Shelah's main gap

Shelah's main gap is a result in stability theory (part of [[model theory]]), which seeks to identify those [[first-order theories]] whose [[models]] admit some sort of reasonable classification. The 'main gap' describes a fundamental dichotomy between theories.

Roughly speaking, a countable first-order [[nlab:theory]] belongs to one of the two classes:

* (intractable:) has the maximal number of models in all
sufficiently large cardinalities

* every model can be decomposed as a tree of countable models

There are some generalizations of this dichotomy beyond first-order theories.

* L. Harrington, M. Makkai, _An exposition of Shelah's ``main gap'': counting uncountable models of ω-stable and superstable theories_, Notre Dame J. Formal Logic 26 (1985), no. 2, 139--177 [doi](http://dx.doi.org/10.1305/ndjfl/1093870822) [euclid](http://projecteuclid.org/euclid.ndjfl/1093870822)


Sheldon Glashow

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

## Selected writings

Co-introducing the theory of the [[electroweak field]]:

* [[Sheldon Glashow]], *Partial-symmetries of weak interactions*, Nuclear Physics **22** 4 (1961) 579-588 [doi:10.1016/0029-5582(61)90469-2]

Introducing [[SU(5)]]-[[grand unified theory]]:

* {#GeorgiGlashow74} [[Howard Georgi]], [[Sheldon Glashow]], _Unity of All Elementary-Particle Forces_, Phys. Rev. Lett. **32** (1974) 438 [[doi:10.1103/PhysRevLett.32.438](https://doi.org/10.1103/PhysRevLett.32.438)]

On [[superstring theory]]:

* [[Paul Ginsparg]], Sheldon Glashow, _Desperately Seeking Superstrings_, Phys.Today 86N5 (1986) 7-9 ([arXiv:physics/9403001](https://arxiv.org/abs/physics/9403001), [doi:10.1063/1.2814991](https://physicstoday.scitation.org/doi/10.1063/1.2814991))

## Interviews

* David Zierler, [Interview with Sheldon Glashow, (3rd June 2020)](https://www.aip.org/history-programs/niels-bohr-library/oral-histories/47288), in the "Oral History"-series of the American Institute of Physics

On the effect of awarding prizes to otherwise respectable scientists:

> By the late 1970s I began to think of myself as a Nobel contender. But I was under the impression that my old friend [[Steven Weinberg]] was doing everything in his power to keep the prize for himself and [[Abdus Salam|Salam]]. In particular -- at a conference that he attended in Tokyo -- he went out of his way to avoid mentioning my name at all while presenting the history of weak interaction theory. I got very upset by that omission. It was the issue which terminated our friendship. In the summer of 1979, I was invited to a meeting in Stockholm, to discuss the current state of physics ideas and others. Prior to the meeting, I sent a transcript of my talk to Steve. He was violently against my giving the talk. Because it examined various alternatives to what was then known as Weinberg/Salam theory. In fact, it was an open-minded talk in which I was discussing whether their -- or more properly -- our theory was a correct one or not. But it was such a heated discussion that I eventually had to simply hang up on him, because I had no intention of revising my talk. And I did not.

## Related $n$Lab entries

* [[quantum field theory]]

* [[GUT]], [[gauge coupling unification]]

category: people


Sheldon Goldstein

* [webpage](http://www.math.rutgers.edu/~oldstein/)

category: people


Sheldon Katz

* [webpage](http://www.math.uiuc.edu/~katz/)

## Selected writings

On [[geometric engineering of quantum field theory]]:

* {#KatzKlemmVafa97} [[Sheldon Katz]], [[Albrecht Klemm]], [[Cumrun Vafa]], _Geometric Engineering of Quantum Field Theories_, Nucl.Phys. B497 (1997) 173-195 ([arXiv:hep-th/9609239](http://arxiv.org/abs/hep-th/9609239))

* {#KatzKlemm96} [[Sheldon Katz]], [[Cumrun Vafa]], _Geometric Engineering of $N=1$ Quantum Field Theories_, Nucl.Phys. B497 (1997) 196-204 ([arXiv:hep-th/9611090](http://arxiv.org/abs/hep-th/9611090))

Discussion of the [[Hilbert schemes of points]] of [[ADE-singularities]]:

* [[Ron Donagi]], [[Sheldon Katz]], [[Eric Sharpe]], _Spectra of D-branes with Higgs vevs_, Adv.Theor.Math.Phys. 8 (2005) 813-859 ([arXiv:hep-th/0309270](https://arxiv.org/abs/hep-th/0309270))

On [[enumerative geometry]] and [[string theory]]/[[Gromov-Witten theory]]:

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

On [[quantum sheaf cohomology]]:

* [[Ron Donagi]], [[Josh Guffin]], [[Sheldon Katz]], [[Eric Sharpe]], *A Mathematical Theory of Quantum Sheaf Cohomology*, Asian J. Math. **18** (2014) 387-418 [[arXiv:1110.3751](https://arxiv.org/abs/1110.3751), [doi:10.4310/AJM.2014.v18.n3.a1](https://doi.org/10.4310/AJM.2014.v18.n3.a1)]

category: people


shelf

# Contents
* table of contents
{:toc}

## Idea

A *shelf* is a set with a binary operation that distributes over itself. Shelves are similar to [[racks]] (and there are forgetful functors from racks to shelves), but shelves are axiomatically simpler.

## Definitions

A **left shelf** is a set with a binary operation $\triangleright$ obeying the left self-distributive law

$$ a \triangleright (b \triangleright c) = (a \triangleright b)\triangleright (a \triangleright  c)   .$$

Similarly a set with a binary operation $\triangleleft$ obeying the right self-distributive law is called a **right shelf**.

A unital left shelf (meaning a left shelf together with an element that acts as an [[identity element|identity]] on both the left and the right) is the same as a graphic monoid: for a proof see [[graphic category]].

## Examples

Of course all the usual examples of [[racks]] and [[quandles]] are *a fortiori* shelves. But there are notable examples not of this type.

### Infinite braid group

Let $B_n$ be the $n^{th}$ [[braid group]]. With the usual inclusion $B_n \to B_{n+1}$ by appending a strand to the end of a braid on $n$ strands, the colimit of the chain $B_0 \to B_1 \to B_2 \to \ldots$ is the *infinite braid group* $B_\infty$. Let $sh: B_\infty \to B_\infty$ be the homomorphism that sends $\sigma_i$ to $\sigma_{i+1}$. Then there is a left distributive operation on $B_\infty$ where

$$a \triangleright b \coloneqq a sh(b) \sigma_1 sh(a)^{-1}.$$

One may verify the left distributivity by a [[string diagram]] calculation (which appears on page 29 of this [Google book](https://books.google.com/books?id=XfsHCAAAQBAJ&pg=PA27&lpg=PA27&dq=braid+exponentiation&source=bl&ots=4ncxE2f7oT&sig=Mn-YDL97W9NWE0baexJEWrD4J4k&hl=en&sa=X&ved=0ahUKEwiB4saR3frKAhXBSiYKHbWaBEoQ6AEIHTAA#v=onepage&q=braid%20exponentiation&f=false), [Dehornoy3](#Deh3)).

### In set theory

Shelves make an appearance in set theory via [[large cardinal]] axioms. Let $(V, \in)$ be a model of [[ZFC]], and let $V_\lambda \subseteq V$ be the collection of elements of rank less than an [[ordinal]] $\lambda$ of $V$. One (rather strong) large cardinal axiom on (limit ordinals) $\lambda$ is:

> There exists an [[elementary embedding]] $j: V_\lambda \to V_\lambda$ on the [[structure]] $(V_\lambda, \in)$ that is not the identity.

Then, for $A \subseteq V_\lambda$, put

$$j(A) \coloneqq \bigcup_{\alpha \lt \lambda} j(A \cap V_\alpha).$$

If we regard $A$ as an unary relation on $V_\lambda$, then $j$ induces an elementary embedding $(V_\lambda, \in, A) \to (V_\lambda, \in, j(A))$. In particular, if $k$ is any elementary embedding $(V_\lambda, \in) \to (V_\lambda, \in)$, which as a set of ordered pairs we may regard as a subset of $V_\lambda$, then $j(k)$ as a set of ordered pairs is also an elementary self-embedding of $(V_\lambda, \in)$. We get in this way a binary operation $(j, k) \mapsto j(k)$ on elementary embeddings, which we denote as $(j, k) \mapsto j \cdot k$, and it is not difficult to verify that $\cdot$ is left self-distributive.

Let $F_1$ denote the [[free object|free]] left shelf generated by 1 element. If $E_\lambda$ denotes the collection of elementary embeddings on the structure $(V_\lambda, \in)$, then the preceding observations imply that $E_\lambda$ is a left shelf, so any $j \in E_\lambda$ induces a shelf homomorphism

$$\phi_j: F_1 \to E_\lambda.$$

+-- {: .num_theorem}
###### Theorem
**(Laver)**
If $j \in E_\lambda$ is not the identity, then $\phi_j$ is injective.
=--

The famous *Laver tables* (derived from set-theoretic considerations which we omit for now) describe certain [[finite set|finite]] quotients of $F_1$. Letting $x$ denote the generator of $F_1$, define $x_n$ by $x_1 = x$ and $x_{n+1} = x_n \cdot x$. The quotient of $F_1$ by the single relation $x_{m+1} = x$ is a shelf of cardinality $2^k$, the largest power of $2$ dividing $m$; it is denoted $A_k$. It can be described alternatively as the unique left shelf on the set $\{1, 2, \ldots, 2^k\}$ such that $p \cdot 1 = p + 1 \mod 2^k$ (here $p$ represents the image of $x_p$ under the quotient $F_1 \to A_k$).

The "multiplication table" of an $A_k$ is called a Laver table. The behavior of Laver tables is largely not understood, but we mention a few facts. The first row consisting of entries $1 \cdot p$ is periodic (of some order dividing $2^k$). Under the large cardinal assumption that a nontrivial elementary self-embedding on a $V_\lambda$ exists, this period $f(k)$ tends to $\infty$ as $k$ does, but whether it does as a consequence of ZFC is not known. What *is* known is that this period, even if it increases to $\infty$, does so quite slowly: if we define $g(m)$ to be the smallest $k$ such that $f(k) \geq m$, then $g$ grows more quickly than say the Ackermann function.

## Action of positive braid monoid

Let $B_n^+$ be the [[monoid]] of positive [[braid group|braids]], which as a monoid is presented by generators $\sigma_1, \ldots, \sigma_{n-1}$ subject to the braid relations

$$\sigma_i \sigma_{i+1} \sigma_i = \sigma_{i+1} \sigma_i \sigma_{i+1}, \qquad \sigma_i \sigma_j = \sigma_j \sigma_i\; if \; {|i-j|} \gt 1.$$

+-- {: .num_prop}
###### Proposition
If $(X, \triangleright)$ is a shelf, then there is a monoid homomorphism $B_n^+ \to \hom(X^n, X^n)$ whose transform to an action $B_n^+ \times X^n \to X^n$ is described by the equations

$$\sigma_i(x_1, \ldots, x_i, x_{i+1}, \ldots, x_n) = (x_1, \ldots, x_i \triangleright x_{i+1}, x_i \ldots, x_n);$$

conversely, if $\triangleright$ is any binary operation, then these equations describe an action of $B_n^+$ only if $\triangleright$ is left distributive.
=--

## Related concepts

* [[rack]]

## References

These are some general references:

* [[Alissa Crans]], _Lie 2-Algebras_, Chapter 3.1: Shelves, Racks, Spindles and Quandles, Ph.D. thesis, U.C. Riverside, 2004.  ([pdf](https://arxiv.org/abs/math/0409602)).

* [[Patrick Dehornoy]], _Braids and Self-Distributivity_, Progress in Mathematics 192, Birkhäuser Verlag, 2000.

These develop the connection between the free shelf on one generator and elementary embeddings in set theory:

* Richard Laver, The left distributive law and the freeness of an algebra of elementary embeddings, Adv. Math. 91 (1992), 209–231.

* Richard Laver, On the algebra of elementary embeddings of a rank into itself, Adv. Math. 110 (1995), 334–346.

* Randall Dougherty and Thomas Jech, Finite left distributive algebras and embedding algebras, Adv. Math. 130 (1997), 201–241.

* Randall Dougherty, Critical points in an algebra of elementary embeddings, Ann. Pure Appl. Logic 65 (1993), 211–241.

* Randall Dougherty, Critical points in an algebra of elementary embeddings, II</>.

For a popularized account of this material, see:

* John Baez, [Shelves and the infinite](https://johncarlosbaez.wordpress.com/2016/05/06/shelves-and-the-infinite/).

[[!redirects shelves]]
[[!redirects left shelf]]
[[!redirects left shelves]]
[[!redirects right shelf]]
[[!redirects right shelves]]
[[!redirects Laver table]]
[[!redirects Laver tables]]


shell

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Variational calculus
+-- {: .hide}
[[!include variational calculus - contents]]
=--
#### Physics
+-- {: .hide}
[[!include physicscontents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In [[physics]] the [[critical locus]] of the [[action functional]], hence the [[solution]] space to the [[partial differential equation|partial differential]] [[equations of motion]] is often called the _shell_, or the _physical shell_ (the [[diffiety]] corresponding to the PDE of motion).

This terminology derives from the case of the free [[relativistic particle]] propagating in [[Minkowski spacetime]], for which the space of solutions is the hyperbola of vectors whose [[Minkowski metric|Minkowski norm square]] is the [[mass]] of the particle. This hyperbola is naturally called the _mass shell_.

It is common to use the adjective "on-shell" for statements that apply after restriction to the shell, and "off-shell" for statements that apply generally.

For instance [[Noether's theorem]] gives for each symmetry of a [[local Lagrangian]] an _on-shell [[conserved current]]_, namely a differential form on spacetime, depending on the field configurations, which is closed (only, in general) if the given field configuration satisfies its equations of motion. Such a differential form which is closed even if the equations of motion do not hold would be called an _off-shell conserved current_.

## Details

In terms of [[variational calculus]] the field configurations are encoded by a [[field bundle]] $E$ over [[spacetime]]/[[worldvolume]] $\Sigma$ as being the [[sections]] of this bundle. The shell then is encoded by a sub-bundle

$$
  \array{
    \mathcal{E} &&\hookrightarrow && J^\infty_\Sigma E
    \\
    & \searrow && \swarrow
    \\
    && \Sigma
  }
$$

of the [[jet bundle]] $J^\infty_\Sigma E$ of $E$, namely the subbundle of those pointwise field configurations with those jets (derivatives) that do solve the equations of motion (see at _[[diffiety]]_).

This way a field configuration given by a section $\phi$ of $E$ is a solution to the equations of motion precisely if its [[jet prolongation]], being a section of $J^\infty_\Sigma E$ comes from a section of $\mathcal{E}$ under this inclusion.

If one thinks of all this not as happening in the [[category]] of [[bundles]] over $\Sigma$ but in the category $PDE_\Sigma$ of [[partial differential equations]] over $\Sigma$, then the above inclusion becomes simply $\iota \colon \mathcal{E} \hookrightarrow E$, where now $E$ is thought of as the trivial partial differential equation on its sections, the one for which each section is a solution. Then a field configuration is just a morphism $\phi : \Sigma \longrightarrow E$ in $PDE_\Sigma$, and this is a solution precisely if it factors through the inclusion of the shell:

$$
  \array{
    && \mathcal{E}
    \\
    & {}^\mathllap{\phi_{sol}}{\nearrow} & \downarrow^{\mathrlap{\iota}}
    \\
    \Sigma &\underset{\phi}{\longrightarrow}& E
  }
$$

Accordingly, a [[variational bicomplex|horizontal form]] $J \in \Omega^{p}_H(E) \hookrightarrow \Omega^{p+1}(J^\infty_\Sigma E)$ is an _on-shell [[conserved current]]_ if it becomes horizontally closed after restriction to the shell:

$$
  \iota^\ast d_H J = 0
  \,.
$$

## Properties

For $(E,\mathbf{L})$ a [[Lagrangian field theory]], let $\mathcal{E}^\infty \hookrightarrow J^\infty_\Sigma(E)$ be the prolonged shell.

The following is intuitively obvious but not entirely trivial to prove:

+-- {: .num_prop #InfinitesialSymmetryOfTheLagrangianIsAlsoSymmetryOfTheShell}
###### Proposition
**([[infinitesimal symmetry of the Lagrangian]] is also symmetry of the shell)**

Let $(E,\mathbf{L})$ be a [[Lagrangian field theory]]. Then if $\hat v$ is the prolongation of an [[evolutionary vector field]] which is an [[infinitesimal symmetry of the Lagrangian]] in that the [[Lie derivative]] of $\mathbf{L}$ along $\hat v$ is [[horizontal derivative|horizontally exact]], then the [[flow]] of $\hat v$ preserves the prolonged shell.

=--

([Olver 95, theorem 5.53](#Olver95))

## References

* _[Higher prequantum geometry II: The principle of extremal action -- comonadically](https://www.physicsforums.com/insights/higher-prequantum-geometry-ii-principle-extremal-action-comonadically/)_

* {#Olver95} [[Peter Olver]], _Applications of Lie groups to differential equations_, Springer; _Equivalence, invariants, and symmetry_, Cambridge Univ. Press (1995) [[doi:10.1007/978-1-4684-0274-2](https://doi.org/10.1007/978-1-4684-0274-2)]

[[!redirects shells]]

[[!redirects on-shell]]
[[!redirects off-shell]]

[[!redirects mass shell]]
[[!redirects mass shells]]

[[!redirects prolonged shell]]


Sheng San Li

* [Researchgate page](https://www.researchgate.net/profile/Sheng-Li-21)

## Selected writings

On the [[solid state physics]] of [[semiconductors]]:

* [[Sheng San Li]], *Semiconductor Physical Electronics*, Springer (2006)  61-104 $[$[doi:10.1007/0-387-37766-2](https://link.springer.com/book/10.1007/0-387-37766-2)$]$

On [[electron band structure]] in [[solid state physics]]:

* [[Sheng San Li]], *Energy Band Theory*, in S. S. Li, (ed.): *Semiconductor Physical Electronics*, Springer (2006)  61-104 $[$[doi:10.1007/0-387-37766-2_4](https://doi.org/10.1007/0-387-37766-2_4)$]$

category: people

[[!redirects Sheng S. Li]]
[[!redirects S. S. Li]]


Shenglin Zhu

* [Researchgate page](https://www.researchgate.net/profile/Zhu-Shenglin)

## Selected writings

On [[Frobenius functors]] (and introducing that terminology):

* {#CaenepeelMilitaruZhu97}  [[Stefaan Caenepeel]], [[Gigel Militaru]], [[Shenglin Zhu]], *Doi-Hopf modules, Yetter-Drinfel’d modules and Frobenius type properties*, Trans. Amer. Math. Soc. **349** (1997) 4311-4342 [[1997-349-11/S0002-9947-97-02004-7](https://www.ams.org/journals/tran/1997-349-11/S0002-9947-97-02004-7), [pdf](https://www.ams.org/journals/tran/1997-349-11/S0002-9947-97-02004-7/S0002-9947-97-02004-7.pdf)]

* {#CGN98} F. Castaño Iglesias, [[José Gómez-Torrecillas]], C. Nastasescu, _Frobenius functors: applications_, Comm. Alg. __27__ 10  (1998) 4879-4900 [[doi:10.1080/00927879908826735](https://doi.org/10.1080/00927879908826735)]

* {#CaenepeelDeGrooMilitaru02} [[Stefaan Caenepeel]], E. De Groot, [[Gigel Militaru]], *Frobenius Functors of the second kind*, Comm. Algebra **30** (2002) 5359-5391 [[arXiv:math/0106109](https://arxiv.org/abs/math/0106109), [doi:10.1081/AGB-120015657](https://doi.org/10.1081/AGB-120015657)]

* [[Stefaan Caenepeel]], [[Gigel Militaru]], [[Shenglin Zhu]], _Frobenius and separable functors for generalized module categories and nonlinear equations_, Springer Lec. Notes in Math. __1787__ (2002) [[gBooks](http://books.google.hr/books?id=6erTEpKIUaYC&lpg=PA89&ots=-TGKqXRstB&dq=Frobenius%20functors&hl=en&pg=PA89#v=onepage&q&f=false)]

category: people


shift map

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Functional analysis
+--{: .hide}
[[!include functional analysis - contents]]
=--
#### Topology
+--{: .hide}
[[!include topology - contents]]
=--
=--
=--

# Contents

* tic
{: toc}

## Idea

There are many situations in [[algebraic topology]] in which one wants to work in a "Really Big Space".  Often, it is not that important which space is used, so long as it has some basic properties.  A common feature that one wants of these spaces is that certain derived spaces be contractible.  In many cases, it is possible to write down an actual contraction (rather than arguing from general nonsense) and in those cases, the contraction often uses the ability to "shift" pieces of the space from one place to another.  This leads us to consider the notion of a **shift map**, and its cousin a **split map**, which provide enough structure to define the required contractions.

In short, a **shift map** is a generalisation of the obvious shift map $\ell^0 \to \ell^0$ given by $(x_1,x_2,x_3,\dots) \mapsto (0,x_1,x_2,x_3,\dots)$.  It is an inclusion (even an embedding) and its [[eventual image]], $\bigcap_k \im S^k$, is zero.
Note that a shift map on $V$ induces an isomorphism $V \cong \mathbb{R} \oplus V$, but the existence of a shift map is a stronger condition than that.

A **split map** is similar, except that the induced decomposition is $V \cong V \oplus V$.

## Definition

+-- {: .num_definition #defshift}
###### Definition

Let $V$ be a [[LCTVS|locally convex topological vector space]] over $\mathbb{R}$.
Let $k \in \mathbb{N}$.
A **shift map** of order $k$ on $V$ is a continuous linear map $S \colon V \to V$ with the following properties:

1. $S$ is an embedding of $V$ onto a closed subspace of $V$ of codimension $k$ (so that $V \cong \mathbb{R}^k \oplus V$), and
2. $\bigcap_k \im S^k = \{0\}$.

A locally convex topological vector space that admits a shift map will be called a **shiftable space**.  The pair $(V,S)$ will be called a **shift space**.
=--

+-- {: .num_definition #defsplit}
###### Definition

Let $V$ be a [[LCTVS|locally convex topological vector space]] over $\mathbb{R}$.
A **split map** on $V$ is a continuous linear map $S \colon V \to V$ with the following properties:

1. $S$ is an embedding of $V$ onto a closed subspace of $V$ with complement also isomorphic to $V$, (so that $V \cong V \oplus V$), and
2. $\bigcap_k \im S^k = \{0\}$.

A locally convex topological vector space that admits a split map will be called a **splittable space**.  The pair $(V,S)$ will be called a **split space**.
=--

There are obvious generalisations for other fields than $\mathbb{R}$.

## Note on Notation

This is new terminology, invented to give a consistent way to refer to these spaces with their properties.  At time of writing [no existing name for this was known](http://mathoverflow.net/questions/82108/is-there-a-standard-notation-for-a-shift-space-in-functional-analysis).

[[!redirects shift map]]
[[!redirects shift maps]]
[[!redirects Shift map]]
[[!redirects Shift maps]]
[[!redirects shift space]]
[[!redirects shift spaces]]
[[!redirects Shift space]]
[[!redirects Shift spaces]]
[[!redirects shiftable space]]
[[!redirects shiftable spaces]]
[[!redirects Shiftable space]]
[[!redirects Shiftable spaces]]

[[!redirects split map]]
[[!redirects split maps]]
[[!redirects Split map]]
[[!redirects Split maps]]
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[[!redirects splittable space]]
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[[!redirects Splittable space]]
[[!redirects Splittable spaces]]


shifted C-field flux quantization

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

#Contents#
* table of contents
{:toc}

## Idea

In [[string theory]]/[[M-theory]], the _shifted C-field flux quantization condition_ is a [[charge quantization]]-condition on the [[supergravity C-field]] expected in [[M-theory]].

## For the magnetic $G_4$-flux

For the magnetic $G_4$-flux, the shifted flux quantization says that the [[real cohomology]] class of the [[flux density]] ([[field strength]]) [[differential 4-form]] $G_4 \in \Omega^4(X)$ on [[spacetime]] $X$ becomes integral after shifted by one quarter of the [[first Pontryagin class]], hence the condition that with the shifted 4-flux density defined as

\[
  \label{Shiftwed}
  \widetilde G_4
  \;\coloneqq\;
  G_4 + \tfrac{1}{4}p_1(\nabla_{T X})
  \;\in\;
  \Omega^4(X)
\]

(for $\nabla_{T X}$ any [[affine connection]] on [[spacetime]], in particular the [[Levi-Civita connection]]) we have (using the [[de Rham theorem]] to translate from [[de Rham cohomology]] to [[real cohomology]]) that $\widetilde G_4$ represents an [[integral cohomology]]-class:

$$
  [\widetilde G_4]
  \;\in\;
  H^4(X, \mathbb{Z})
  \overset{H^4(X, \mathbb{Z}\hookrightarrow \mathbb{R})}{\longrightarrow}
  H^4(X, \mathbb{R})
  \,.
$$

This condition was originally argued for in ([Witten 96a](#Witten96a), [Witten 96b](#Witten96b)) as a sufficient condition for ensuring that the [[prequantum line bundle]] for the [[7d Chern-Simons theory]] on an [[M5-brane]] [[worldvolume]] is divisible by 2.

Proposals for encoding this condition by a [[Wu class]]-shifted variant of [[generalized (Eilenberg-Steenrod) cohomology theory|stable]] [[ordinary differential cohomology]] were considered in [Hopkins-Singer 02](#HopkinsSinger02), [Diaconescu-Freed-Moore 03](#DiaconescuFreedMoore03), [FSS 12](#FSS12).

It turns out that the shifted flux quantization condition on the C-field is naturally implied ([FSS1 19b, Prop. 4.12](#FSS19b)) by the requirement that $G_4$ is the differential form datum underlying, via [[Sullivan model|Sullivan's theorem]], a [[cocycle]] in unstable [[J-homomorphism|J-]] [[twisted Cohomotopy]] in degree 4 ([[Hypothesis H]]).

## For the electric $G_7$-flux
 {#ForElectricG7Flux}

In the presence of non-vanishing [[C-field]] [[flux]] $G_4$, the electric [[flux density]] of [[M2-branes]] is not $G_7 \coloneqq \star G_4$ alone, but receives corrections, first due to the quadratic [[C-field]] self-interaction in [[D=11 supergravity]], but then also due to the [[shifted C-field flux quantization]] expected in [[M-theory]]:

\linebreak

**The [[11d supergravity]] literature** states the corrected 7-flux to be the following combination, also known as the _[[Page charge]]_ (due to [Page 1983 (8)](M2-brane#Page83), [Duff & Stelle 1991, (43)](M2-brane#DuffStelle91), reviewed e.g. in [BLMP 13, p. 21](M2-brane#BaggerLambertMukhiPapageorgakis13)):

\[
  \label{FirstCorrectionToG7}
  \widetilde G'_7
  \;\coloneqq\;
  G_7 + \tfrac{1}{2} C_3 \wedge G_4
  \,,
\]

where the second term subtracts the electric flux induced by the self-intersection of the [[field]], and also ensures that the full expression is a [[closed differential form]] if the naive [[11d supergravity]] [[equations of motion]] hold:

$$
  d \widetilde G_7
  \;=\;
  0
  \,.
$$

But in fact (eq:FirstCorrectionToG7) does not quite make general sense, for two reasons:

1. In general $G_4 = 0$ is not an admissible condition and is not the actual vanshing of the [[C-field]], due to the [[shifted C-field flux quantization]].

1. Even if $G_4$ happens to be intregrally quantizaed (if $\tfrac{1}{4}p_1$ is integral) the appearance of a globally defined [[C-field]] potential $C_3$ in (eq:FirstCorrectionToG7),means that the total [[flux]] actually does vanish after all.

\linebreak

**Charge-quantized $\widetilde G_7$-flux with [[shifted C-field flux quantization]]** ([FSS 19b, Prop. 4.3](#FSS19b), [FSS 19c, Section 4](#FSS19c))

Both of these issues are solved if the [[C-field]] is taken to be [[charge quantization|charge quantized]] in [[J-homomorphism|J-]][[twisted Cohomotopy]] ([[Hypothesis H]]). This gives the corrected formula

\[
  \label{FullCorrectionToG7}
  \widetilde G_7
  \;\coloneqq\;
  h^\ast G_7 + \tfrac{1}{2} H_3 \wedge h^\ast \widetilde G_4
\]

where

1. the expression lives on the [[homotopy pullback]] of the [[Sp(2)]]-[[parametrized homotopy theory|parametrized]] [[quaternionic Hopf fibration]]

   $$
     h \coloneqq h_{\mathbb{H}}\sslash Sp(2)
   $$

   to [[spacetime]], along the [[twisted Cohomotopy]]-[[cocycle]] that represents the [[C-field]] under [[Hypothesis H]];

1. $\widetilde G_4 \coloneqq h^\ast G_4 + \tfrac{1}{4}h^\ast p_1(\nabla)$ is the integral [[shifted C-field flux quantization|shifted C-field]] pulled back to that 3-[[spherical fibration]] over spacetime;

1. $d H_3 = h^\ast G_4 - \tfrac{1}{4}h^\ast p_1(\nabla)$ trivializes not the C-field itself, but its pullback, and not absolutely but relative to the background charge implied by [[shifted C-field flux quantization]].

With the corrected 7-flux in [[twisted Cohomotopy]] it becomes true that

1. the integral of $G_7$ around the [[7-sphere]] linking a [[black brane|black]] [[M2-brane]] is always [[integer]] ([FSS 19c, Theorem 4.6](#FSS19c));

1. this integer satisfies the [[C-field tadpole cancellation condition]] ([FSS 19b, Section 4.6](#FSS19b)).

## Related concepts

* [[C-field tadpole cancellation]]

## References

### General

The suggestion originates in

* {#Witten96a} [[Edward Witten]], _On Flux Quantization In M-Theory And The Effective Action_, J. Geom. Phys. **22** 1   (1997) 1-13 [[arXiv:hep-th/9609122](https://arxiv.org/abs/hep-th/9609122), doi:10.1016/S0393-0440(96)00042-3]

* {#Witten96b} [[Edward Witten]], *Five-Brane Effective Action In M-Theory*, J. Geom. Phys. **22** 2 (1997) 103-133 [[arXiv:hep-th/9610234](https://arxiv.org/abs/hep-th/9610234), doi:10.1016/S0393-0440(97)80160-X]

Proposals to model the condition by a [[Wu class]]-shifted variant of [[ordinary differential cohomology]] include

* {#HopkinsSinger02} [[Michael Hopkins]], [[Isadore Singer]], _[[Quadratic Functions in Geometry, Topology, and M-Theory]]_  J. Differential Geom. Volume 70, Number 3 (2005), 329-452.  ([arXiv:math.AT/0211216](http://arxiv.org/abs/math.AT/0211216), [euclid:1143642908](https://projecteuclid.org/euclid.jdg/1143642908))

* {#DiaconescuFreedMoore03} [[Duiliu-Emanuel Diaconescu]], [[Dan Freed]], [[Greg Moore]],  _The $M$-theory 3-form and $E_8$-gauge theory_, chapter in [[Haynes Miller]], [[Douglas Ravenel]] (eds.) _Elliptic Cohomology Geometry, Applications, and Higher Chromatic Analogues_, Cambridge University Press (2007) [[arXiv:hep-th/0312069](http://arxiv.org/abs/hep-th/0312069), [doi:10.1017/CBO9780511721489](https://doi.org/10.1017/CBO9780511721489)]

* {#FreedMoore04} [[Dan Freed]], [[Greg Moore]], _Setting the [[quantum integrand]] of M-theory_, Communications in Mathematical Physics **263** 1 (2006) 89-132 [[arXiv:hep-th/0409135](http://arxiv.org/abs/hep-th/0409135), [doi:10.1007/s00220-005-1482-7](https://doi.org/10.1007/s00220-005-1482-7)]

* {#Moore04} [[Greg Moore]], _Anomalies, Gauss laws, and Page charges in M-theory_, Comptes Rendus Physique **6** (2005) 251-259 [[arXiv:hep-th/0409158](http://arxiv.org/abs/hep-th/0409158), [doi:10.1016/j.crhy.2004.12.005](https://doi.org/10.1016/j.crhy.2004.12.005)]

* {#FSS12} [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], _[[schreiber:The moduli 3-stack of the C-field]]_, Communications in Mathematical Physics,  Volume 333, Issue 1 (2015), Page 117-151 ([arXiv:1202.2455](http://arxiv.org/abs/1202.2455), [doi:10.1007/s00220-014-2228-1](http://link.springer.com/article/10.1007%2Fs00220-014-2228-1))

Suggestion that an actual [[E8]]-[[principal bundle]] on 11d spacetime plays a role here:

* D. Diaconescu, [[Gregory Moore]], [[Edward Witten]], _$E_8$ Gauge Theory, and a Derivation of K-Theory from M-Theory_, Adv. Theor. Math. Phys. 6:1031-1134, 2003 ([arXiv:hep-th/0005090](https://arxiv.org/abs/hep-th/0005090))

* [Diaconescu-Freed-Moore 03, Section 3](#DiaconescuFreedMoore03)

* Allan Adams, [[Jarah Evslin]], _The loop group of $E_8$ and K-theory from 11d_, JHEP 02 (2003) ([arXiv:hep-th/0203218](https://arxiv.org/abs/hep-th/0203218))

* [[Hisham Sati]], _$E_8$ Gauge Theory and Gerbes in String Theory_, Adv. Theor. Math. Phys. 14:1-39, 2010 ([arXiv:hep-th/0608190](https://arxiv.org/abs/hep-th/0608190))

The observation that the condition is implied by [[supergravity C-field|C-field]] [[charge quantization]] in [[J-homomorphism|J-]][[twisted Cohomotopy]] ([[Hypothesis H]]):

* {#FSS19b} [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], _[[schreiber:Twisted Cohomotopy implies M-theory anomaly cancellation]]_,
Comm. Math. Phys. **377** (2020) 1961-2025 ([doi:10.1007/s00220-020-03707-2](https://doi.org/10.1007/s00220-020-03707-2), [arXiv:1904.10207](https://arxiv.org/abs/1904.10207))

* {#FSS19c} [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], _[[schreiber:Twisted Cohomotopy implies M5 WZ term level quantization]]_, Comm. Math. Phys. **384** 403–432 (2021) ([doi:10.1007/s00220-021-03951-0](https://doi.org/10.1007/s00220-021-03951-0), [arXiv:1906.07417](https://arxiv.org/abs/1906.07417))

surveyed in:

* {#FSS20} [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], _[[schreiber:The Character Map in Twisted Non-Abelian Cohomology|The Character Map in Nonabelian Cohomology --- Twisted, Differential, Generalizaed]]_, World Scientific (2023) [[arXiv:2009.11909](https://arxiv.org/abs/2009.11909), [doi:10.1142/13422](https://doi.org/10.1142/13422)]

* [[Hisham Sati]], [[Urs Schreiber]], *[[schreiber:Introduction to Hypothesis H]]* ([here](/schreiber/show/Introduction+to+Hypothesis+H#OverviewStatementOfHypothesisH))

Discussion of the [[Page charge]] in relation to the [[Myers effect]] in [[M-theory]] for [[M2-branes]] polarizing into [[M5-branes]] of [[fuzzy 3-sphere|fuzzy]] [[3-sphere]]-shape:

* [[Krzysztof Pilch]], Alexander Tyukov, [[Nicholas Warner]], _Flowing to Higher Dimensions: A New Strongly-Coupled Phase on M2 Branes_, JHEP11 (2015) 170 ([arXiv:1506.01045](https://arxiv.org/abs/1506.01045))

### Relation to Freed-Witten anomaly

On relating the [[Freed-Witten anomaly]] to the [[shifted C-field flux quantization]]:

On [[D4-branes]]:

* [[Edward Witten]], Section 2 of: _Duality Relations Among Topological Effects In String Theory_, JHEP 0005:031, 2000 [[arXiv:hep-th/9912086](https://arxiv.org/abs/hep-th/9912086), [doi:10.1088/1126-6708/2000/05/031](https://doi.org/10.1088/1126-6708/2000/05/031)]

On [[D6-branes]]:

* [[Sergei Gukov]], [[James Sparks]], p. 21 of: _M-Theory on $Spin(7)$ Manifolds_, Nucl. Phys. B **625** (2002) 3-69 [[arXiv:hep-th/0109025](https://arxiv.org/abs/hep-th/0109025), doi:10.1016/S0550-3213(02)00018-4]

* [[James Sparks]], _Global Worldsheet Anomalies from M-Theory_, JHEP 0408:037, 2004 ([arXiv:hep-th/0310147](https://arxiv.org/abs/hep-th/0310147))

### In F-theory

Discussion in [[F-theory]]:

* [[Andres Collinucci]], [[Raffaele Savelli]], *On Flux Quantization in F-Theory*,  J. High Energ. Phys. **2012** 15 (2012)   [[arXiv:1011.6388](https://arxiv.org/abs/1011.6388), doi:10.1007/JHEP02(2012)015]

* [[Andres Collinucci]], [[Raffaele Savelli]], *On Flux Quantization in F-Theory II: Unitary and Symplectic Gauge Groups*,  J. High Energ. Phys. **2012** 94 (2012) [[arXiv:1203.4542](https://arxiv.org/abs/1203.4542), doi:10.1007/JHEP08(2012)094]

* [[Raffaele Savelli]], *Flux Quantization in F-theory and Freed-Witten anomaly*, talk at StringMath (2012) [[pdf](https://www.hcm.uni-bonn.de/fileadmin/stringmath2012/parallel_talks/savelli.pdf), [[Savelli-FTheoryFluxQuantization.pdf:file]]]

[[!redirects shifted C-field flux quantization condition]]

[[!redirects Page charge]]
[[!redirects Page charges]]

[[!redirects C-field flux quantization]]


shifted symplectic structure

+-- {: .rightHandSide}
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###Context###
#### Higher geometry
+--{: .hide}
[[!include higher geometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The [[derived geometry]]-version of [[symplectic geometry]].

## Related concepts

* [[symplectic structures]]

* [[derived algebraic geometry]]

* [[AKSZ sigma-model]]

* [[BV-BRST formalism]]

## References

* [[Tony Pantev]], [[Bertrand Toen]], [[Michel Vaquie]], [[Gabriele Vezzosi]], _Shifted symplectic structures_, Publ. Math. IHES 117:1 (2013) 271--328 [arXiv:1111.3209](http://arxiv.org/abs/1111.3209)

* [[Damien Calaque]], _Lagrangian structures on mapping stacks and semi-classical TFTs_, [arXiv:1306.3235](http://arxiv.org/abs/1306.3235).

* [[Damien Calaque]], _Three lectures on derived symplectic geometry and topological field theories_, 2014,
[pdf](https://imag.umontpellier.fr/~calaque/lecturenotes-DerivedTFT) - see also [these notes](https://imag.umontpellier.fr/~calaque/lecturenotes-LagStr)

* [[Pavel Safronov]], _Quasi-Hamiltonian reduction via classical Chern--Simons theory_, [arXiv:1311.6429](http://arxiv.org/abs/1311.6429)

* [[James Wallbridge]], _Derived smooth stacks and prequantum categories_, [arxiv/1610.00441](https://arxiv.org/abs/1610.00441)
* Xiaojun Chen, Farkhod Eshmatov, _Calabi-Yau algebras and the shifted noncommutative symplectic structure_, [arXiv:1802.07423](https://arxiv.org/abs/1802.07423)

[[Darboux theorem]]:

* [[Chris Brav]], [[Vittoria Bussi]], [[Dominic Joyce]], _A Darboux theorem for derived schemes with shifted symplectic structure_, 2013, [arXiv:1305.6302](http://arxiv.org/abs/1305.6302).

* [[Oren Ben-Bassat]], [[Christopher Brav]], [[Vittoria Bussi]], [[Dominic Joyce]], _A 'Darboux Theorem' for shifted symplectic structures on derived Artin stacks, with applications_, [arXiv:1312.0090](http://arxiv.org/abs/1312.0090)

* [[Pavel Safronov]], [[Dominic Joyce]], _A Lagrangian Neighbourhood Theorem for shifted symplectic derived schemes_, [arXiv:1506.04024](http://arxiv.org/abs/1506.04024)

Shifted Poisson structures

* [[Damien Calaque]], [[Tony Pantev]], [[Bertrand Toen]], [[Michel Vaquie]], [[Gabriele Vezzosi]], _Shifted Poisson structures and deformation quantization_, Journal of topology 10.2 (2017) 483--584 [arXiv:1506.03699](http://arxiv.org/abs/1506.03699)

* Valerio Melani, _Poisson bivectors and Poisson brackets on affine derived stacks_, [arXiv:1409.1863](http://arxiv.org/abs/1409.1863)

* [[J. P. Pridham]], _Shifted Poisson and symplectic structures on derived N-stacks_, [arXiv:1504.01940](http://arxiv.org/abs/1504.01940)

* [[J. P. Pridham]], _Shifted bisymplectic and double Poisson structures on noncommutative derived prestacks_, [arXiv:2008.11698](https://arxiv.org/abs/2008.11698)

* [[J. P. Pridham]], _Shifted symplectic structures on derived analytic moduli of l-adic local systems and Galois representations_, [arXiv:2205.02292](https://arxiv.org/abs/2205.02292)

* [[Pavel Safronov]], _Poisson reduction as a coisotropic intersection_, [arXiv:1509.08081](http://arxiv.org/abs/1509.08081)

Review:

* [[Damien Calaque]], *Derived symplectic geometry*, [[Encyclopedia of Mathematical Physics 2nd ed]] [[arXiv:2308.04210](https://arxiv.org/abs/2308.04210)]

[[!redirects shifted symplectic structures]]
[[!redirects derived symplectic geometry]]
[[!redirects shifted Poisson structure]]


shifted tangent bundle

+-- {: .rightHandSide}
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###Context###
#### Super-Geometry
+--{: .hide}
[[!include supergeometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea ##

The _shifted tangent bundle_ or _odd tangent bundle_ $\Pi T X$ of a [[manifold]] $X$ is an incarnation of the ordinary [[tangent bundle]] of $X$ as a [[supermanifold]] with the underlying manifold in even degree and the tangent vectors in odd degree.

## Definition ##

For $X$ a smooth [[manifold]] the [[supermanifold]] $\Pi T X$ is defined to be the one specified by the fact that its [[superalgebra]] of functions is the graded-commutative [[exterior algebra]] of [[differential forms]]

$$
  C^\infty(\Pi T X) := \Omega^\bullet(X)
  = \wedge^\bullet_{C^\infty(X)} \Gamma(T^* X)
  \,.
$$

More precisely, for each open $U \subset X$ the value of the [[structure sheaf]] of $\Pi T X$ is $\Omega^\bullet(U)$.

For more details see at _[[geometry of physics -- supergeometry]]_.

## Alternative characterizations ##

### As an NQ-supermanifold ###

In the context of [[supergeometry]] the algebra $\Omega^\bullet(X)$ is regarded as a $\mathbb{Z}_2$-graded algebra, but of course this $\mathbb{Z}_2$-grading lifts to an $\mathbb{N}$-grading in the obvious way.

Moreover, there is a canonical odd vector field $v_d$ on $\Pi T X$, which as an odd derivation on the function algebra $\Omega^\bullet(X)$ is just the [[deRham complex|deRham differential]].

Equipped with this structure $\Pi T X$ is naturally an [[NQ-supermanifold]].

### As the tangent Lie algebroid ###

The [[dg-algebra]] $(\Omega^\bullet(X), d_{dR})$ may also be regarded as the [[Chevalley-Eilenberg algebra]] of the [[tangent Lie algebroid]] of $X$, which identifies the shifted tangent bundle in its refinement to an [[NQ-supermanifold]] with the [[tangent Lie algebroid]] of $X$.

From this perspective, the fact that the vectors are regarded as being in degree one in $\Pi T X$ corresponds to the fact that these are the tangents to the [[k-morphism|1-morphism]]s of the [[fundamental groupoid]] of $X$. (Which is denoted $\Pi(X)$ but with the "$\Pi$" of completely different meaning than the "$\Pi$" as used here, which just indicates degree shift).

### As an internal hom object ###

With $\mathbb{R}^{0|1}$ denoting the [[odd line]], i.e. the [[supermanifold]] with function algebra the algebra of [[dual number]]s, one finds that

$$
  \Pi T X = [\mathbb{R}^{0|1}, X]
$$

is the [[internal hom]] object in the category of [[supermanifold]]s of maps from $\mathbb{R}^{0|1}$ to $X$. More precisely this means that the [[internal hom]] which exists in the [[closed monoidal structure on presheaves]] on the category of supermanifolds, between the presheaves [[representable functor|represented by]] $\mathbb{R}^{0|1}$ and $X$, is itself [[representable functor|representable]] and is represented by $\Pi T X$.

The existence of the structure of an [[NQ-supermanifold]] is from this point of view a consequence of the fact that $[\mathbb{R}^{0|1},X]$ naturally carries an action of the [[endomorphism]] object $End(\mathbb{R}^{0|1})$.  For more on this see [[NQ-supermanifold]].

## References

* [[geometry of physics -- supergeometry]] the section _[Super mapping spaces](geometry+of+physics+--+supergeometry#SuperMappingSpaces)_

[[!redirects shifted tangent bundles]]

[[!redirects odd tangent bundle]]
[[!redirects odd tangent bundles]]


Shigefumi Mori

* [institute page](https://kuias.kyoto-u.ac.jp/e/profile/mori/)

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

## Selected writings

On [[birational geometry]]:

* [[János Kollár]], [[Shigefumi Mori]], _Birational geometry of algebraic varieties_, Cambridge Tracts in Mathematics __134__, Cambridge University Press 2010 ([doi:10.1017/CBO9780511662560](https://doi.org/10.1017/CBO9780511662560))

category: people


Shigeki Sugimoto

* [webpage](http://www2.yukawa.kyoto-u.ac.jp/~shigeki.sugimoto/index_e.html)

## Selected writings

The [[Sugimoto string]]:

* {#Sug99} [[Shigeki Sugimoto]]. *Anomaly cancellations in type I D-9 - anti-D-9 system and the USp(32) string theory*. Prog. Theor. Phys. **102** (1999) 685-699 [[hep-th/9905159](https://arxiv.org/abs/hep-th/9905159), [doi:10.1143/PTP.102.685](https://doi.org/10.1143/PTP.102.685)]

On [[orientifolds]], [[O-planes]] and [[RR-field tadpole cancellation]]:

* Yoshifumi Hyakutake, Yosuke Imamura, [[Shigeki Sugimoto]], _Orientifold Planes, Type I Wilson Lines and Non-BPS D-branes_, JHEP 0008 (2000) 043 ([arXiv:hep-th/0007012](https://arxiv.org/abs/hep-th/0007012))

* {#BergmanGimonSugimoto01} [[Oren Bergman]], [[Eric Gimon]], [[Shigeki Sugimoto]], _Orientifolds, RR Torsion, and K-theory_, JHEP 0105:047, 2001 ([arXiv:hep-th/0103183](https://arxiv.org/abs/hep-th/0103183))

Towards an enhancement of the [[BFSS matrix model]] by [[topological K-theory]] to account for the [[K-theory classification of D-brane charge]] ([[K-matrix model]]):

* {#AsakawaSugimotoTerashima01} [[Tsuguhiko Asakawa]], [[Shigeki Sugimoto]], [[Seiji Terashima]], *D-branes, Matrix Theory and K-homology*, JHEP 0203 (2002) 034 [[arXiv:hep-th/0108085](https://arxiv.org/abs/hep-th/0108085), [doi:10.1088/1126-6708/2002/03/034](https://doi.org/10.1088/1126-6708/2002/03/034)]

On the [[Sakai-Sugimoto model]] for [[holographic QCD]] ([[geometric engineering of QFT|geometric engineering]] of [[QCD]] via [[D4-D8-brane bound state]] [[intersecting D-brane models]]):

* {#SakaiSugimoto04} [[Tadakatsu Sakai]], [[Shigeki Sugimoto]], _Low energy hadron physics in holographic QCD_, Prog.Theor.Phys.113:843-882, 2005 ([arXiv:hep-th/0412141](https://arxiv.org/abs/hep-th/0412141))

* {#SakaiSugimoto05} [[Tadakatsu Sakai]], [[Shigeki Sugimoto]], _More on a holographic dual of QCD_, Prog.Theor.Phys.114:1083-1118, 2005 ([arXiv:hep-th/0507073](https://arxiv.org/abs/hep-th/0507073))

* Hiroyuki Hata, [[Tadakatsu Sakai]], [[Shigeki Sugimoto]], Shinichiro Yamato, _Baryons from instantons in holographic QCD_, Prog.Theor.Phys.117:1157, 2007 ([arXiv:hep-th/0701280](https://arxiv.org/abs/hep-th/0701280))

* [[Koji Hashimoto]], [[Tadakatsu Sakai]], [[Shigeki Sugimoto]], _Holographic Baryons : Static Properties and Form Factors from Gauge/String Duality_, Prog.Theor.Phys.120:1093-1137, 2008 ([arXiv:0806.3122](https://arxiv.org/abs/0806.3122))

Review:

* [[Shigeki Sugimoto]], _Holographic QCD -- Status and perspectives_, 2012 ([[Sugimoto12.pdf:file]])

Specifically on [[skyrmions]]:

* {#Sugimoto16} [[Shigeki Sugimoto]], _Skyrmion and String theory_, chapter 15 in [[Mannque Rho]], [[Ismail Zahed]] (eds.) _[[The Multifaceted Skyrmion]]_, World Scientific 2016 ([doi:10.1142/9710](https://doi.org/10.1142/9710))

On 3d [[Chern-Simons theory]] in [[intersecting D-brane models]]:

* Mitsutoshi Fujita, Charles M. Melby-Thompson, Rene Meyer, [[Shigeki Sugimoto]], _Holographic Chern-Simons Defects_, JHEP06 (2016) 163 ([arxiv:1601.00525](https://arxiv.org/abs/1601.00525))

On the [[nuclear matrix model]]:

* Yasuhiro Hayashi, Takahiro Ogino, [[Tadakatsu Sakai]], [[Shigeki Sugimoto]], _Stringy excited baryons in holographic QCD_, Prog Theor Exp Phys (2020) ([arXiv:2001.01461](https://arxiv.org/abs/2001.01461))

Computing the [[D-term]] of [[hadrons]] via [[holographic QCD]]:

* {#FHSU22} Mitsutoshi Fujita, Yoshitaka Hatta, [[Shigeki Sugimoto]], Takahiro Ueda, *Nucleon D-term in holographic QCD* $[$[arXiv:2206.06578](https://arxiv.org/abs/2206.06578)$]$

## Related $n$Lab entries

* [[AdS-QCD correspondence]]

* [[skyrmion]]

* [[hadron]],

  [[meson]], [[baryon]]

category: people


Shigeo Sasaki

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

## selected writings

Precursors to [[Sasakian geometry]]:

* [[Shigeo Sasaki]], _On differentiable manifolds with certain structures which are closely related to almost contact structure I_, Tohoku Math. J. (2) 12 (1960), 459-476 ([euclid:1178244407](https://projecteuclid.org/euclid.tmj/1178244407))

* [[Shigeo Sasaki]], Y. Hatakeyama, _On differentiable manifolds with contact metric structures_, J. Math. Soc. Japan 14 (1962), 249-271 ([euclid:1261060580](https://projecteuclid.org/euclid.jmsj/1261060580))

An early set of lecture notes:

* [[Shigeo Sasaki]], _Almost contact manifolds, Part 1_, Lecture Notes, Mathematical Institute, Tohoku University (1965).

* [[Shigeo Sasaki]], _Almost contact manifolds, Part 2_, Lecture Notes, Mathematical Institute, Tohoku University (1967).

* [[Shigeo Sasaki]], _Almost contact manifolds, Part 3_, Lecture Notes, Mathematical Institute, Tohoku University (1968).

> Even today, after 40 years, the breath, depth and the relative completeness of the Sasaki lectures is truly quite remarkable. $[...]$ As it is, the notes are not easily available and, consequently, not well-known. ([Boyer-Galicki 07, p. 1](Sasakian+manifold#BoyerGalicki07))

## related $n$Lab entries

* [[Sasakian geometry]]

category: people


Shigeru Mukai

**Shigeru Mukai** is a Japanese algebraic geometer. He discovered a first historical example of a nontrivial automorphism of the derived category of coherent sheaves on an abelian variety. Generlizations of this example are studied under the name of Fourier-Mukai transforms, or integral transforms of derived categories.

Mukai's textbook (advanced undergraduate level) on moduli spaces has been translated from  Japanese to English and is one of the masterpieces of exposition in algebraic geometry, in accessible manner but with content to the point.

* [homepage](http://www.kurims.kyoto-u.ac.jp/~mukai)

category:people


Shigeru Yamagami

* [personal page](https://www.math.nagoya-u.ac.jp/~yamagami/)

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

## Selected writings

On a conceptual formulation of Haagerup's definition of [[L^p-spaces]] of (noncommutative) [[von Neumann algebras]]:

* [[Shigeru Yamagami]], _Algebraic aspects in modular theory_, Publications of the Research Institute for Mathematical Sciences 28:6 (1992), 1075-1106.  [doi](http://dx.doi.org/10.2977/prims/1195167738).

* [[Shigeru Yamagami]], _Modular theory for bimodules_, Journal of Functional Analysis 125:2 (1994), 327-357.  [doi](http://dx.doi.org/10.1006/jfan.1994.1127).

On what came to be called *[[Tambara-Yamagami categories]]*:

* {#Tambara1998tensor} [[Daisuke Tambara]], [[Shigeru Yamagami]], *Tensor categories with fusion rules of self-duality for finite abelian groups*, Journal of Algebra **209**  (1998)  692-707 [[doi:10.1006/jabr.1998.7558](https://doi.org/10.1006/jabr.1998.7558)]

category: people


Shih-Han Hung

* [personal page](https://hungshihhan.github.io/)

* [institute page](https://www.cs.utexas.edu/people/faculty-researchers/shih-han-hung)

## Selected writings

On [[quantum circuit]] [[certified programming|certification]] with the [[quantum programming language]] *[[SQIR]]*:

* [[Kesha Hietala]], [[Robert Rand]], [[Shih-Han Hung]], Xiaodi Wu, [[Michael Hicks]], *A verified optimizer for Quantum circuits*, Proceedings of the ACM on Programming Languages **5** Issue POPL 37 (2021) 1–29 [[doi:10.1145/3434318](https://doi.org/10.1145/3434318)]

* [[Kesha Hietala]], [[Robert Rand]], [[Shih-Han Hung]], Liyi Li, [[Michael Hicks]], *Proving Quantum Programs Correct*, in *12th International Conference on Interactive Theorem Proving (ITP 2021)*, Leibniz International Proceedings in Informatics (LIPIcs) **193** (2021) [[arXiv:2010.01240](https://arxiv.org/abs/2010.01240)]

category: people


Shiing-Shen Chern

* [Wikipedia entry](https://en.wikipedia.org/wiki/Shiing-Shen_Chern)

## Selected writings

On [[Euler forms]] and the [[Gauss-Bonnet theorem]]:

* {#Chern44} [[Shiing-Shen Chern]], _A Simple Intrinsic Proof of the Gauss-Bonnet Formula for Closed Riemannian Manifolds_, Annals of Mathematics Second Series, Vol. 45, No. 4 (1944), pp. 747-752 ([jstor:1969302](https://www.jstor.org/stable/1969302))

On the [[Chern-Weil homomorphism]] and introducing [[Chern-Weil theory]]:

* {#Chern50} [[Shiing-shen Chern]], _Differential geometry of fiber bundles_, in: Proceedings of the International Congress of Mathematicians, Cambridge, Mass., (August-September 1950), vol. 2, pages 397-411,  Amer. Math. Soc., Providence, R. I. (1952) ([[Chern-DifferentialGeometryOfFiberBundles.pdf:file]], [full proceedings vol 2 pdf](https://www.mathunion.org/fileadmin/ICM/Proceedings/ICM1950.2/ICM1950.2.ocr.pdf))

Introducing [[Chern-Simons forms]]:

* [[Shiing-Shen Chern]], [[James Simons]], Section 2 of: _Characteristic Forms and Geometric Invariants_, Annals of Mathematics Second Series, Vol. 99, No. 1 (Jan., 1974), pp. 48-69 ([jstor:1971013](https://www.jstor.org/stable/1971013))

## related $n$Lab entries

* [[Chern class]]

* [[Chern-Weil theory]]

* [[Chern-Simons form]]

* [[Euler form]]

* [[Gauss-Bonnet theorem]]

category: people


Shiing-shen Chern

* [Wikipedia entry](http://en.wikipedia.org/wiki/Shiing-Shen_Chern)

## Selected writings

On [[principal bundles]], their [[characteristic classes]] and introducing the [[Chern-Weil homomorphism]]:

* [[Shiing-shen Chern]], _Differential geometry of fiber bundles_, in: Proceedings of the International Congress of Mathematicians, Cambridge, Mass., 1950, vol. 2, pages 397-411,  Amer. Math. Soc., Providence, R. I. (1952) ([[Chern-DifferentialGeometryOfFiberBundles.pdf:file]], [full proceedings vol 2 pdf](https://www.mathunion.org/fileadmin/ICM/Proceedings/ICM1950.2/ICM1950.2.ocr.pdf))

## Quotes
 {#Quotes}

\begin{imagefromfile}
  "file_name": "YangNotJustJoy.jpg",
  "width": 320,
  "float": "right",
  "margin": {
    "top": -40,
    "right": 10,
    "bottom": 10,
    "left": 20,
    "unit": "px"
  }
\end{imagefromfile}

[[Chen Ning Yang]] writes in  _C. N. Yang, Selected papers, 1945-1980, with commentary_, W. H. Freeman and Company, San Francisco, 1983, on p. 567,

recalled in: D. Z. Zhang,  _C. N. Yang and contemporary mathematics_ The Mathematical Intelligencer 15, 13–21 (1993) ([doi:10.1007/BF03024319](https://doi.org/10.1007/BF03024319)):

> In 1975, impressed with the fact that [[gauge fields]] are [[connection on a bundle|connections]] on [[fiber bundles]], I drove to the house of [[Shiing-shen Chern|S. S. Chern]] in El Cerrito, near Berkeley... I said I found it amazing that [[gauge theory]] are exactly [[connection on a bundle|connections]] on [[fiber bundles]], which the mathematicians developed without reference to the physical world. I added: "this is both thrilling and puzzling, since you mathematicians dreamed up these concepts out of nowhere." He immediately protested: "No, no. These concepts were not dreamed up. They were natural and real."

## Related entries

* [[Chern-Weil theory]]

  * [[Chern-Weil homomorphism]]

  * [[Chern-Gauss-Bonnet theorem]]

* [[secondary characteristic class]]

  * [[Chern-Simons form]]

* [[CR manifold]]

[[!redirects Chern]]

category: people


Shimon Yankielowicz

* [webpage](https://physics.tau.ac.il/profile/shimonya)

## Selected writings

On [[Seiberg-Witten curves]] understood as transversal factors of [[M5-brane]]-[[worldvolumes]]:

* A. Brandhuber, [[Jacob Sonnenschein]], [[Stefan Theisen]], [[Shimon Yankielowicz]], *M Theory And Seiberg-Witten Curves: Orthogonal and Symplectic Groups*, Nucl. Phys. B **504** (1997) 175-188 [[arXiv:hep-th/9705232](https://arxiv.org/abs/hep-th/9705232), doi:10.1016/S0550-3213(97)00531-2]

On [[supergravity]], [[D-branes]] and the [[large N limit]] of [[super Yang-Mills theories]]:

* [[Nissan Itzhaki]], [[Juan Maldacena]], [[Jacob Sonnenschein]], [[Shimon Yankielowicz]], _Supergravity and The Large $N$ Limit of Theories With Sixteen Supercharges_, Phys. Rev. D 58, 046004 (1998) ([arXiv:hep-th/9802042](https://arxiv.org/abs/hep-th/9802042))

On [[heterotic M-theory on ADE-orbifolds]]:

* {#KSTY99} [[Vadim Kaplunovsky]], [[Jacob Sonnenschein]], [[Stefan Theisen]], [[Shimon Yankielowicz]], _On the Duality between Perturbative Heterotic Orbifolds and M-Theory on $T^4/Z_N$_, Nuclear Physics B Volume 590, Issues 1–2, 4 December 2000, Pages 123-160 Nuclear Physics B ([arXiv:hep-th/9912144](https://arxiv.org/abs/hep-th/9912144), doi:10.1016/S0550-3213(00)00460-0)

* {#GKSTY02} E. Gorbatov, [[Vadim Kaplunovsky]], [[Jacob Sonnenschein]], [[Stefan Theisen]], [[Shimon Yankielowicz]], _On Heterotic Orbifolds, M Theory and Type I' Brane Engineering_, JHEP 0205:015, 2002 ([arXiv:hep-th/0108135](https://arxiv.org/abs/hep-th/0108135))

On [[baryons]] in [[holographic QCD]]:

* {#BISY98} A. Brandhuber, N. Itzhaki, [[Jacob Sonnenschein]], [[Shimon Yankielowicz]],  _Baryons from Supergravity_, JHEP 9807:020, 1998 ([arxiv:hep-th/9806158](https://arxiv.org/abs/hep-th/9806158))

category: people


Shimshon Amitsur

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

## Related entries

* [[Amitsur complex]]

[[!redirects Amitsur]]

category: people


Shimura variety

#Contents#
* table of contents
{:toc}

## Idea

A _Shimura variety_ is a higher-dimensional analog of a [[modular curve]].

## Definition

A _Shimura datum_ is a pair $(G,X)$ where $G$ is a reductive group and $X$ is a $G(\mathbb{R})$-conjugacy class of homorphisms $h:\mathbb{S}\to G_{\mathbb{R}}$, where $\mathbb{S}=\Res_{\mathbb{C}/\mathbb{R}}\mathbb{G}_{m}$.

Given a Shimura datum $G(X)$ and a sufficiently small compact open subgroup $K$ of $G(\mathbb{A}_{f})$ (where $\mathbb{A}_{f}=\prod'\mathbb{Q}_{p}$ is the finite adeles), the quotient

$$
  \Sh_{K}(G,X)
  \;\coloneqq\;
  G(\mathbb{Q}) \backslash G(\mathbb{A}_{f}) \times X/K
$$

can be realized as a quasi-projective variety over $\mathbb{C}$ (and actually over a canonical number field called the _reflex field_ that depends on the Shimura datum). The inverse limit $\Sh(G,X):=(\Sh_{K}(G,X))_{K}$ over all sufficiently small compact subgroups is called a _Shimura variety_.

## Applications to the Langlands program

Since the [[l-adic cohomology]] of the Shimura variety $\Sh(G,X)$ possesses an action of $G(\mathbb{A})$, as well as an action of the [[absolute Galois group]] $\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$, the theory of Shimura varieties finds application in the [[Langlands program]]. See also [Youcis](#Youcis).

## Related concepts

* [[Siegel modular form]]

* [[Hilbert modular form]]

* [[Rapoport-Zink space]]

## References

### General

Surveys:

* [[James Milne]], _Introduction to Shimura Varieties_, 2004, ([pdf](https://www.jmilne.org/math/articles/2005aX.pdf))

* Kai-Wen Lan, _An Example-Based Introduction to Shimura Varieties_, ([pdf](https://www-users.cse.umn.edu/~kwlan/articles/intro-sh-ex.pdf))

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

Introductory discussion of PEL Shimura varieties with an eye towards the definition of [[topological automorphic forms]] is in

* {#Lawson08} [[Tyler Lawson]], _An overview of abelian varieties in homotopy theory_ ([arXiv:0810.0507](https://arxiv.org/abs/0810.0507))

Discussion in [[GAGA]] include

* Kai-Wen Lan, _Comparison between analytic and algebraic constructions of toroidal compactifications of PEL-type Shimura varieties_ [pdf](http://math.umn.edu/~kwlan/articles/cpt-ana-alg.pdf)

Discussion relating to [[moduli space of Calabi-Yau spaces]] includes

* Alice Garbagnati, Bert van Geemen, _Examples of Calabi-Yau threefolds parametrised by Shimura varieties_ ([arXiv:1005.0478](https://arxiv.org/abs/1005.0478))

Discussion related to the Langlands program can be found in

* {#Youcis} Alex Youcis, _The cohomology of Shimura varieties and the Langlands correspondence_ ([pdf](https://alex-youcis.github.io/MatsushimaNotes.pdf))

### Model theoretic aspects

Discussion in [[model theory]]:

* [[Boris Zilber]], _Model theory of special subvarieties and Schanuel-type conjectures_, Annals of Pure and Applied Logic __167__:10 (2016) 1000--1028 [doi](https://doi.org/10.1016/j.apal.2015.02.002)

* C. Daw, A. Harris, _Categoricity of modular and Shimura curves_, J. Inst. Math. Jussieu, 16(5) (2017) 1075--1101 [doi](https://doi.org/10.1017/S1474748015000365)

[[!redirects Shimura varieties]]

[[!redirects PEL Shimura variety]]
[[!redirects PEL Shimura varieties]]


Shin'ichirō Tomonaga

* [Wikipedia entry](https://en.wikipedia.org/wiki/Shin%27ichir%C5%8D_Tomonaga)

## related $n$Lab entries

* [[perturbative quantum field theory]]

* [[Tomonaga-Schwinger equation]]

* [[Schwinger-Tomonaga-Feynman-Dyson]]

category: people

[[!redirects Tomonaga]]


Shin-ya Katsumata

* [personal page](https://group-mmm.org/~s-katsumata/index-e.html)

## Selected writings

On [[comonads]] and [[graded modalities]] [[comonads in computer science|in computer science]];

* {#GKOBU16} Marco Gaboardi, [[Shin-ya Katsumata]], [[Dominic Orchard]], Flavien Breuvart, [[Tarmo Uustalu]], *Combining effects and coeffects via grading*, ICFP 2016: Proceedings of the 21st ACM SIGPLAN International Conference on Functional Programming (2016) 476–489 [[doi:10.1145/2951913.2951939](https://doi.org/10.1145/2951913.2951939), [talk abstract](https://icfp16.sigplan.org/details/icfp-2016-papers/31/Combining-Effects-and-Coeffects-via-Grading), [video rec](https://www.youtube.com/watch?v=l1ZNMT3fQCo)]

* {#KRU20} [[Shin-ya Katsumata]], Exequiel Rivas, [[Tarmo Uustalu]], LICS (2020) 604-618 *Interaction laws of monads and comonads* [[arXiv:1912.13477](https://arxiv.org/abs/1912.13477), [doi:10.1145/3373718.3394808](https://doi.org/10.1145/3373718.3394808)]

category: people


Shing-Tung Yau

* [yauc.seu page](https://yauc.seu.edu.cn/27627/list.psp)

* [physics.harvard page](https://www.physics.harvard.edu/people/facpages/yau)

* [MathGenealogy page](https://www.genealogy.math.ndsu.nodak.edu/id.php?id=18758)

* [Wikipedia entry](http://en.wikipedia.org/wiki/Shing-Tung_Yau)

* [GoogleScholar page](https://scholar.google.com/citations?user=Apxg6tgAAAAJ&hl=en)

* [MacTutor biography page](https://mathshistory.st-andrews.ac.uk/Biographies/Yau/)

* [Yau Mathematical Science Center](https://ymsc.tsinghua.edu.cn/en/)

## Selected writings

Establishing what came to be called *[[Calabi-Yau manifolds]]*:

* [[Shing-Tung Yau]], *On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampére equation*, Communications on Pure and Applied Mathematics,
**31** 3 (1978) 339-411 [[doi:10.1002/cpa.3160310304](https://doi.org/10.1002/cpa.3160310304)]

On the [[mirror map]]:

* S. Hosono, [[Albrecht Klemm]], [[Stefan Theisen]], [[Shing-Tung Yau]], _Mirror symmetry, mirror map and applications to Calabi-Yau hypersurfaces_, Commun. Math. Phys. __167__ (1995) 301-350 [[doi:10.1007/BF02100589](https://doi.org/10.1007/BF02100589), [arXiv:hep-th/9308122](https://arxiv.org/abs/hep-th/9308122)
]

On [[moduli spaces of curves]]/[[moduli spaces of Riemann surfaces]] with emphasis on their [[orbifold]]-structure:

* Lizhen Ji, [[Shing-Tung Yau]], _Transformation Groups and Moduli Spaces of Curves_, International Press of Boston 2011 ([ISBN:9781571462237](https://www.intlpress.com/site/pub/pages/books/items/00000353/index.php))

On [[p-adic string theory]] and the [[Riemann zeta function]]:

* {#HuangStoicaYau19} [[An Huang]], [[Bogdan Stoica]], [[Shing-Tung Yau]], _General relativity from $p$-adic strings_ ([arXiv:1901.02013](https://arxiv.org/abs/1901.02013))

On a rigorous [[proof]] of [[moduli stabilization]] of vacuum [[spacetimes]], refuting [Penrose 03](moduli+stabilization#Penrose03) (for $\geq 10$-dimensions with a [[Killing spinor]] on the [[compact topological space|compact]] [[fiber]] and for [[Schwarzschild spacetime|Schwarzschild]]-asymptotics):

* {#AndersonBlueWyattYau20} [[Lars Andersson]], [[Pieter Blue]], [[Zoe Wyatt]], [[Shing-Tung Yau]], _Global stability of spacetimes with supersymmetric compactifications_ ([arXiv:2006.00824](https://arxiv.org/abs/2006.00824))

On [[mirror map]]

* [[Bong H. Lian]], [[Shing-Tung Yau]], _Arithmetic properties of mirror map and quantum coupling_, Commun. Math. Phys. __176__ (1996) 163--191 [doi](https://doi.org/10.1007/BF02099367)
* B. Lian, S-T. Yau, _Mirror symmetry, rational curves on algebraic manifolds and hypergeometric series_, 163--183 in: XIth International Congress of Mathematical Physics (Paris 1994), Daniel Iagolnitzer (eds.), Intern. Press 1995
* [[Bong H. Lian]], [[Shing-Tung Yau]], _Mirror maps, modular relations and hypergeometric series I_, arXiv:[hep-th/9507151](https://arxiv.org/pdf/hep-th/9507151)
* [[Bong H. Lian]], [[Shing-Tung Yau]], _Mirror maps, modular relations and hypergeometric series II_, arXiv:[hep-th/9507153](https://arxiv.org/abs/hep-th/9507153) Nuclear Physics B - Proc. Suppl. __46__:1--3 (1996) 248--262 doi

## related $n$Lab entries

* [[Calabi-Yau manifold]]

* [[geometrization conjecture]]

* [[p-adic string theory]], [[Riemann zeta function]]

* [[mirror symmetry]]...

category: people


Shinichi Mochizuki

\tableofcontents

\section{Introduction}

**Shinichi Mochizuki** is a Japanese algebraic/arithmetic geometer, developing Grothendieck's program of so-called [[anabelian geometry]]. His preprint on [[inter-universal Teichmüller theory]] claims to prove the [[abc conjecture]].

* [Wikipedia page](http://en.wikipedia.org/wiki/Shinichi_Mochizuki)
* [homepage](http://www.kurims.kyoto-u.ac.jp/~motizuki/top-english.html) at RIMS
* CV [pdf](http://www.kurims.kyoto-u.ac.jp/~motizuki/Curriculum%20Vitae.pdf)

\section{Related nLab pages}

* [[Mochizuki's corollary 3.12]]
* [[anabelioid]]
* [[initial Θ-data]]
* [[semi-graph]]

category: people


Shinji Shimasaki

* [inspire page](https://inspirehep.net/authors/1025128)

## Selected writings

On [[M2-M5-brane bound states]] in the [[BMN matrix model]]:

* {#AIST17b} [[Yuhma Asano]], [[Goro Ishiki]], [[Shinji Shimasaki]], [[Seiji Terashima]], _Spherical transverse M5-branes from the plane wave matrix model_, JHEP 02 (2018) 076 [[arXiv:1711.07681](https://arxiv.org/abs/1711.07681), doi:10.1007/JHEP02(2018)076]

* {#AIST17a} [[Yuhma Asano]], [[Goro Ishiki]], [[Shinji Shimasaki]], [[Seiji Terashima]], _On the transverse M5-branes in matrix theory_, Phys. Rev. D **96** 126003 (2017) [[arXiv:1701.07140](https://arxiv.org/abs/1701.07140), [doi:10.1103/PhysRevD.96.126003](https://doi.org/10.1103/PhysRevD.96.126003)]

and in a limit where aspects of [[little string theory]] on [[NS5-branes]] becomes visible:

* [[Yuhma Asano]], [[Goro Ishiki]], [[Takaki Matsumoto]], [[Shinji Shimasaki]], [[Hiromasa Watanabe]], *On the existence of the NS5-brane limit of the plane wave matrix model*, PTEP 2023 (2023) 4, 043B01 $[$[arXiv:2211.13716](https://arxiv.org/abs/2211.13716), [doi:10.1093/ptep/ptad042](https://doi.org/10.1093/ptep/ptad042)$]$

category: people


Shinsei Ryu

* [webpage](http://physics.illinois.edu/people/directory/profile/ryuu)

## Selected writings

On the [[Ryu-Takayanagi formula]] for [[holographic entanglement entropy]]:

* {#RyuTakayanagi06a} [[Shinsei Ryu]], [[Tadashi Takayanagi]], _Holographic Derivation of Entanglement Entropy from AdS/CFT_, Phys. Rev. Lett. 96:181602, 2006 ([arXiv:hep-th/0603001](https://arxiv.org/abs/hep-th/0603001))

* {#RyuTakayanagi06a} [[Shinsei Ryu]], [[Tadashi Takayanagi]], _Aspects of Holographic Entanglement Entropy_, JHEP 0608:045, 2006 ([arXiv:hep-th/0605073](https://arxiv.org/abs/hep-th/0605073))

* {#NishiokaRyuTakayanagi09} Tatsuma Nishioka, [[Shinsei Ryu]], [[Tadashi Takayanagi]], _Holographic Entanglement Entropy: An Overview_, J.Phys.A42:504008,2009 ([arXiv:0905.0932](https://arxiv.org/abs/0905.0932))

On [[Chern-Simons theories]] arising from the [[higher WZ term]] on [[super p-brane]] [[worldvolumes]] (notably on [[D2-branes]]):

* Mitsutoshi Fujita, Wei Li, [[Shinsei Ryu]], [[Tadashi Takayanagi]], _Fractional Quantum Hall Effect via Holography: Chern-Simons, Edge States, and Hierarchy_, JHEP 0906:066, 2009 ([arXiv:0901.0924](https://arxiv.org/abs/0901.0924))

On the [[K-theory classification of topological phases of matter]] translating under [[AdS/CFT duality in solid state physics]]  to the [[K-theory classification of D-brane charge]] in [[string theory]], allowing a dual description of the topological phases even at [[non-perturbative effects|strong coupling]] via [[AdS/CFT duality]]:

* [[Shinsei Ryu]], [[Tadashi Takayanagi]], _Topological Insulators and Superconductors from D-branes_, Phys. Lett. B693:175-179, 2010 ([arXiv:1001.0763](https://arxiv.org/abs/1001.0763))

* [[Shinsei Ryu]], [[Tadashi Takayanagi]], _Topological Insulators and Superconductors from String Theory_, Phys. Rev. D82:086014, 2010 ([arXiv:1007.4234](https://arxiv.org/abs/1007.4234))

On ([[symmetry protected topological phase|symmetry protected]]) [[topological phases of matter]]:

* [[Ching-Kai Chiu]], [[Jeffrey C.Y. Teo]], [[Andreas P. Schnyder]] [[Shinsei Ryu]], *Classification of topological quantum matter with symmetries*, Rev. Mod. Phys. **88** 035005 (2016) $[$[arXiv:1505.03535](https://arxiv.org/abs/1505.03535), [doi:10.1103/RevModPhys.88.035005](https://doi.org/10.1103/RevModPhys.88.035005)$]$

## Related $n$Lab entries

* [[holographic entanglement entropy]]

category: people


Shira Chapman

* [personal page](https://sites.google.com/view/shira-chapman/home)

* [Institute page](https://in.bgu.ac.il/en/Pages/Shira-Chapman.aspx)

## Selected writings

* [[Shira Chapman]], [[Giuseppe Policastro]], *Quantum Computational Complexity -- From Quantum Information to Black Holes and Back*, The European Physical Journal C **82** 128 (2022)  [[arXiv:2110.14672](https://arxiv.org/abs/2110.14672), [doi:10.1140/epjc/s10052-022-10037-1](https://doi.org/10.1140/epjc/s10052-022-10037-1)]

* [[Shira Chapman]], _Holographic Complexity and de Sitter Space_, talk at *[[Strings 2022]]* [[indico:4940811](https://indico.cern.ch/event/1085701/contributions/4940811), [slides](https://indico.cern.ch/event/1085701/contributions/4940811/attachments/2481532/4260404/Slides_Chapman.pdf), [video](https://ustream.univie.ac.at/media/core.html?id=e3ddea36-2f95-42ac-8b26-98c41f1d800d)]

category: people


Shiraz Minwalla

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

## Selected writings

Introducing the [[fluid/gravity correspondence]]:

* [[Sayantani Bhattacharyya]], [[Veronika Hubeny]], [[Shiraz Minwalla]], [[Mukund Rangamani]], _Nonlinear Fluid Dynamics from Gravity_, JHEP 0802:045, 2008 ([arXiv:0712.2456](https://arxiv.org/abs/0712.2456))

More on the [[fluid/gravity correspondence]]:

* [[Veronika Hubeny]], [[Shiraz Minwalla]], [[Mukund Rangamani]], _The fluid/gravity correspondence_ ([arXiv:1107.5780](https://arxiv.org/abs/1107.5780))

## Related $n$Lab entries

* [[supersymmetry]], [[SCFT]]

category: people


Shlomo Gelaki

__Shlomo Gelaki__ is a mathematician at Iowa State University ([web](https://faculty.sites.iastate.edu/gelaki)). Much of his work is on the theory of [[tensor categories]].

## Selected writings

On [[tensor categories]]:

* [[Pavel Etingof]], [[Shlomo Gelaki]], [[Dmitri Nikshych]], [[Victor Ostrik]], _Topics in Lie theory and Tensor categories -- 9 Tensor categories_, Lecture notes (spring 2009) ([pdf](http://ocw.mit.edu/courses/mathematics/18-769-topics-in-lie-theory-tensor-categories-spring-2009/lecture-notes/MIT18_769S09_lec09.pdf) [web](http://ocw.mit.edu/courses/mathematics/18-769-topics-in-lie-theory-tensor-categories-spring-2009/lecture-notes/))

* [[Pavel Etingof]], [[Shlomo Gelaki]], [[Dmitri Nikshych]], [[Victor Ostrik]], *Tensor Categories*, AMS Mathematical Surveys and Monographs **205** (2015) $[$[ISBN:978-1-4704-3441-0](https://bookstore.ams.org/surv-205)$]$

On [[braided monoidal category|braided]] [[fusion categories]]:

* [[Vladimir Drinfeld]], [[Shlomo Gelaki]], [[Dmitri Nikshych]], [[Victor Ostrik]], *On braided fusion categories I*, Selecta Mathematica. New Series 16 (2010), no. 1, 1–119 ([doi:10.1007/s00029-010-0017-z](https://doi.org/10.1007/s00029-010-0017-z))

## Related $n$Lab entries

* [[Deligne's theorem on tensor categories]]

category: people


Shlomo Razamat

* [Research group page](https://phsites.technion.ac.il/razamat/)

* [Institute page](https://phys.technion.ac.il/en/people/faculty/person/741)

* [GoogleScholar page](https://scholar.google.com/citations?user=N2gbP-IAAAAJ&hl=en)

## Selected writings

Survey of [[SCFTs]] in higher dimensions ([[D=4 SYM]], [[D=5 SYM]], [[D=6 N=(1,0) SCFT|D=6 N=1]], [[D=6 N=(2,0) SCFT|D=6, N=2]]) their [[moduli]] ([[Higgs branches]]/[[Coulomb branches]]) and their [[duality in string theory|dualities]] ([[KK-compactification]], [[AGT correspondence]], [[AdS/CFT duality]], ...):

* [[Shlomo Razamat]], *News from the higher $\mathcal{D}$ SQFT frontier*, talk at *[[Strings 2022]]* [[indico:4940856](https://indico.cern.ch/event/1085701/contributions/4940856), [slides](https://indico.cern.ch/event/1085701/contributions/4940856/attachments/2482369/4261653/Shlomo%20Razamat%20Tuesday-Review-Razamat.pdf), [video](https://ustream.univie.ac.at/media/core.html?id=ad1f1a6b-5018-4b69-9e38-e499daabda1c)]

category: people


Shlomo Sternberg

* [webpage](http://www.math.harvard.edu/~shlomo)

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

## Selected writings

On [[differential geometry]] with focus of [[geometric optics]] and [[current (distribution theory)|currents]]:

* [[Victor Guillemin]], [[Shlomo Sternberg]], *Geometric asymptotics*, Mathematical Surveys and Monographs **14**, AMS (1977) [[ams:surv-14](https://bookstore.ams.org/surv-14)]

On [[supersymmetry]] and the [[equivariant de Rham complex]]:

* {#GuilleminSternberg99} [[Victor Guillemin]], [[Shlomo Sternberg]], _Supersymmetry and equivariant de Rham theory_, Springer, (1999) ([doi:10.1007/978-3-662-03992-2](https://link.springer.com/book/10.1007/978-3-662-03992-2))

On [[differential geometry]] (including [[G-structures]]):

* {#Sternberg64} [[Shlomo Sternberg]], _Lectures on differential geometry_, Prentice-Hall (1964), 2nd edition AMS 1983 ([ISBN:978-0-8218-1385-0](https://bookstore.ams.org/chel-316))

On [[group theory]] and [[physics]]:

* {#Sternberg94} [[Shlomo Sternberg]], *Group Theory and Physics*, Cambridge University Press 1994 ([ISBN:9780521558853](https://www.cambridge.org/gb/academic/subjects/mathematics/algebra/group-theory-and-physics?format=PB&isbn=9780521558853))

## Related $n$Lab entries

* [[differential geometry]], [[G-structure]]

* [[Guillemin-Sternberg geometric quantization conjecture]]

* [[Weil algebra]]

* [[equivariant de Rham cohomology]]

[[!redirects S. Sternberg]]

category: people


Shmuel Elitzur

* [institute page](http://old.phys.huji.ac.il/~hep/shmuel.elitzur.html)

* [inspire page](https://inspirehep.net/authors/1010842)

## Selected writings

On the [[CP^1 sigma-model|$\mathbb{C}P^1$ sigma-model]]:

* [[Shmuel Elitzur]], A. Forge, [[Eliezer Rabinovici]], *On effective theories of topological strings*, Nuclear Physics B **388** 1 (1992) 131-155 [doi:10.1016/0550-3213(92)90548-P]

* [[Shmuel Elitzur]], [[Yaron Oz]], [[Eliezer Rabinovici]], [[Johannes Walcher]], *Open/Closed Topological $\mathbb{C}P^1$ Sigma Model Revisited*,  J. High Energ. Phys. **2012** 101 (2012) [[arXiv:1106.2967](https://arxiv.org/abs/1106.2967)]

On [[NS5-brane]]-[[brane intersections|intersections]] with [[D-branes]]:

* {#EGKRS00} [[Shmuel Elitzur]], [[Amit Giveon]], [[David Kutasov]], [[Eliezer Rabinovici]], [[Gor Sarkissian]], *D-Branes in the Background of NS Fivebranes*, JHEP 0008 (2000) 046 [[arXiv:hep-th/0005052](https://arxiv.org/abs/hep-th/0005052), [doi:10.1088/1126-6708/2000/08/046](https://doi.org/10.1088/1126-6708/2000/08/046)]

On the [[Riemann hypothesis and physics]]:

* {#ACER11} Carlo Angelantonj, Matteo Cardella, [[Shmuel Elitzur]], [[Eliezer Rabinovici]], *Vacuum stability, string density of states and the Riemann zeta function*, JHEP **1102** 024 (2011) [[arXiv:1012.5091](http://arxiv.org/abs/1012.5091)]

category: people


Shmuel Weinberger

* [webpage](http://www.math.uchicago.edu/~shmuel/)

category: people


Sho Saito

* [webpage](http://www.math.nagoya-u.ac.jp/~m09019h/)

## related $n$Lab entries

* [[algebraic K-theory]]

category: people


Shor's algorithm

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###Context###
#### Quantum systems
+--{: .hide}
[[!include quantum systems -- contents]]
=--
#### Constructivism, Realizability, Computability
+-- {: .hide}
[[!include constructivism - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A [[prime factorization]] [[algorithm]] for [[quantum computers]].

## Related concepts

* [[Grover's algorithm]]

* [[Deutsch-Jozsa algorithm]]

* [[quantum complexity theory]]

* [[quantum supremacy]]

## References
 {#References}

Due to:

* {#Shor97} [[Peter W. Shor]], *Algorithms for quantum computation: discrete logarithms and factoring*,  Proceedings 35th Annual Symposium on Foundations of Computer Science, IEEE Comput. Soc. Press: 124-134 (1994) [[doi:10.1109/SFCS.1994.365700](https://doi.org/10.1109/SFCS.1994.365700)]

* {#Simon97} [[Daniel R. Simon]], *On the power of quantum computation*, SIAM Journal on Computing **26** 5 (1997) [[doi:10.1137/S0097539796298637](https://doi.org/10.1137/S0097539796298637), [pdf](https://courses.cs.washington.edu/courses/cse599/01wi/papers/simon_qc.pdf)]

Early review:

* [[Yuri I. Manin]], *Classical computing, quantum computing, and Shor's factoring algorithm*, Astérisque, **266** Séminaire Bourbaki 862 (2000) 375-404 [[arXiv:quant-ph/9903008](https://arxiv.org/abs/quant-ph/9903008), [numdam:SB_1998-1999__41__375_0](http://www.numdam.org/item/?id=SB_1998-1999__41__375_0)]

Further review:

* [[Daniel Simon]], *[Exploring Simon’s Algorithm with Daniel Simon](https://aws.amazon.com/blogs/quantum-computing/simons-algorithm/)* (Oct 2021)

  > A bit of amusing history: I submitted my algorithm to a theoretical computer science conference (STOC 1993), but it was rejected. However, Peter Shor was on the program committee for that conference, and immediately saw the potential (that I had had absolutely no inkling of, to be honest) for applying the same general algorithm structure to concrete problems like factoring and discrete logarithms. After trying unsuccessfully to persuade the committee to accept my paper, he got to work on his own, and submitted it to the next major theoretical computer science conference (FOCS 1993)—in parallel with mine, resubmitted. We had in fact agreed to merge the papers if only his was accepted, but the committee fortunately agreed to accept them both, giving us each credit for our respective portions of the resulting achievement.”

* Renato Portugal, *Basic Quantum Algorithms* [[arXiv:2201.10574](https://arxiv.org/abs/2201.10574)]

Textbook account:

* [[Giuliano Benenti]], [[Giulio Casati]], [[Davide Rossini]], Section 3.14 of: *Principles of Quantum Computation and Information*, World Scientific 2018  ([doi:10.1142/10909](https://doi.org/10.1142/10909), [2004 pdf](http://mmrc.amss.cas.cn/tlb/201702/W020170224608149307696.pdf))

Lecture notes:

* [[Scott Aaronson]], Chapters 18-21 of: *Introduction to Quantum Information Science* (2018) [[pdf](https://www.scottaaronson.com/qclec.pdf), [webpage](https://www.scottaaronson.com/cs378/)]

Implementation in [[Quipper]]:

* Safat Siddiqui, Mohammed Jahirul Islam, Omar Shehab, §4.3, §4.5 in: *Five Quantum Algorithms Using Quipper* [[arXiv:1406.4481](https://arxiv.org/abs/1406.4481)]

See also:

* Wikipedia, *[Shor's algorithm](https://en.wikipedia.org/wiki/Shor%27s_algorithm)*

* Wikipedia, *[Simon's problem](https://en.wikipedia.org/wiki/Simon's_problem)*

On compiling Shor's algorithm to [[su(2)-anyon]] [[braid representation|braid]] [[quantum gates]] (i.e. implementation in [[topological quantum computation]]):

* M. Baraban, Nicholas E. Bonesteel, Steven H. Simon, *Resources required for topological quantum factoring*, Phys. Rev. A **81** 062317 (2010) [[doi:10.1103/PhysRevA.81.062317](https://doi.org/10.1103/PhysRevA.81.062317), [arXiv:1002.0537](https://arxiv.org/abs/1002.0537)]

[[!redirects Shor algorithm]]

[[!redirects Simon's algorithm]]
[[!redirects Simon algorithm]]


short exact sequence of vector bundles

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### Context
#### Bundles
+-- {: .hide}
[[!include bundles - contents]]
=--
#### Linear algebra
+-- {: .hide}
[[!include homotopy - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

Give a [[topological field|topological]] [[ground field]] $k$ and a  base [[topological space]] $B$, a [[short exact sequence]] of topological $k$-[[vector bundles]] over $B$ is a sequence of topological $k$-vector bundle homomorphisms over $B$ (i.e. [[continuous functions]] which are [[fiber]]-wise $k$-[[linear maps]])

\[
  \label{ShortExactSequenceOfTopologicalVectorBundles}
  0
  \to
  \mathcal{V}_L
    \overset{
      \;\;
        i
      \;\;}{\longrightarrow}
  \mathcal{V}
    \overset{
      \;\;
      p
      \;\;
     }{\longrightarrow}
  \mathcal{V}_R
  \to
  0
  \;\;\;\;
  \in
  k VectorBundles_B
\]

(where $0$ denotes the [[rank of a vector bundle|rank]]-[[zero]] bundle) such that
$p$ is a [[surjection]]
and
$i = ker_B(p)$ in the [[injection]] of its [[fiber]]-wise [[kernel]],
hence such that over each point $b \colon \ast \overset{}{\longrightarrow} B$ we have a [[short exact sequence]] of $k$-[[vector spaces]]:

$$
  \underset{
    b \in B
  }{\forall}
  \;\;\;
  0
  \to
  b^\ast \mathcal{V}_L
    \overset{
      \;\;
      b^\ast i
      \;\;
    }{\longrightarrow}
  b^\ast \mathcal{V}
    \overset{
      \;\;
      b^\ast p
     \;\;
    }{\longrightarrow}
  b^\ast \mathcal{V}_R
  \to
  0
  \;\;\;
  \in
  \;
  k VectorSpaces
  \,.
$$

## Properties

### Splitting

\begin{prop}
  **(over [[paracompact topological spaces]] [[short exact sequences]] of [[real vector bundles]] [[split exact sequence|split]])**
\linebreak

If

1. the [[ground field]] is the [[real numbers]] $k = \mathbb{R}$,

1. the base space $B$ is a [[paracompact Hausdorff space]],

1. the [[rank of a vector bundle|ranks]] are all [[finite number|finite]],

then every short exact sequence of topological vector bundles (eq:ShortExactSequenceOfTopologicalVectorBundles) [[split exact sequence|splits]] and exhibits the middle item as the [[direct sum of vector bundles]], over $B$, of the left and the right item:

$$
  \mathcal{V}
  \;\simeq\;
  \mathcal{V}_L
    \oplus_B
  \mathcal{V}_R
  \,.
$$
\end{prop}
(e.g. [Hatcher, Prop. 1.3](#Hatcher), [Freed, Lemma 5.6](#Freed))

\begin{proof}
Sketch: Under the assumption on $B$, there exists (by [this Prop.](inner+product+on+vector+bundles#ExistenceOfInnerProductOfTopologicalVectorBundlesOverParacompactHausdorffSpaces)) a [[inner product on vector bundles|fiberwise inner product]] on $\mathcal{V}$. With this the splitting follows by th usual splitting of short exact sequences of [[real vector spaces]], applied fiberwise: $\mathcal{V}_R$  is fiberwise identified with the [[orthogonal complement]] of $\mathcal{V}_L$. \end{proof}

## Related concepts

* [[vector bundle]]

  * [[real vector bundle]]

  * [[complex vector bundle]]

    * [[holomorphic vector bundle]], [[pseudoholomorphic vector bundle]]

  * [[universal vector bundle]]

  * [[dual vector bundle]]

  * [[module bundle]]

  * [[direct sum of vector bundles]]

  * [[connection on a vector bundle]]

  * [[flat vector bundle]]

  * [[real vector bundle]], [[complex vector bundle]]

  * [[super vector bundle]]

## References

Textbook accounts:

* {#Hatcher} [[Allen Hatcher]], Prop. 1.3 in: _Vector bundles and K-Theory_ ([webpage](https://pi.math.cornell.edu/~hatcher/VBKT/VBpage.html), [pdf](https://pi.math.cornell.edu/~hatcher/VBKT/VB.pdf))

Lecture notes:

* {#Freed} [[Daniel Freed]], _More on stabilization_, Lecture 5 in _[Bordism: Old and New](https://web.ma.utexas.edu/users/dafr/M392C-2012/)_ ([pdf](https://jfdmath.sitehost.iu.edu/teaching/m529/vector_bundles.pdf))

[[!redirects short exact sequences of vector bundles]]


short linear map

# Short linear maps
* table of contents
{: toc}

## Idea

The short linear maps are the [[morphisms]] of the category [[Ban]] of [[Banach spaces]] (or its [[full subcategory]] [[Hilb]] of [[Hilbert spaces]]), at least if one wishes to recover [[isometric isomorphism]]s as the abstract [[isomorphisms]] of this [[category]].

## Definition

Given [[Banach spaces]] $V$ and $W$, a __short linear map__ form $V$ to $W$ is a (total) [[function]] $f\colon V \to W$ that is both [[short map|short]] and [[linear map|linear]], equivalently a [[bounded linear operator]] whose [[norm]] is at most $1$.  These are also called __contractive__ (or __weakly contractive__) linear maps; they may also be called short (or contractive or weakly contractive) __operators__ (with linearity usually assumed but sometimes mentioned), or even simply (weak) __contractions__.

## On Hilbert spaces

Short linear maps from a [[Hilbert space]] to itself (that is [[endomorphisms]] in [[Hilb]]) are discussed (as simply 'contractions') in [[operator theory]]; see [Wikipedia](https://en.wikipedia.org/wiki/Contraction_%28operator_theory%29).

[[!redirects short linear map]]
[[!redirects short linear maps]]
[[!redirects short linear function]]
[[!redirects short linear functions]]
[[!redirects short linear operator]]
[[!redirects short linear operators]]
[[!redirects short operator]]
[[!redirects short operators]]
[[!redirects contractive linear function]]
[[!redirects contractive linear functions]]
[[!redirects weakly contractive linear function]]
[[!redirects weakly contractive linear functions]]
[[!redirects contractive linear map]]
[[!redirects contractive linear maps]]
[[!redirects weakly contractive linear map]]
[[!redirects weakly contractive linear maps]]
[[!redirects contractive operator]]
[[!redirects contractive operators]]
[[!redirects weakly contractive operator]]
[[!redirects weakly contractive operators]]


short map

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### Context
#### Analysis
+-- {: .hide}
[[!include analysis - contents]]
=--
=--
=--

# Short maps
* table of contents
{: toc}

## Idea

A _short map_ is a well-behaved sort of [[morphism]] of [[metric spaces]] (or a generalisation of metric spaces, such as the extended quasi-pseudo-versions, or [[gauge spaces]] or [[prometric spaces]]).

\begin{remark}
Short maps go by many names in the literature, such as _non-expansive functions_, _distance-nonincreasing maps_ and _weak contractions_.
\end{remark}

## Definition

\begin{definition}
A [[function]] $f\colon X \to Y$ is __short__ if $d_Y\bigl(f(a),f(b)\bigr) \leq d_X(a,b)$ for every $a,b \in X$ where $d_X$ and $d_Y$ are respectively the [[metrics]] (or generalisations of metrics) on $X$ and $Y$.
\end{definition}

If $X$ and $Y$ are (or may be) [[gauge spaces]] and so have several (pseudo)metrics on them, then we require that, for every gauging distance $d$ on $X$, there exists a gauging distance $e$ on $Y$ such that the above inequality holds for all $a$ and $b$.

## The category of metric spaces

We define $\mathsf{Met}$ to be the [[category]] whose [[objects]] are [[Lawvere metric spaces]] and whose [[morphisms]] are short maps.

$\mathsf{Met}$ is [[complete category|complete]] and admits a [[faithful functor]] from [[Ban]] (the category of [[Banach spaces]] and short [[linear operators]]).

$\mathsf{Met}$ may be made into an $\mathcal{M}$-[[M-category|category]] by taking short maps as the tight morphisms and some more generous notion of maps (such as [[continuous maps]]) as the loose morphisms.

If instead we use ordinary metric spaces, which includes the axioms of finiteness $d(x, y) \lt \infty$, symmetry $d(x, y) = d(y, x)$, and separability ($d(x, y) = 0$ implies $x = y$), then the category $\mathsf{Met}_{ord}$ (with short maps as morphisms) is not so [[dichotomy between nice objects and nice categories|nice]] as $\mathsf{Met}$. For example, $\mathsf{Met}_{ord}$ fails to have arbitrary products $\prod_{i \in I} X_i$, on account of the finiteness axiom where the putative distance

$$d\bigl((x_i), (y_i)\bigr) = \sup_i d(x_i, y_i)$$

may not exist as a finite number. However, if all the ordinary metric spaces $X_i$ are uniformly bounded in diameter, then this formula does give the product. Note well though that the topology induced by this product will not be the same as the product topology (cf. the discussion below).

### Use for denotational semantics

$\mathsf{Met}$ is used to give a [[denotational semantics]] to Fuzz, a [[functional programming language|functional language]] with a [[linear type]] system.

## Justification

There are many other kinds of maps between metric spaces; [[continuous maps]] and [[uniformly continuous map]]s are more general, while [[isometries]] and contractions are more restrictive.  What\'s so special about short maps that we consider them the proper morphisms between metric spaces?

One answer is to look at [[Lawvere]]\'s characterisation of metric spaces as certain [[enriched categories]]; see [[Lawvere metric space]].  Then the short maps are precisely the [[enriched functors]] between metric spaces.

Another answer is to consider what the category-theoretic [[isomorphisms]] between metric spaces are; by the definition of metric spaces as [[structured sets]], these are the [[global isometries]].  So for a good notion of morphism, we need to recover global isometries as isomorphisms.  Using continuous or uniformly continuous maps, we recover [[homeomorphisms]] or uniform homeomorphisms as isomorphisms, which are too general; this really gives us the category of [[metrisable topological spaces]] or of [[metrisable]] [[uniform spaces]] rather than the category of metric spaces.  Using contractions, we do not even get a category; the [[identity function]] is not a contraction.  We could still use global isometries themselves as morphisms, but since this defines a [[groupoid]], we should look for a more general notion of morphism that still gives global isometries as isomorphism.  And short maps do that.

Short maps give the [[Met|category of metric spaces]] some nice properties.  In particular, $\mathsf{Met}$ is [[complete category|complete]], which does not hold using either global isometries or distance-preserving maps as morphisms.  This interacts with the properties of the category of [[Banach spaces]]; as a Banach space may be defined as a set with compatible vector-space and metric-space structures, so a Banach space morphism is a function that is both linear and short: the [[short linear maps]].

## Injective objects in $\mathsf{Met}_{ord}$

In $\mathsf{Met}_{ord}$ every object $X$ admits an [[injective hull]] $\varepsilon(X)$. The space $\varepsilon(X)$ is [[compact topological space|compact]] if $X$ is [[compact topological space|compact]]. Furthermore every [[compact topological space|compact metric space]] has an __injective boundary__ that is the smallest subspace $A$ of $X$ such that $\varepsilon(A) = X$.

## Related pages

* [[Lipschitz map]]
* [[metric space]]

## References

* Arthur Azevedo de Amorim, Marco Gaboardi, Justin Hsu, Shin-ya Katsumata and Ikram Cherigui, _A Semantic Account of Metric Preservation_ (2017) ([arXiv:1702.00374](https://arxiv.org/abs/1702.00374)).

Injective objects $\mathsf{Met}_{ord}$ have been studied in

* Aronszajn, Nachman, and Prom Panitchpakdi. "Extension of uniformly continuous transformations and hyperconvex metric spaces." _Pacific Journal of Mathematics_ 6.3 (1956): 405-439.

* Isbell, John R. "Six theorems about injective metric spaces." _Commentarii Mathematici Helvetici_ 39.1 (1964): 65-76.

* [[Anton Petrunin]], "Lectures on metric geometry" 2020, [PDF](https://anton-petrunin.github.io/metric-geometry/tex/lectures.pdf).

An overview of results is included in

* Culbertson, Jared, Dan P. Guralnik, and Peter F. Stiller. _Injective metrizability and the duality theory of cubings_, Expositiones Mathematicae, ***37*** 4 (2019) 410-453 ([arXiv: 1502.00126](https://arxiv.org/abs/1502.00126)).

[[!redirects short map]]
[[!redirects short maps]]
[[!redirects short function]]
[[!redirects short functions]]
[[!redirects distance-nonincreasing map]]
[[!redirects distance-nonincreasing maps]]
[[!redirects distance-nonincreasing function]]
[[!redirects distance-nonincreasing functions]]
[[!redirects weak contraction]]
[[!redirects weak contractions]]
[[!redirects contractive map]]
[[!redirects contractive maps]]
[[!redirects non-expansive map]]
[[!redirects non-expansive maps]]
[[!redirects non-expansive function]]
[[!redirects non-expansive functions]]

[[!redirects category of metric spaces]]
[[!redirects the category of metric spaces]]
[[!redirects Met]]


short-range entanglement

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### Context
#### Solid state physics
+-- {: .hide}
[[!include solid state physics -- contents]]
=--
#### Topological physics
+--{: .hide}
[[!include topological physics -- contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In [[solid state physics]], where the nature of [[quantum materials]] depends crucially on the [[entanglement]] of their [[ground states]],
it has been proposed ([Kitaev 11](#Kitaev11), [13](#Kitaev13)) that "*short-range entanglement*" (SRE) in a quantum material means or implies that the "ground state is non-degenerate", hence that the [[Hilbert space]] of [[quantum states]] with the same lowest [[energy]] has [[dimension of a vector space|dimension]] $1$. See also [Chen, Gu & Wen 10, Sec. II](#ChenGuWen2010).

If such an SRE material is in a [[topological phase of matter]], with the ground state separated by an [[energy]] gap from the excited states, then this means ([Freed 14](#Freed14), cf. [G & JF 19](#GJF19), [ftn 2](https://arxiv.org/pdf/1712.07950.pdf#page=4)) that the ([[extended topological field theory|extended]]) [[topological field theory]] which describes the [[topological phase of matter|topological phase]] is an *[[invertible topological field theory]]*, hence one also speaks of an *invertible topological phase*.

In particular, short-range entanglement thus implies the *absence* of "[[topological order]]" (at least by two of three common definitions of the latter, see [there](topological+order#RelationBetweenDefiningConditions)).

In contrast, *[[long-range entanglement]]* in the ground state is supposedly the hallmark of [[topological order]] ([Chen, Gu & Wen 10, Sec. V](#ChenGuWen2010))

## Related concepts

* [[entanglement entropy]]

* [[long-range entanglement]]

## References

### Short-range entanglement

The notion was introduced (only) in a series of talks:

* {#Kitaev11} [[Alexei Kitaev]], *Toward Topological Classification of Phases with Short-range Entanglement*, talk at KITP (2011) $[$[video](https://online.kitp.ucsb.edu/online/topomat11/kitaev/)$]$

* {#Kitaev13} [[Alexei Kitaev]], *On the Classification of Short-Range Entangled States*, talk at Simons Center (2013) $[$[video](https://scgp.stonybrook.edu/archives/7874)$]$

Formalization in terms of [[invertible topological field theories]]:

* {#Freed14} [[Daniel S. Freed]], *Short-range entanglement and invertible field theories* $[$[arXiv:1406.7278](https://arxiv.org/abs/1406.7278) $]$

See also around [footnote 2](https://arxiv.org/pdf/1712.07950.pdf#page=4) of

* {#GJF19} [[Davide Gaiotto]], [[Theo Johnson-Freyd]], *Symmetry Protected Topological phases and Generalized Cohomology*, J. High Energ. Phys. **2019** 7 (2019) $[$[arXiv:1712.07950](https://arxiv.org/abs/1712.07950), [doi:10.1007/JHEP05%282019%29007](https://doi.org/10.1007/JHEP05%282019%29007)  $]$

### Long-range entanglement

The opposite notion of *[[long-range entanglement]]*:

* {#ChenGuWen2010} [[Xie Chen]], [[Zheng-Cheng Gu]], [[Xiao-Gang Wen]], Section V of: _Local unitary transformation, long-range quantum entanglement, wave function renormalization, and topological order_ Phys. Rev. B **82** 155138 (2010) $[$[arXiv:1004.3835](http://arxiv.org/abs/1004.3835)$]$

Further discussion:

* Tsung-Cheng Lu, Sagar Vijay, *Characterizing Long-Range Entanglement in a Mixed State Through an Emergent Order on the Entangling Surface* $[$[arXiv:2201.07792](https://arxiv.org/abs/2201.07792)$]$

[[!redirects short range entanglement]]


Shoshichi Kobayashi

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

## Selected writings

On [[differential geometry]]:

* [[Shoshichi Kobayashi]], [[Katsumi Nomizu]], _Foundations of differential geometry_, Volume 1 (1963), Volume 2 (1969) Interscience Publishers, reprinted by Wiley Classics Library (1996) [[ISBN:978-0-470-55558-3](https://www.wiley.com/en-us/Foundations+of+Differential+Geometry%2C+2+Volume+Set-p-9780470555583), [Wikipedia entry](https://en.wikipedia.org/wiki/Foundations_of_Differential_Geometry)]

On [[differential geometry]] with emphasis on [[G-structures]];

* [[Shoshichi Kobayashi]], _Transformation Groups in Differential Geometry_ 1972, reprinted as: Classics in Mathematics Vol. 70, Springer 1995 ([doi:10.1007/978-3-642-61981-6](https://link.springer.com/book/10.1007/978-3-642-61981-6))

## Related $n$Lab entries

* [[differential geometry]]

* [[G-structure]]

category: people


Shotaro Shiba

* [webpage](https://www.ias.edu/scholars/shotaro-shiba)

## related $n$Lab entries

* [[M2-brane]], [[M5-brane]]

category: people


Shou-Cheng Zhang

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

* [[Biao Lian]] et al (eds). *Memorial Volume for Shoucheng Zhang*, World Scientific (2021) [[doi:10.1142/12146](https://doi.org/10.1142/12146)]

## Selected writings

Predicting a [[quantum spin Hall effect]] in [[topological insulators]] based on mercury telluride (HgTe quantum wells):

* [[Andrei Bernevig]], [[Shou-Cheng Zhang]], _Quantum Spin Hall Effect_, Phys. Rev. Lett. 96, 106802 (2006) ([arXiv:cond-mat/0504147](https://arxiv.org/abs/cond-mat/0504147), [doi:10.1103/PhysRevLett.96.106802](https://doi.org/10.1103/PhysRevLett.96.106802))

* [[Andrei Bernevig]], [[Taylor Hughes]], [[Shou-Cheng Zhang]], _Quantum spin Hall effect and topological phase transition in HgTe quantum wells_, Science 15 December 2006: __314__, n. 5806, pp. 1757-1761 ([doi:10.1126/science.1133734](http://dx.doi.org/10.1126/science.1133734), [arXiv:cond-mat/0611399](https://arxiv.org/abs/cond-mat/0611399))

On [[topological phases of matter]]:

* {#WangZhang17} Jing Wang, [[Shou-Cheng Zhang]], *Topological states of condensed matter*, Nature Materials **16** (2017) 1062–1067 [[doi:10.1038/nmat5012](https://doi.org/10.1038/nmat5012)]

On [[Chern-Simons theory]] potentially applicable to a form of [[Berry curvature]] on the [[Brillouin torus]] of 3d [[topological semimetals]], with nodal lines playing the role of [[Wilson lines]]:

* [[Biao Lian]], [[Cumrun Vafa]], [[Farzan Vafa]], [[Shou-Cheng Zhang]], *Chern-Simons theory and Wilson loops in the Brillouin zone*, Phys. Rev. B **95** (2017) 094512 [[doi:10.1103/PhysRevB.95.094512](https://doi.org/10.1103/PhysRevB.95.094512)]

category: people

[[!redirects Shoucheng Zhang]]


Shozo Uehara

* [inspire page](https://inspirehep.net/authors/985313)

## Selected writings

On [[M-brane]] [[sigma-models]] on [[exceptional geometry|exceptional]] [[exceptional generalized geometry|generalized geometric]] [[target spacetimes]]:

* [[Yuho Sakatani]], [[Shozo Uehara]], _Branes in Extended Spacetime: Brane Worldvolume Theory Based on Duality Symmetry_, Phys. Rev. Lett. **117** 191601 (2016) [[arXiv:1607.04265](https://arxiv.org/abs/1607.04265), [talk slides](http://www2.yukawa.kyoto-u.ac.jp/~qft.web/2016/slides/sakatani.pdf)]

* [[Yuho Sakatani]], [[Shozo Uehara]], _Exceptional M-brane sigma models and $\eta$-symbols_ [[arXiv:1712.10316](https://arxiv.org/abs/1712.10316)]

category: people


Shrawan Kumar

* [webpage](http://www.unc.edu/math/Faculty/kumar/)

## related $n$Lab entries

* [[loop group]]

* [[flag variety]]

category: people


Shreya Vardhan

* [Institute page](https://physics.mit.edu/faculty/shreya-vardhan/)

## Selected writings

On [[quantum error correcting codes]] and [[black hole entropy]]:

* [[Chris Akers]], [[Netta Engelhardt]], [[Daniel Harlow]], [[Geoff Penington]], [[Shreya Vardhan]], *The black hole interior from non-isometric codes and complexity* [[arXiv:2207.06536](https://arxiv.org/abs/2207.06536)]

category: people


shrinkable map

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#### Topology
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[[!include topology - contents]]
=--
=--
=--

**shrinkable map** $\Rightarrow$ [[Dold fibration]]

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

## Definition

A _shrinkable map_ $p:X \to Y$ in the [[category]] [[Top]] is a map with a section $s:Y \to X$ such that $s\circ p:X \to X$ is vertically homotopic to $id_X$ (i.e. is homotopic to $\id_X$ in the slice category $Top/Y$). In particular, a shrinkable map is a homotopy equivalence.

## Properties

Every shrinkable map is a [[Dold fibration]]. This result follows from a theorem of Dold about [[locally homotopy trivial map]]s being the same as Dold fibrations. It should be obvious that a shrinkable map is globally homotopy trivial, with trivial fibre.

## Examples

**Example:**(Segal) Let $U_i \to Y$ be a [[numerable open cover]]. Then the [[geometric realization]] of the [[Cech nerve]] $N\check{C}(U_i)$ comes with a canonical map $|N\check{C}(U_i)| \to Y$ which is shrinkable.

There are extensions of this to other categories with a notion of homotopy.

## References

This definition is due to Dold, in his 1963 _Annals_ paper _Partitions of unity in the theory of fibrations_.

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