Urs Schreiber

***

$\,$

#contents#
* table of contents
{:toc}

## position

I work in

the [Mathematics Division of Science](https://nyuad.nyu.edu/en/academics/divisions/science/academic-programs/math.html)

of [New York University in Abu Dhabi](https://nyuad.nyu.edu/en/)

in the group of Prof. [[Hisham Sati]] leading the

[Research Center for Quantum and Topological Systems](https://nyuad.nyu.edu/en/research/faculty-labs-and-projects/cqts.html)

(see at *[[CQTS]]* for latest activity).

* [page at NYU AD](https://nyuad.nyu.edu/en/research/faculty-labs-and-projects/cqts/urs-schreiber.html)

## contact

* <us13@nyu> or <urs.schreiber@gmail.com>

## research

I am researching mathematical and fundamental physics (e.g. on *[[Hypothesis H]]* and *[[schreiber:Quantum Certification via Linear Homotopy Types|Quantum Systems]]*).

For latest talk notes see [here](http://ncatlab.org/schreiber/show/differential+cohomology+in+a+cohesive+topos#References).

## selected talks

* **[[schreiber:Topological Quantum Gates from M-Theory|Topological Quantum Gates from M-Theory]]**

  [talk at](M-Theory+and+Mathematics#Schreiber2023): *[M-Theory and Mathematics 2023](M-Theory+and+Mathematics#2023)*

  video: [YT](https://www.youtube.com/watch?v=Ob9tOgWumPI)

> $\phantom{------}$ (logo adapted from [JMP **62** (2021) 042301](https://arxiv.org/abs/2002.11093))

\linebreak

* **[[schreiber:Topological Quantum Programming in TED-K]]**

  [talk at](https://planqc2022.hotcrp.com/paper/33): *[PlanQC 2022](https://icfp22.sigplan.org/home/planqc-2022)*

  video: [YT](https://www.youtube.com/watch?v=hDwXQVcloB8&feature=emb_logo)

* **[[schreiber:Topological Quantum Gates in Homotopy Type Theory|Topological Quantum Gates in Homotopy Type Theory]]**

  [talk at](https://ncatlab.org/nlab/show/Center+for+Quantum+and+Topological+Systems#SchreiberMar2023): *[QFT and Cobordism 2023](https://ncatlab.org/nlab/show/Center+for+Quantum+and+Topological+Systems#ConferencesMar2023)*

  video: [YT](https://www.youtube.com/watch?v=pu5bpJ263X0)

* **[[schreiber:Higher Topos Theory in Physics]]**

  [talk at](https://ncatlab.org/nlab/show/Center+for+Quantum+and+Topological+Systems#SchreiberOct2023): *[Workshop on Homotopy Theory and Applications](https://ncatlab.org/nlab/show/Center+for+Quantum+and+Topological+Systems#oct_2023)*

  video: [YT](https://youtu.be/wNhoAiITJQs)

  and talk at: [Wolfram Science Winter School 2024](https://education.wolfram.com/winter-school/)

  video: [YT](https://youtu.be/GD20W6vxMI4)

* **[Topological Quantum Programming with Linear Homotopy Types](/schreiber/show/Topological+Quantum+Gates+in+Homotopy+Type+Theory#HoTTEST2024)**

  talk at: *[Homotopy Type Theory Electronic Seminar,](https://www.uwo.ca/math/faculty/kapulkin/seminars/hottest.html)*, 02 Feb 2024

  video: [YT](https://youtu.be/Wnm3yCUzNb0)

## publications

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

* & [on the arXiv](https://arxiv.org/find/math/1/au:+Schreiber_U/0/1/0/all/0/1) preprint server

* & those about *[Hypothesis H](https://ncatlab.org/nLab/show/Hypothesis%20H#References)*

recent monograph:

* **[[schreiber:The Character Map in Non-Abelian Cohomology]]**

  (with [[Domenico Fiorenza|D. Fiorenza]] & [[Hisham Sati|H. Sati]])

  World Scientific (2023)

  [[doi:10.1142/13422](https://doi.org/10.1142/13422)]

> {#Blurb} **Cover blurb:** This book presents a novel development of fundamental and fascinating aspects of [[algebraic topology]] and [[mathematical physics]]: “[[Whitehead-generalized cohomology|extra-ordinary]]” and further [[generalized cohomology theories]] enhanced to "[[twisted cohomology|twisted]]" and [[differential cohomology|differential]]-geometric form with focus on their [[rational homotopy theory|rational approximation]] by [[Chern-Dold character|generalized Chern character]] maps and on the resulting [[charge quantization]] laws in [[higher gauge theory|higher $n$-form gauge field theories]] appearing in [[string theory]] and in the [[K-theory classification of topological phases of matter|classification of]] [[topological phase of matter|topological]] [[quantum materials]].

> Motivation for the conceptual re-development is the observation, laid out in the introductory chapter, that famous and famously elusive effects in strongly interacting (“[[non-perturbative physics|non-perturbative]]”) [[quantum field theory|physics]] demand "[[non-abelian cohomology|non-abelian]]” generalization of much of established [[generalized cohomology theory]]. But the relevant higher [[non-abelian cohomology]] theory ("[[principal infinity-bundle|higher gerbes]]") has an esoteric reputation and has remained underdeveloped.

> The book's theme is that variously [[generalized cohomology theories]] are [[cohomology|best viewed]] through their [[classifying spaces]] -- possibly but not necessarily [[infinite-loop spaces]] -- from which perspective the [[Chern-Dold character|character map]] is really an incarnation of the [[fundamental theorem of dg-algebraic rational homotopy theory|fundamental theorem of rational homotopy theory]], thereby uniformly subsuming not only the classical [[Chern character]] and a multitude of scattered variants that have been proposed, but now seamlessly applying in the previously elusive generality of ([[twisted cohomology|twisted]], [[differential cohomology|differential]] and) [[non-abelian cohomology]].

> In laying out this result with plenty of examples, we provide modernized introduction and review of fundamental classical topics: 1. abstract [[homotopy theory]] via [[model categories]], 2. [[generalized cohomology]] in [[homotopy theory|homotopical]] incarnation, 3. [[dg-algebra|dg-algebraic]] [[rational homotopy theory]], whose [[fundamental theorem of dg-algebraic rational homotopy theory|fundamental theorem]] we re-cast as a (twisted) non-abelian [[de Rham theorem]] which naturally induces the (twisted) non-abelian character map.

* further monographs on the [[nLab]]:

  * _[[schreiber:Introduction to Homological algebra]]_

  * _[[Introduction to Stable homotopy theory]]_

  * [[geometry of physics -- categories and toposes]]

* other:

  * _[[schreiber:Higher Prequantum Geometry]]_, in: _[[nLab:New Spaces for Mathematics and Physics]]_, Cambr. Univ. Press (2021) $[$[doi:10.1017/9781108854429](https://doi.org/10.1017/9781108854429)$]$

  * *[[schreiber:Mathematical Foundations of Quantum Field and Perturbative String Theory]]*, Proceedings of Symposia in Pure Mathematics **83** Amer. Math. Soc (2011) $[$[ISBN:978-0-8218-5195-1](http://bookstore.ams.org/PSPUM-83)$]$

## research wiki

I like to put math/phys information into context.
 For that purpose I had started (in Nov 2008)

* *[[Home Page|the $n$Lab research wiki project]]*

with edit logs and discussion being had on:

* [the nForum discussion board](https://nforum.ncatlab.org/).

For more on the $n$Lab, see:

* me on: _[[schreiber:What is... the nLab|What is... the nLab?]]_

  (presentation at: *[Big Data in Pure Mathematics 2022](http://math.bu.edu/people/matschke/BigDataPureMath2022/)*)

* Wikipedia on: _[nLab](http://en.wikipedia.org/wiki/NLab)_.

## more web logs

For logs of further activity see:

* [my feed at Twitter](https://twitter.com/UrsSchreiber) (new)

* [my old feed at Twitter](https://twitter.com/SchreiberUrs) (lost my password...)

* [my channel on YouTube](https://www.youtube.com/@UrsSchreiber)

while [[nLab|$n$Lab]]-edits are announced

* *[on the $n$Forum](https://nforum.ncatlab.org/)*.

For links to technical discussions about math and physics see my pages:

* *[at MathOverflow](http://mathoverflow.net/users/381/urs-schreiber)*

* *[at PhysicsOverflow](http://www.physicsoverflow.org/user/Urs+Schreiber)*

* *[at Physics.SE](http://physics.stackexchange.com/users/5603/urs-schreiber)*

* *[at QuantumComputing.SE](https://quantumcomputing.stackexchange.com/users/19363/urs-schreiber).

## teaching

See behind the links for detailed lecture notes that I wrote:

| date |  lecture notes |
|------|--------|
| winter $\,\,$ 2017 | _[[schreiber:Mathematical Quantum Field Theory]]_ |
| summer 2017 | _[[schreiber:Topological K-Theory]]_ |
| summer 2017 | _[[Introduction to Topology|Topology]]_ |
| winter $\,\,$ 2016 | _[[schreiber:From the Superpoint to T-Folds|Super Cartan Geometry]]_ |
| summer 2016 | _[[Introduction to Stable homotopy theory -- S|Complex oriented cohomology theory]]_ |
| summer 2016 | _[[Introduction to Stable homotopy theory|Stable Homotopy Theory]]_ |
| summer 2015 | _[[schreiber:Structure Theory for Higher WZW Terms]]_ |
| summer 2015 | _[[schreiber:Higher Cartan Geometry]]_  |
| summer 2014 | _[[schreiber:HAI|Homological Algebra]]_ |

A list of further teaching in the past is _[here](http://ncatlab.org/schreiber/show/HomePage#Teaching)_.

## column
 {#column}

I used to write an irregular column at [PhysicsForums Insights](https://www.physicsforums.com/insights/). Articles in the series include these:

**on pre-quantum field theory**

* _[Higher prequantum geometry I: The need for prequantum geometry](https://www.physicsforums.com/insights/higher-prequantum-geometry-need-prequantum-geometry/)_

* _[Higher prequantum geometry II: The principle of extremal action -- Comonadically](https://www.physicsforums.com/insights/higher-prequantum-geometry-ii-principle-extremal-action-comonadically/#toggle-id-1)_

* _[Higher prequantum geometry III: The global action functional -- Cohomologically](https://www.physicsforums.com/insights/higher-prequantum-geometry-iii-global-action-functional-cohomologically/)_

* _[Higher prequantum geometry IV: The covariant phase space -- Transgressively](https://www.physicsforums.com/insights/higher-prequantum-geometry-iv-covariant-phase-space-transgressively/)_

* _[Higher prequantum geometry V: The local observables -- Lie theoretically](https://www.physicsforums.com/insights/higher-prequantum-geometry-v-local-observables-lie-theoretically/)_

* _[Examples of Prequantum field theories I: Gauge fields](https://www.physicsforums.com/insights/examples-prequantum-field-theories-gauge-fields/)_

* _[Examples of Prequantum field theories II: Higher gauge fields](https://www.physicsforums.com/insights/examples-prequantum-field-theories-ii-higher-gauge-fields/)_

* _[Examples of Prequantum field theories III: Chern-Simons-type theories](https://www.physicsforums.com/insights/examples-prequantum-field-theories-iii-chern-simons-type-theories/)_

* _[Examples of Prequantum field theories IV: Wess-Zumino-Witten-type theories](https://www.physicsforums.com/insights/examples-prequantum-field-theories-iv-wess-zumino-witten-type-theories/)_

**on perturbative quantum field theory**

* _[Introduction to Perturbative Quantum Field Theory](https://www.physicsforums.com/insights/paqft-idea-references/)_

**on string theory**

* _[It was 20 years ago today -- the M-theory conjecture](https://www.physicsforums.com/insights/20-years-ago-today-m-theory-conjecture/)_

* _[Homotopy Lie n-algebras in Supergravity](https://www.physicsforums.com/insights/homotopy-lie-n-algebras-supergravity/)_

* _[Emergence from the superpoint](https://www.physicsforums.com/insights/emergence-from-the-superpoint/)_

* _[Why supersymmetry? Because of Deligne's theorem](https://www.physicsforums.com/insights/supersymmetry-delignes-theorem/)_

* _[11d Gravity from just the Torsion Constraint](https://www.physicsforums.com/insights/11d-gravity-just-torsion-constraint/)_

* _[Spectral Standard Model and String Compactifications](https://www.physicsforums.com/insights/spectral-standard-model-string-compactifications/)_

* _[Why Higher Category Theory in Physics?](https://www.physicsforums.com/insights/higher-category-theory-physics/)_

* _[Super p-Brane Theory from Super Homotopy Theory](http://www.physicsforums.com/insights/super-p-brane-theory-emerging-super-homotopy-theory/)_

## copyright statement
 {#CopyrightStatement}

> To the extent that it matters, my contributions to the [[nLab:HomePage|nLab]] are copyrighted according to [CC BY-SA 3.0](https://creativecommons.org/licenses/by-sa/3.0/).

\linebreak
\linebreak

***

\linebreak
\linebreak

category: people

[[!redirects Urs]]


Urs Stammbach

* [webpage](https://people.math.ethz.ch/~stammb/)

## Selected writings

On [[homological algebra]]:

* {#HiltonStammbach71} [[Peter Hilton]],  [[Urs Stammbach]], _A course in homological algebra_, Springer-Verlag, New York, 1971, Graduate Texts in Mathematics, Vol. 4 ([doi:10.1007/978-1-4419-8566-8](https://link.springer.com/book/10.1007/978-1-4419-8566-8), [pdf](https://web.math.rochester.edu/people/faculty/doug/otherpapers/hilton-stammbach.pdf))

category: people


Urtzi Buijs

* [Mathematics Genealogy page](http://www.genealogy.ams.org/id.php?id=97618)

## Selected writngs

On [[rational models of mapping spaces]]:

* [[Urtzi Buijs]], [[Aniceto Murillo]], _Basic constructions in rational homotopy theory of function spaces_,  Annales de l'Institut Fourier, Volume 56 (2006) no. 3, p. 815-838 ([doi:10.5802/aif.2201](https://doi.org/10.5802/aif.2201))

* [[Urtzi Buijs]], [[Aniceto Murillo]], _The rational homotopy Lie algebra of function spaces_, Comment. Math. Helv. 83 (2008), 723–739 ([pdf](https://pdfs.semanticscholar.org/d404/657ccc24b0c06434086485d3e528e0316e26.pdf))

* {#BuijsFelixMurillo12} [[Urtzi Buijs]], [[Yves Félix]], [[Aniceto Murillo]], _$L_\infty$-rational homotopy of mapping spaces_,  published as _$L_\infty$-models of based mapping spaces_,  J. Math. Soc. Japan Volume 63, Number 2 (2011), 503-524 ([arXiv:1209.4756](https://arxiv.org/abs/1209.4756), [doi:10.2969/jmsj/06320503](https://doi.org/10.2969/jmsj/06320503))

On [[Whitehead L-infinity algebras|Whitehead $L_\infty$-algebras]] in [[rational homotopy theory]]:

* [[Francisco Belchí]], [[Urtzi Buijs]], [[José M. Moreno-Fernández]], [[Aniceto Murillo]], _Higher order Whitehead products and $L_\infty$ structures on the homology of a DGL_, Linear Algebra and its Applications, **520** (2017) 16-31 [[arXiv:1604.01478](https://arxiv.org/abs/1604.01478), [doi:10.1016/j.laa.2017.01.008](https://doi.org/10.1016/j.laa.2017.01.008)]

On [[rational homotopy theory]] with general [[fundamental groups]]:

* [[Urtzi Buijs]], [[Yves Félix]], [[Aniceto Murillo]], [[Daniel Tanré]], _Homotopy theory of complete Lie algebras and Lie models of simplicial sets_, Journal of Topology (2018) 799-825 ([arXiv:1601.05331](https://arxiv.org/abs/1601.05331), [doi:10.1112/topo.12073](https://doi.org/10.1112/topo.12073))

## related $n$Lab entries

* [[rational homotopy theory]]

* [[L-infinity algebra]]

* [[Whitehead product]]

category: people


Urysohn metrization theorem

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

#Contents#
* table of contents
{: toc}

## Statement and proof

[[!include Urysohn metrization theorem - main theorem]]

The **proof** can be divided into three parts. Recall that "second-countable" means having a [[countable set|countable]] [[topological base|base]]. All topological spaces considered here are assumed to be [[T1|$T_1$]] (points are closed), so that "[[regular topological space|regular]]" means "regular [[Hausdorff space|Hausdorff]]".

1. A [[regular space]] with a [[countable set|countable]] [[topological base|base]] is [[normal space|normal]].

1. A normal space $X$ is a [[completely regular space]], i.e., given a point $x \in X$ and an open set $U \subseteq X$ containing $x$, there exists a continuous function $f: X \to [0, 1]$ such that $f(x) = 1$ and $f(y) = 0$ for $y \notin U$.

1. A completely regular space $X$ with a countable base can be embedded in the [[Hilbert cube]] $[0, 1]^\mathbb{N}$. Since $[0, 1]^\mathbb{N}$ is metrizable, so is $X$.

The second assertion being proved at [[Urysohn lemma]], we prove the first and third assertions.

+-- {: .num_prop}
###### Proposition
A regular space $X$ with a countable base is normal.
=--

+-- {: .proof}
###### Proof
Let $A, B$ be disjoint [[closed sets]] of $X$. The collection of [[open sets]] $U$ such that $U \cap A$ is inhabited and $\widebar{U} \cap B = \emptyset$ is, by regularity, an open covering of $A$. By second-countability, we may index it as $U_1, U_2, \ldots$. Similarly, there is a countable open covering $V_1, V_2, \ldots$ of $B$ such that $\widebar{V_k} \cap A = \emptyset$.

Now form open sets $Y_n \coloneqq U_n \cap \bigcap_{k=1}^n \neg \widebar{V_k}$ and $Z_n = V_n \cap \bigcap_{k=1}^n \neg \widebar{U_k}$. It is clear that the $Y_n$ cover $A$ and the $Z_n$ cover $B$. Moreover, $Y_m \cap Z_n = \emptyset$ for all $m, n$. For if $m \geq n$ say, then

$$Y_m \cap Z_n \subseteq Y_m \cap V_n \subseteq Y_m \cap \widebar{V_n} \subseteq U_m \cap \neg \widebar{V_n} \cap \widebar{V_n} = \emptyset.$$

It follows that $\bigcup_m Y_m$ and $\bigcup_n Z_n$ are disjoint open sets containing $A$ and $B$, respectively. This completes the proof.
=--

+-- {: .num_prop}
###### Proposition
A completely regular space $X$ with countable base can be embedded in $[0, 1]^\mathbb{N}$.
=--

+-- {: .proof}
###### Proof
$X$ has a countable base $\mathcal{B}$. The set $S = \{(U, V) \in \mathcal{B} \times \mathcal{B}: \widebar{U} \subseteq V\}$ is countable. For each $s = (U, V) \in S$ there is by the Urysohn lemma a continuous map $g_s: X \to [0, 1]$ such that $g_s$ is identically $1$ on $\widebar{U}$ and identically $0$ on $\neg V$. The map

$$g: X \to [0,1]^S: x \mapsto (s \mapsto g_s(x))$$

is a continuous injection, since if $x \neq y$, we can find a pair $s = (U, V) \in S$ with $x \in U$ and $y \notin V$, so that $g_s(x) = 1$ differs from $g_s(y) = 0$, whence $g(x) \neq g(y)$. The subspace topology on $X$ induced from the monomorphism $g$ is contained in the given topology of $X$, simply by continuity of $g$. On the other hand, if $W$ is an open neighborhood of $x$ in $X$, there exists a smaller open neighborhood $V \in \mathcal{B}$ and $s  = (U, V) \in S$ such that $g_s(x) = 1$ and $g_s$ is identically $0$ outside $V$. Provided that $g_s(x) - g_s(y) \lt 1$, we see $g_s(y) \neq 0$, so $y \in V$. In other words, letting $\pi_s: [0, 1]^S \to [0, 1]$ be the evident projection, we have $g_s = \pi_s \circ g$, so that the inverse image under $g$ of the subbase element $\pi_s^{-1}((0, 1])$ is $g_s^{-1}((0, 1]) \subseteq V \subseteq W$. This shows that the subspace topology induced by $g$ contains the topology of $X$. It follows that $g: X \to [0, 1]^S \cong [0, 1]^\mathbb{N}$ is an embedding into the [[Hilbert cube]].
=--

## Related theorems

* [[metrization theorem]]
* [[Nagata-Smirnov metrization theorem]]

## Related concepts

* [[topological space]]

* [[metrisable topological space]]

* [[second-countable space]]

## References

Named after [[Pavel Urysohn]].

Textbook accounts:

* {#Kelley55} [[John Kelley]], Ch. 4, Th. 16 in: *General topology*, D. van Nostrand, New York (1955), reprinted as: Graduate Texts in Mathematics, Springer (1975) [[ISBN:978-0-387-90125-1](https://www.springer.com/gp/book/9780387901251)]

* {#Munkres75} [[James Munkres]], Ch. 4, Thm 34.1 _Topology_, Prentice Hall (1975, 2000) [ISBN:0-13-181629-2, [pdf](http://mathcenter.spb.ru/nikaan/2019/topology/4.pdf)]

* [[Richard E. Hodel]], *The Alexandroff-Urysohn metrization theorem revisited*, in: *Set-Theoretic Topology*, Academic Press (1977) 239-253 [[doi:10.1016/B978-0-12-584950-0.50021-3](https://doi.org/10.1016/B978-0-12-584950-0.50021-3)

See also:

* ProofWiki, *[Urysohn's_Metrization_Theorem](https://proofwiki.org/wiki/Urysohn%27s_Metrization_Theorem)*


Urysohn metrization theorem - main theorem

\begin{theorem}
Every [[second-countable space|second-countable]] [[regular Hausdorff space|regular Hausdorff]] [[topological space|space]] $X$ is [[metrizable space|metrizable]]
\end{theorem}


Urysohn topological space

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

#Contents#
* table of contents
{:toc}

## Definition

An [[topological space]] is called an _Urysohn space_ if it satisfies the [[separation axiom]] $T_{2\tfrac{1}{2}}$ which demands that for very pair of distinct points in the space they have [[closed neighbourhoods]] which are disjoint.

## Related concepts

* [[Kolmogorov topological space]]

* [[Hausdorff topological space]]

## References

* Wikipedia, _[Urysohn and completely Hausdorff spaces](https://en.wikipedia.org/wiki/Urysohn_and_completely_Hausdorff_spaces)_

[[!redirects Urysohn topological spaces]]

[[!redirects Urysohn space]]
[[!redirects Urysohn spaces]]


Urysohn's lemma

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

#Contents#
* table of contents
{:toc}

## Idea

_Urysohn's lemma_ (prop. \ref{UrysohnLemma} below) states that on a [[normal topological space]] disjoint [[closed subsets]] may be separated by [[continuous functions]] in the sense that a continuous function exists which takes value 0 on one of the two subsets and value 1 on the other (called an "Urysohn function", def. \ref{UrysohnFunction}) below. In fact the  existence of such functions is equivalent to a space being normal (remark \ref{Equivalence} below).

Urysohn's lemma is a key ingredient for instance in the proof of the [[Tietze extension theorem]] and in the proof of the existence of [[partitions of unity]] on [[paracompact topological spaces]].  See the list of implications [below](#Implications).

## Statement

+-- {: .num_defn #UrysohnFunction}
###### Definition
**(Urysohn function)**

Let $X$ be a [[topological space]], and let $A,B \subset X$ be disjoint [[closed subsets]]. Then an _Urysohn function_ for this situation is

* a [[continuous function]] $f \colon X \to [0,1]$

to the [[closed interval]] equipped with its [[Euclidean space|Euclidean]] [[metric topology]], such that

* it takes the value 0 on $A$ and the value 1 on $B$:

  $$
    f(A) = \{0\}
    \phantom{AAA}
    \text{and}
    \phantom{AAA}
    f(B) = \{1\}
    \,.
  $$

=--

+-- {: .num_prop #UrysohnLemma}
###### Proposition
**(Urysohn's lemma)**

Assuming [[excluded middle]] then:

Let $X$ be a [[normal topological space|normal]] (or $T_4$) [[topological space]], and let $A,B \subset X$ be two disjoint [[closed subsets]] of $X$. Then there exists an Urysohn function (def. \ref{UrysohnFunction}).

=--

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

Beware that the function in prop. \ref{UrysohnLemma} may take the values 0 or 1 even outside of the two subsets. The condition that the function takes value 0 or 1, respectively, _precisely_ on the two subsets corresponds to _[[perfectly normal spaces]]_.

=--

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

It is immediate that, conversely, the existence of an Urysohn function (def. \ref{UrysohnFunction}) implies that the topological space is [[normal topological space|normal]]. For let $A, B \subset X$ be disjoint closed subsets, and consider a continuous function $f \colon X \to [0,1]$ with $f(A) = \{0\}$ and $f(B) = \{1\}$ then

$$
  U_A \coloneqq f^{-1}([0,1/3)
  \phantom{AAA}
  U_B \coloneqq f^{-1}((2/3,1])
$$

are disjoint open neighbourhoods of these subsets.

Hence Urysohn's lemma shows that a topological space being normal is _equivalent_ to it admitting Urysohn functions.

=--

## Proof
 {#Proof}

+-- {: .proof}
###### Proof
of Urysohn's lemma, prop. \ref{UrysohnLemma}

Set

$$
  C_0 \coloneqq A
  \phantom{AAA}
  U_1 \coloneqq X \backslash B
  \,.
$$

Since by assumption

$$
  A \cap B = \emptyset
  \,.
$$

we have

$$
  C_0 \subset U_1
  \,.
$$

Notice that (by [this lemma](separation+axioms#T4InTermsOfTopologicalClosures)) if a space is normal then every open neighbourhood $U \supset C$ of a closed subset $C$ contains a smaller neighbourhood $V$ together with its closure $Cl(V)$

$$
  C \subset V \subset Cl(V) \subset U
  \,.
$$

Apply this fact successively to the above situation to obtain the following infinite sequence of nested open subsets $U_r$ and closed subsets $C_r$

$$
  \array{
    C_0 &&  &&  &&  &\subset&  &&  &&  && U_1
    \\
    C_0 &&  &\subset&  && U_{1/2} &\subset& C_{1/2} &&  &\subset&  && U_1
    \\
    C_0 &\subset& U_{1/4} &\subset& C_{1/4} &\subset& U_{1/2} &\subset& C_{1/2} &\subset& U_{3/4} &\subset& C_{3/4} &\subset& U_1
  }
$$

and so on, labeled by the [[dyadic rational numbers]] $\mathbb{Q}_{dy} \subset \mathbb{Q}$ within $(0,1]$

$$
 \{ U_{r} \subset X \}_{r \in (0,1] \cap \mathbb{Q}_{dy}}
$$

with the property

$$
  \underset{r_1,r_2 \in (0,1] \cap \mathbb{Q}_{dy}}{\forall}
  \left(
    \left(
       r_1 \lt r_2
    \right)
     \Rightarrow
    \left(
      U_{r_1} \subset Cl(U_{r_1}) \subset U_{r_2}
    \right)
  \right)
  \,.
$$

Define then the function

$$
  f \;\colon\; X \longrightarrow [0,1]
$$

to assign to a point $x \in X$ the [[infimum]] of the labels of those open subsets in this sequence that contain $x$:

$$
  f(x)
     \coloneqq
  \underset{U_r \supset \{x\}}{\lim} r
$$

Here the [[limit of a net|limit]] is over the [[directed set]] of those $U_r$ that contain $x$, ordered by reverse inclusion.

This function clearly has the property that $f(A) = \{0\}$  and $f(B) = \{1\}$. It only remains to see that it is continuous.

To this end, first observe that

$$
  \array{
    (\star)
    &&
    \left(
      x \in Cl(U_r)
    \right)
      &\Rightarrow&
    \left(
       f(x) \leq r
    \right)
    \\
    (\star\star)
    &&
    \left(
      x \in U_r
    \right)
      &\Leftarrow&
    \left(
      f(x) \lt r
    \right)
  }
  \,.
$$

Here it is immediate from the definition that $(x \in U_r) \Rightarrow (f(x) \leq r)$ and that $(f(x) \lt r) \Rightarrow (x \in U_r \subset Cl(U_r))$. For the remaining implication, it is sufficient to observe that

$$
  (x \in \partial U_r) \Rightarrow (f(x) = r)
  \,,
$$

where $\partial U_r \coloneqq Cl(U_r) \backslash U_r$ is the [[boundary]] of $U_r$.

This holds because the [[dyadic numbers]] are [[dense subset|dense]] in $\mathbb{R}$. (And this would fail if we stopped the above decomposition into $U_{a/2^n}$-s at some finite $n$.) Namely, in one direction, if $x \in \partial U_r$ then for every small positive real number $\epsilon$ there exists a dyadic rational number $r'$ with $r \lt r' \lt r + \epsilon$, and by construction $U_{r'} \supset Cl(U_r)$ hence  $x \in U_{r'}$. This implies that $\underset{U_r \supset \{x\}}{\lim} = r$.

{#PreimagesOfTheSubbaseOpens} Now we claim that for all $\alpha \in [0,1]$ then

1. $f^{-1}(\,(\alpha, 1]\,) = \underset{r \gt \alpha}{\cup} \left( X \backslash Cl(U_r) \right)$

1. $f^{-1}(\,[0,\alpha)\,) = \underset{r \lt \alpha}{\cup} U_r$

Thereby $f^{-1}(\,(\alpha, 1]\,)$ and $f^{-1}(\,[0,\alpha)\,)$ are exhibited as unions of open subsets, and hence they are open.

Regarding the first point:

$$
  \begin{aligned}
     & x \in f^{-1}( \,(\alpha,1]\, )
     \\
     \Leftrightarrow\,
     &
     f(x) \gt \alpha
     \\
     \Leftrightarrow\,
     &
     \underset{r \gt \alpha}{\exists} (f(x) \gt r)
     \\
     \overset{(\star)}{\Rightarrow}\,
     &
     \underset{r \gt \alpha}{\exists}
     \left( x \notin Cl(U_r) \right)
     \\
     \Leftrightarrow\,
     &
     x  \in \underset{r \gt \alpha}{\cup} \left(X \backslash Cl(U_r)\right)
  \end{aligned}
$$

and

$$
  \begin{aligned}
     & x  \in \underset{r \gt \alpha}{\cup} \left(X \backslash Cl(U_r)\right)
     \\
     \Leftrightarrow\,
     &
     \underset{r \gt \alpha}{\exists}
     \left( x \notin Cl(U_r) \right)
     \\
     \Rightarrow\,
     &
     \underset{r \gt \alpha}{\exists}
     \left( x \notin U_r \right)
     \\
     \overset{(\star \star)}{\Rightarrow}\,
     &
     \underset{r \gt \alpha}{\exists}
     \left(
       f(x) \geq r
     \right)
     \\
     \Leftrightarrow\,
     &
     f(x) \gt \alpha
     \\
     \Leftrightarrow\,
     &
     x \in f^{-1}(\, (\alpha,1] \,)
  \end{aligned}
  \,.
$$

Regarding the second point:

$$
  \begin{aligned}
    &
    x \in f^{-1}(\, [0,\alpha) \,)
    \\
    \Leftrightarrow\,
    &
    f(x) \lt \alpha
    \\
    \Leftrightarrow\,
    &
    \underset{r \lt \alpha}{\exists}( f(x) \lt r )
    \\
    \overset{(\star \star)}{\Rightarrow}\,
    &
    \underset{r \lt \alpha}{\exists }( x \in U_r )
    \\
    \Leftrightarrow\,
     &
    x \in \underset{r \lt \alpha}{\cup} U_r
  \end{aligned}
$$

and

$$
  \begin{aligned}
     &
    x \in \underset{r \lt \alpha}{\cup} U_r
    \\
    \Leftrightarrow\,
    &
    \underset{r \lt \alpha}{\exists }( x \in U_r )
    \\
    \overset{}{\Rightarrow}\,
    &
    \underset{r \lt \alpha}{\exists }( x \in Cl(U_r) )
    \\
    \overset{(\star)}{\Rightarrow}\,
     &
    \underset{r \lt \alpha}{\exists }( f(x) \leq r )
    \\
    \Leftrightarrow\,
     &
    f(x) \lt \alpha
    \\
    \Leftrightarrow\,
    &
    x \in f^{-1}(\, [0,\alpha) \,)
  \end{aligned}
 \,.
$$

(In these derivations we repeatedly use that $(0,1] \cap \mathbb{Q}_{dy}$ is dense in $[0,1]$, and we use the [[contrapositions]] of $(\star)$ and $(\star \star)$, by [[excluded middle]].)

Now since the subsets $\{ [0,\alpha), (\alpha,1]\}_{\alpha \in [0,1]}$
form a [[topological subbase|sub-base]] for the Euclidean metric topology on $[0,1]$, it follows that all pre-images of $f$ are open, hence that $f$ is continuous.

=--

## Implications
 {#Implications}

Urysohn's lemma is key in the proof of many other theorems, for instance

* [[Tietze extension theorem]]

* [[paracompact Hausdorff spaces equivalently admit subordinate partitions of unity]]

## Related statements

* [[paracompact Hausdorff spaces are normal]]

## References

Due to [[Pavel Urysohn]].

* [[Terence Tao]], _[245B, Notes 12: Continuous functions on locally compact Hausdorff spaces](https://terrytao.wordpress.com/2009/03/02/245b-notes-12-continuous-functions-on-locally-compact-hausdorff-spaces/#more-1844)_

* [Proof at planetmath](http://planetmath.org/proofofurysohnslemma)

* Wikipedia, _[Urysohn's lemma](https://en.wikipedia.org/wiki/Urysohn%27s_lemma)_

Lectures notes include

* Tarun Chitra, section 2.1 of _The Stone-Cech Compactification_ 2009 [pdf](httP://www.math.cornell.edu/~riley/Teaching/Topology2009/essays/chitra.pdf)

[[!redirects Urysohn's extension theorem]]
[[!redirects Urysohn's theorem]]
[[!redirects Urysohn theorem]]
[[!redirects Urysohn lemma]]

[[!redirects Urysohn function]]
[[!redirects Urysohn functions]]


Utsav Choudhury

**Utsav Choudhury** is an Indian [[algebraic geometry|algebraic geometer]], specialized in [[motives]].

* [web](http://utsavchoudhury.weebly.com/)

* [[motives]]
  * [[motivic Galois groups]]
* [[motivic homotopy theory]]


Uwe Jannsen

* [Wikipedia entry](http://de.wikipedia.org/wiki/Uwe_Jannsen)

category: people


Uwe Petersen

* [personal page](https://asfpg.de/17303/petersen.html)

> Seine Hoffnung geht tatsächlich so weit, dass seine Arbeit die Metaphysik (in der Form spekulativer Logik) den Philosophen aus der Hand nehmen und in die Mathematik verlagern wird.

> [Engl.] His hope actually goes as far that his work will take metaphysics (in the form of speculative logic) out of the hands of the philosophers, relocating it into mathematics.

* [online publications](https://independent.academia.edu/UwePetersen)

## Selected writings

On a proposal for a [[formal logic|logical formalization]] of [[Hegel|Hegelian]] [[dialectics]]:

* {#Petersen09} [[Uwe Petersen]], *35 Thesen zur Grundlegung einer formalen dialektischen Logik — nebst Kommentaren*, DILEMMATA: Jahrbuch der ASFPG **4** (2009) 125–207 [[pdf](https://asfpg.de/english/17303/Jahrbuecher/JB09/jb09-petersen.pdf)]

* [[Uwe Petersen]], _Logic Without Contraction as Based on Inclusion and Unrestricted Abstraction_, Studia Logica 64, 365–403 (2000). [doi:10.1023/A:1005293713265](https://doi.org/10.1023/A:1005293713265)

His _opus magnum_:

* {#Petersen02} [[Uwe Petersen]], _[[Diagonal Method and Dialectical Logic]] 3 vols._, Der Andere Verlag, Osnabrück (2002).

category: people


Uwe Semmelmann

* [webpage](http://www.mi.uni-koeln.de/~semmelma/)

## Selected writings

On [[quaternion-Kähler manifolds]]:

* {#Moroianu} [[Andrei Moroianu]], [[Uwe Semmelmann]], _Killing Forms on Quaternion-Kähler Manifolds_, Annals of Global Analysis and Geometry, November 2005, Volume 28, Issue 4, pp 319–335 ([arXiv:math/0403242](https://arxiv.org/abs/math/0403242), [doi:10.1007/s10455-005-1147-y](https://doi.org/10.1007/s10455-005-1147-y))

category: people


V-manifold

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

> under construction

#Contents#
* table of contents
{:toc}

## Idea

In the [[axiom|axiomatics]] of [[differential cohesion]] one may [[synthetic differential geometry|synthetically]] formulate a concept of _[[manifolds]]_ locally modeled on a [[group object]] $V$. In the [[interpretation]] in an [[differential cohesion|differentially]] [[cohesive (infinity,1)-topos]] these are [[étale infinity-groupoids]].

For exposition see at _[[geometry of physics -- manifolds and orbifolds]]_ and _[[geometry of physics -- supergeometry]]_.

## $V$-Manifolds
 {#VManifolds}

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

Given $X,Y\in \mathbf{H}$ then a morphism $f \;\colon\; X\longrightarrow Y$ is a _[[local diffeomorphism]]_ if its naturality square of the [[infinitesimal shape modality]]

$$
  \array{
    X &\longrightarrow& \Im X
    \\
    \downarrow^{\mathrlap{f}} && \downarrow^{\mathrlap{\Im f}}
    \\
    Y &\longrightarrow& \Im Y
  }
$$

is a [[homotopy pullback]] square.

=--

Let now $V \in \mathbf{H}$ be given, equipped with the structure of a [[group]] ([[∞-group]]).

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

A _[[V-manifold]]_ is an $X \in \mathbf{H}$ such that there exists a _$V$-[[atlas]]_, namely a [[correspondence]] of the form

$$
  \array{
    && U
    \\
    & \swarrow && \searrow
    \\
    V && && X
  }
$$

with both morphisms being [[local diffeomorphisms]], def. \ref{LocalDiffeomorphismsInDifferentialCohesion}, and the right one in addition being an [[epimorphism]], hence an [[atlas]].

=--

## Frame bundles
 {#FrameBundles}

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

For $X \in \mathbf{H}$ an object and $x \colon \ast \to X$ a point, then we say that the _[[infinitesimal neighbourhood]] of_, or the _[[infinitesimal disk]] at_ $x$ in $X$ is the [[homotopy fiber]] $\mathbb{D}^X_x$ of the [[unit of a monad|unit]] of the [[infinitesimal shape modality]] at $x$:

$$
  \array{
    \mathbb{D}^X_x &\longrightarrow& X
    \\
    \downarrow && \downarrow
    \\
    \ast &\stackrel{x}{\longrightarrow}& \im X
  }
  \,.
$$

=--

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

For $X$ any object in differential cohesion, its _infinitesimal disk bundle_ $T_{inf} X \to X$ is the [[homotopy pullback]]

$$
  \array{
    T_{inf} X &\stackrel{ev}{\longrightarrow}& X
    \\
    \downarrow^{\mathrlap{p}} && \downarrow
    \\
    X &\longrightarrow& \Im X
  }
$$

of the [[unit of a monad|unit]] of its [[infinitesimal shape modality]] along itself.

=--

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

By the [[pasting law]], the [[homotopy fiber]] of the infinitesimal disk bundle, def. \ref{FormalDiskBundle},  over any point $x \in X$ is the infinitesimal disk $\mathbb{D}^X_x$ in $X$ at that point, def.\ref{InfinitesimalDisk}. Nevertheless, for general $X$ the infinitesimal disk bundle need not be an [[fiber ∞-bundle]] with typical fiber (the infinitesimal disks at different points need not be equivalent, and even if they are, the bundle need not be locally trivial). Below in prop. \ref{FormalDiskBundleOfRegularManifoldsTrivializesOverCover} we see that for $X$ a $V$-manifold modeled on a group object $V$, then its infinitesimal disk bundle is indeed an [[fiber ∞-bundle]], and hence is the [[associated ∞-bundle]] to some [[principal ∞-bundle]]. That principal bundle is the [[frame bundle]] of $X$.

=--

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

The [[Atiyah groupoid]] of $T_{inf} X$ is the [[jet groupoid]] of $X$.

=--

+-- {: .num_lemma #EtalePullbackOfFormalDiskBundleIsFormalDiskBundle}
###### Lemma

If $\iota \colon U \to X$ is a local diffeomorphism, def. \ref{LocalDiffeomorphismsInDifferentialCohesion}, then

$$
  \iota^\ast T_{inf} X \simeq T_{inf}U
  \,.
$$

=--

+-- {: .proof}
###### Proof

By the definition of local diffeos and using the [[pasting law]] we have an equivalence of [[pasting diagrams]] of [[homotopy pullbacks]] of the following form:

$$
  \array{
     \iota^\ast T_{inf} X &\longrightarrow& T_{inf} &\longrightarrow& X
     \\
     \downarrow && \downarrow && \downarrow
     \\
     U &\longrightarrow& X &\longrightarrow& \Im X
  }
  \;\;\;\;
  \simeq
  \;\;\;\;
  \array{
      T_{inf} U &\longrightarrow& U &\longrightarrow& X
       \\
       \downarrow && \downarrow && \downarrow
       \\
       U &\longrightarrow& \Im U &\longrightarrow& \Im X
  }
$$

=--

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

For $V$ an object, a **[[framing]]** on $V$ is a trivialization of its infinitesimal disk bundle, def. \ref{FormalDiskBundle}, i.e. an object $\mathbb{D}^V$ -- the typical [[infinitesimal disk]] or [[formal disk]], def. \ref{InfinitesimalDisk}, -- and a (chosen) [[equivalence]]

$$
  \array{
    T_{inf} V && \stackrel{\simeq}{\longrightarrow} && V \times \mathbb{D}^n
    \\
    & \searrow && \swarrow_{\mathrlap{p_1}}
    \\
    && V
  }
  \,.
$$

=--

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

For $V$ a framed object, def. \ref{Framing}, we write

$$
  GL(V) \coloneqq \mathbf{Aut}(\mathbb{D}^V)
$$

for the [[automorphism ∞-group]] of its typical [[infinitesimal disk]]/[[formal disk]].

=--

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

When the [[infinitesimal shape modality]] exhibits first-order infinitesimals, such that $\mathbb{D}(V)$ is the first order [[infinitesimal neighbourhood]] of a point, then $\mathbf{Aut}(\mathbb{D}(V))$ indeed plays the role of the [[general linear group]]. When $\mathbb{D}^n$ is instead a higher order or even the whole [[formal neighbourhood]], then $GL(n)$ is rather a [[jet group]]. For order $k$-jets this is sometimes written $GL^k(V)$ We nevertheless stick with the notation "$GL(V)$" here, consistent with the fact that we have no index on the [[infinitesimal shape modality]]. More generally one may wish to keep track of a whole tower of infinitesimal shape modalities and their induced towers of concepts discussed here.

=--

This class of examples of framings is important:

+-- {: .num_prop #DifferentialCohesiveInfinityGroupIsCanonicallyFramed}
###### Proposition

Every differentially cohesive [[∞-group]] $G$ is canonically framed (def. \ref{Framing}) such that the horizontal map in def. \ref{FormalDiskBundle} is given by the left action of $G$ on its [[infinitesimal disk]] at the neutral element:

$$
  ev \;\colon\; T_{inf}G \simeq G \times \mathbb{D}^G_e \stackrel{\cdot}{\longrightarrow} G
  \,.
$$

=--

+-- {: .proof}
###### Proof

By the discussion at _[[Mayer-Vietoris sequence]]_ in the section _[Over an ∞-group](Mayer-Vietoris%20sequence#OverAGroupObject)_ and using that the [[infinitesimal shape modality]] preserves group structure, the defining [[homotopy pullback]] of $T_{inf} G$, def. \ref{FormalDiskBundle}, is equivalent to the pasting of pullback diagrams

$$
  \array{
    T_{inf} G &\stackrel{}{\longrightarrow}&
    \mathbb{D}^G_e
    &\stackrel{}{\longrightarrow}& \ast
    \\
    \downarrow &&  \downarrow && \downarrow
    \\
    G \times G &\stackrel{(-)\cdot (-)^{-1}}{\longrightarrow}& G
    &\stackrel{}{\longrightarrow}&
    \Im G
  }
$$

where the right square is the defining pullback for the [[infinitesimal disk]] $\mathbb{D}^G$. Finally for the left square we find by [this proposition](Mayer-Vietoris%20sequence#HTTArgumentForPullback) that $T_{inf} G \simeq G\times \mathbb{D}^G$ and that the top horizontal morphism is as claimed.

=--

By lemma \ref{EtalePullbackOfFormalDiskBundleIsFormalDiskBundle} it follows that:

+-- {: .num_prop #FormalDiskBundleOfRegularManifoldsTrivializesOverCover}
###### Proposition

For $V$ a framed object, def. \ref{Framing}, let $X$ be a $V$-manifold, def. \ref{VManifold}. Then the infinitesimal disk bundle, def. \ref{FormalDiskBundle}, of $X$ canonically trivializes over any $V$-cover $V \leftarrow U \rightarrow X$ , i.e. there is a [[homotopy pullback]] of the form

$$
  \array{
     U \times \mathbb{D}^V &\longrightarrow& T_{inf} X
     \\
     \downarrow && \downarrow
     \\
     U &\longrightarrow& X
  }
  \,.
$$

This exhibits $T_{inf} X\to X$ as a $\mathbb{D}^V$-[[fiber ∞-bundle]].

=--

+-- {: .num_prop #ModulatingMapOfFormalDiskBundle}
###### Remark

By [this discussion](fiber+infinity-bundle#Properties) this fiber [[fiber ∞-bundle]] is the [[associated ∞-bundle]] of an essentially uniquely determined $\mathbf{Aut}(\mathbb{D}^V)$-[[principal ∞-bundle]] $Fr(X)$, i.e. there exists a [[homotopy pullback]] diagram of the form

$$
  \array{
     T_{inf} X \simeq
     &
   Fr(X) \underset{\mathbf{Aut}(\mathbb{D}^V)}{\times} \mathbb{D}^V
     &\longrightarrow&
     V//\mathbf{Aut}(\mathbb{D}^V)
     \\
     & \downarrow && \downarrow
     \\
     & X &\stackrel{}{\longrightarrow}& \mathbf{B}\mathbf{Aut}(\mathbb{D}^V)
  }
  \,.
$$

=--

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

Given a $V$-manifold $X$, def. \ref{VManifold}, for framed $V$, def. \ref{Framing}, then its _[[frame bundle]]_

$$
  \array{
     Fr(X)
     \\
     \downarrow
     \\
     X
  }
$$

is the $GL(V)$-[[principal ∞-bundle]] given by prop. \ref{FormalDiskBundleOfRegularManifoldsTrivializesOverCover} via remark \ref{ModulatingMapOfFormalDiskBundle}.

=--

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

As in remark \ref{OrderOfInfinitesimalDisks}, this really axiomatizes in general [[higher order frame bundles]] with the order implicit in the nature of the [[infinitesimal shape modality]].

=--

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

By prop. \ref{EtalePullbackOfFormalDiskBundleIsFormalDiskBundle} the construction of frame bundles in def. \ref{FrameBundleMap} is functorial in [[formally étale maps]] between $V$-manifolds.

=--

This provides all the necessary structure to now set up an axiomatic theory of [[G-structure]] and [[higher Cartan geometry]]. This is discussed further at _[[geometry of physics -- G-structure and Cartan geometry]]_.

## References

The concept is due to

* {#Schreiber} [[Urs Schreiber]], _[[schreiber:differential cohomology in a cohesive topos]]_

* [[Igor Khavkine]], [[Urs Schreiber]], _[[schreiber:Synthetic variational calculus|Synthetic geometry of differential equations Part I -- Jets and comonad structure]]_ ([arXiv:1701.06238](https://arxiv.org/abs/1701.06238))

Formalization in [[modal type theory|modal]] [[homotopy type theory]] is in

* [[Felix Wellen]], _[[schreiber:thesis Wellen|Formalizing Cartan Geometry in Modal Homotopy Type Theory]]_, 2017 ([pdf](http://www.math.kit.edu/iag3/~wellen/media/diss.pdf))

* [[Felix Wellen]], _Cartan Geometry in Modal Homotopy Type Theory_ ([arXiv:1806.05966](https://arxiv.org/abs/1806.05966))

[[!redirects V-manifolds]]


V_cb puzzle

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Fields and quanta
+--{: .hide}
[[!include fields and quanta - table]]
=--
#### Physics
+-- {: .hide}
[[!include physicscontents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In [[particle physics]], what is called the _$V_{cb}$-puzzle_ (status updates in [GJS 19](#GambinoJungSchacht19), [Ricciardi 21](#Ricciardi21)) is an [[observation]] about the [[semileptonic decay]] of [[B-mesons]] (by the [[Belle collaboration]] and other [[experiments]]) which implies a value of the element $\left\vert V_{c b}\right\vert$ in the [[CKM matrix]] that differs from the apparent prediction of the [[standard model of particle physics]] with a [[statistical significance]] of about $3 \sigma$. But the [[form factor (QFT)|form factors]] involved in this prediction suffer from theoretical uncertainties of [[quark]] contributions, and further [[lattice QCD]]-computation is expected to be required to clarify the actual prediction of the [[standard model of particle physics]] ([GJS 19, last sentence ](#GambinoJungSchacht19)).

The $V_{c b}$-puzzle is different from, but related to, the [[flavour anomalies]] ([GJS 19 (6)](#GambinoJungSchacht19))

## Related concepts

* [[flavor problem]]

* [[flavour anomaly]]

* [[Cabibbo anomaly]]

* [[(g-2)-anomaly]]

## References

* {#GambinoJungSchacht19} Paolo Gambino, Martin Jung, Stefan Schacht, _The $V_{c b}$ puzzle: an update_ ([arXiv:1905.08209](https://arxiv.org/abs/1905.08209))

* {#Ricciardi21} Giulia Ricciardi, _Semileptonic B decays and $\vert V_{x b}\vert$ update_ ([arXiv:2103.06099](https://arxiv.org/abs/2103.06099))

With emphasis on [[lattice QCD]]-computations:

* Matthew Wingate, _Quark flavor physics and lattice QCD_ ([arXiv:2103.17224](https://arxiv.org/abs/2103.17224))


vacua -- contents

**[[vacuum]]**

* [[vacuum state]], [[Hadamard state]]

* [[interacting vacuum]]

* [[vacuum expectation value]], [[vacuum amplitude]], [[vacuum fluctuation]]

* [[vacuum energy]]

* [[thermal vacuum]], [[KMS state]]

* [[vacuum diagram]]

* [[vacuum stability]]

* [[false vacuum]], [[tachyon]], [[Coleman-De Luccia instanton]]

* [[theta vacuum]]

* [[perturbative string theory vacuum]]

  * [[non-geometric string theory vacuum]]

  * [[landscape of string theory vacua]]


vacuum

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

#Contents#
* table of contents
{:toc}

## Idea

### In quantum field theory
 {#InQuantumFieldTheory}

In [[quantum field theory]] a _vacuum state_ is supposed to be a [[quantum state]] that expresses the absence of any [[particle]] excitations of the fields.

On [[Minkowski spacetime]] the _[[vacuum state]]_ for a [[free field theory]] is the standard [[Hadamard state]]. On general [[globally hyperbolic spacetimes]] there are always [[Hadamard states]], and they do play the role of the vacuum state in  the construction of [[AQFT on curved spacetimes]], see at _[[locally covariant perturbative AQFT]]_. Notably the choice of such a Hadamard state fixes the _[[Feynman propagator]]_, hence the [[time-ordered product]] of [[quantum observables]] and thus the [[perturbative quantum field theory|perturbative]] [[S-matrix]] away from coinciding interaction points (the [[extension of distributions|extension of these distributions]] to coinciding interaction points is the process of [[renormalization]]).

However, since on a general [[globally hyperbolic spacetime]] there is no globally well-defined concept of [[particles]], there is in general no concept of vacuum state. But under good conditions (such as existence of suitable [[timelike]] [[Killing vectors]]) one may identify Hadamard states which deserve to be thought of as vacuum states ([Brum-Fredenhagen 13](vacuum+state#BrumFredenhagen13)).

### In string theory

As mentioned [above](#InQuantumFieldTheory), in [[perturbative quantum field theory]] a choice of [[vacuum state]] (more generally: [[Hadamard state]]) defines the [[Feynman propagator]], hence the [[time-ordered product]] of [[quantum observables]] and thus the perturbative [[S-matrix]] away from coinciding interaction points and [[extension of distributions|extended]] to coincing interaction points via a choice of [[renormalization]].

Now [[perturbative string theory]] is not a [[local field theory]], in which the [[S-matrix]] is derived (up to choice of [[vacuum state]] and [[renormalization]]) from a [[local Lagrangian density]], but is _defined_ by _declaring_ the [[string perturbation series]] (obtained by other means, namely by [[2d CFT]] [[correlators]]) to be the [[scattering matrix]].

Hence while a vacuum in [[perturbative string theory]] cannot be a [[quantum state]] as in [[local field theory]] (since there is a priori just no [[free field]] [[Wick algebra]] that it could be a [[quantum state]] on), it makes sense to define a vacuum of perturvative string theory to be whatever data it takes to define the stringy [[S-matrix]], namely the [[string perturbation series]]. This datum is that of a [[2d SCFT]] (of [[central charge]] 15).

Hence in [[perturbative string theory]] the vacua of the [[effective QFT|effective background QFT]] are at the same time identified with certain 2-dimensional [[CFT]]s -- which for "geometric vacua" are the [[string]] [[sigma-models]] which have the given [[effective QFT]] as their [[second quantization]].

Perturbative string theory is _defined_ as the [[string perturbation series]] of these [[sigma-models]] about these vacua.

The [[moduli space]] of these vacua -- which is hardly understood -- has come to be called the _[[landscape of string theory vacua]]_ .

## Examples

* _[Higgs field -- Mass and Vacuum (in-)stability](Higgs+field#MassAndVacuumInstability)_

## Related concepts

* [[QFT with defects|topological defects]] due to [[spontaneous symmetry breaking]]

  * [[domain wall]], [[cosmic string]], [[monopole]]

* [[Yang-Mills instanton]]

  * [[BPST instanton]]

  * [[instanton in QCD]], [[QCD vacuum]]

* [string theory FAQ -- What does it mean to say that string theory has a "landscape of solutions"?](http://ncatlab.org/nlab/show/string+theory+FAQ#WhatDoesItMeanToSayStringTheoryHasALandscapeOfSolutions)

[[!include states and observables -- content]]

[[!redirects vacua]]


vacuum amplitude

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#### Algebraic Quantum Field Theory
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=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In [[quantum field theory]] (and [[string theory]]) the [[scattering amplitudes]] ([[string scattering amplitudes]]) "where nothing external scatters," hence for no incoming and no outgoing states, are called _[[vacuum]] amplitudes_.

## Properties

### As generating functionals for all other amplitudes

As functions of [[source]] fields, the vacuum amplitudes, or rather the _[[vacuum energy]]_, serve as the [[generating functionals]] for all [[correlators]]/[[n-point functions]] (e.g. [Scrucca, 1.6](#Scrucca)).

### One loop contribution and zeta functions
 {#OneLoopContributionAndZetaFunction}

The 1-loop vacuum amplitudes are [[regularization (physics)|regularized]] [[traces]] over [[Feynman propagators]]/[[Dirac propagators]]. These are the incarnations of [[special values of L-functions|special values]] of [[zeta functions]], [[L-functions]] and [[eta functions]] in [[physics]]:

[[!include zeta-functions and eta-functions and theta-functions and L-functions -- table]]

### Vanishing of the vacuum amplitude and supersymmetry

In the presence of [[supersymmetry]] 1-loop vacuum amplitudes are typically supposed to vanish.

For the [[type II superstring]], see e.g. ([Palti](#Palti)). For the [[heterotic superstring]] see e.g. [Han 89](#Han89).

In view of the [above](#OneLoopContributionAndZetaFunction) relation of 1-loop vacuum amplitudes to [[special values of L-functions]] such vanishing reminds one of the [[Riemann hypothesis]]. See ([ACER 11](#ACER11)).

## Related concepts

[[!include states and observables -- content]]

\linebreak

* [[Riemann hypothesis and physics]]

## References

### For particles

Discussions for [[particles]] includes

* {#Scrucca} Claudio Scrucca, _Advanced quantum field theory_ [pdf](http://itp.epfl.ch/webdav/site/itp/users/181759/public/aqft.pdf)

* Ori Yudilevich, _Calculating Massive One-Loop Amplitudes in QCD_, Utrecht 2009 ([pdf](http://testweb.science.uu.nl/ITF/teaching/2009/Ori%20Yudilevich.pdf))

* Robbert Rietkerk, _One-loop amplitudes in perturbative  quantum field theory_, Utrecht 2012 ([pdf](http://www.nikhef.nl/pub/theory/masters-theses/robbert_rietkerk.pdf))

### For strings

Lecture notes for 1-loop vacuum amplitudes for [[strings]] (vacuum [[string scattering amplitudes]]) include

* {#Edelstein13} [[José Edelstein]], _Lecture 8: 1-loop closed string vacuum amplitude_, 2013 ([pdf](http://www-fp.usc.es/~edels/Strings/Lect8Str.pdf))

* {#Palti} [[Eran Palti]], _The IIA/B superstring one-loop vacuum amplitude_ ([pdf](http://www.thphys.uni-heidelberg.de/~palti/Stringcourse/problemset11.pdf))

* {#Han89} Seung Kee Han, _Vanishing vacuum amplitude of four-dimensional heterotic string theory compactified on N=2 superconformal field theory_, Phys. Rev. D 39, 2322 – Published 15 April 1989 ([web](http://journals.aps.org/prd/abstract/10.1103/PhysRevD.39.2322))

And with relation to the [[Riemann hypothesis]]:

* {#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))

[[!redirects vacuum amplitudes]]

[[!redirects 1-loop vacuum amplitude]]
[[!redirects 1-loop vacuum amplitudes]]


vacuum diagram

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#### Algebraic Quantum Field Theory
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=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A [[Feynman diagram]] without external edges is called a _vacuum diagram_.

For example the simplest vacuum diagram obtained from the [[electron-photon interaction]] in [[quantum electrodynamics]] is

In the computation of [[scattering amplitudes]] in a [[vacuum stability|stable vacuum]] the contribution of the vacuum diagrams to the [[Feynman perturbation series]] cancels out.

For details see at _[[S-matrix]]_ [this section](S-matrix#VacuumDiagrams)

## Related concepts

* **[[vacuum]]**

  * [[vacuum state]], [[Hadamard vacuum state]]

  * [[interacting vacuum]]

  * [[vacuum expectation value]], [[vacuum amplitude]], [[vacuum fluctuation]]

  * [[vacuum energy]]

  * [[vacuum stability]]

  * [[false vacuum]], [[tachyon]], [[Coleman-De Luccia instanton]]

  * [[theta vacuum]]

  * [[perturbative string theory vacuum]]

  * [[landscape of string theory vacua]]

[[!redirects vacuum diagrams]]


vacuum energy

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#### Theta functions
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[[!include theta functions - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

Given a [[Feynman propagator]] $\frac{1}{H}$, then the corresponding _vacuum energy_ $Z$ is the [[logarithm]] of the [[functional determinant]] $det_{reg}$ of $H$

$$
  Z \coloneqq -log\,det_{reg} H
  \,.
$$

## Properties

### In terms of special values of the zeta function
 {#InTermsOfSpecialValuesOfZetaFunctions}

The vacuum energy is equivalently the [[special values of L-functions|special value]] of the  [[zeta function of an elliptic differential operator|zeta function]] $\zeta_H$ of $H$ given by the [[derivative]] at 0:

$$
  Z = -\frac{1}{2}\zeta_H^\prime(0)
  \,.
$$

See at _[zeta function of an elliptic differential operator -- Functional determinant](zeta+function+of+an+elliptic+differential+operator#FunctionalDeterminant)_ for more.

### In terms of the path integral and relation to generating function

Traditionally the vacuum energy is expressed in terms of a hypothetical [[path integral]]. (As opposed to the [above](#InTermsOfSpecialValuesOfZetaFunctions) zeta-function formalization this is not rigorous, but it serves to give the idea of _why_ this is the vacuum energy and the the zeta-function expression may be taken to be the rigorous definition of the path integral heuristics.)

By [[analogy]] with finite-dimensional [[Gaussian integrals]] (see at _[Feynman diagram -- For finitely many degrees of freedom](Feynman+diagram#ForFinitelyManyDegreesOfFreedom)_) one expects that the [[Wick rotation|Wick rotated]] [[vacuum amplitude]] version of the [[path integral]] (no [[field (physics)|field]] insertions, no [[boundary field theory|boundary conditions]]) is

$$
  \underset{\phi \in \mathbf{Fields}}{\int}
   \exp(- S_H(\phi))
  D\phi
  =
  (det_reg H)^{-1/2}
  \,.
$$

Therefore

$$
  log
    \underset{\phi \in \mathbf{Fields}}{\int}
   \exp(- S_H(\phi))
  D\phi
  =
  -\frac{1}{2}log\, det_{reg} H
$$

is the [[generating functional]] for [[n-point functions]]. (...)

e.g. ([Scrucca, section 1.6](#Scrucca), [Edelstein 13, page 2](#Edelstein13))

### As holomorphic potential for Determinant line bundle

Regard

$$
  h \coloneqq \frac{1}{2} det_{reg}H
$$

as a hermitian structure on a [[holomorphic line bundle]], hence, locally, as the [[absolute value]]-squared of the unit [[section]] $\phi_i$ of a [[holomorphic line bundle]] with respect to a local trivializing section (see at [[Chern connection]]).

$$
  h|_{U_i} = {\Vert \phi_i \Vert}^2
  \,.
$$

Then this line bundle is the [[determinant line bundle]] of $H$.
([Quillen 85](#Quillen85)), review includes ([Freed 87, p. 18](#Freed87), [Qiu 12, section 2.8.1](#Qiu12)).

The [[Chern connection]] is

$$
  A = \partial log(h) = \partial \frac{1}{2} det_{reg}H = \partial Z
$$

and the [[curvature]] [[differential 2-form]] is

$$
  F = i \bar \partial\partial Z
  \,.
$$

> eh? Something wrong with the factors of $1/2$ here...

## Applications

### The solution to the strong CP problem via axions

In the solution to the [[strong CP problem]] via [[axions]]
it is analysis of the dependency of the vacuum energy of [[Yang-Mills theory]] on the [[theta-angle]] $\theta$ which is argued to show that the [[axion]] expectation value $\theta = \langle a \rangle$ vanishes ([Vafa-Witten 84](#VafaWitten84)).

See at _[axion -- As a solution to the strong CP-problem](axion#AsASolutionToTheStrongCPProblem)_.

## Related concepts

[[!include states and observables -- content]]

## References

For instance

* {#Scrucca} Claudio Scrucca, section 1.6 in _Advanced quantum field theory_ [pdf](http://itp.epfl.ch/webdav/site/itp/users/181759/public/aqft.pdf)

* {#Edelstein13} [[José Edelstein]], page 2 of _Lecture 8: 1-loop closed string vacuum amplitude_, 2013 ([pdf](http://www-fp.usc.es/~edels/Strings/Lect8Str.pdf))

* {#Quillen85} [[Daniel Quillen]], _Determinants of Cauchy-Riemann Operators over a Riemann Surface_, Functional Anal.
Appl. 19 (1985) 31.

* {#Freed87} [[Daniel Freed]], _On determinant line bundles_, Math. aspects of [[string theory]], ed. S. T. Yau, World Sci. Publ. 1987,  (revised [pdf](http://www.math.utexas.edu/~dafr/Index/determinants.pdf), [dg-ga/9505002](http://arxiv.org/abs/dg-ga/9505002))

* {#Qiu12} Jia Qiu, section 2.8 of _Lecture notes on topological field theory_ [arXiv:1201.5550](http://arxiv.org/abs/1201.5550)

The application to the [[axion]] solution to the [[strong CP problem]] is due to

* {#VafaWitten84} [[Cumrun Vafa]], [[Edward Witten]], _Parity Conservation in Quantum Chromodynamics_ Phys. Rev. Lett. 53, 535 (1984) ([publisher](http://journals.aps.org/prl/abstract/10.1103/PhysRevLett.53.535))


vacuum expectation value

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#Contents#
* table of contents
{:toc}

## Idea

In [[quantum field theory]] the _vacuum expectation value_ (vev) of an [[observable]] $A$ is its value $\omega(A) \in \mathbb{C}$ ("[[expectation value]]") in a given [[vacuum state]] $\omega$.

## Related concepts

* [[condensate]]

* [[n-point function]]

\linebreak

[[!include propagators - table]]

\linebreak

[[!include states and observables -- content]]

## References

See also

* Wikipedia, _[Vacuum expectation value](https://en.wikipedia.org/wiki/Vacuum_expectation_value)_

[[!redirects vacuum expectation values]]

[[!redirects vev]]
[[!redirects vevs]]

[[!redirects VEV]]
[[!redirects VEVs]]


vacuum polarization

#Contents#
* table of contents
{:toc}

## Idea

(...)

## Related concepts

[[!include states and observables -- content]]

## References

See the references at _[[Schwinger effect]]_

See also

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

[[!include holographic Schwinger effect -- references]]

[[!redirects vacuum polarizations]]


vacuum stability

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=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In [[quantum field theory]], a given [[vacuum state]] is called _stable_ if in a suitable sense it does not evolve into or from any orthogonal state.

In [[perturbative quantum field theory]] with specified [[S-matrix]] $\mathcal{S}(g S_{int})$ it makes sense to say that a vacuum state is stable if there is vanishing [[quantum probability]] for the vacuum state to scatter into a non-vacuum state, or for a non-vacuum state to scatter into the vacuum state. (e.g. [Nikolic 01, p. 6](#Nikolic01))

## Definition

+-- {: .num_defn #VacuumStability}
###### Definition
**([[vacuum stability]])**

Let $(E_{\text{BV-BRST}}, \mathbf{L}', \Delta_H )$ be a [[relativistic field theory|relativistic]] [[free field theory|free]] [[vacuum]] according to ([this def.](S-matrix#VacuumFree)), let $\mathcal{S}$ be a corresponding [[S-matrix]] scheme according to ([this. def.](S-matrix#LagrangianFieldTheoryPerturbativeScattering)), and let $g S_{int} \in LocObs(E_{\text{BV-BRST}})[ [ \hbar, g] ]\langle g \rangle$ be a [[local observable]], regarded as an [[adiabatic switching|adiabatically switched]] [[interaction]] [[action functional]].

We say that the given [[Hadamard vacuum state|Hadamard]] [[vacuum state]] ([this prop.](Wick+algebra#WickAlgebraCanonicalState))

$$
  \langle - \rangle
  \;\colon\;
  PolyObs(E_{\text{BV-BRST}})_{mc}[ [ \hbar , g, j ] ]
  \longrightarrow
  \mathbb{C}[ [ \hbar, g, j ] ]
$$

is _[[vacuum stability|stable]]_ with respect to the [[interaction]] $S_{int}$, if for all elements of the [[Wick algebra]]

$$
  A \in PolyObs(E_{\text{BV-BRST}})_{mc}[ [ \hbar, g] ]
$$

we have

$$
  \left\langle
    A \mathcal{S}(g S_{int})
  \right\rangle
  \;=\;
  \left\langle
     \mathcal{S}(g S_{int})
  \right\rangle
  \,
  \left\langle
    A
  \right\rangle
  \phantom{AA}
  \text{and}
  \phantom{AA}
  \left\langle
    \mathcal{S}(g S_{int})^{-1}
    A
  \right\rangle
  \;=\;
  \frac{1}
  {
    \left\langle
      \mathcal{S}(g S_{int})
    \right\rangle
   }
   \left\langle
     A
   \right\rangle
$$

=--

## Properties

+-- {: .num_example #ScatteringAmplitudeFromInteractingFieldObservables}
###### Example
**([[scattering amplitudes]] as [[vacuum expectation values]] of [[interacting field observables]])**

Let $(E_{\text{BV-BRST}}, \mathbf{L}', \Delta_H )$ be a [[relativistic field theory|relativistic]] [[free field theory|free]] [[vacuum]] according to [this def.](S-matrix#VacuumFree), let $\mathcal{S}$ be a corresponding [[S-matrix]] scheme according to [this def.](S-matrix#LagrangianFieldTheoryPerturbativeScattering), and let $g S_{int} \in LocObs(E_{\text{BV-BRST}})[ [ \hbar, g ] ]\langle g \rangle$ be a [[local observable]] regarded as an [[adiabatic switching|adiabatically switched]] [[interaction]]-[[action functional|functional]],
such that the [[vacuum state]] is [[vacuum stability|stable]] with respect to $g S_{int}$ (def. \ref{VacuumStability}).

Consider [[local observables]]

$$
  \array{
    A_{in,1}, \cdots, A_{in , n_{in}},
    \\
    A_{out,1}, \cdots, A_{out, n_{out}}
  }
  \;\;\in\;\;
  LocObs(E_{\text{BV-BRST}})[ [ \hbar, g] ]
$$

whose spacetime support satisfies the following [[causal ordering]]:

$$
  A_{out, i_{out} }
  {\gt\!\!\!\!\lt}
  A_{out, j_{out}}
  \phantom{AAA}
  A_{out, i_{out} }
    {\vee\!\!\!\wedge}
  S_{int}
    {\vee\!\!\!\wedge}
  A_{in, i_{in}}
  \phantom{AAA}
  A_{in, i_{in} }
  {\gt\!\!\!\!\lt}
  A_{in, j_{in}}
$$

for all $1 \leq i_{out} \lt j_{out} \leq n_{out}$ and $1 \leq  i_{in} \lt j_{in} \leq n_{in}$.

Then the [[vacuum expectation value]] of the [[Wick algebra]]-product of the corresponding [[interacting field observables]] ([this def.](S-matrix#InteractingFieldObservables))
is

$$
  \begin{aligned}
   &
   \left\langle
     {\, \atop \,}
     (A_{out, 1})_{int}
     \cdots
     (A_{out,n_{out}})_{int}
     \,
     (A_{in, 1})_{int}
     \cdots
     (A_{in,n_{in}})_{int}
     {\, \atop \,}
   \right\rangle
   \\
   & =
   \left\langle
     {\, \atop \,}
       A_{out,1}
       \cdots
       A_{out,n_{out}}
    \right|
       \;
       \mathcal{S}(g S_{int})
       \;
    \left|
       A_{in,1}
       \cdots
       A_{in, n_{in}}
     {\, \atop \,}
   \right\rangle
   \\
   & \coloneqq
   \frac{1}{
     \left\langle \mathcal{S}(g S_{int}) \right\rangle
   }
   \left\langle
     {\, \atop \,}
       A_{out,1}
       \cdots
       A_{out,n_{out}}
       \;
       \mathcal{S}(g S_{int})
       \;
       A_{in,1}
       \cdots
       A_{in, n_{in}}
     {\, \atop \,}
   \right\rangle
   \,.
  \end{aligned}
$$

These [[vacuum expectation values]] are interpreted, in the [[adiabatic limit]] where $g_{sw} \to 1$, as _[[scattering amplitudes]]_ (see [this remark](S-matrix#FromAxiomaticSMatrixScatteringAmplitudes)).

=--

For **proof** see at _[[S-matrix]]_, [this prop.](S-matrix#ScatteringAmplitudeFromInteractingFieldObservables).

## Examples

* _[Higgs field -- Mass and Vacuum (in-)stability](Higgs+field#MassAndVacuumInstability)_

## Related concepts

[[!include states and observables -- content]]

## References

* {#Scharf96} [[Günter Scharf]], _Vacuum stability in quantum field theory_, Nuovo Cim. A109 (1996) 1605-1607 ([spire:432208](http://inspirehep.net/record/432208/))

Discussion for [[QED]]:

* {#Nikolic01} [[Hrvoje Nikolić]], _Physical stability of the QED vacuum_, 2001 ([arXiv:hep-ph/0105176](https://arxiv.org/abs/hep-ph/0105176))

Discussion for [[Higgs field]]:

* J.R. Espinosa, G. Giudice, A. Riotto, _Cosmological implications of the Higgs mass measurement_, JCAP 0805:002, 2008 ([arXiv:0710.2484](https://arxiv.org/abs/0710.2484))

* [[John Ellis]], J.R. Espinosa, G.F. Giudice, A. Hoecker, A. Riotto, _The Probable Fate of the Standard Model_, Phys. Lett. B679:369-375, 2009 ([arXiv:0906.0954](https://arxiv.org/abs/0906.0954))

* Dario Buttazzo, Giuseppe Degrassi, Pier Paolo Giardino, [[Gian Giudice]], Filippo Sala, Alberto Salvio, Alessandro Strumia, _Investigating the near-criticality of the Higgs boson_ ([arXiv:1307.3536](https://arxiv.org/abs/1307.3536))

* Anson Hook, John Kearney, Bibhushan Shakya, Kathryn M. Zurek, _Probable or Improbable Universe? Correlating Electroweak Vacuum Instability with the Scale of Inflation_, J. High Energ. Phys. (2015) 2015: 61 ([arXiv:1404.5953](https://arxiv.org/abs/1404.5953))

* Jose R. Espinosa, Gian F. Giudice, Enrico Morgante, Antonio Riotto, Leonardo Senatore, Alessandro Strumia, Nikolaos Tetradis, _The cosmological Higgstory of the vacuum instability_ ([arXiv:1505.04825](https://arxiv.org/abs/1505.04825))

* William E. East, John Kearney, Bibhushan Shakya, Hojin Yoo, Kathryn M. Zurek, _Spacetime Dynamics of a Higgs Vacuum Instability During Inflation_, Phys. Rev. D 95, 023526 (2017) ([arXiv:1607.00381](https://arxiv.org/abs/1607.00381))

* {#Kane18} [[Gordon Kane]], _Exciting Implications of LHC Higgs Boson Data_ ([arXiv:1802.05199](https://arxiv.org/abs/1802.05199))


vacuum state

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=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In [[quantum field theory]] a _vacuum state_ is supposed to be a [[quantum state]] that expresses the absence of any [[particle]] excitations of the fields.

On [[Minkowski spacetime]] the _vacuum state_ for a [[free field theory]] is the standard [[Hadamard state]]. On general [[globally hyperbolic spacetimes]] there are always [[Hadamard states]], and they do play the role of the vacuum state in  the construction of [[AQFT on curved spacetimes]], see at _[[locally covariant perturbative AQFT]]_. Notably the choice of such a Hadamard state fixes the _[[Feynman propagator]]_, hence the [[time-ordered product]] of [[quantum observables]] and thus the [[perturbative quantum field theory|perturbative]] [[S-matrix]] away from coinciding interaction points (the [[extension of distributions|extension of these distributions]] to coinciding interaction points is the process of [[renormalization]]).

However, since on a general [[globally hyperbolic spacetime]] there is no globally well-defined concept of [[particles]], there is in general no concept of vacuum state. But under good conditions (such as existence of suitable [[timelike]] [[Killing vectors]]) one may identify Hadamard states which deserve to be thought of as vacuum states ([Brum-Fredenhagen 13](#BrumFredenhagen13)).

## Related concepts

[[!include states and observables -- content]]

## References

* Stephen J. Summers, _Yet More Ado About Nothing: The Remarkable Relativistic Vacuum State_ ([arXiv:0802.1854] (http://xxx.uni-augsburg.de/abs/0802.1854))

* {#BrumFredenhagen13} Marcos Brum, [[Klaus Fredenhagen]], _"Vacuum-like" Hadamard states for quantum fields on curved spacetimes_ ([arXiv:1307.0482](https://arxiv.org/abs/1307.0482))

* {#Duetsch18} [[Michael Dütsch]], def. 2.12 in _[[From classical field theory to perturbative quantum field theory]]_, 2018

[[!redirects vacuum states]]


Vadim Gorin

* [personal page](https://people.math.wisc.edu/~vadicgor/)

## Selected writings

On [[character of a linear representation|characters]] on the [[symmetric group]] depending only on [[Cayley distance]]:

* {#GnedinGorinKerov11} [[Alexander Gnedin]], [[Vadim Gorin]], [[Sergei Kerov]], *Block characters of the symmetric groups*, Journal of Algebraic Combinatorics, 38, no. 1 (2013), 79-101 ([arXiv:1108.5044](https://arxiv.org/abs/1108.5044), [doi:10.1007/s10801-012-0394-9](https://doi.org/10.1007/s10801-012-0394-9))

category: people


Vadim Kaplunovsky

* [webpage](https://web2.ph.utexas.edu/~vadim/home.html)

## Selected writings

On [[M-theory on S1/G_HW times H/G_ADE]]:

* [[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 [[sigma-mesons]] and attractive [[nuclear forces]] between [[baryons]] in [[holographic QCD]] ([[Witten-Sakai-Sugimoto model]] on [[D4-D8 brane intersections]]):

* [[Vadim Kaplunovsky]], [[Jacob Sonnenschein]], _Searching for an Attractive Force in Holographic Nuclear Physics_, JHEP 05 (2011) 058 ([arXiv:1003.2621](https://arxiv.org/abs/1003.2621))

category: people


Vadim Knizhnik

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

## related entries

* [[Knizhnik-Zamolodchikov equation]]

category: people


Vadim Schechtman

* [webpage](http://www.math.univ-toulouse.fr/~schechtman/)

## Selected writings

On the [hypergeometric integral construction](Knizhnik-Zamolodchikov+equation#BraidRepresentationsViaTwisteddRCohomologyOfConfigurationSpaces) of solutions to the [[Knizhnik-Zamolodchikov equation]]:

* [[Vadim Schechtman]], [[Alexander Varchenko]], *Integral representations of N-point conformal correlators in the WZW model*,  Max-Planck-Institut für Mathematik, (1989) Preprint MPI/89- $[$[cds:1044951](http://cds.cern.ch/record/1044951)$]$

* [[Vadim Schechtman]], [[Alexander Varchenko]], *Hypergeometric solutions of Knizhnik-Zamolodchikov equations*, Lett. Math. Phys. **20** (1990) 279–283 $[$[doi:10.1007/BF00626523](https://doi.org/10.1007/BF00626523)$]$

* [[Vadim Schechtman]], [[Alexander Varchenko]], *Arrangements of hyperplanes and Lie algebra homology*, Inventiones mathematicae  **106** 1 (1991) 139-194 $[$[dml:143938](https://eudml.org/doc/143938), [[SchechtmanVarchenko_HyperplaneArrangements.pdf:file]]$]$

On the [hypergeometric integral construction](Knizhnik-Zamolodchikov+equation#BraidRepresentationsViaTwisteddRCohomologyOfConfigurationSpaces) of [[conformal blocks]] for the [[sl(n)|$\mathfrak{sl}(n)$]] [[WZW model]]:

* {#FeiginSchechtmanVarchenko90} [[Boris Feigin]], [[Vadim Schechtman]], [[Alexander Varchenko]], *On algebraic equations satisfied by correlators in Wess-Zumino-Witten models*,  Lett Math Phys **20** (1990)  291–297  $[$[doi:10.1007/BF00626525](https://doi.org/10.1007/BF00626525)$]$

* {#FeiginSchechtmanVarchenko94} [[Boris Feigin]], [[Vadim Schechtman]], [[Alexander Varchenko]], *On algebraic equations satisfied by hypergeometric correlators in WZW models. I*, Commun. Math. Phys. **163** (1994) 173–184 $[$[doi:10.1007/BF02101739](https://doi.org/10.1007/BF02101739)$]$

  > (for [[sl(2)|$\mathfrak{sl}(2, \mathbb{C})$]])

* [[Boris Feigin]], [[Vadim Schechtman]], [[Alexander Varchenko]], *On algebraic equations satisfied by hypergeometric correlators in WZW models. II*, Comm. Math. Phys. **170** 1 (1995) 219-247  $[$[euclid:cmp/1104272957](https://projecteuclid.org/journals/communications-in-mathematical-physics/volume-170/issue-1/On-algebraic-equations-satisfied-by-hypergeometric-correlators-in-WZW-models/cmp/1104272957.full)$]$

On [[chiral algebras]] in relation to [[quantum cohomology]]:

* [[Fyodor Malikov]], [[Vadim Schechtman]], *Deformations of chiral algebras and quantum cohomology of toric varieties* [[arXiv:math/0001170](https://arxiv.org/abs/math/0001170)]

## Related entries

* [[chiral de Rham complex]]

category: people


Vadim Vologodsky

* [web](http://pages.uoregon.edu/vvologod/)

[[!redirects V. Vologodsky]]

category:people


Vagn Lundsgaard Hansen

* [webpage](http://www2.mat.dtu.dk/people/V.L.Hansen/)

## Selected writings

On [[Cohomotopy]] [[cocycle spaces]] (i.e. [[spaces of maps]] into an [[n-sphere]]):

* {#Hansen74} [[Vagn Lundsgaard Hansen]], _The homotopy problem for the components in the space of maps on the $n$-sphere_, Quart. J. Math. Oxford Ser. (3) 25 (1974), 313-321 ([DOI:10.1093/qmath/25.1.313](https://doi.org/10.1093/qmath/25.1.313))

* {#Hansen81} [[Vagn Lundsgaard Hansen]], _On Spaces of Maps of $n$-Manifolds Into the $n$-Sphere_, Transactions of the American Mathematical Society
Vol. 265, No. 1 (May, 1981), pp. 273-281 ([jstor:1998494](https://www.jstor.org/stable/1998494))

category: people


Vaia Patta

* [Institute page](https://www.cs.ox.ac.uk/people/vaia.patta/)

## Selected writings

On the [[category]] of [[matroids]] with *strong maps* between them:

* [[Chris Heunen]], [[Vaia Patta]], *The category of matroids*, Applied Categorical Structures **26** (2018) 205–237 [[arXiv:1512.01390](https://arxiv.org/abs/1512.01390), [doi:10.1007/s10485-017-9490-2](https://doi.org/10.1007/s10485-017-9490-2)]

category: people


valence bundle

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Solid state physics
+-- {: .hide}
[[!include solid state physics -- contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

### General

For a [[crystal|crystalline material]], [[Bloch-Floquet theory]] shows that the [[Hilbert space]] [[space of quantum states|of quantum states]] of the [[electrons]] (in the [[background field|background]] of the [[atoms]]/[[nuclei]] at the crystal sites) is [[natural transformation|naturally]] the [[space of sections]] of a [[separable Hilbert space]]-bundle over the [[Brillouin torus]], and the [[Hamiltonian]] is the [[direct integral]] of [[Bloch Hamiltonians]] on its [[fibers]], whose [[eigenvalues]] constitute the [[electronic energy bands]].  If the [[chemical potential]] of the electrons lies in a gap between two such [[electronic energy bands]], then this Hilbert space bundle has a well-defined [[sub-bundle]] of Bloch states with energy below the chemical potential.

One says that the vectors in this vector bundle are the electron [[quantum states]] which are "filled" and one says that the vector bundle over the [[Brillouin torus]] which these form is the *valence bundle* of the system, and that the [[electronic band structure|band]] formed by (the [[graph of a function|graph]]) of the corresponding [[energy]] [[eigenvalues]] is the *valence band*.

The orthogonal complement of the valence bundle inside the ambient Bloch Hilbert bundle would be known as the *conduction bundle*, though this term is not widely used. But the (lowest) [[electronic band structure|band]] which is constituted by the corresponding [[energy]] [[eigenvalues]] is commonly known as the *conduction band*.

(graphics from [[schreiber:Topological Quantum Computation in TED-K|SS 22]])

\begin{remark}
Sometimes the valence bundle is referred to as a "Bloch bundle", eg. [Panati 06](#Panati06). This can be confusing, since the latter term would more naturally apply to the full Hilbert bundle of all Bloch states.
\end{remark}

\begin{remark}
For [[metals]] the chemical potential is not inside a gap, and a valence sub-bundle is not really well-defined, though the valence *band* is still defined. For [[semi-metals]] there is a well-defined valence bundle on the [[complement]] of its nodal points in the [[Brillouin torus]].
\end{remark}

### Classification

As a [[sub-bundle]] of the full a Hilbert space bundle of a quantum system, the valence bundle is naturally identified not up to [[isomorphism]] but up to ambient isomorphism, hence up to [[Murray-von Neumann equivalence]]. In fact, it has been argued to be physically classified by its class in [[topological K-theory]] (i.e. the [[Grothendieck group completion]] of the [[semigroup]] of Murray-von Neumann equivalence classes under [[direct sum of vector bundles]]), or rather by the [[twisted equivariant K-theory]] reflecting the material's [[quantum symmetries]]. This is the statement of the *[[K-theory classification of topological phases of matter]]*.

## Related concepts

* [[fiber bundle in physics]]

## References

The valence/conduction *bands* are discussed in any text on [[solid state physics]].

The valence *bundle* is sometimes called the *Bloch bundle*:

* {#Panati06} [[Gianluca Panati]], *Triviality of Bloch and Bloch-Dirac bundles*, Ann. Henri Poincaré **8** (2007) 995-1011. ([arXiv:math-ph/0601034](https://arxiv.org/abs/math-ph/0601034), [doi:10.1007/s00023-007-0326-8](https://doi.org/10.1007/s00023-007-0326-8))

For mathematical discussion of the actual valence *bundle* see:

* {#FreedMoore13} [[Daniel Freed]], [[Gregory Moore]], Prop. D.13 in: *Twisted equivariant matter*, Ann. Henri Poincaré (2013) 14: 1927 ([arXiv:1208.5055](https://arxiv.org/abs/1208.5055))

[[!redirects valence bundles]]

[[!redirects conduction bundle]]
[[!redirects conduction bundles]]

[[!redirects valence band]]
[[!redirects valence bands]]

[[!redirects conduction band]]
[[!redirects conduction bands]]


Valentin Goranko

* [personal page](https://www2.philosophy.su.se/goranko/)

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

## Selected writings

On [[semantics]]/[[model theory]] of [[modal logic]] ([[possible worlds semantics]], [[Kripke frames]]):

* [[Valentin Goranko]], [[Martin Otto]], *Model Theory of Modal Logic*, in Section 5 in: *Handbook of Modal Logic*, Studies in Logic and Practical Reasoning **3** (2007) 249-329 [[pdf](https://www2.mathematik.tu-darmstadt.de/~otto/papers/mlhb.pdf),  [book webpage](https://cgi.csc.liv.ac.uk/~frank/MLHandbook/), [publisher page](https://www.sciencedirect.com/bookseries/studies-in-logic-and-practical-reasoning/vol/3/suppl/C)]

category: people


Valentin Perrelle

* [dl.acm page](https://dl.acm.org/profile/81496689096)

## Selected writings

On [[dynamic lifting]] (of [[quantum measurement]]-results) in [[quantum programming]]/[[quantum computing]]:

* [[Dongho Lee]], Sebastien Bardin, [[Valentin Perrelle]], [[Benoît Valiron]], *Formalization of a Programming Language for Quantum Circuits with Measurement and Classical Control*, talk at *[Journées Informatique Quantique 2019](https://quantcert.github.io/GT-IQ'19/)* (Nov 2019) [[pdf](https://quantcert.github.io/GT-IQ%2719/slides/Lee.pdf), [[LBPV_QCWithClassicalControl.pdf:file]]]

* [[Dongho Lee]], [[Valentin Perrelle]], [[Benoît Valiron]], Zhaowei Xu, *Concrete Categorical Model of a Quantum Circuit Description Language with Measurement*, Leibniz International Proceedings in Informatics **213** (2021) 51:1-51:20 [[arXiv:2110.02691](https://arxiv.org/abs/2110.02691), [doi:10.4230/LIPIcs.FSTTCS.2021.51](https://doi.org/10.4230/LIPIcs.FSTTCS.2021.51)]

category: people


Valentin Zakharevich

* [personal page](https://math.jhu.edu/~vzakhar2/)

## Selected writings

On the [[Verlinde ring]] via [[twisted equivariant K-theory]]/[[orbifold K-theory]] ([[twisted ad-equivariant K-theory]]):

* [[Valentin Zakharevich]], _K-Theoretic Computation of the Verlinde Ring_, thesis 2018 ([hdl:2152/67663](http://hdl.handle.net/2152/67663), [pdf](http://www.math.jhu.edu/~vzakharevich/research/Dissertation.pdf), [[ZakharevichKTheoryAndVerlindeRing.pdf:file]])

category: people


Valentina B. Petkova

* [GoogleScholar page](https://scholar.google.com/citations?user=9TFxuUUAAAAJ&hl=en)

## Selected writings

On [[WZW models]] at [[rational number|fractional]] [[level (Chern-Simons theory)|level]]:

* P. Furlan, [[A. Ch. Ganchev]], R. Paunov, [[Valentina B. Petkova]], *Solutions of the Knizhnik-Zamolodchikov Equation with Rational Isospins and the Reduction to the Minimal Models*, Nucl. Phys. B394 (1993) 665-706 ([arXiv:hep-th/9201080](https://arxiv.org/abs/hep-th/9201080), doi:10.1016/0550-3213(93)90227-G)

* [[A. Ch. Ganchev]], [[Valentina B. Petkova]], G. M. T. Watts, *A note on decoupling conditions for generic level $\widehat{s l}(3)_k$ and fusion rules*, Nucl. Phys. B **571** (2000) 457-478 $[$doi:10.1016/S0550-3213(99)00745-2$]$

category: people

[[!redirects Valentina Petkova]]


Valentine Bargmann

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

## Selected writings

On [[Wigner's theorem]]:

* [[Valentine Bargmann]], _Note on Wigner's theorem on symmetry transformations_, Journal of Mathematical Physics **5** 7 (1964) 862-868 [[doi:10.1063/1.1704188](https://doi.org/10.1063/1.1704188), [pdf](http://materias.df.uba.ar/t2a2019c2/files/2012/07/Bergman-1964.pdf), [[Bargmann-WignerTheorem.pdf:file]]]

category: people


Valentine S. Kulikov

* [MathematicsGenealogy page](https://www.genealogy.math.ndsu.nodak.edu/id.php?id=119514)

## Selected writings

On [[Gauss-Manin connections]] and [[mixed Hodge structures]]:

* {#Kulikov98} [[Valentine S. Kulikov]], *Mixed Hodge Structures and Singularities*, Cambridge University Press (1998) $[$[doi:10.1017/CBO9780511758928](https://doi.org/10.1017/CBO9780511758928)$]$

category: people

[[!redirects Valentine Kulikov]]


Valentino Foit

* [spire page](https://inspirehep.net/authors/1395407?ui-citation-summary=true)

* [NYU Abu Dhabi Stochastics Seminar page](https://wp.nyu.edu/adps/people/)

## Selected writings

Relating the [[Brownian loop soup]] to [[conformal field theory]]:

* [[Federico Camia]], [[Valentino Foit]], [[Alberto Gandolfi]], [[Matthew Kleban]], *Scalar conformal primary fields in the Brownian loop soup* [[arXiv:2109.12116](https://arxiv.org/abs/2109.12116)]

* [[Federico Camia]], [[Valentino Foit]], [[Alberto Gandolfi]], [[Matthew Kleban]], *The Brownian loop soup stress-energy tensor* [[arXiv:2112.00074](https://arxiv.org/abs/2112.00074), doi:10.1007/JHEP11(2022)009]

Review:

* [[Valentino Foit]], *Brownian loops and conformally invariant systems*, talk at *[[CQTS]]* (Nov 2022) [[[Foit_CQTS-Nov2022.pdf:file]], video: [rec](https://nyu.zoom.us/rec/play/DxmoQSXAGXA331jrD4isyOJrmpvTdgmRnaDLeD3gAabO1v6zlTjYNcxPT74SS0VDtWzbiRyYQTltw9vp.dI1nqe74UCpaouu_?continueMode=true&_x_zm_rtaid=kQZZ5EfLRZiMesvQXmFYBg.1669295362714.dd968befceb1ba32e41f0349dfb7ca0f&_x_zm_rhtaid=14)]

category: people

[[!redirects Valentino F. Foit]]


Valeri Frolov

* [webpage](https://sites.ualberta.ca/~vfrolov/)

## selected writings

* {#FrolovZelnikov11} [[Valeri Frolov]], Andrei Zelnikov,  _Introduction to black hole physics_, Oxford 2011

## related $n$Lab entries

* [[black hole]], [[black hole thermodynamics]]

* [[thermal quantum field theory]]

category: people


Valeria de Paiva

* [webpage (old)](http://www.cs.bham.ac.uk/~vdp/)

* [webpage (new)](http://vcvpaiva.github.io/)

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

## Selected writings

On [[categorical semantics]] for (the [[exponential modality]] in) [[linear logic]]:

* {#depaiva89} [[ Valeria de Paiva]], §2 of: *The Dialectica Categories*, in *Categories in Computer Science and Logic*, Contemporary Mathematics **92** (1989)  [[ISBN:978-0-8218-5100-5](https://bookstore.ams.org/conm-92), [doi:10.1090/conm/092](https://doi.org/10.1090/conm/092)]

* {#deP1988} [[Valeria de Paiva]], *The Dialectica Categories*, PhD thesis, technical report 213, Computer Laboratory, University of Cambridge (1991) [[pdf](https://www.cl.cam.ac.uk/techreports/UCAM-CL-TR-213.pdf), [[dePaivaDialectica.pdf:file]]]

* {#BBP92} [[Nick Benton]], [[Gavin Bierman]], [[Valeria de Paiva]], *Term assignment for intuitionistic linear logic*, Technical report 262, Computer Laboratory, University of Cambridge (August 1992) [[pdf](https://www.cl.cam.ac.uk/techreports/UCAM-CL-TR-262.pdf), [[BentonBiermanDePaiva-TermAssignment.pdf:file]]]

  > (published abridged as [BBPH92](#BBPH92))

* {#BBPH92} [[Nick Benton]], [[Gavin Bierman]], [[Valeria de Paiva]], [[Martin Hyland]], *Linear $\lambda$-Calculus and Categorical Models Revisited*, in *Computer Science Logic. CSL 1992*, Lecture Notes in Computer Science **702**, Springer (1993) [[doi:10.1007/3-540-56992-8_6](https://doi.org/10.1007/3-540-56992-8_6)]

* {#HylandPaiva93} [[Martin Hyland]], [[Valeria de Paiva]], _Full Intuitionistic Linear Logic_ (extended abstract). Annals of Pure and Applied Logic, 64(3), pp.273-291, 1993. ([pdf](http://www.cs.bham.ac.uk/~vdp/publications/fill.pdf))

A [[natural deduction]]-formulation of [[modal logic]] ([[S4 modal logic|S4]]/[[S5 modal logic|S5]]) and making explicit the [[necessity]] [[modal operator]] as a [[comonad]]:

* {#BiermanDePaiva96} [[Gavin M. Bierman]], [[Valeria de Paiva]], *Intuitionistic necessity revisited*, School of Computer Science research reports-University of Birmingham CSR (1996) [[researchgate](https://www.researchgate.net/publication/2810611_Intuitionistic_Necessity_Revisited), [[Bierman-dePaiva-NecessityRevisited.pdf:file]]]

* [[Gavin M. Bierman]], [[Valeria de Paiva]], _On an Intuitionistic Modal Logic_ Studia Logica **65** (2000) 383–416 [[doi:10.1023/A:1005291931660](https://doi.org/10.1023/A:1005291931660), [pdf] (https://www.researchgate.net/profile/Valeria-De-Paiva/publication/226515897_On_An_Intuitionistic_Modal_Logic/links/00b4951ed416906ccc000000/On-An-Intuitionistic-Modal-Logic.pdf)]

* Natasha Alechina, Michael Mendler, [[Valeria de Paiva]], Eike Ritter: *Categorical and Kripke Semantics for Constructive S4 Modal Logic*, in *Computer Science Logic. CSL 2001* Lecture Notes in Computer Science **2142**, Springer (2001) 292 [[doi:10.1007/3-540-44802-0_21](https://doi.org/10.1007/3-540-44802-0_21)]

See also:

* {#BentonBiermanPaiva98} [[Nick Benton]], [[Gavin Bierman]], [[Valeria de Paiva]], *Computational Types from a Logical Perspective*, Journal of Functional Programming **8** 2  (1998) 177-193 [[doi:10.1017/S0956796898002998](https://doi.org/10.1017/S0956796898002998), [web](http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.36.5778)]

On Natural Language Inference:

* {#KCdP2020} A.-L. Kalouli, R. Crouch, [[Valeria de Paiva]]. *Hy-NLI: a Hybrid system for Natural Language Inference*. Proceedings of the 28th International Conference for Computational Linguistics (COLING). Barcelona - Spain (online), 01 December 2020.

## Related pages

* Gödel's [[Dialectica interpretation]]

* [[explicit substitution]]

* [[linear type theory]]

* [[computational type theory]]

category: people


Valerij G. Bardakov

* [Institute page](https://research.nsu.ru/en/persons/vbardakov)

* [MathematicsGenealogy page](https://www.mathgenealogy.org/id.php?id=214847)

## Selected writings

On [[extension|extending]]  [[braid representations]] to [[automorphism groups]] of [[free groups]]:

* {#Bardakov05} [[Valerij G. Bardakov]], *Extending representations of braid groups to the automorphism groups of free groups*, Journal of Knot Theory and Its Ramifications **14** 08 (2005) 1087-1098 [[arXiv:math/0408330](https://arxiv.org/abs/math/0408330), [doi:10.1142/S0218216505004251](https://doi.org/10.1142/S0218216505004251)]

On relation between skew-[[brace]]s and [[Rota-Baxter algebra|Rota-Baxter]] groups

* Valeriy G. Bardakov, Vsevolod Gubarev, _Rota-Baxter groups, skew left braces, and the Yang-Baxter equation_, [arXiv:2105.00428](https://arxiv.org/abs/2105.00428)

[[!redirects Valerij Bardakov]]

category: people


Valerio Toledano Laredo

* [homepage](http://www.math.neu.edu/toledano)

category: people

[[!redirects Valerio Toledano-Laredo]]


Valery Isaev

* [webpage](https://research.jetbrains.org/researchers/valis)

## related $n$Lab entries

* [[relation between category theory and type theory]]

category: people


Valery Lunts

**Valery A. Lunts** is a professor of mathematics at the Indiana University at Bloomington ([faculty profile](https://math.indiana.edu/about/faculty/lunts-valery.html)). His interests include [[algebraic geometry]] (including [[noncommutative algebraic geometry|noncommutative]]), [[representation theory]], [[D-modules]], [[deformation theory]], [[sheaf theory]] and [[homological algebra]].

Some works of relevance for the $n$lab:

* J. Bernstein, V. Lunts, _On nonholonomic irreducible D-modules_, Invent. Math. __94__, (1988), no. 2, 223--243 [doi](https://doi.org/10.1007/BF01394325)

* J. Bernstein, V. Lunts, _Equivariant sheaves and functors_, Springer Lecture Notes in Math. __1578__ (1994) MR 95k:55012

* V. A. Lunts, [[Dmitri Orlov|D. O. Orlov]], _Uniqueness of enhancement for triangulated categories_, J. Amer. Math. Soc. __23__  (2010), 853-908, [journal](http://www.ams.org/journals/jams/2010-23-03/S0894-0347-10-00664-8/home.html), [arXiv:0908.4187](http://arxiv.org/abs/0908.4187), [doi](https://doi.org/10.1090/S0894-0347-10-00664-8)

* V. A. Lunts, [[Alexander Rosenberg|A. L. Rosenberg]], _Differential operators on noncommutative rings_, Selecta Math. (N.S.)  __3__  (1997),  no. 3, 335--359 ([doi](https://doi.org/10.1007/s000290050014)); sequel: _Localization for quantum groups_, Selecta Math. (N.S.) __5__ (1999), no. 1, pp. 123--159 ([doi](https://doi.org/10.1007/s000290050044)).

* V. A. Lunts, [[Alexander Rosenberg|A. L. Rosenberg]], _Differential calculus in noncommutative algebraic geometry I. D-calculus on noncommutative rings_, MPI 1996-53 [pdf](http://www.mpim-bonn.mpg.de/preblob/3894), _II. D-Calculus in the braided case. The localization of quantized enveloping algebras_, MPI 1996-76 [pdf](http://www.mpim-bonn.mpg.de/preblob/3916)

* [[Alexander I. Efimov]], Valery A. Lunts, [[Dmitri Orlov|Dmitri O. Orlov]], _Deformation theory of objects in homotopy and derived categories_
   * _I: general theory_ [arXiv:math/0702838](http://arxiv.org/abs/math/0702838);
   * _II: pro-representability of the deformation functor_ [arXiv:math/0702839](http://arxiv.org/abs/math/0702839);
   * _III: abelian categories_ [arXiv:math/0702840](http://arxiv.org/abs/math/0702840)

* Valery A. Lunts, _Coherent sheaves on configuration schemes_, J. Algebra __244__:2 (2001) 379-406 [doi](https://doi.org/10.1006/jabr.2001.8866) [math.AG/0012061](https://arxiv.org/abs/math/0012061)

* Valery A. Lunts, _Categorical resolution of singularities_, J. Algebra __323__:10 (2010) 2977-3003 [doi](https://doi.org/10.1016/j.jalgebra.2009.12.023) [math.AG/0905.4566](https://arxiv.org/abs/0905.4566)

* Valery A. Lunts, _Formality of DG algebras (after Kaledin)_
[math.AG/0712.0996](https://arxiv.org/abs/0712.0996)

* A. I. Bondal, M. Larsen, V. A. Lunts, _Grothendieck ring of pretriangulated categories_,  Intern. Math. Res. Notices __29__ (2004) 1461--1495 [doi](https://doi.org/10.1155/S1073792804140385)
[math.AG/0401009](https://arxiv.org/abs/math/0401009)

* Valery A. Lunts, _Deformations of quasicoherent sheaves of algebras_, J. Algebra __259__:1 (2003) 59-86 <a href="https://doi.org/10.1016/S0021-8693(02)00555-0">doi </a>, [math.AG/0105006](https://arxiv.org/abs/math/0105006)
* [[Valery A. Lunts]], Olaf M. Schnürer, _Matrix factorizations and semi-orthogonal decompositions for blowing-ups_, J. Noncommut. Geom. __10__:3 (2016) 907--979 [doi](https://10.4171/JNCG/252)

category: people

[[!redirects V. A. Lunts]]
[[!redirects V. Lunts]]
[[!redirects Valery A. Lunts]]


validity

# Validity
* table of contents
{:toc}

## Idea

In [[logic]], an argument is said to be **valid** relative to a specified logic if its form in that logic is such that whenever the premises are true then the conclusion must be true.

A logical [[formula]] is said to be valid if it true under every [[interpretation]].

## Validity in modal logic

To judge whether a [[modal logic]] gives a good encoding of the properties of a particular relational structure, we need to put aside the valuations and to concentrate on the [[frame (modal logic)|frames]].  A formula will be **valid** on a frame $\mathfrak{F}$ if it is true at every state in every model that can be built from $\mathfrak{F}$.  More formally:

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

*  A formula $\phi$ is said to be _valid at a state_ $w$ is a frame $\mathfrak{F}= (W,R_1,\ldots, R_n)$ (notation $\mathfrak{F},w \models \phi$) if $\phi$ is true at $w$ in every model $\mathfrak{M} = (\mathfrak{F},V)$, based ion $\mathfrak{F}$.

*  A formula, $\phi$, is _valid in a frame_ $\mathfrak{F}$ (notation $\mathfrak{F}\models \phi$) if it is valid at every state of $\mathfrak{F}$.

*  If $\mathbb{F}$ is a class of frames (all of the same relational signature), then we say $\phi $ is _valid in $\mathbb{F}$_ (notation $\mathbb{F}\models \phi$ if $\mathfrak{F}\models \phi$ for all $\mathfrak{F}\in \mathbb{F}$, and is **valid**, $\models\phi$, if it is valid on the class of all frames.
=--

Fixing a given set $P$ of atomic formulae and the [[signature (in logic)|relational signature]] that we are considering:

+-- {: .un_prop}
###### Proposition

The set of formulae that are valid on a class,$\mathbf{F}$, of frames of that signature forms a logic.
=--

(We restrict to Kripke frames (i.e. to binary relations) for simplicity of exposition, and refer to the sources, such as Blackburn et al, for a fuller discussion.)

We denote the logic thus specified by $\Lambda_\mathbf{F}$.

## References

Generally this entry is based on

* P. [[Blackburn]], M. de Rijke and Y. [[Venema]], _Modal Logic_, Cambridge Tracts in Theoretical Computer Science, vol. 53, 2001,

(any mistakes or errors of interpretation are due to ....!)

[[!redirects validity]]
[[!redirects valid]]


Valter Moretti

Valter Moretti is a mathematical physicists who deals with several mathematical aspects of quantum and quantum-relativistic theories.

* [webpage](https://moretti.maths.unitn.it/home.html)

## Selected writings

On [[AQFT on curved spacetimes]] and its [[Hadamard states]]:

* {#KhavkineMoretti15}  [[Igor Khavkine]], [[Valter Moretti]], *Algebraic QFT in Curved Spacetime and quasifree Hadamard states: an introduction*, Chapter 5 in: [[Romeo Brunetti]] et al. (eds.) *Advances in Algebraic Quantum Field Theory* Springer (2015) [[doi:10.1007/978-3-319-21353-8](https://doi.org/10.1007/978-3-319-21353-8)]

On [[quantum mechanics]] and [[spectral theory]]:

* [[Valter Moretti]], *Mathematical Foundations of Quantum Mechanics: An Advanced Short Course*, Int. J. Geom. Methods Mod. Phys. **13** Supp. 1 (2016) 1630011 [[arXiv:1508.06951](http://arxiv.org/abs/1508.06951), [doi:10.1142/S0219887816300117](https://doi.org/10.1142/S0219887816300117)]

* [[Valter Moretti]], _Spectral Theory and Quantum Mechanics -- Mathematical Foundations of Quantum Theories, Symmetries and Introduction to the Algebraic Formulation_, Springer (2017) [[doi:10.1007/978-3-319-70706-8](https://doi.org/10.1007/978-3-319-70706-8)]

* [[Valter Moretti]], *Fundamental Mathematical Structures of Quantum Theory* -- *Spectral Theory, Foundational Issues, Symmetries, Algebraic Formulation*, Springer (2020) [[doi:10.1007/978-3-030-18346-2](https://doi.org/10.1007/978-3-030-18346-2)]

On [[quantum observables]] and [[quantum probability]]:

* Nicolò Drago, [[Valter Moretti]], _The notion of observable and the moment problem for $\ast$-algebras and their GNS representations_ [[doi:1903.07496](https://arxiv.org/abs/1903.07496), [spire:1725528](http://inspirehep.net/record/1725528)]

On [[classical mechanics]]:

* [[Valter Moretti]], *Analytical Mechanics* -- *Classical, Lagrangian and Hamiltonian Mechanics, Stability Theory, Special Relativity*, Springer (2023) [[doi:10.1007/978-3-031-27612-5](https://doi.org/10.1007/978-3-031-27612-5)]

## Related entries

* [[quantum field theory]]

* [[zeta function regularization]]

* [[Kochen-Specker theorem]]

* [[Gleason's theorem]]

* [[Hadamard distribution]]

* [[spectral theory]]

* [[quaternionic quantum mechanics]]

* [[Gelfand-Naimark-Segal construction]]

* [[Wigner theorem]]

* [[quantum probability theory]]

* [[state on a star-algebra]]

* [[hidden variable theory]]

* [[quantum contextuality]]

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