq I$), so the latter must itself be nonzero, as claimed.

\end{proof}

It is now straightforward to recover more familiar forms of Nakayama's lemma as a corollary of the above.

\begin{proposition}\label{FirstSpecialVersion}
**(special version 1)**
\linebreak
Let $R$ be a local ring with maximal ideal $\mathfrak{m}\subseteq R$ and $M$ be a finitely generated $R$-module. Then if $M$ is nonzero, so too is $M \otimes_R R/\mathfrak{m}$.
\end{proposition}
\begin{proof}
Indeed, $\mathfrak{m}$ is practically tautologically in (and so $\mathcal{V}(\mathfrak{m})$ practically tautologically intersects) the support of any nonzero $R$-module; the conditions of Prop. \ref{GeneralNakayamaLemma}, and we conclude that $M \otimes_R R/\mathfrak{m}$ is nonzero as advertised.
\end{proof}

Sometimes (for instance in the [Stacks Project](https://stacks.math.columbia.edu/tag/07RC)), Nakayama's lemma is stated instead as follows:

\begin{proposition}\label{SecondSpecialVersion}
**(special version 2)**
\linebreak
Let $R$ be a commutative ring, $M$ be a finitely generated $R$-module, and $I\subseteq R$ be an ideal with $I M = M$. Then there exists $f\in R$ such that $f = 1\ (mod\ I)$ and $f M=0$.
\end{proposition}
\begin{proof}
Of course, $I M = M\ \iff\ M \otimes_{R} R/I \simeq 0$; it follows from Prop. \ref{GeneralNakayamaLemma} that $I$ is comaximal with $Ann(M)$. The [Chinese Remainder theorem](https://stacks.math.columbia.edu/tag/00DT) then asserts the existence of $f\in R$ such that $f = 1\ (mod\ I)$ and $f = 0\ (mod\ Ann(M))$; the desired $f$.
\end{proof}

## Examples and consequences

\begin{example}
Suppose $f \colon N \to M$ is an $R$-module map, giving rise to an exact sequence

$$N \stackrel{f}{\to} M \stackrel{p}{\to} M/N \to 0.$$

Tensoring with $k$ is a right exact functor, so we have an exact sequence

$$k \otimes_R N \stackrel{k \otimes_R f}{\to} k \otimes_R M \to k \otimes_R M/N \to 0.$$

Nakayama's lemma says that if $k \otimes_R M/N \cong 0$, then $M/N \cong 0$. Equivalently, that if $k \otimes_R f$ is epic, then $f$ is epic. In particular, to check whether a finite set of elements $v_1, \ldots, v_n$ generates $M$, it suffices to check whether the residue classes $v_i \mod \mathfrak{m}M$ generate the vector space $M/\mathfrak{m}M$, which is a linear algebra calculation.
\end{example}

\begin{example}
Suppose $O$ is a [[Noetherian ring|Noetherian]] local ring. A typical example is the stalk at a point $p$ of a Noetherian scheme as locally ringed space, and we will write as if we were in that situation. Being Noetherian, its maximal ideal $\mathfrak{m}$ is finitely generated. Suppose $k \otimes_O \mathfrak{m} \cong \mathfrak{m}/\mathfrak{m}^2$ -- the cotangent space -- is a vector space of dimension $n$. We would like to know whether a collection of functions $f_1, \ldots, f_n$ that vanish at $p$ form a local coordinate system.

For this, it suffices to check whether the differentials $d f_1, \ldots, d f_n$ at $p$, belonging to the cotangent space $\mathfrak{m}/\mathfrak{m}^2$, are linearly independent. (For then they span the cotangent space, and one concludes from Nakayama that the $f_i$ generate $\mathfrak{m}$ as an $O$-module, thereby forming a local coordinate system at $p$.) In this way, Nakayama's lemma operates as a kind of "inverse function theorem".
\end{example}

To cement this further, the following statement is offered in [Harris](#Harris) as a corollary of Nakayama's lemma (corollary 14.10, page 179):

+-- {: .num_prop}
###### Proposition
**(Inverse Function Theorem)**
A map between complex projective varieties of dimension $n$ which is a bijection and has injective derivative at every point is an isomorphism.
=--

[[!redirects Nakayama lemma]]


naked singularity

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## Idea

In [[gravity]] ([[general relativity]]) a [[singularity]] (such as that of a [[black hole]]) that is not behind an [[event horizon]] is called a _naked singularity_.

Naked singulrities appear for "extremal" black holes, with maximum [[charge]]/[[angular momentum]] at the given [[mass]] (see at [[charged and rotating black holes -- table]]).

Accoringly, in [[supergravity]] naked singularities appear for [[BPS state]] [[black branes]].

## References

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

[[!redirects naked singularities]]


Nakwoo Kim

* [institute page](http://string.khu.ac.kr/people.html)

## Selected writings

On [[D=11 N=1 supergravity]] solutions for the [[near horizon geometry]] of [[black brane|black]] [[M5-branes]] [[wrapped brane|wrapped]] on [[supersymmetric cycles]]:

* [[Jerome Gauntlett]], [[Nakwoo Kim]], [[Daniel Waldram]],  _M-Fivebranes Wrapped on Supersymmetric Cycles_, Phys. Rev. D63 (2001) 126001 ([arXiv:hep-th/0012195](https://arxiv.org/abs/hep-th/0012195))

Discussion of [[pp-wave spacetimes]] as [[Penrose limits]] of [[anti de Sitter spacetime|AdSp]] [[product manifold|x]] [[n-sphere|S^q]] spacetimes and of the corresponding limit of the [[AdS5-CFT4 correspondence]]:

* [[Nakwoo Kim]], Ari Pankiewicz, Soo-Jong Rey, [[Stefan Theisen]], _Superstring on PP-Wave Orbifold from Large-N Quiver Gauge Theory_, Eur. Phys. J. C25:327-332, 2002 ([arXiv:hep-th/0203080](https://arxiv.org/abs/hep-th/0203080))

On the [[BMN matrix model]]:

* [[Nakwoo Kim]], [[Jan Plefka]], _On the Spectrum of PP-Wave Matrix Theory_, Nucl. Phys. B643:31-48, 2002 ([arXiv:hep-th/0207034](https://arxiv.org/abs/hep-th/0207034))

On the [[BMN matrix model]] as a [[KK-compactification]] of [[D=4 N=4 super Yang-Mills theory]] on the [[3-sphere]]:

* [[Nakwoo Kim]], [[Thomas Klose]], [[Jan Plefka]], _Plane-wave Matrix Theory from N=4 Super Yang-Mills on $\mathbb{R} \times S^3$_, Nucl. Phys. B671:359-382, 2003 ([arXiv:hep-th/0306054](https://arxiv.org/abs/hep-th/0306054))

Specifically on [[D=11 N=1 supergravity]] solutions for the [[near horizon geometry]] of [[black brane|black]] [[M5-branes]] [[wrapped brane|wrapped]] on [[hyperbolic 3-manifolds]] $\Sigma^3 = H^3/\Gamma$:

* Aristomenis Donos, [[Jerome Gauntlett]], [[Nakwoo Kim]], Oscar Varelam, _Wrapped M5-branes, consistent truncations and AdS/CMT_, JHEP 1012:003, 2010 ([arXiv:1009.3805](https://arxiv.org/abs/1009.3805))

On the [[3d-3d correspondence]]:

* {#GangKimLee14a} [[Dongmin Gang]], [[Nakwoo Kim]], Sangmin Lee, _Holography of Wrapped M5-branes and Chern-Simons theory_, Physics Letters B
Volume 733, 2 June 2014, Pages 316-319 ([arXiv:1401.3595](https://arxiv.org/abs/1401.3595))

* {#GangKimLee14b} [[Dongmin Gang]], [[Nakwoo Kim]], Sangmin Lee, _Holography of 3d-3d correspondence at Large $N$_, JHEP04(2015) 091 ([arXiv:1409.6206](https://arxiv.org/abs/1409.6206))

* {#GangKim18} [[Dongmin Gang]], [[Nakwoo Kim]], _Large $N$ twisted partition functions in 3d-3d correspondence and Holography_, Phys. Rev. D 99, 021901 (2019) ([arXiv:1808.02797](https://arxiv.org/abs/1808.02797))

* {#GangKimPandoZayas19} [[Dongmin Gang]], [[Nakwoo Kim]], Leopoldo A. Pando Zayas, _Precision Microstate Counting for the Entropy of Wrapped M5-branes_ ([arXiv:1905.01559](https://arxiv.org/abs/1905.01559))

Discussion of the [[volume conjecture]] by combining the [[3d-3d correspondence]] with [[AdS/CFT]] for background of [[M5-branes]] [[wrapped brane|wrapped]] on [[hyperbolic 3-manifolds]]:

* [Gang-Kim-Lee 14b, Section 3.2](#GangKimLee14b)

* [Gang-Kim 18, around (21)](#GangKim18)

Enhanced to a [[defect field theory]]:

* [[Dongmin Gang]], [[Nakwoo Kim]], Mauricio Romo, Masahito Yamazaki, _Aspects of Defects in 3d-3d Correspondence_, J. High Energ. Phys. (2016) ([arXiv:1510.05011](https://arxiv.org/abs/1510.05011))

category: people


Nalini Joshi

Nalini Joshi is an Australian mathematician born in Burma.
Her research is mainly about [[integrable system]]s (and [[special function]]s appearing there).

* [wikipedia](https://en.wikipedia.org/wiki/Nalini_Joshi)

## Selected writing

On [[Painlevé transcendent]]s:

* Martin D. Kruskal, Nalini Joshi, Rod Halburd, _Analytic and asymptotic methods for nonlinear singularity analysis: a review and extensions of tests for the Painlevé property_, 1996 [pdf](http://www.ucl.ac.uk/~ucahrha/Publications/Pond-97.pdf)

* [[Nalini Joshi]], Pieter Roffelsen, _On the monodromy manifold of $q$-Painlevé VI and its Riemann-Hilbert problem_, [arXiv/2202.10597](https://arxiv.org/abs/2202.10597)

category: people


Nambu mechanics

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> under construction

#Contents#
* table of contents
{:toc}

## Idea

What is called _Nambu mechanics_ is a generalization of the formulation of [[classical mechanics]] ([[prequantum geometry]]) via [[Poisson manifold|Poisson brackets]] to bracket-operations of arity $(n+1)$ higher than 2 -- the Nambu brackets --, with an "[[n-Lie algebra|(n+1)-Lie algebra]]"-structure (see the discussion there for distinction with proper [[Lie n-algebras]]).

## Examples

### The membrane

One example that naturally gives rise to Nambu mechanics is the relativistic [[membrane]], see at _[[membrane matrix model]]_ and at _[[BLG model]]_.

## Related concepts

* [[higher Poisson structure]]

  * [[Poisson n-algebra]]

  * [[Poisson bracket Lie n-algebra]]

* [[membrane]]

## References

### General

Nambu mechanics and Nambu brackets were introduced in

* [[Yoichiro Nambu]], _Generalized Hamiltonian dynamics_, Phys. Rev. D (3) 7 (1973), p. 2405-2412 [MR455611](http://www.ams.org/mathscinet-getitem?mr=455611) [doi](http://dx.doi.org/10.1103/PhysRevD.7.2405) [Zbl](http://www.zentralblatt-math.org/zmath/en/search/?q=an:1027.70503)

Detailed analysis is in

* [[Tamiaki Yoneya]], _Generalized Hamilton-Jacobi theory of Nambu Mechanics_ ([arXiv:1612.08509](https://arxiv.org/abs/1612.08509))

* [[Tamiaki Yoneya]], _Lectures on Higher-Gauge Symmetries from Nambu Brackets and Covariantized M(atrix) Theory_,  lectures delivered in the workshop "[Higher Structure in String Theory and M-theory](http://www.tfc.tohoku.ac.jp/event/4123.html)", TFC Thematic Program, Fundamental Problems on Quantum Physics, Tohoku University (March 7-11, 2016), ([arXiv:1612.08513](https://arxiv.org/abs/1612.08513))

Review:

* [[Thomas L. Curtright]], [[Cosmas K. Zachos]], _Classical and Quantum Nambu Mechanics_, Phys. Rev. D **68** 085001 (2003) [[arXiv:hep-th/0212267](http://arxiv.org/abs/hep-th/0212267), [doi:10.1103/PhysRevD.68.085001](https://doi.org/10.1103/PhysRevD.68.085001)]

* Wikipedia, _[Nambu mechanics](http://en.wikipedia.org/wiki/Nambu_mechanics)_

### Interpretation in higher symplectic geometry
Comments on interpreting Nambu mechanics in [[2-plectic geometry]] (and in view of the [[IKKT matrix model]]) appear in

* [[Christian Saemann]], [[Patricia Ritter]], _Lie 2-algebra models_ ([arXiv:1308.4892](http://arxiv.org/abs/1308.4892))

For background on this see also the discussion at _[[3-Lie algebra]]_ on how these are given by [[Lie 2-algebras]] (as [metric Lie 2-algebras](#n-Lie+algebra#SaemannRitter13))

[[!redirects Nambu bracket]]
[[!redirects Nambu brackets]]


Nambu-Goto action

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## Definition

The _Nambu-Goto action_ is an [[action functional]] for [[sigma-models]] with [[target space]] a ([[pseudo-Riemannian manifold|pseudo]]) [[Riemannian manifold]] $(X,g)$: it is the induced volume functional

$$
  S_{NG}
  \;\colon\;
  (\Sigma \stackrel{\gamma}{\to} X)
    \mapsto
  T \int_\Sigma dvol(\gamma^* g)
  \,,
$$

where $dvol(\gamma^* g)$ is the [[volume form]] of the pullback $\gamma^* g$ of the metric tensor from $X$ to $\Sigma$, and where $T$ (the [[brane]]-"tension", e.g. the [[string tension]] for $dim(\Sigma) = 2$) is an inverse [[unit]] of [[length]] to the power the [[dimension]] $dim(\Sigma)$.

## Definition

Let

* $p \in \mathbb{N}$ (for [[p-brane]] dynamics);

* $(X,g)$ a [[pseudo-Riemannian manifold]] ([[target spacetime]]);

* $\Sigma$ a [[compact topological space|compact]] [[smooth manifold]] of dimension $(p+1)$ ([[worldvolume]]).

* $[\Sigma,X]$ the [[diffeological space]] of [[smooth functions]] $\Sigma \to X$.

For $\phi \colon \Sigma \longrightarrow X$ the induced "proper volume" or _Nambu-Goto action_ of $\phi$ is the [[integral]] over $\phi$ of the [[volume form]] of the pullback of the target space metric $g$ to $\Sigma$.

$$
  S_{NG}(\phi) \coloneqq \int_{\Sigma} dvol(\phi^\ast(g))
  \,.
$$

Notice that the rank-2 [[tensor]] $\phi^\ast g\in \Gamma(T* \Sigma \oplus T* \Sigma)$ is in general not non-degenerate (unless $\phi$ is an [[embedding]]), hence is in general not, strictly speaking a [[pseudo-Riemannian metric]] on $\Sigma$, but nevertheless it induces a [[volume form]] by the standard formula, only that this allowed to vanish pointwise (and even globally, for instance if $\phi$ is constant on a single point). In the literature $dvol(\phi^\ast g)$ is usually written as $\sqrt{-g}d^{p+1}\sigma$.

## Properties

### Relation to the Polyakov action

The NG is classically equivalent to the [[Polyakov action]] with "[[worldvolume]] [[cosmological constant]]". See at _[Polyakov action -- Relation to Nambu-Goto action](Polyakov+action#RelationToNambuGotoAction)_.

## Applications

The NG-action serves as the kinetic [[action functional]] of the [[sigma-model]] that described a fundamental [[brane]] propagating on $X$. For
$dim \Sigma = 1$ this is the [[relativistic particle]], for $dim \Sigma = 2$ the [[string]], for $dim \Sigma = 3$ the [[membrane]].

## Related concepts

* [[minimal surface]]

* [[Polyakov action]]

* [[Dirac-Born-Infeld action]]

* [[Veneziano amplitude]]

* [[particle]], [[string]], [[membrane]]

* [[membrane instanton]]

* for more on brane tension see also _[Worldsheet and brane instantons](non-perturbative+effect#WorldsheetAndBraneInstantons)_

## References

The Nambu-Goto action functional originates as a proposal for the dynamics of [[strings]] meant to explain the "dual resonance model" for [[hadron]] [[bound states]] ([[quantum hadrodynamics]], cf. *[[Polyakov gauge-string duality]]*):

* {#Nambu70} [[Yoichiro Nambu]], *Duality and Hadrodynamics*, Notes prepared for the Copenhagen High Energy Symposium (1970) [[doi:10.1142/9789812795823_0026](https://doi.org/10.1142/9789812795823_0026), [[Nambu-DualityAndHadrodynamics.pdf:file]]]

* {#Gotō71} [[Tetsuo Gotō]], *Relativistic Quantum Mechanics of One-Dimensional Mechanical Continuum and Subsidiary Condition of Dual Resonance Model*, Progress of Theoretical Physics **46** 5 (1971) 1560–1569 [[doi:10.1143/PTP.46.1560](https://doi.org/10.1143/PTP.46.1560)]

Historical review:

* [[Joël Scherk]], *An introduction to the theory of dual models and strings*, Rev. Mod. Phys. **47** 123 (1975) [[doi:10.1103/RevModPhys.47.123](https://doi.org/10.1103/RevModPhys.47.123)]

* [[Hiroshi Itoyama]], *Birth of String Theory*, Progress of Theoretical and Experimental Physics **2016** 6 (2016) 06A103 [[arXiv:1604.03701](http://arxiv.org/abs/1604.03701), [doi:10.1093/ptep/ptw063](https://doi.org/10.1093/ptep/ptw063)]

Detailed discussion of the relation to the [[Polyakov action]] and the [[Dirac-Born-Infeld action]] is in

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

One [[string theory]] textbook that deals with the Nambu-Goto action in a bit more detail than usual is

* [[Barton Zwiebach]], _A first course in string theory_ , Cambridge (2009)

Discussion of the Nambu-Goto action and [[Polyakov action]] on [[worldsheets]] with [[manifold with boundary|boundary]] (i.e. in the generality of [[open strings]]) and cast in [[BV-BRST formalism]]:

* {#MartinoliSchiavina22} S. Martinoli, [[Michele Schiavina]], *BV analysis of Polyakov and Nambu–Goto theories with boundary*,  Lett. Math. Phys. **112** 35 (2022) [[doi:10.1007/s11005-022-01526-1](https://doi.org/10.1007/s11005-022-01526-1), [arXiv:2106.02983](https://arxiv.org/abs/2106.02983)]

[[!redirects Nambu-Gotō action]]

[[!redirects tension]]
[[!redirects tensions]]

[[!redirects Nambu-Goto action functional]]
[[!redirects Nambu-Goto action functionals]]

[[!redirects Nambu-Gotō action functional]]
[[!redirects Nambu-Gotō action functionals]]


name binding

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## Idea

(...)

## Related concepts

* [[programming language]]

* [[type theory]], [[syntax]]

* [[Schanuel topos]]

## References

See also

* Wikipedia, _[Name binding](https://en.m.wikipedia.org/wiki/Name_binding)_

[[!redirects name bindings]]

[[!redirects name-binding]]
[[!redirects name-bindings]]


name of the relation

##Definition##

Take a topos $\mathcal{E}$ and a [monic](http://ncatlab.org/nlab/show/monomorphism) $\rho \colon R \hookrightarrow B \times A$ (thought of as a _relation_ between $A$ and $B$). The _name_ of $R$ is then the transpose $r \colon B \rightarrow \Omega^A$ of the characteristic map $\phi_{\rho} \colon B \times A \rightarrow \Omega$ of $\rho$ in $\mathcal{E}$.

##References##
**Peter Johnstone**, [Sketches of an Elephant](http://ncatlab.org/nlab/show/Elephant) Section A2.1


Naming and Necessity

The page is about

* [[Saul A. Kripke]]

  \linebreak

  **Naming and Necessity**

  \linebreak

  Harvard University Press

  1980

  \linebreak

  [pdf](http://people.exeter.ac.uk/sp344/naming_and_necessityocr.pdf)

  ([Wikipedia entry](https://en.wikipedia.org/wiki/Naming_and_Necessity))

on the [[philosophy of language]].

## Related entries

* [[modal logic]]

* [[possible worlds semantics]]

* [[multiverse]]

* [[Meaning and Necessity]]

category: reference


Nancy Hingston

* [Wikipedia entry](https://en.wikipedia.org/wiki/Nancy_Hingston)

* [Institute page](https://science.tcnj.edu/school-information/women-in-science/dr-nancy-hingston/)

* [Mathematics Genealogy page](https://science.tcnj.edu/school-information/women-in-science/dr-nancy-hingston/)

## Selected writings

On the [[cyclic loop spaces]] of [[n-spheres]]:

* [[Nancy Hingston]], *An Equivariant Model for the Free Loop Space of $S^N$*, American Journal of Mathematics **114** 1 (1992) 139-155 ([doi:10.2307/2374740](https://doi.org/10.2307/2374740), [jstor:2374740](https://www.jstor.org/stable/2374740))

category: people


nanometer

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## Idea

The _nanometer_ is a [[physical unit]] of [[length]].

## Definition

A **nanometer** (abbreviated $nm$) is one $10^{9}$th of a [[meter]]:

$$
  nm \coloneqq 10^{-9} m
  \,.
$$

The [[scale]] between about a nanometer and a [[micrometer]] is the scale of [[mesoscopic physics]].

## Related concepts

* [[meter]]

* [[micrometer]]

* [[femtometer]]

* [[Planck length]]

## References

See also

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

[[!redirects nanometers]]

[[!redirects nanometre]]
[[!redirects nanometres]]

[[!redirects nm]]


Naoto Nagaosa

* [research group page](http://cems.riken.jp/en/laboratory/sctrg)

* [Wikipedia entry (de)](https://de.wikipedia.org/wiki/Naoto_Nagaosa)

## Selected writings

On [[quantum field theory]]-methods (such as [[path integrals]]) in [[condensed matter theory]]:

* [[Naoto Nagaosa]], *Quantum Field Theory in Condensed Matter Physics*, Texts and Monographs in Physics, Springer (1999) [[doi:10.1007/978-3-662-03774-4_2](https://doi.org/10.1007/978-3-662-03774-4_2)]

category: people


Narain group

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## Idea

The [[orthogonal group]] $O(n,n)$ for signature $(n,n)$ is sometimes called the _Narain group_ or _generalized T-duality group_ for the role that it plays in [[T-duality]] of [[type II string theory]]. See also at _[[type II geometry]]_.

## Properties

### Structure group of generalized tangent bundle

For $X$ a [[smooth manifold]], the [[generalized tangent bundle]] $T X \oplus T^* X$ has as [[structure group]] the Narain group.

[[!include Spin(8)-subgroups and reductions -- table]]

### Maximal compact subgroup

The [[maximal compact subgroup]] of the Narain group is the product group $O(n) \times O(n)$ (e.g. [Ooguri-Yin 96, p. 44](#OoguriYin96)).

A [[reduction of the structure group]]  of the [[generalized tangent bundle]] along the inclusion defines a [[type II geometry]].

## Related concepts

[[!include table of orthogonal groups and related]]

## References

* {#OoguriYin96} [[Hirosi Ooguri]], Zheng Yin, _TASI Lectures on Perturbative String Theories_ ([arXiv:hep-th/9612254](https://arxiv.org/abs/hep-th/9612254))


Narasimhan–Seshadri theorem

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## Idea

The _Narasimhan–Seshadri theorem_ ([Narasimhan-Seshadri 65](#NarasimhanSeshadri65)) identifies certain
[[moduli spaces of flat connections]] over a ([[compact topological space|compact]]) [[Riemann surface]] $\Sigma$ with ([[compact topological space|compact]]) [[complex manifolds]] of [[stable vector bundles|stable]] [[holomorphic vector bundles]] over $\Sigma$.

For the special case of [[line bundles]] this may be viewed as a special case of the _[[Hodge-Maxwell theorem]]_, also of Deligne's characterization of the [[intermediate Jacobian]], see there at _[Examples -- Picard variety](intermediate+Jacobian#ExamplePicard)_. The analogue of the theorem for higher dimensional [[complex manifolds]] is the _[[Kobayashi-Hitchin correspondence]]_. The special case of that for [[Kähler manifolds]] is the _[[Donaldson-Uhlenbeck-Yau theorem]]_.

## Statement

An indecomposable Hermitian [[holomorphic vector bundle]] $E$ on a [[Riemann surface]] $(\Sigma,g)$ is [[stable vector bundle|stable]] precisely if there is a compatibly [[unitary group|unitary]] [[connection on a bundle|connection]] $\nabla$ on $E$ with constant [[center|central]] [[curvature]]

$$
  \star F_\nabla = - \mu(E)
$$

equal to (minus) the [[slope of a coherent sheaf|slope]] of $E$ (where $\star$ is the [[Hodge star]] operator).

e.g. ([Evans, p. 2](#Evans))

([Atiyah-Bott 83 (8.1)](#AtiyahBott83))

## Applications

### In Quantum field theory

In [[gauge field theories]] such as notably [[Chern-Simons theory]], moduli spaces of flat connections may appear as [[phase spaces]]. The Narasimhan–Seshadri theorem then provides a [[complex structure]] on these phase spaces and thus a [[spin^c structure]] (see there). This is the data required to obatain the [[geometric quantization by push-forward]] of the phase space. See also at _[[Hitchin connection]]_.

## Related concepts

* [[Hitchin connection]]

* [[quantization of Chern-Simons theory]]

* [[equivariant elliptic cohomology]]

Related theorems

* [[Harder-Narasimhan theorem]]

* [[Koszul-Malgrange theorem]]

* [[Donaldson-Uhlenbeck-Yau theorem]]

## References

The original article is

* {#NarasimhanSeshadri65} [[Mudumbai Narasimhan]], [[C. Seshadri]], _Stable and unitary vector bundles on a compact Riemann surface_, Annals of Mathematics. Second Series 82: 540–567, (1965) ([JSTOR]( http://www.jstor.org/stable/1970710))

and another proof appeared in

* [[Simon Donaldson]], _A new proof of a theorem of Narasimhan and Seshadri_,  J. Differential Geom. Volume 18, Number 2 (1983), 269-277. ([EUCLID](http://projecteuclid.org/euclid.jdg/1214437664))

A good general survey and re-discussion is in

* {#AtiyahBott83} [[Michael Atiyah]], [[Raoul Bott]], section 8 of _The Yang-Mills equations over Riemann surfaces_, Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences
Vol. 308, No. 1505 (Mar. 17, 1983), pp. 523-615 ([jstor](http://www.jstor.org/stable/37156), [lighning summary](http://math.stackexchange.com/a/295505/58526))

A recording of a review talk is here

* {#Narasimhan16} [[Mudumbai Narasimhan]], _Moduli of vector bundles on compact Riemann surfaces_, talk Dec 2016, Bangalore ([video](https://youtu.be/7SuzfmwR6Z0))

Lecture notes include

* {#Evans} [[Jonathan Evans]], _[Aspects of Yang-Mills theory](http://www.homepages.ucl.ac.uk/~ucahjde/yangmills.htm)_, lecture notes,
_Narasimhan-Seshadri theorem_ ([lecture 13 pdf](http://www.homepages.ucl.ac.uk/~ucahjde/YM-lectures/lecture13.pdf), [lecture 14 pdf](http://www.homepages.ucl.ac.uk/~ucahjde/YM-lectures/lecture14.pdf), [lecture 15 pdf](http://www.homepages.ucl.ac.uk/~ucahjde/YM-lectures/lecture15.pdf), [lecture 16 pdf](http://www.homepages.ucl.ac.uk/~ucahjde/YM-lectures/lecture16.pdf))

A textbook providing much of the background definitions involved is

* {#HuybrechtsLehn96} [[Daniel Huybrechts]], [[Mafred Lehn]], _The Geometry of the Moduli Spaces of Sheaves_, 1996 ([[HuybrechtsLehn.pdf:file]])

Related discussion in the context of [[Hitchin connections]] is in

* {#ScheinostSchottenloher96} Peter Scheinost, [[Martin Schottenloher]], _Metaplectic quantization of the moduli spaces of flat and parabolic bundles_, J. reine angew. Mathematik, 466 (1996) ([web](https://eudml.org/doc/153753))

[[!redirects Narasimhan–Seshadri theorem]]
[[!redirects Narasimhan-Seshadri theorem]]


Narciso Román-Roy

* [webpage](https://web.mat.upc.edu/narciso.roman/)

## Selected writings

On [[multisymplectic geometry]] and the [[De Donder-Weyl-Hamilton equation]]:

* [[Narciso Román-Roy]], _Multisymplectic Lagrangian and Hamiltonian Formalisms of Classical Field Theories_, SIGMA 5 (2009), 100 ([journal](http://www.emis.de/journals/SIGMA/2009/100/), [arXiv:math-ph/0506022](http://arxiv.org/abs/math-ph/0506022))

category: people


Narthana Epa

* [webpage](http://users.math.yale.edu/public_html/People/nse5.html)

## Selected writings

On [[Platonic 2-groups]]:

* {#Epa10} [[Narthana Epa]], _Platonic 2-groups_, 2010 ([[Epa_Platonic2Groups.pdf:file]])

* {#EpaGanter16} [[Narthana Epa]], [[Nora Ganter]], _Platonic and alternating 2-groups_, Higher Structures 1(1):122-146, 2017 ([arXiv:1605.09192](http://arxiv.org/abs/1605.09192), [hs:30](https://journals.mq.edu.au/index.php/higher_structures/article/view/30))

category: people


Nash embedding theorem

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## Statement

Every [[Riemannian manifold]] admits a [[isometry|isometric]] [[embedding of smooth manifolds|embedding]] into a [[Euclidean space]].

## Related concepts

* The analogous statement for plain [[differentiable manifolds]] is the [[Whitney embedding theorem]]

## References

See also

* Wikipedia, _[Nash embedding theorem](https://en.wikipedia.org/wiki/Nash_embedding_theorem)_


Nash equilibrium

Related $n$Lab entries: [[game theory]], [[mathematical economics]]

* John Nash, _Equilibrium points in n-person games_, Proceedings of the National Academy of Sciences 36(1):48-49 )1950); _Non-Cooperative Games_ The Annals of Mathematics 54(2):286-295 (1951)

* wikipedia: [Nash equilibrium](http://en.wikipedia.org/wiki/Nash_equilibrium)


Natalia Saulina

* [website](http://www.perimeterinstitute.ca/index.php?option=com_content&task=view&id=30&Itemid=72&pi=5801)

## Selected writings

On [[Rozansky-Witten theory]] as a [[boundary field theory]]:

* [[Anton Kapustin]], [[Lev Rozansky]], [[Natalia Saulina]],
_Three-dimensional topological field theory and symplectic algebraic geometry I_ ([arXiv:0810.5415](https://arxiv.org/abs/0810.5415))

On the relation between [[Rozansky-Witten theory]] and [[Chern-Simons theory]]:

* [[Anton Kapustin]], [[Natalia Saulina]], _Chern-Simons-Rozansky-Witten topological field theory_, Nucl. Phys. B823 (2009) 403-427 ([arXiv:0904.1447](https://arxiv.org/abs/0904.1447), [spire:817599/](http://inspirehep.net/record/817599/))

category: people


Natalie Paquette

[[!redirects Natalie M. Paquette]]

* [webpage](https://phys.washington.edu/people/natalie-paquette)

## Selected writings

On [[holography as Koszul duality]]:

* [[Natalie Paquette]], _Koszul Duality  in Field Theory and Holography_ 2019 ([pdf](https://www.math.ksu.edu/research/m-center/paquette_koszulduality.pdf), [[PaquetteKoszulDuality.pdf:file]])

On [[holography as Koszul duality]] for [[AdS3/CFT2 duality]]:

* [[Kevin Costello]], [[Natalie Paquette]], _Twisted Supergravity and Koszul Duality: A case study in $AdS_3$_ ([arXiv:2001.02177](https://arxiv.org/abs/2001.02177))

and with emphasis on [[QFT with defects]]:

* [[Natalie Paquette]], Brian R. Williams, *Koszul duality in quantum field theory* ([arXiv:2110.10257](https://arxiv.org/abs/2110.10257))

On [[scattering amplitudes]] and [[chiral correlators]] in 4D [[conformal field theories]]:

* [[Kevin Costello]], [[Natalie Paquette]], _Celestial holography meets twisted holography: 4d amplitudes from chiral correlators_ ([arXiv:2201.02595](https://arxiv.org/abs/2201.02595))

On [[operator product expansions]] in 2D [[conformal field theories]]:

* [[Kevin Costello]], [[Natalie Paquette]], _On the associativity of one-loop corrections to the celestial OPE_ ([arXiv:2204.05301](https://arxiv.org/abs/2204.05301))

* [[Natalie Paquette]], _4d physics from 2d chiral correlators_,  talk at *[[Strings 2022]]* [[slides](https://indico.cern.ch/event/1085701/contributions/4940742/attachments/2481322/4259658/Natalie%20Paquette%20Monday-Research-Paquette.pdf), [video](https://ustream.univie.ac.at/media/core.html?id=967d7949-dc4e-47e8-befb-8aa53c589ba9)]

On [[celestial holography]] via [[topological strings]] on [[twistor space]]:

* [[Kevin Costello]], [[Natalie M. Paquette]], [[Atul Sharma]], *Top-down holography in an asymptotically flat spacetime* [[arXiv:2208.14233](https://arxiv.org/abs/2208.14233)]

* [[Kevin Costello]], [[Natalie M. Paquette]], [[Atul Sharma]], *Burns space and holography* [[arXiv:2306.00940](https://arxiv.org/abs/2306.00940)]

* Niklas Garner, [[Natalie M. Paquette]], *Twistorial monopoles & chiral algebras* [[arXiv:2305.00049](https://arxiv.org/abs/2305.00049)]

## Related entries

* [[Moonshine]]

category: people


Nathan A. McMahon

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

## Selected writings

On [[holographic tensor network]] [[quantum error correcting codes]]:

* [[Robert J. Harris]], [[Nathan A. McMahon]], [[Gavin K. Brennen]], [[Thomas M. Stace]], *Calderbank-Steane-Shor Holographic Quantum Error Correcting Codes*, Phys. Rev. A 98, 052301 (2018) ([arXiv:1806.06472](https://arxiv.org/abs/1806.06472))

* [[Nathan A. McMahon]], [[Gavin K. Brennen]], [[Thomas M. Stace]], [[Robert J. Harris]], Elliot Coupe, *Decoding Holographic Codes with an Integer Optimisation Decoder* ([arXiv:2008.10206](https://arxiv.org/abs/2008.10206))

* [[Terry Farrelly]], [[Robert J. Harris]], [[Nathan A. McMahon]], [[Thomas M. Stace]], *Tensor-network codes* ([arXiv:2009.10329](https://arxiv.org/abs/2009.10329))

category: people

[[!redirects Nathan McMahon]]


Nathan Berkovits

* [webpage](https://www.ift.unesp.br/#!/en/professors-homepages/nathan-jacob-berkovits/)

## Selected writings

On [[twistor string theory]]:

* [[Nathan Berkovits]], *An Alternative String Theory in Twistor Space for $N = 4$ Super-Yang-Mills*, Phys. Rev. Lett. 93 (2004) 011601 ([arXiv:hep-th/0402045](https://arxiv.org/abs/hep-th/0402045))

On the [[pure spinor]]-formulation of the [[superparticle]] in [[D=11 supergravity]]-backgrounds:

* [[Nathan Berkovits]], Eduardo Casali, [[Max Guillen]], Lionel Mason, _Notes on the $D=11$ pure spinor superparticle_ ([arXiv:1905.03737](https://arxiv.org/abs/1905.03737))

On the relation between the [[NSR superstring]] and the [[pure spinor]] [[Berkovits superstring]]:

* [[Nathan Berkovits]], *Manifest Spacetime Supersymmetry and the Superstring* ([arXiv:2106.04448](https://arxiv.org/abs/2106.04448))

Review:

* [[Nathan Berkovits]], [[Carlos R. Mafra]], *Pure spinor formulation of the superstring and its applications*, in *[[Handbook of Quantum Gravity]]* (2023) [[arXiv:2210.10510](https://arxiv.org/abs/2210.10510)]

On the relation to $D=5$ [[holomorphic Chern-Simons theory]]:

* [[Nathan Berkovits]], *$D=5$ Holomorphic Chern-Simons and the Pure Spinor Superstring* [[arXiv:2211.06731](https://arxiv.org/abs/2211.06731)]

and to [[semi-topological 4d Chern-Simons theory]]:

* [[Nathan Berkovits]], Rodrigo S. Pitombo, *4D Chern-Simons and the pure spinor $AdS_5 \times S^5$ superstring* [[arXiv:2401.03976](https://arxiv.org/abs/2401.03976)]

## Related entries

* [[Berkovits superstring]]

category: people


Nathan Geer

* [webpage](https://www.usu.edu/math/geer/)

* [webpage](https://www.usu.edu/math/directory/faculty/geer-nathan)

## Selected writings

On [[super Chern-Simons theory]] and [[Rozansky-Witten theory]]:

* [[Nathan Geer]], [[Matthew B. Young]], *Three dimensional topological quantum field theory from $U_q(\mathfrak{gl}(1\vert1))$ and $U(1\vert 1)$ Chern--Simons theory* [[arXiv:2210.04286](https://arxiv.org/abs/2210.04286)]

category: people


Nathan Rosen

* [Wikipedia entry](https://en.wikipedia.org/wiki/Nathan_Rosen)

## Selected writings

Introducing the idea of [[spacetime]] [[wormholes]] in [[gravity]] ([[Einstein-Rosen bridges]]):

* [[Albert Einstein]], [[Nathan Rosen]], *The Particle Problem in the General Theory of Relativity*, Phys. Rev. **48** 73 (1935) ([doi:10.1103/PhysRev.48.73](https://doi.org/10.1103/PhysRev.48.73))

Introducing the [[Einstein-Podolsky-Rosen paradox]] concerning [[quantum entanglement]]:

* {#EPR} [[Albert Einstein]], [[Boris Podolsky]], [[Nathan Rosen]], _Can the Quantum-Mechanical Description of Physical Reality be Considered Complete?_ Physical Review 47 (10): 777–780. (1935) ([doi:10.1103/PhysRev.47.777](https://doi.org/10.1103/PhysRev.47.777))

category: people


Nathan Seiberg

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

## Selected writings

Introducing [[mirror symmetry]] of [[D=3 N=4 super Yang-Mills theory]]:

* {#IntriligatorSeiberg96} [[Ken Intriligator]], [[Nathan Seiberg]], _Mirror Symmetry in Three Dimensional Gauge Theories_, Phys. Lett. B 387:513-519,1996 ([arXiv:hep-th/9607207](http://arxiv.org/abs/hep-th/9607207))

On the [[D=6 N=(1,0) SCFT]] on [[heterotic M5-branes]] and their [[KK-compactifications]]:

* [[Ori Ganor]], [[David Morrison]], [[Nathan Seiberg]], _Branes, Calabi-Yau Spaces, and Toroidal Compactification of the $N=1$ Six-Dimensional $E_8$ Theory_, Nucl. Phys. B487:93-127, 1997 ([arXiv:hep-th/9610251](https://arxiv.org/abs/hep-th/9610251))

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 longitudinal [[M5-branes]] in the [[BFSS matrix model]]:

* [[Tom Banks]], [[Nathan Seiberg]], [[Stephen Shenker]], _Branes from Matrices_, Nucl. Phys. B490:91-106, 1997 ([arXiv:hep-th/9612157](https://arxiv.org/abs/hep-th/9612157))

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 [[generalized global symmetries]]:

* {#GKSW14} [[Davide Gaiotto]], [[Anton Kapustin]], [[Nathan Seiberg]], [[Brian Willett]], *Generalized Global Symmetries*  J. High Energ. Phys. **2015** 172 (2015) [[arXiv:1412.5148](https://arxiv.org/abs/1412.5148), doi:10.1007/JHEP02(2015)172]

On the [[theta angle]] in [[QCD]]:

* [[Davide Gaiotto]], [[Anton Kapustin]], [[Zohar Komargodski]], [[Nathan Seiberg]], _Theta, Time Reversal, and Temperature_, JHEP05(2017)091 ([arXiv:1703.00501](https://arxiv.org/abs/1703.00501))

## Related entries

* [[Seiberg-Witten theory]]

* [[BFSS matrix model]]

category: people

[[!redirects N. Seiberg]]