numerable open cover

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

#Contents#
* table of contents
{:toc}

## Idea

A _numerable open cover_ (alias _normal cover_) is an [[open cover]] of a [[topological space]] that admits a subordinate [[partition of unity]].

## Definition

Let $\{U_i\}$ be an open [[cover]] of the [[topological space]] $X$ (actually [Dold 1963](#Dold63) doesn't always require open, see discussion below). It is said to be **numerable** if there is a collection of [[continuous functions]] $\phi_i \colon X \to [0,1]$ (to the [[closed interval]]) such that:

1. the [[support]] of the $i$th function is contained in the $i$th subset:

   $Supp(\phi_i) \subset U_i$;

1. at each point $x\in X$, only a [[finite number]] of the $\phi_i$ are non-zero;

1. $\underset{x \in X}{\forall}  \sum_i \phi_i(x) \equiv 1  $.

(The functions $\{\phi_i\}$ are a [[partition of unity]].)

The collection of [[preimages]] $\phi_i^{-1}(0,1]$ then constitute a [[locally finite cover]] that [[refinement|refines]] $\{U_i\}$.

## Alternative characterizations

\begin{proposition}
**([[Ernest Michael]], [[Kiiti Morita]], [[Arthur H. Stone]], see [Morita 1962, Thm. 1.2](#Morita62))**
\linebreak
For an [[open cover]] $\{U_i\}_{i\in I}$ of a [[topological space]] $X$ the following properties are equivalent.

* $U$ is a numerable cover, i.e., it admits a compatible positive partition;

* ([Michael 1953, Prop. 2](#Michael53)) $U$ admits a subordinate locally finite [[partition of unity]];

* ([Stone, Thms. 1, 2](#Stone).) $U$ is a normal cover, meaning there is a sequence $W_0=U$, $W_1$, $W_2$, $\ldots$ of open covers of $X$ such that $W_{n+1}$ is a [[star refinement]] of $W_n$ for all $n\ge0$;

* $U$ can be refined by the [[inverse image]] of an open cover of $Y$ under some continuous map $X\to Y$, where $Y$ is a [[metrizable topological space]];

* ([Mardešić & Segal 1982, Lemma I.6.1](#MardesicSegal82)) $U$ can be refined by the inverse image of an open cover of $Y$ under some continuous map $X\to Y$, where $Y$ is an absolute neighborhood retract, i.e., a metrizable topological space $Y$ such that any closed embedding $Y\to Z$ into a metrizable topological space $Z$ factors through an open subset $U\subset Z$ such that there is a map $U\to Y$ for which the composition $Y\to U\to Y$ is identity;

* $U$ can be refined by a locally finite normal open cover;

* $U$ can be refined by a locally finite open cover consisting of cozero sets;

* ([Michael 1953, Thm. 1](#Michael53), [Morita 1964, Thm. 1.2](#Morita64).) $U$ can be refined by an open cover given by the union of a countable collection of locally finite families of cozero sets.

* ([Michael 1953, Proposition 1](#Michael53), [Hoshina 1989, Theorem 1.1 and 1.2](#Hoshina89)) $U$ can be refined by an open cover given by the union of a countable collection of discrete families of cozero sets.

\end{proposition}

## Properties

### As a coverage, as a site

Numerable open covers form a [[site]] called the **numerable site**. More precisely, numerable open covers are a [[coverage]] on the category [[Top]] of topological spaces (this is essentially given in [Dold's lectures, A.2.17](#DoldLectures), but not using this terminology).

For [[paracompact topological spaces]], numerable covers are [[cofinal functor|cofinal]] in open covers, so that the numerable site is equivalent to the open cover site for such spaces.

There is also a 1944 result by Dieudonnne that numerable covers are cofinal in [[locally finite open cover|locally finite covers]] of [[normal spaces]] — need to add this! See, eg, Theorem 6.3 of Howes' _Modern analysis and topology_.

### Relation to numerable bundles

Many classical theorems concerning [[fiber bundles]] are stated for the numerable site. For example, the [[classifying space]] $\mathcal{B}G$ actually classifies [[bundles]] which trivialise over a numerable cover. (References? Dold for [[Milnor's classifying space]], and tom Dieck I think for Segal's) These are called [[numerable bundles]]. This is because the standard constructions of the universal bundle by Minor and Segal both are trivialisable over a numerable cover.

## References ##

* {#Stone} [[Arthur H. Stone]], _Paracompactness and product spaces_, Bulletin of the American Mathematical Society 54:10 (1948), 977–983 $[$[doi:10.1090/S0002-9904-1948-09118-2](https://doi.org/10.1090/S0002-9904-1948-09118-2)$]$

* {#Michael53} [[Ernest Michael]], _A note on paracompact spaces_, Proceedings of the American Mathematical Society 4:5 (1953) 831–838  $[$[doi:10.1090/S0002-9939-1953-0056905-8](https://doi.org/10.1090/S0002-9939-1953-0056905-8)$]$

* {#Morita62} [[Kiiti Morita]], _Paracompactness and product spaces_, Fundamenta Mathematicae 50:3 (1962), 223–236 $[$[doi:10.1090/S0002-9939-1991-1054165-6](https://doi.org/10.1090/S0002-9939-1991-1054165-6)$]$

* {#Dold63} [[Albrecht Dold]], _Partitions of unity in the theory of fibrations_, Ann. of Math. **78** 2 (1963) 223–255 $[$[MR0155330](http://www.ams.org/mathscinet-getitem?mr=155330) [jstor:1970341](http://www.jstor.org/stable/1970341) [doi:10.2307/1970341](http://dx.doi.org/10.2307/1970341)$]$

* {#Morita64} [[Kiiti Morita]], _Products of normal spaces with metric spaces_, Mathematische Annalen 154:4 (1964), 365–382 $[$[doi:10.1007/BF01362570](https://doi.org/10.1007/BF01362570)$]$

* {#MardesicSegal82} [[Sibe Mardešić]], [[Jack Segal]], _Shape Theory_,  North-Holland Mathematical Library 26 (1982).

* {# Hoshina89} Takao Hoshina, _Extensions of mappings II_, Topics in General Topology, Elsevier (1989).

The appendix of

* {#DoldLectures} [[Albrecht Dold]], _Lectures on algebraic topology_

talks about "stacked covers": these are useful for 'decomposing' numerable covers of products to a sort of parameterised version depending on a numerable cover of the first factor. This is important in looking at concordance of numerable bundles.

Detailed review of numerable covers and discussion of [[descent]] over them for [[simplicial presheaves]] via a [[Brown-Gersten property]]:

* [[Dmitri Pavlov]], *Numerable open covers and representability of topological stacks*, Topology and its Applications **318** (2022) 108203 [[arXiv:2203.03120](https://arxiv.org/abs/2203.03120), [doi:10.1016/j.topol.2022.108203](https://doi.org/10.1016/j.topol.2022.108203)]

[[!redirects numerable open covers]]

[[!redirects numerable cover]]
[[!redirects numerable covers]]


numeral

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

# Numerals
* table of contents
{: toc}

## Definition
 {#Definition}

In informal mathematical speech and writing, a **numeral** refers to any notation or terms (for instance [[string (computer science)|strings]] of digits) which *denote* a [[natural number]]. In [[computer science]] one may say that a numeral is [[syntax]] whose [[denotational semantics]] is a [[number]] [[Scott & Strachey (1971, §0)](#ScottStrachey71)].

Usually, in mathematics, this refers to [[decimal number|base ten place-value notation]], so that for instance $0$, $2$, $13$, and $890$ are numerals.

In [[type theory]], a **numeral** generally means (e.g. [Martin-Löf (1973)](#MartinLoef73)) a [[term]] whose [[type]] is the [[natural numbers type]] and which is of [[canonical form]].  The meaning of "canonical form" may vary with the formal theory, but with the usual presentation of the [[natural numbers]] as an [[inductive type]] generated by [[zero]] $0$ and the [[successor]] operation $s$, the numerals are the terms of the form

$$ s(s(\cdots s(s(0))\cdots )). $$

Often, the numeral representing the natural number $n$ --- which is to say, the term with $s$ applied $n$ times to $0$ --- is denoted by $\underline{n}$.  Thus, for instance, $\underline{2}$ means $s(s(0))$.  It is important to note that $\underline{2}$ is not (usually) a term *inside* the formal system being considered; it is a "meta-notation" which stands for the term $s(s(0))$.  (One might say that the underline converts *informal* numerals to *formal* ones.)  In particular, any statement which quantifies over a natural number $n$ that occurs in a term $\underline{n}$ can only be expressed in the [[metatheory]].

## Canonicity

Not every term of natural number type $\mathbb{N}$ is a numeral; consider for instance $\underline{2}+\underline{2}$.  However, good formal systems have the property of [[canonicity]], which in this context means that every term of type $\mathbb{N}$ *computes to*, or is *provably [[equal]] to*, a numeral.  In our example, if $+$ is defined by [[recursion]], there is a sequence of [[beta-reduction]] steps leading from $\underline{2}+\underline{2}$ to $\underline{4}$.  (Canonicity is about terms in the [[empty context]]; in a [[context]] with [[free variables]] of type $\mathbb{N}$, then of course there will be more terms of type $\mathbb{N}$, built out of these variables.)

If we add to such a formal system an [[axiom]] using an [[existential quantification|existential statement]], then this is equivalent to adding to the language an additional term for a natural number that is not (and may not provably be equal to) any canonical numeral.  For example, in $PA + \neg{Con(PA)}$ ([[Peano arithmetic]] plus the axiom that Peano arithmetic is [[inconsistent]]), we have the axiom
$$ \exists n, (n \vdash_{PA} \bot) ,$$
stating the existence of a number $n$ that is the [[Goedel numbering|Gödel number]] of a [[proof]] in $PA$ of a [[falsehood]].  If we instead extend the language of $PA$ with a new symbol $※$ and add the axiom
$$ ※ \vdash_{PA} \bot ,$$
then (assuming that $PA$ and so $PA + \neg{Con(PA)}$ is in fact consistent) one can prove
$$ \underline{n} \neq ※ $$
in this system for every natural number $n$, but one cannot prove
$$ \forall n,\, n \neq ※ $$
(which is actually trivially refutable).

## Related concepts

* [[number]]
* [[decimal number]] (which really should be called decimal numerals)

## References

Discussion in motivation of [[denotational semantics]] and [[domain theory]]:

* {#ScottStrachey71} [[Dana S. Scott]], [[Christopher Strachey]], §0 of: *Toward a Mathematical Semantics for Computer Languages*, Oxford University Computing Laboratory, Technical Monograph PRG-6 (1971) [[pdf](https://www.cs.ox.ac.uk/files/3228/PRG06.pdf), [[ScottStrachey-MathematicalSemantics.pdf:file]]]

Discussion in [[type theory]]:

* {#MartinLoef73} [[Per Martin-Löf]], §1.7 in: *An intuitionistic theory of types: predicative part*, in: H. E. Rose, J. C. Shepherdson (eds.), *Logic Colloquium '73, Proceedings of the Logic Colloquium*, Studies in Logic and the Foundations of Mathematics **80**, Elsevier (1975) 73-118 [doi:10.1016/S0049-237X(08)71945-1, [CiteSeer](http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.131.926)]

where it says:

> A closed normal term with type symbol $N$, which obviously must have the form $s(s(...s(0)...))$, is called a *numeral*.

[[!redirects numeral]]
[[!redirects numerals]]


numerical analysis

[[!redirects numerical methods]]

* related $n$Lab entries include [[symplectic integrator]], [[Runge-Kutta method]], [[applied mathematics]], [[preconditioner]], [[sampling theory]], [[homological algebra in the finite element method]]

* Simon Širca, Martin Horvat, _Computational methods for physicists : compendium for students_, Springer Graduate texts in physics, 2013

category: applications


numerical motive

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

#Contents#
* table of contents
{:toc}

## Idea

A [[pure motive]] defined by applying [[numerical equivalence]] of [[algebraic cycles]].

## Properties

### Semi-simple abelian category

The category of numerical motives is a [[semisimple category|semisimple]] [[abelian category]].

In fact, numerical equivalence is precisely the only [[adequate equivalence relation]] relation that gives the [[category of pure motives]] this property.

([Jannsen 92](#Jannsen92))

## Related concepts

* [[motivic Galois group]]

## References

A standard account is in

* U. Jannsen, S. Kleiman and J.-P. Serre, _Motives_, Proceedings of the AMS-IMS-SIAM Joint
Summer Research Conference held at the University of Washington, Seattle, Washington,
July 20–August 2, 1991. Proceedings of Symposia in Pure Mathematics 55, part 1 and 2.
American Mathematical Society, Providence, RI, 1994.

The fact that the category of numerical motives is semisimple abelian is in

* [[Uwe Jannsen]], _Motives, numerical equivalence, and semi-simplicity_, Invent. math. 107 (1992) ([pdf](http://epub.uni-regensburg.de/26642/1/jannsen10.pdf))
 {#Jannsen92}

A brief review of numerical [[noncommutative motives]] is in section 4 of

* [[Goncalo Tabuada]], _A guided tour through the garden of noncommutative motives_, ([arxiv1108.3787](http://arxiv.org/abs/1108.3787));
 {#Tabuada11}

[[!redirects numerical motives]]

[[!redirects numerical equivalence]]


Nuno Reis

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

## Selected writings

On the [[basic bundle gerbe|basic]] [[bundle gerbe]] over a [[compact Lie group|compact]] [[simple Lie group]] in relation to the [[WZW model]]:

* [[Krzysztof Gawędzki]], [[Nuno Reis]], *WZW branes and gerbes*, Rev. Math. Phys. **14** (2002) 1281-1334 [[arXiv:hep-th/0205233](https://arxiv.org/abs/hep-th/0205233), [doi:10.1142/S0129055X02001557](https://doi.org/10.1142/S0129055X02001557)]

* [[Krzysztof Gawędzki]], [[Nuno Reis]], _Basic gerbe over non-simply connected compact groups_, Journal of Geometry and Physics **50** 1-4 (2004) 28–55 [[arXiv:0307010](https://arxiv.org/abs/math/0307010), [doi:10.1016/j.geomphys.2003.11.004](http://dx.doi.org/10.1016/j.geomphys.2003.11.004)]

[[!redirects Nuno M. Reis]]


NuPRL

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Constructivism, Realizability, Computability
+-- {: .hide}
[[!include constructivism - contents]]
=--
#### Type theory
+-- {: .hide}
[[!include type theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

An [[extensional type theory]] [[proof management system]].

The dialect of [[intuitionistic type theory]] used is also referred to as _[[computational type theory]]_.

The PRL part of the name evidently stands for "Proof Refinement Logic".

## Descendants

* [MetaPRL](http://metaprl.org/): A logical framework implementation that includes (an old version of) Nuprl's logic as a theory, and focuses on an efficient, modular tactic engine for proof automation.

* [RedPRL](http://redprl.org/): An experimental proof assistant for a [[cubical type theory|cubical]] variant of computational type theory.

* [[CompLF|Prototype CompLF]]: A tentative design for a modern, minimalistic variant of Nuprl's logic, with an emphasis on internally reflecting the semantic judgments.

## Related entries

[[!include proof assistants and formalization projects -- list]]

## References

* _[PRL Project](http://www.nuprl.org/)_ .

* [[Robert Constable]], Stuart F. Allen, H. M. Bromley, W. R. Cleaveland, J. F. Cremer, R. W. Harper, Douglas J. Howe, T. B. Knoblock, N. P. Mendler, P. Panangaden, James T. Sasaki, and Scott F. Smith. _Implementing Mathematics with the Nuprl Proof Development System_. Prentice-Hall, NJ, 1986.
 {#ConstableEtAl86}

* _Innovations in Computational Type Theory using Nuprl_ ([pdf](http://www.cs.uni-potsdam.de/ti/kreitz/PDF/05jal-nuprl.pdf))
 {#NuPRL05}

* [[Robert Constable]], _The Triumph of Types: Creating a Logic of Computational Reality_ ([[ConstableTriumphOfTypes.pdf:file]])
 {#Constable}

[[!redirects NuPrl]]

[[!redirects Nuprl]]


nyan-cat > history

< [[nyan-cat]]


Néron-Severi group

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Complex geometry
+--{: .hide}
[[!include complex geometry - contents]]
=--
#### Cohomology
+--{: .hide}
[[!include cohomology - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

Given an [[algebraic variety]] or [[scheme]] $X$, then its _Néron-Severi group_ is the [[quotient]] of its [[Picard scheme]] $Pic_X$ by the connected component $Pic_X^0$ of the neutral element.

## Related concepts

* A choice of an [[ample sheaf|ample]] element in the Néron-Severi group is called a choice of [[polarized algebraic variety|polarization]].

[[!redirects Neron-Severi group]]


O'Nan moonshine

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

#Contents#
* table of contents
{:toc}

##Idea

**O'Nan moonshine** concerns the existence of a graded infinite-dimensional module for the [[sporadic finite simple group]] of O'Nan, a so-called [[pariah group]], for which the McKay-Thompson series are weight 3/2 mock modular forms. This form of [[moonshine]] relates congruences of McKay-Thompson series and the Selmer and Tate-Shafarevich groups of certain optimal elliptic curves.

##Related concepts

* [[Moonshine]]
* [[Mathieu moonshine]]
* [[umbral moonshine]]

##References

* [[John Duncan]], [[Michael Mertens]], [[Ken Ono]], _O'Nan moonshine and arithmetic_, [arXiv:1702.03516](https://arxiv.org/abs/1702.03516)

For an informal introduction,

* [[John Duncan]], [[Michael Mertens]], [[Ken Ono]], _Pariah moonshine_, Nature Communications 8: 670 (2017), [article](https://www.nature.com/articles/s41467-017-00660-y), [arXiv:1709.08867](https://arxiv.org/abs/1709.08867)

For a popular summary,

* Erica Klarreich, _Moonshine Link Discovered for Pariah Symmetries_, [Quanta](https://www.quantamagazine.org/moonshine-link-discovered-for-pariah-symmetries-20170922/)


o-minimal structure

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

# Contents
* table of contents
{: toc}

## Definition

Let $(R, \lt)$ be a [[dense linear order|dense]] [[linear order]] without [[bounded order|endpoints]]. An order-minimal or **o-minimal structure** on $R$ is a [[structure (model theory)|structure]] $\mathcal{S}$ on $R$ such that

* The relation $\lt$ belongs to $\mathcal{S}_2$;

* The elements of $\mathcal{S}_1$ are precisely finite unions of points and intervals in $R$.

Here an interval can mean a set of the form $I_{a, b} = \{x \in R: a \lt x \lt b\}$, or $I_{\downarrow a} = \{x \in R: x \lt a\}$, or $I_{\uparrow a} = \{x \in R: a \lt x\}$.

## Commentary

A structure on a set $R$ can be thought of as the collection $\mathcal{S} = \bigcup_n \mathcal{S}_n$ of sets that are definable with respect to a one-sorted first-order language $L$ with a given interpretation in $R$. Thus $\mathcal{S}_n$ is the collection of subsets of $R^n$ which are defined by $n$-ary predicates in $L$. The definition of o-minimal structure supposes that $L$ contains a relation symbol $\lt$, and that $\lt$ is interpreted in $R$ as a dense linear order without endpoints.

The o-minimality condition places a sharp restriction on which subsets of $R$ can be defined in the language. Essentially, it means that the only definable subsets of $R$ are those which are definable in terms of constants and the predicates $\lt$ and $=$.

The archetypal example of an o-minimal structure is that of [[semi-algebraic set]]s defined over $\mathbb{R}$ (which form a structure due to the [[semialgebraic set|Tarski-Seidenberg theorem]]).

Remarkably, quite a lot can be said about the structure of definable sets in an o-minimal structure over $\mathbb{R}$, and this is a very active area of model theory. The notion of o-minimal structure has been proposed as a reasonable candidate for an axiomatic approach to Grothendieck's hoped-for "[[tame topology]]" ( _topologie modérée_ ).

## O-minimal theories

A [[theory]] is o-minimal if every model $M$ of $T$ is an o-minimal structure.

## Relation to number theory and arithmetic geometry

The theory of o-minimal structures has applications to [[number theory]] and [[arithmetic geometry]], for example via the Pila-Wilkie theorem, which is used in proofs of Lang's conjecture, the Manin-Mumford conjecture, and the Andre-Oort conjecture (see [Tsimerman19](#Tsimerman19)).

## Links

* wikipedia [o-minimal theory](http://en.wikipedia.org/wiki/O-minimal_theory)

## References

General:

* Michel Coste, _An introduction to O-minimal geometry_ , Lecture notes Pisa 1999. ([pdf](http://perso.univ-rennes1.fr/michel.coste/polyens/OMIN.pdf))

* [[Lou van den Dries]], _Exponential rings, exponential polynomials and exponential functions_ , Pacific Journal of Mathematics **113** no.1 (1984) pp.51–66. ([pdf](http://projecteuclid.org/download/pdf_1/euclid.pjm/1102709376))

* Lou van den Dries, _Tame topology and O-minimal structures_, London Math. Soc. Lecture Notes Series 248, Cambridge U. Press 1998.

* [[Alexandre Grothendieck]], _Esquisse d'un Programme_, section 5. English translation available in Geometric Galois Actions I (edited by L. Schneps and P. Lochak), LMS Lecture Notes Ser. 242, CUP 1997.

* Alessandro Berarducci, _Definable groups in o-minimal structures_, [pdf](http://www.dm.unipi.it/~dinasso/marian2004/berarducci.pdf); _Cohomology of groups in o-minimal structures: acyclicity of the infinitesimal subgroup_, J. Symbolic Logic __74__:3 (2009), 891-900, [MR2548466](http://www.ams.org/mathscinet-getitem?mr=2548466), [euclid](http://projecteuclid.org/euclid.jsl/1245158089), [doi](https://doi.org/10.2178/jsl/1245158089).

* Alessandro Berarducci, _O-minimal spectra, infinitesimal subgroups and cohomology_, J. Symbolic Logic __72__ (2007), no. 4, pp. 1177--1193, [MR2371198](http://www.ams.org/mathscinet-getitem?mr=2371198), [euclid](http://projecteuclid.org/euclid.jsl/1203350779 ), [doi](https://doi.org/10.2178/jsl/1203350779).

* M. Edmundo, G. O. Jones, N. J. Peatfield, _Sheaf cohomology in o-minimal structures_, J. Math. Logic __6__ (2006), no. 2, pp. 163--179, [MR2317425](http://www.ams.org/mathscinet-getitem?mr=2317425), [doi](https://doi.org/10.1142/S0219061306000566)

* Mario J. Edmundo, Luca Prelli, _Invariance of o-minimal cohomology with definably compact supports_, Confluentes Mathematici 7, n. 1 (2015) 35-53 [arxiv/1205.6124](http://arxiv.org/abs/1205.6124) [doi](https://doi.org/10.5802/cml.17)

* M. J. Edmundo, N. J. Peatfield, _O-minimal Čech cohomology_, (2006) [pdf](https://repositorioaberto.uab.pt/bitstream/10400.2/2839/1/ConstCech.pdf)

* Ricardo Bianconi, Rodrigo Figueiredo, _O-minimal de Rham cohomology_, [arxiv/1904.05485](https://arxiv.org/abs/1904.05485)

* Olivier Le Gal, Jean-Philippe Rolin, _An o-minimal structure which does not admit $C^{\infty }$ cellular decomposition_, Annales de l'institut Fourier __59__:2 (2009), p. 543-562, [MR2521427](http://www.ams.org/mathscinet-getitem?mr=2521427) [Zbl 1193.03065](http://www.zentralblatt-math.org/zmath/en/search/?q=an:1193.03065) [numdam](http://www.numdam.org/item?id=AIF_2009__59_2_543_0)

* Thomas Scanlon, _Algebraic differential equations from covering maps_, Adv. Math. __330__ (2018) 1071-1100 [doi](https://doi.org/10.1016/j.aim.2018.03.008)
* Benjamin Bakker, Yohan Brunebarbe, Jacob Tsimerman, _o-minimal GAGA and a conjecture of Griffiths_, [arxiv/1811.12230](https://arxiv.org/abs/1811.12230)

* [[Reid Barton]], Johan Commelin, _Model categories for o-minimal geometry_ ([arXiv:2108.11952](https://arxiv.org/abs/2108.11952))

* {#Tsimerman19} Jacob Tsimerman, _Functional transcendence results and arithmetic applications_, Proceedings of the International Congress of Mathematicians (ICM 2018), pp. 435-454 (2019) [doi](https://doi.org/10.1142/9789813272880_0062) [pdf] (https://eta.impa.br/dl/091.pdf)

In Physics, the concept of [[tame topology]] and [[o-minimal structures]] appears in a proposal for a finiteness condition in [[QFT]] in

* [[Michael Douglas]], [[Thomas Grimm]], and Lorenz Schlechter. *The Tameness of Quantum Field Theory, Part I--Amplitudes.* (2022)([arXiv:2210.10057](https://arxiv.org/abs/2210.10057)).

* [[Michael Douglas]], [[Thomas Grimm]], and Lorenz Schlechter. *The Tameness of Quantum Field Theory, Part II--Structures and CFTs* (2023) ([arXiv:2302.04275] (https://arxiv.org/abs/2302.04275)).

[[!redirects O-minimal structure]]
[[!redirects o-minimal structures]]
[[!redirects order-minimal structure]]
[[!redirects order-minimal structures]]
[[!redirects o-minimal theory]]
[[!redirects o-minimal theories]]


O4-plane

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

#Contents#
* table of contents
{:toc}

## Idea

In [[type IIA string theory]] an _O4-plane_ is an [[orientifold plane]] ("[[O-plane]]") of [[dimension]] $4+1$, hence of the same dimension as [[D4-branes]], but carrying opposite [[D-brane charge]].

## Properties

### Lift to the MO5-plane in M-theory

Under [[duality between M-theory and type IIA string theory]], for [[KK-compactification]] along a [[circle]]-[[fiber]] parallel to an [[MO5-plane]], a plain [[MO5-plane]] in [[M-theory]] becomes the [[O4-plane]], specifically the $O^- 4$-plane, while an MO5 with an [[M5-brane]] on top of it becomes the $O^+ 4$-plane ([Gimon 9, Sec. III8](#Gimon98),[Hanany-Kol 00, Sec. 3.1](#HananyKol00)).

## References

### General

(...)

### Relation to MO5-planes

Relation of the [[MO5-plane]] to the [[O4-plane]] under [[duality between M-theory and type IIA string theory]]:

* {#Gimon98} Eric G. Gimon, _On the M-theory Interpretation of Orientifold Planes_ ([arXiv:hep-th/9806226](https://arxiv.org/abs/hep-th/9806226), [spire:472499](http://inspirehep.net/record/472499))

* {#HananyKol00} [[Amihay Hanany]], [[Barak Kol]], Section 3.1 in: _On Orientifolds, Discrete Torsion, Branes and M Theory_, JHEP 0006 (2000) 013 ([arXiv:hep-th/0003025](https://arxiv.org/abs/hep-th/0003025))

### String phenomenology with intersecting D-branes

[[string phenomenology]] on [[intersecting D-brane model|intersecting]] [[D4-branes]] on [[O4-planes]] [[geometric engineering of QFTs|geometrically engineering]] [[QFTs]] close to the [[standard model of particle physics]]:

* {#BKL01} D. Bailin, G. V. Kraniotis, A. Love, _Standard-like models from intersecting D4-branes_, Phys. Lett. B530 (2002) 202-209 ([arXiv:hep-th/0108131](https://arxiv.org/abs/hep-th/0108131))

* {#KataokaShimojo01} H. Kataoka, M. Shimojo, _$SU(3) \times SU(2) \times U(1)$ Chiral Models from Intersecting D4-/D5-branes_, Progress of Theoretical Physics, Volume 107, Issue 6, June 2002, Pages 1291–1296 ([arXiv:hep-th/0112247](https://arxiv.org/abs/hep-th/0112247), [doi:10.1143/PTP.107.1291](https://doi.org/10.1143/PTP.107.1291))

* {#Bailin02} D. Bailin, _Standard-like models from D-branes_, J Phys (2003) 60: 199 ([arXiv:hep-th/0210227](https://arxiv.org/abs/hep-th/0210227))

* D. Bailin, G. V. Kraniotis, A. Love, _New Standard-like Models from Intersecting D4-Branes_, Phys. Lett. B547 (2002) 43-50 ([arXiv:hep-th/0208103](https://arxiv.org/abs/hep-th/0208103))

(...)

[[!redirects O4-planes]]


Oberwolfach

In the context of [[mathematics]], **Oberwolfach** refers to the

* [Mathematisches Forschungsinstitut Oberwolfach](http://www.mfo.de/)

  ( "Mathematics Research Institute Oberwolfach" ),

one of the more pleasant places on this planet.

# Tagungsbericht

A series of workshops on [[category theory]] have taken place in Oberwolfach over the years. "Tagungsbericht" means "conference report", and comprises various notes on discussions that took place during the workshops.

- 1966: [link](https://oda.mfo.de/handle/mfo/1809)
- 1972: [link](https://oda.mfo.de/handle/mfo/1575)
- 1973: [link](https://oda.mfo.de/handle/mfo/1527)
- 1974: [link](https://oda.mfo.de/handle/mfo/1474)
- 1975: [link](https://oda.mfo.de/handle/mfo/1432)
- 1977: [link](https://oda.mfo.de/handle/mfo/1328)
- 1979: [link](https://oda.mfo.de/handle/mfo/1230)
- 1995: [link](https://oda.mfo.de/handle/mfo/365)

(Note that the above list is likely incomplete.)

# Workshop notes#

There are [weekly workshops](http://www.mfo.de/programme/workshops) in Oberwolfach. The $n$Lab features workshop notes on the following events

* [[Oberwolfach Workshop, June 2009 -- Strings, Fields, Topology]]

  * [[Oberwolfach Workshop, June 2009 -- Abstracts|List of Abstracts]]

category: reference


Oberwolfach HoTT-Coq tutorial

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

At an Oberwolfach workshop in 2011 on [[homotopy type theory]], [[Andrej Bauer]] and [[Peter LeFanu Lumsdaine]] created a tutorial that walks the reader through the proof that the [[univalence axiom]] implies [[functional extensionality]], formalized entirely in [[Coq]]-proof assistant code.

For a pdf-version of the tutorial see

* [[Andrej Bauer]], [[Peter LeFanu Lumsdaine]], _A Coq proof that Univalence Axioms implies Functional Extensionality_ (2011) ([[BauerLumsdaineUnivalence.pdf:file]]).

In order to follow the proof explicitly in [[Coq]]-itself, download the accompanying source files here:

* [https://github.com/andrejbauer/Homotopy/tree/master/OberwolfachTutorial](https://github.com/andrejbauer/Homotopy/tree/master/OberwolfachTutorial)

and feed them into your [[Coq]]. See the README file there for further hints.

A rephrasing of the Coq-proofs given there in terms of ordinary language is given in

* Carlo Angiuli, _Univalence implies function extensionality_ ([blog](http://homotopytypetheory.org/2011/12/05/hott-in-prose/), [pdf](http://hottheory.files.wordpress.com/2011/12/hott2.pdf))

category: reference


Oberwolfach Workshop, June 2009 -- Abstracts

[[!redirects Oberwolfach 2009 - Abstracts]]

Here are abstracts of talks given at

* [[Oberwolfach Workshop, June 2009 -- Strings, Fields, Topology]]

## Thomas Schick: _Smooth cohomology theories_ ##
*  See also [[Oberwolfach Workshop, June 2009 -- Tuesday, June 9|notes from the conference]]

The basic idea of [[differential cohomology]] (also called _smooth cohomology_ ) is to combine generalized [[cohomology]] (e.g. ordinary integral cohomology, or K-theory, or tmf, or ...) and differential forms. Hopkins and Singer showed that a smooth refinement exists for any generalized cohomology theory, but didn't provide an (easy for mere mortals to understand) explicit construction. In this talk, Thomas Schick advertised geometric models for multiplicative smooth cohomology with $S^1$-integration for $K$-theory (joint with Ulrich Bunke) and MU-bordism (joint with Ulrich Bunke, Schröder? and Wethamp?). They have proved a uniqueness theorem, stating that under certain assumptions any two smooth refinements of a generalized cohomology theory are naturally isomorphic.

## Ullrich Bunke: _Smooth K-theory_ ##
*  See also [[Oberwolfach Workshop, June 2009 -- Tuesday, June 9|notes from the conference]]

In this talk, which led on from the previous one by Thomas Schick, Ulrich Bunke gave an explicit description of their smooth $K$-theory model in terms of geometric cocycles. Basically a cocycle consists of a pair $(\mathcal{E}, \rho)$ where $\mathcal{E}$ is a 'geometric family', namely a smooth proper submersion $E \rightarrow B$ with a fiberwise metric, a Clifford bundle $W$ and some connection of some sort. The object $\rho$ is an element of $\Omega(B, K) = C^\infty(B, \Lambda^*T^* B \otimes K^*)$. He also described smooth $K$-orientatiosn and the push-forward map, with an application to the $e$-invariant of Adams. There is also a Riemann-Roch theorem.

## Christoph Schweigert and Ingo Runkel: _CFT and algebra in braided tensor categories I and II_ ##
*  See also [[Oberwolfach Workshop, June 2009 -- Monday, June 8|notes from the conference]]

This talk explains how correlators for a rational two-dimensional [[conformal field theory]] can be constructed in the [[FQFT|functorial TFT]] formalism. Based on a [[modular tensor category]] $C$, decoration data have been introduced in terms of special symmetric Frobenius algebras in $C$ and the correlator, as an element of $tft_C (\hat{X})$ with $\hat{X}$ a double cover of the surface $X$, has been expressed in terms ofthe invariant of a decorated 3-manifold $M_X$ with $\partial M_X = \hat{X}$. The correlators are invariant under the [[mapping class group]]s and obey the [[sewing constraint]].

[[Morita equivalence|Morita equivalent]] special symemtric Frobenius algebras lead to an equivalent description of the correlators. A Morita invariant formulation is provided by the notion of a module category $M$ over the module tensor category $C$. The [[worldsheet|world sheet]] is now decorated by categories, functors between module categories and natural transformations.

## Kevin Costello: _Factorization algebras in perturbative QFT_ ##
*  [[Oberwolfach Workshop, June 2009 -- Monday, June 8|Talk 1]], [[Oberwolfach Workshop, June 2009 -- Tuesday, June 9|Talk 2]], [[Oberwolfach Workshop, June 2009 -- Tuesday, June 9|Talk 3]]

A [[factorization algebra]] on a manifold $M$ is an object which associates to every ball $B \subseteq M$ a vector space (or cochain complex) $F(B)$; and to every collection $B_1 \coprod \cdots \coprod B_n \subset B_{n+1}$ of disjoint balls in a larger ball, a map $F(B_1) \otimes \cdots \otimes F(B_n) \rightarrow F(B_{n+1})$. These structures are the $C^\infty$ analogues of chiral algebras, as introduced by Beilinson and Drinfeld; they are closely related to [[algebra over an operad|algebras]] for the [[little n-disk operad]] $E_n$.

In this series of lectures, Kevin Costello set up a framework to construct [[factorization algebra|factorization algebras]] from [[perturbative quantum field theory|perturbative]] [[quantum field theory]]. The set up is analagous to the [[deformation quantization]] picture of [[quantum mechanics]]. Just as the [[observable]]s of [[quantum mechanics]] are encoded in an associative algebra, Costello argues that the observables of a QFT on $M$ are encoded in a factorization algebra on $M$, similar to but slightly different from the waqy it works in (euclidean) [[AQFT]]. This factorization algebra arises by quantizing a commutative factorization algebra associated to classical field theory.

This series of lectures finished with the statement of a theorem allowing one to quantize the commutative factorization algebra associated to a classical field theory in a range of situations, including situations of physical interest.  This is joint work with Owen Gwilliam.

## [[Gabriel C. Drummond-Cole]]: _$\infty$-Operads, $BU_\infty$ and $Hypercomm_\infty$_ ##
*  See also [[Oberwolfach Workshop, June 2009 -- Wednesday, June 10|notes from the conference]]

The speaker applies the machinery of the [[model category]] of [[operad]]s to extend and explain the Barannikov-Kontsevich passage from differential BV algebras satisfying the $\partial-\overbar{\partial}$ lemma to hypercommutative algebras (Frobenius manifolds). This is joint work with Bruno Valette. The following theorem was proved:

+-- {: .un_thm}
###### Theorem

Let $V$ be a differential BV-algebra over a field of characteristic zero. Let $H$ be its homology. Then:
1. If $V$ satisfies the noncommutative Hodge to de Rham degeneration condition, then there exists a hypercommutative $\infty$-structure on $H$.
2. If $V$ satisfies Park's semiclassical condition then this structure is unique up to $Hypercomm_\infty$ quasi-isomorphism.
=--

## Alex Kahle: _Superconnections and index theory_ ##
*  See also [[Oberwolfach Workshop, June 2009 -- Wednesday, June 10|notes from the conference]]

In this talk, Alex Kahle described results from his thesis work (his supervisor was Dan Freed) about [superconnections and index theory](http://arxiv.org/abs/0810.0820). The idea is to prove a local index theorem using [[superconnection]]s via direct geometric-analytical methods (i.e. not stochastic) for use with family index problems, determinant line bundles, etc. Just like [[connection on a bundle|ordinary connection]]s on spinor buundles, a superconnection gives rise to a [[Dirac operator]] on a [[spinor bundle]]. The usual local index theorem (that the trace of the heat kernel converges as $t \rightarrow 0$ to a certain differential form) needs to be modified though, because forms of different degrees have different scaling behaviour (recall that a [[superconnection]] involves forms of different degrees). Alex worked out how to scale everything correctly so that one indeed gets a local index theorem for superconnections, leading to many potential applications.

## [[Chris Schommer-Pries]]: _Topological defects, $D$-branes, and the classifications of TFTs in low dimensions_ ##
*  [Slides on webpage](http://sites.google.com/site/chrisschommerpriesmath/Home/Slides-MFO-6-11-09.pdf?attredirects=0)

Chris explained his classification result mentioned at [[(infinity,n)-category of cobordisms]] from his thesis of extended (he suggested calling them _local_) 2d TQFT's via the explicit generators and relations he obtained on $2Cob$. He also showed how the [[higher category theory|higher-categorical viewpoint]] unites the following two ideas: the 'open-closed' theories and the 'field theories with defects' from Ingo Runkel and Christoph Schweigert's talk. He showed how both these concepts are particular examples of a single notion, namely that of a collection of natural transformations between a restriction of [[FTQFT]] $n$-functors (called "unnatural" or "supernatural" transformations to indicate that it is a transformation not of the entire functor, but just of its restriction to the 2-category of 2-manifolds with only inverttible 2-morphisms).

This is precisely the kind of viewpoint that Urs Schreiber and Jens Fjelstad [suggested](http://golem.ph.utexas.edu/category/2007/08/dbranes_from_tin_cans_part_x.html) at the [n-category cafe](http://golem.ph.utexas.edu/category/) under the slogal [D-branes from tin-cans](http://golem.ph.utexas.edu/category/2006/10/dbranes_from_tin_cans_arrow_th.html) (a D-[[brane]] is a boundary condition ordefect, a "tin-can diagram" is the naturality diagram for a higher natural transformation).

By the way, there's much more to come from Chris, he is working on the 3d theory and much besides!

## Dan Freed: _Geometry and topology of orientifolds_ ##
*  See also [[Oberwolfach Workshop, June 2009 -- Monday, June 8|notes from the conference]]

Freed described recent work with [[Jacques Distler]] and Greg Moore on a certain background structure in [[string theory|superstring theory]] called an [[orientifold], which is a [[bundle gerbe]] on a $\mathbb{Z}_2$ [[orbifold]] with a peculiar "twisted" [[equivariance]] condition. There is apparantly a beautiful string theory formula to calculate the '[[RR charge]]', namely something like
\[
 RR charge = \pm 2^{something} i_* \sqrt\left( \frac{L'(F)}{L'(\nu)} \right)
\]
involving a certain normal bundle and the 'Bott element in K-theory', which looks similar to the Hirzebruch formula for the L-genus. This uses some kind of twisted [[differential cohomology]] version of KR theory. The whole thing can be viewed as a new sort of anomaly, having to do with an exotic orientation of some kind.

## Scott Wilson: _Some algebra relat4ed to mapping spaces and applications_ ##
*  See also [[Oberwolfach Workshop, June 2009 -- Wednesday, June 10|notes from the conference]]

Wilson defined a _partial algebra_ as a lax [[monoidal functor]] from the category of finite sets to [[chain complex|chain complexes]], such that the map $A(j \coprod k) \rightarrow A(j) \otimes A(k)$ is a quasi-isomorphism. He showed how these pop up in [[homotopy theory]] all the time.

(Remark: this formula is the basis for [[Jacob Lurie]]'s discussion of [[commutative algebra in an (infinity,1)-category]] in general and of [[symmetric monoidal (infinity,1)-category]] in particular).

For instance, if $Y$ is any space and $A$ a partial algebra, the resulting total complex can be seen as a generalization of [[Hochschild cohomology]]. One really interesting example (for [[Bruce Bartlett|Bruce]]) was where $Y$ is the interval, and $A$ is the partial algebra of forms on a Riemannian manifold $M$. It turns out that a certain canonical equation which pops out is precisely the Navier-Stokes equation. Gulp!

## [[Urs Schreiber]]: _Background fields in twisted differential nonabelian cohomology_ ##
*  See also [[Oberwolfach Workshop, June 2009 -- Thursday, June 11|notes from the conference]]
*  detailed notes are at [[schreiber:Background fields in twisted differential nonabelian cohomology]]

What can I ([[Bruce Bartlett|Bruce]]) say about Urs's talk? Many mere mortals might think it impressive to pass from ordinary cohomology to twisted cohomology, or to differential cohomology, or to nonabelian cohomology. Urs does this all in one step! Yes, we're talking about _twisted differential nonabelian cohomology_. As we all know and love, Urs has developed some $\infty$-machinery based on all sorts of work which enables him to 'turn the crank' and output all the cohomology theories which mathematicians and physicists currently are bumping into. A great application of his machinery is an understanding of the 'Green-Schwarz anomaly cancellation' mechanism in terms of twisted nonabelian String-gerbes with connection and the Chern-Simons 2-gerbe. Unfortunately Urs didn't have time to quite get to that point, but he did at least mention other examples like how his machinery naturally produces the 'twisted Bianchi identities' in twisted flat differential cohomology, as well as nonabelian gerbes. Urs ended his talk by explaining how nonabelian cohomology on $X$ is related to _twisted_ abelian cohomology on $X$. This ties in with Konrad's talk, where the abelian viewpoint is taken, but the nonabelian viewpoint is essential for some applications.

## André Henriques: _Invertible conformal nets_ ##
*  See also [[Oberwolfach Workshop, June 2009 -- Thursday, June 11|notes from the conference]]

In this talk, Henriques described joint work with Chris Douglas and Arthur Bartels on the program of establishing an equivalence between full conformal field theories and 'conformal nets'. Preliminary write-ups of this work is available on his [webpage](http://www.math.uu.nl/people/henrique/). A conformal net is something like a [[factorization algebra]] (in the sense of Costello's talk) on 1-dimensional manifolds taking values in the bicategory of Von Neumann algebras with [[bimodule]]s as 1-morphisms. The $\mu$_index $\mu(A)$ of a conformal net $A$ is defined to be the statistical dimension of the vacuum sector $H_0$, viewed as some kind of bimodule involving things which $A$ assigns to arcs on the unit circle. He stated the following result.

+-- {: .un_theorem}
###### Theorem

A [[conformal net]] is invertible if and only if $\mu(A)=1$, and it is fully dualizable (in the sense of Lurie) if and only if $\mu(A) \lt \infty$.
=--

## [[Konrad Waldorf]]: _String connections and Chern-Simons 2-gerbes_ ##
*  See also [[Oberwolfach Workshop, June 2009 -- Thursday, June 11|notes from the conference]]

Konrad described the [[Chern-Simons 2-gerbe]] $CS_P$ which is a geometric object living on any manifold $M$ which is equipped with a Spin-bundle $P$. Its definition involves some kind of [[pullback]] of the basic bundle gerbe over the Spin group. He demonstrates that a [[string structure]] $S$ on $P$ is a _trivialization_ of this 2-gerbe, in the sense of a morphism from the trivial 2-gerbe living over $M$ to $CS_P$. Such a trivialization only exists precisely if the first fractional Pontryagin class of the bundle vanishes, $\frac{1}{2} p_1(P) = 0$, otherwise the Chern-Simons 2-gerbe is globally nontrivial and a string structure doesn't exist. He defined a  _string connection_ on a string structure $S$ (a connection on $P$ is needed here) on $P$ as a connection on this trivialization $S$ which is compatible with various things.

Basically, the data of a string structure together with a string connection on $P$ is a class in the _$[CS_P]$-twisted_ differential cohomology of $M$ in Urs's sense, for vanishing twist (the twist is the nontriviality of the String structure). Konrad described various results about string connections, such as the fact that they form a contractible space, confirming a conjecture by Stolz and Teichner. See his recent [arXiv article](http://arxiv.org/abs/0906.0117).

## Kevin Walker: _Blob homology_. ##
*  Slides available on [webpage](http://canyon23.net/math)

Kevin Walker described a new way to think about [[FQFT|extended TQFTs]]. Namely to an n-manifold $M$ and an [[n-category]] $C$ he defined a '[[blob complex]]' $B_*(M, C)$. The construction produces a kind of 'derived version' of an extended TQFT. The idea is that the usual quantum invariants $Z(M)$ to manifolds of dimension $\leq n$ are produced out of this chain complex, with the Atiyah-Segal axioms following as corollaries of the setup. He listed a number of examples of the TQFT's he has in mind in the beginning, such as the finite group model, one based on a *-algebra, one based on a [[pivotal category]], one based on a [[braided monoidal category|braided]] [[ribbon category]], and even one based on contact structures.

## Ralph Cohen: _String topology, field theories and Fukaya categories ##
*  No notes!?

This talk was about relating the subject of [[string topology]], which is now 10 years old, with other field theories, especially in the light of recent work by Costello, Hopkins and Lurie. [[string theory|String theory]] was described as a 'homological [[conformal field theory]]'. The slogan was that string topology simplifies when one applies Poincaré duality. A relation was sketched between string topology and [[Gromov-Witten theory|Gromov-Witten symplectic field theory]]. Ultimately it was conjectured that the symplectic field theory of $T^*M$ is equivalent to string topology of $M$.

## Corbett Redden: _String structures, 3-forms, and tmf classes_ ##
*  See also [[Oberwolfach Workshop, June 2009 -- Thursday, June 11|notes from the conference]]

Corbett began my defining a [[string structure]] on a manifold $M$ as a certain lift of a classifying map; string structures up to homotopy are canonically isomorphic with things called 'string classes' on $M$ - namely, elements in the third integral cohomology of the total space of the bundle which behave on the fibers as the generator of the third cohomology of the Spin group. The point about string structures is that they transgress to give a spin structure on the loop space of the manifold, which is what one wants to study things like the Witten genus. He showed a beautiful way to obtain string structures from a metric on $M$, using Hodge representatives for the forms, and then passing to the 'adiabatic limit', to obtain a certain 3-form $H$. A theorem of Mazzeo-Melrose, Dai and Forman then says that the kernel of the Hodge laplacian extends smoothly in the adiabatic limit, and comes from a filtration in the Serre spectral sequence (something like that). He gave a hypothesis that if a manifold $M^n$ admits a spin and a string structure and a Riemannian metric $g$ such that the Ricci curvature of $g$ is positive and that $H=0$, then a certain invariant $\sigma(M, S)=0$ inside $tmf^{-n}(pt)$. He showed that all these conditions are necessary!

## Mike Hopkins: _The Kervaire invariant_ ##
*  See also [[Oberwolfach Workshop, June 2009 -- Thursday, June 11|notes from the conference]]

Hopkins gave a nice story of the history and origins of the [[Kervaire invariant]] in the classification of [[manifold]]s. He spoke about Pontryagin's work about the cobordism groups of stably framed manifolds in the 1930's, and the mistake he made in dimension $n=2$! Basically he thought a certain function was linear, when in fact it was quadratic. This led to the Kervaire invariant being introduced. The problem about when the Kervaire invariant vanishes has been solved in dramatic fashion by Hopkins, Hill and Ravenel very recently... if a manifold has Kervaire invariant $1$, then its dimension must be either 2, 6, 14, 30, 62 or 126, with the final of these remaining open! See the notes for the great story. Hopkins ended with a nice geometric description of the Kervaire invariant on spheres in dimension 2 and 6, relating it to complex structures and exceptional Lie groups. He wondered if these things feature in dimension 126? See also Andrew Ranicki's [http://www.maths.ed.ac.uk/~aar/exotic.htm](webpage) on exotic spheres and Douglas Ravenel's [webpage](http://www.math.rochester.edu/u/faculty/doug/).


Oberwolfach Workshop, June 2009 -- Friday, June 12

Talk notes for the session of

* Friday June 12

of

* [[Oberwolfach Workshop, June 2009 -- Strings, Fields, Topology]].

## Corbett Redden: String structures and 3-forms ##

* notational conventions

  * $M^n$ a smooth closed $n$-manifold

  * $g$ a Riem. metric

  * $Spin \to P \to M$ a principal $Spin(n)$-bundle

  * $A$ a connection on $P$

  * $S$ a string class

### Outline ###

* I $\{String\ Structures\}/_{homotopy}$

* II harmonic 4-forms on $P$

* III geometry and tmf?

### I. String structures ###

**definition**

* $B String$ is defined to be the homotopy fiber $B String(n) \to B Spin(n) \stackrel{p_1/2}{\to} K(\mathbb{Z},4)$

* a **[[string structure|String structure]]** on $\pi : P \to M$ is a lift of the classifying map to $B Spin$ through
  $B String \to B Spin$
  $$
    \array{
        P && B String
        \\
        \downarrow &\nearrow& \downarrow
        \\
        M &\to& B Spin
    }
  $$

**Prop**

* String structure exists iff $\frac{1}{2}p_1(P) = 0 \in H^4(X,\mathbb{Z})$ (essentially by definition through homotopy fiber)

* let $i^*_P$ the pullback to the fiber of $P$, then we have

$$
\{ String\ structures \}/_{homotopy} \simeq_{canonical} \{ String\ classes \} := \{\rho \in H^3(P,\mathbb{Z})\ s.t.\ i_P^* \rho = 1 \in H^3(Spin, \mathbb{Z})\}
$$

* $\{string\ classes\}$ is a torsor for $H^3(M, \mathbb{Z})$ under $\rho \mapsto \rho + \pi^ H^3(M)$

Proof. universal example

$$
  \array{
    K(\mathbb{Z},3) &\simeq& \pi^* E Spin &\to & E Spin
    \\
    && B String &\to& B Spin &\to& K(\mathbb{Z}, 4)
  }
$$

>hm, here is my ([[Urs Schreiber|Urs]]) description of the situation:

consider the following pasting diagram of
[[homotopy limit|homtopy pullbacks]]
($P$ is the given $Spin$-bundle, $\hat P$ its String-lift)

$$
  \array{
    String &\to& \hat P &\to& {*}
    \\
    \downarrow && \downarrow && \downarrow
    \\
    Spin &\to& P &\to&  B^2 U(1) &\to& {*}
    \\
    \downarrow && \downarrow && \downarrow && \downarrow
    \\
    {*} &\to& X &\to& B String(n) &\to& B Spin(n)
  }
$$

**Why String structures? **

String structure on $P$ $\stackrel{transgresses}{\mapsto}$
Spin structure on loop space $L Spin \to L P \to L M$

but all the reps of this loop group are projective, so there is
actually a central $S^1$ extension of $L Spin$ in the game

$$
  \hat {L Spin} \to L Spin
$$

Need:

$$
  \array{
    \hat {L Spin} &\to & \hat {L P}
    \\
    \downarrow && \downarrow
    \\
    L Spin &\to& L P
  }
$$

$$
  \rho \in H^3(P, \mathbb{Z}) \mapsto
$$

$$
  \array{
    (\pi_! ev^*) \rho &\in& H^2(L P, \mathbb{Z})
    \\
    \downarrow && \downarrow
    \\
    iniv. ext. && H^2(L G, \mathbb{Z})
  }
$$

(on the right: $S^1$-bundles)

**String orientation** of tmf = topological modular forms

$$
M O\langle 8\rangle^{-n} = M String \stackrel{\sigma}{\to} tmf^{-n}(pt)
$$

so give a String manifold with a String class on $Spin(T M)$

$$
  M , \rho \mapsto [M, \rho] \mapsto \sigma(M,\rho)
$$

$$
  \array{
    && tmf
    \\
    & {}^{\sigma}\nearrow & \downarrow
    \\
    M String &\stackrel{Witten\ genus}{\to}& Mod Forms
  }
$$

$Witten\ genus(M) =$ "$index^{S^1} D_{L M}$"

(by the way, $\sigma$ is surjective on homotopy classes)

**warning**: I think above my $\rho$ should really be an $S$

### II Harmonic representative of $S$ ###

*reminder*

$(M,g) \mapsto \Delta = d d^* + d^* d$

$$
  H^k(M,\mathbb{R}) \simeq_{Hodge} \Delta^*_g \subset \Omega^k(M)
$$

**construction**

start with $(\pi : P  \to M, g_m, A)$

choose a bininvariant metric

$$
  g_\rho := \pi^* g_m \oplus g_{spin}
$$

where the direct sum comes from the splitting of tangent spaces $T_p P$ using
the connection that we have

Introduce scaling factor $\delta \gt 0$

$$
  g_\rho := \pi^* g_m \oplus \delta^2 g_{spin}
$$

take the "adiabatic limit" $\lim_{\delta \to 0}$

so now there is a 1-parameter family of metric on the bundle, and for each
one can look at its Laplacian, so as $\delta$ tends to 0 something becomes
singular and one has to be careful, but fortunately others already did
that for us...

**theorem** (Mazzeo-Melrose, Dai, Forman)

the kernel
$
  Ker \Delta_{g_\rho}
$

* extends smoothly to $\delta = 0$
  *(there is a smooth path in some Grassmannian space)

* and comes from a filtration isomorphic to Serre SS for $(Spin \to P \to M)$

this means that

$$
  \Rightarrow H^k(P, \mathbb{Z})
  \stackrel{\simeq}{\to}
   \lim_{\delta \to 0}
   Ker \Delta_{g_\rho}
   =: H^k(P)
$$

**Theorem** (Redden)

Given $P \stackrel{\pi}{\to} g, A$ and $\frac{1}{2}p_1(P) = 0$

then
$$
  \array{
    H^3(P, \mathbb{Z}) &\to& H^3(P,\mathbb{R}) &\to& H^3(P)
    \\
    S &&\mapsto& CS_3(A)&& CS_3(A) - \pi^* H
  }
$$

here $CS_3(A)$ is the Chern-Simons 3-form of the spin-connection

and recall $H^3(P)$ here denotes _harmonic forms_ on $P$ (should really be
script font

**remark**

in genral

$$
  [\rho]_{g_\delta} =
  CS_3(A) - \pi^* H + O(\delta) \notin \pi^* \Omega^3(M)
$$

if we have a product of two groups we accordingly would get
CS of one connection minus CS of the other.

**What is $H$? **

(first digression)

**theorem** (Chern-Simons,...)

given $(P \to M , A)  \mapsto \widehat{\frac{1}{2}p_1}(A) \hat H^4(M)$

$$
  \array{
    \Omega^3_{\mathbb{Z}} &\to&
    (M)\Omega^3(M) &\stackrel{a}{to}& \hat H^4(M) &\to& H^4(M,\mathbb{Z}) &\to& 0
    \\
    &&
    H &\mapsto& \hat{\frac{1}{2}p_1(A)}
    &\mapsto&
    \frac{1}{2}p_1(P) = 0
    &&
  }
$$

in particular

$$
  \array{
    && \mathbb{R}
    \\
    & {}^{\int H} \nearrow & \downarrow
    \\
    \mathbb{Z}_3(M)
    &\stackrel{\hat {\frac{1}{2}p_1(A)}}{\to}&
    \mathbb{R}/\mathbb{Z}
  }
$$

and secondly  $d^* H = 0$

these two properties determine $H$ uniquely up to
harmonic forms $\mathcal{H}^3_{\mathbb{Z}}(M) = ker \Delta_g$

**Equivariance**

$$
  H_{\rho + \pi^* \psi} = H_\rho + \pi^* H4
$$

where $\psi \in H^3(M, \mathbb{Z})$ $\mapsto H_4 \in \mathcal{H}^3(M)$

over all what this says is that if we go from

$$
 \array{
    Metr(M)\times A(P) \times \{String\ Classes\} & (g,A,S)
    \\
    \downarrow & \downarrow
    \\
    \Omega^3(P) & CS_3(A) - \pi^* H_{q,A,S}
    \\
    \downarrow & \downarrow
    \\
    \Omega^3(M) & H_{g, A \rho}
 }
$$

$$
  \array{
     && tmf^{-n}
     \\
     & \nearrow & \downarrow
    \\
    M String &\stackrel{}{\to}&
    MF
  }
$$

**conjecture** (S. Stolz) if $M$ is String and admits a positive
Ricci curvature metric, then $Witten(M) = 0$

question: also $\sigma(M,\rho) = 0$? no, no way!

**hypothesis**

If $M$ is a Spin manifold that admits a metric and String structure
$(g, S)$ and $A$ is the Levi-Civita connection

such that

* $Ric(g) \gt 0$

* $H_{g,s} = 0 \in \Omega^3(M)$

$\Rightarrow \sigma(M,S) = 0 \in tmf^{-n}(pt)$

**example**  $M = S^3 = SU(2)$

$$
  p_1 \in H^4(S^3) = 0
$$

$$
  H^3(S^3, \mathbb{Z}) = \mathbb{Z} = number\ of\ string\ classes
$$

$$
  d H = d^* H = 0  \Rightarrow H \in H^3(S^3, \mathbb{R}) \simeq \mathbb{R}
$$

$$
  \array{
    String\ Classes
    \\
    \downarrow
    \\
    M String^{-3} = \pi_3^S = tmf^{-3} = \mathbb{Z}/{24}
  }
$$

>(can't type the full diagram...)

consider a 1-parameter family of "berger metrics" on $S^3$

rescaling the fiber in the Hopf fibration $S^1 \to S^3 \to S^2$

## Konrad Waldorf ##

>[[Urs Schreiber|Urs]]: I had to miss that and the following two talks, hopefully somebody else has notes. Konrad's talk is based on his new article

* [[Konrad Waldorf]], _String connections and Chern-Simons theory_ ([arXiv](http://arxiv.org/abs/0906.0117))

## one more##

...

## Mike Hopkins : Kervaire invariant one ##

...

***

[[Oberwolfach Workshop, June 2009 -- Thursday, June 11|Previous day]] --- [[Oberwolfach Workshop, June 2009 -- Strings, Fields, Topology|Main workshop page]] --- No next day


Oberwolfach Workshop, June 2009 -- Monday, June 8

Here are notes from [[Urs Schreiber]] for Monday, June 8, from [[Oberwolfach Workshop, June 2009 -- Strings, Fields, Topology|Oberwolfach]].

## Schweigert: CFT and algebra in braided tensor categories, I##

### 1. Modular tensor categories and rational CFT ###

#### 1.1 ####

* consider a _rational semisimple conformal vertex algebra_ $V$

* $\to$ from this we get a representation category $C$, which is a
  [[braided monoidal category]]

  * that $V$ is semisimple and rarional makes $C$ in fact a [[modular tensor category]]

* $\to$ this gives rise to conformal blocks

**Definition: [[modular tensor category]]**

* [[abelian category]], $\mathbb{C}$-linear (i.e. $Vect_{\mathbb{C}}$ [[enriched category]]), [[semisimple category]] [[monoidal category|tensor category]]

* the tensor unit is a simple object, $I$
  a finite set of representatives of isomorphism classes of simple objects

* [[fusion category]]

* [[braided monoidal category]]

* [[ribbon category]], in particular objects have duals

* **modularity** a non-degeneracy condition on the braiding given by an isomorphism of algebras
  $$
    K(C) \otimes_{\mathbb{Z}} \stackrel{\simeq}{\to} End(Id_C)
  $$
  where
  $$
    [U] \mapsto \alpha_U
  $$
  where the transformation $\alpha_U$ is given on the simple object $V$ by
  $$
    \alpha_U(V) = straight V-line encircled by U-loop
  $$
  (on the right we use string diagram notation)

#### 1.2 ####

**Fact ([[Reshitikhin-Turaev model]])** for any
[[modular tensor category]]
$C$ there is a [[monoidal functor]]
$$
  tft_C : cobord_{3,2}^C \to vect_f(\mathcal{C})
$$

* 1) factorization homomorphism: given a surface $\Sigma_{U,U^*}$ with two marked points labeled by $U$ and $U^*$ consider the 3d cobordism from $\Sigma$ to the result of gluing a handle
to $\Sigma$ that connects the two marked points with a $U$-line running inside the handle.
the corresponding linear map given by the tft
$$
 fact_U := tft_C(\Sigma_{U,U^*} \to \Sigma)
$$
we have
*
$$
   \oplus_{i \in I} fact_{U_i}
   :
   \oplus_{i \in I} tft_C(\Sigma_{U,U^*})
   \stackrel{\simeq}{\to}
   tft_C(X)
  $$

* a representation of the mapping class group $Map(\Sigma)$ on $tft_C(X)$

### 2. CFT correlator ###

#### 2.1 ####

* $X$ a 2-dimensional conformal manifold, either oriented or unoriented
without boundary

* for the purpose of this talk restrict to the oriented case (but unoriented
case has been dealt with, too)

* also, from now on $X$ regarded just as a topological surface, no longer a
conformal one

#### 2.2 ####

strategy

* decorate $X$

* appropriate spaces of "functions" for correlators

**holomorphic factorization**

* pass from $X$ to its $\mathbb{Z}_2$ _orientation bundle_ $\hat X$,
the _orientation double cover_ (identifying points on the boundary, though)
(an orientation twisted version of what is called the __double__ )

* double accounts for what physicists call left- and rightmoving
degrees of freedom

**goal**

* 1) find a decoration for $X$ such that $\hat X \in cobord_{3,2}^C$

* 2) specify the correlator $Cor(X) \in tft_C(\hatt X)$

  * a) $Cor(X)$ invariant under $Map(\hat X)^{\mathbb{Z}_2}$
    **modular invariance**

  * b) compatibility with factorization homomorphism (technical to state, dropped here)

#### 2.3 ####

**Insight.** decoration bicategory of special symmetric Frobenius algebras in $C$

* Frobenius algebra in $C$: an object
  $$
    (A \in Obj(C), \eta : 1 \to A,\epsilon : A \to 1,
    m : A \otimes A \to A, \Delta : A \to A \otimes A)
  $$
  which is a unital associative algebra and counital coassociative coalgebra
  in $C$

* being Frobenius means that the coproduct
  $\Delta : A \to A \otimes A$ is a homomorphism of $A$-bimodules

* being _symmetric_ means that the two obvious nontriviall morphisms $A \to A^*$
  that one can build using unit and counit are equal

* being special means that $m \circ \Delta = Id_A$ and $\epsilon \circ \eta = dim(A) Id_1$

#### 2.4 ####

A typical worldsheet $X$: higher genus surface with defect lines and
marked points drawn on it

**decoration**

* to 2-dimensional cells assign special symmetric Frobenius algebras $A$;

* to 1-dimensional cells

  * to boundary lines a left or right module over the respective Frobenius algebra
    (boundary is oriented);

  * to a defect line: a bimodule over the respective Frobenius algebras

* to 0-dimensional cells

  * to a junction of three defect lines labeled by three bimodules $D_i$, associate
    an element in $Hom_{A_1| A_3}(D_1 \otimes_{A_2} D_2, D_3)$

  * boundary field insertions:
    for marked point on the boundary $x \in \partial X$
    attach a simple object $U \in Obj(C)$ to a marked point
    on a given defect line (or rather, on a junction of two defect lines) and an
    element in $Hom_A(D_1 \otimes U, D_2)$

  * bulk field insertion: for $x$ not in the boundary but in the inside of $X$,
    consider the two preimages in $\hat X$, assign simple objects $U, V$ to these,
    respectively and assign an element in
    $$
      Hom_{A_1|A_2}(U \otimes A_1 \otimes V, A_2)
    $$

  * here the bimodule structure on these tensor products are given by over- and
    underbraiding, respectively (depending on orientation)

#### 2.5 ####

correlators from cobordisms

given $X$ consider the cobordism from the empty surface

$$
  \emptyset \stackrel{M_X}{\to} \hat X
$$
given by
$$
  M_X = (\hat X \times [-1,1])/_{\sigma t \sim -t}
$$
and set
$$
  Cor(X) = tft_C(M_X) 1  \in tft_C(\hat H)
$$

**Example**

$X$ the disk with a defect line running across it from boundary to boundary.

then

* $\hat X$ is the sphere

* $M_X$ is the 3-ball

* $X$ sits inside the equatorial plane of $M_X$: one copy of $[-1,1]$ over each of
  its interior points connectint its two preimages in $\hat X$, in addtion one copy of
  the interval connecting each boundary point to its unique preimage in $\hat X$

>long discussion with audience ensues: audience wants to better understand why
all these constructions are being undertaken. The answer is in the theorem to come:
every assignment of correlators as above (compatible with factorization morphism and
invariant in the above sense under mapping class group (preserving defect decoration))
is obtained by the recipe presented here

## Runkel: CFT and modular tensor categories, II##

* Ingo Runkel, Monday morning 11.15. CFT and algebra in braided tensor categories II; notes by [[Bruce Bartlett]]: [[talk2.pdf:file]]

## Freed ##

joint work with [[Jacques Distler]] and Greg Moore.

(on [[differential cohomology]] of background fields in type II [[string theory]], in particular on [[orientifold]] backgrounds)

* $\Sigma$ in this talk a compact 2-dimensional manifold: the _worldsheet_

  * (called $X$ in the morning talks)

* $X$ smooth 10-dimensional manifold: _spacetime_

* fields $\Sigma \stackrel{\phi}{\to} X$ (see [[sigma-model]] )

* 2-dimensional theory $\to$ 10-dimensional theory

* more generally: $X$ is an [[orbifold]]

* today: a variation of this called _orientiold_ : $X_W \to X$ is a double cover

  * physicists think of $X_W$ as the spacetime equipped with an involution,
    but really spacetime is the $\mathbb{Z}_2$-quotient $X$

* I) $\sigma \in \mathbb{Z}_2$ acts trivially : _type I string_

* II) section $X_W \stackrel{\leftarrow}{\to} X$ _type II string theory_

this project started when Jacques Distler showed Dan Freed a certain formula, namely

let
$$
  i : F \hookrightarrow  X_w
$$
be the fixed point

* set
  $$
    RR-charge = \pm 2^{something} i_*
    ( \sqrt{\frac{L'(F)}{L'(\nu)}} )
  $$
  where
  $$
    L' = \prod \frac{x/4 u}{tanh x/4u}
  $$
  where
  * $u$ is the Bott generator of K-theory
  * $\nu$ is normal bundle of $F \hookrightarrow X_W$
  * with another factor this would be the Hrizebruch
      L-function, this way it is not

what is it?

this led to thinking about the following

* 1) definition of fields/theory

* 2) derive RR-charge formula over $\mathbb{Z}[1/2]$ from $10 d$

* 3) anomaly cancellation in 2d

* there is paper on arXiv with a summary, but this is still work in progress

diff geoemtric structures that one needs to make sense of this:

* differential cohomology

* twistings

### differential cohomology ###

suppose $h$ is any cohomology theory

then $h$ with rational coefficients $h(-; \mathbb{Q}) = H(-;h(pt, \mathbb{Q}))^\bullet$

here the coefficient object

$$
  h_Q := h(pt;\mathbb{Q})
$$

on the right is a graded ring and the bulleted-degree is the
corresponding total degree of that and the cohomology degree

consider the [[homotopy limit]] which gives [[differential cohomology]]

$$
  \array{
    h^{v \bullet} &\to& \Omega(M, h_{\mathbb{R}})_{closed}
    \\
    \downarrow && \downarrow
    \\
    h^\bullet(M) &\to& H(M; h_{\mathbb{R}})
  }
$$

>question from Mike Hopkins: are there choices in the bottom horizontal map; doesn't one
have to choose a basis of cocycles?

we get from this two exact sequences

$$
  0 \to h(M;h_{\mathbb{R}} \otimes \mathbb{R}/\mathbb{Z})^{q-1}
  \stackrel{curv}{\to}\Omega(M;h_{\mathbb{R}})^q_{\mathbb{Z}} \to 0
$$

$$
  0 \to forms \to h^{v \bullet}(M) \to h^q(M) \to 0
$$

* this was introduced for ordinary cohomology by Cheeger and Simons and generalized by
Hopkins and Singer

* we can think of the cohomology group as components of some space
  $$
    h^q(M) = \pi_0(Map(M, h_q))
  $$
  and
  $$
    h^{v q}(M) = \pi_0(--)
  $$

* various people constructed various models for this, such as Bunke and Schick;
  by Gomi for equivariant cohomology, also Szabo and Alessandro

**examples**

$$
    H^{v 1}(M) = Map(M, \mathbb{T})
$$

$$
    H^{1}(M) = \pi_0 Map(M, \mathbb{T})
$$

$$
    H^{v 2}(M) = \{line bundles with connection\}
$$

$$
    H^{v 3}(M) = \{line bundle gerbes with connection\}
$$

**remarks**

* the application to string theory here will have completely topological flavor

* defining certain terms in an action like Wess-Zumino-Witten  term and chern-Simons term, these are nicely understood in terms of differential cohomology: observation goes back to Gawedzki

* in physics the forms are "currents" which say where charges are located, the class in real cohomology is the total charge

* in quantum physics this total charge has to be quantized (sit on a lattice inside the real cohomology)

* so the above pullback diagram says that classical charges are to be combined with quantization condition in order to give physical fields

### twistings of $KR(X_w)$ ###

so consider again $\pi : X_w \to X$ a double cover

* an object in $KR^0(X_W)$ is represented by

  * a $\mathbb{Z}_2$-graded complex
vector bundle $E \to X_W$ (in terms of pseudobundles: even part minus odd part)

  * $\tilde \sigma : \sigma^* \bar E \to E$

recall that $\sigma$ may have fixedpoints

special cases

* $\sigma$ acts trivially: we get just $KO^0(X_w)$-theory

* $X_w \to X$ has a section: $K^0(X)$

**twisting**

pass to a locally equivalent groupoid

$$
  Y_W \to Y
$$

$$
  Y : (Y_0  \stackrel{\leftarrow}{\leftarrow} Y_1 \stackrel{\stackrel{\leftarrow}{\leftarrow}}{\leftarrow}
 \cdots)
$$

notation: $V^\phi = $ $V$ if $\phi = 0$ and $\bar V$ otherwise

**definition**

a twsiting of $KR(X_W)$ is an equivalent thing $Y_w \to Y$ as above

where

$$
  d : Y_0 \to \mathbb{Z}
$$
continuous

$$
  L \to Y_1
$$
hermitian line bundle, $\mathbb{Z}_2$-graded

$$
  \theta: L_g^{\phi(f)} \otimes L_f \to L_{g f}
$$

cocycle condition for $\theta$

recognize these twistings as classified by some cohomology theory

**cohomology group**

For $K(X)$:  $\pi_{\{0,1,2,3\}h \simeq \{\mathbb{Z}, \mathbb{Z}_2, 0 , \mathbb{Z}\}$

for some $h$ that we are not being told about

For $KO(X_W)$: $\pi_{\{0,1,2\}} k_{0 \lt 0..2\gt} \simeq \{\mathbb{Z},  \mathbb{Z}_2, \mathbb{Z}_2\}$

for $KR(X_W)$ the iso classes are

$$
  H^0(X, \mathbb{Z}) \times H^1(X; \mathbb{Z}) \times
   H^{w+3}(X, \mathbb{Z})
$$

as a set

$$
  u \in K^2(pt) \in KR^{\tau_1 +2}(pt)
$$

>speaker is running out of time, coherence is being lost a bit... notetaker misses to take notes on some central statement on these twisted cohomology classes, but see the arXiv article

### string backgrounds ###

the differential cocycles are background data for the 2-d theory and
field data for the 10 d theory (see [[sigma-model]])

**def** an NS-NS superstring background is

* i) a smooth 10d orbifold with metruc and real function (dilaton field)

* ii) $\pi : X_W \to X$ orientifold double cover

* iii) $\beta^v$ a differential twisting of $KR(X_w)$: the $B$-field

* iv) K : $R(\beta) \to \tau^{KO}(T X -2)$ iso of twistings of $KO(X)$: twisted Spin-structure

Bott shift, leading to equivalent theories $\beta^v \to \beta^v + (\tau^v + 2)$

and something else

Stiefel-Whitney classes
$$
  w_1(X) = t w
$$
$$
  w_2(X) =  t w^2 + a w
$$

aim: mix iii) and iv)

2d theory: A worldsheet  $\Sigma$ with metric,
a spin structure on $\hat \Sigma$: the orientation double cover

$$
  \array{
     \hat \Sigma &\to& X_w
     \\
     \downarrow && \downarrow
    \\
    \Sigma &\to& X
  }
$$

$$
  \phi^* w \simeq \hat w
$$

and spme spinor fields on $\hat \Sigma$

in the [[path integral]]: integrate over all of these pieces of data

$\to$ effective action Pfaffian of Dirac operator

$$
  Pfaff  D_{\hat \Sigma}(\phi^*T X - 2)
  \cdot
    exp 2 \pi i \int_{\Sigma/S} \phi^* \beta^v
$$

Pfaffian bit is section of a Pfaffian line bundle over $S$

where?

work over some parameter space $S$

both factors above are sections of a line bundle over $S$

**theorem** (in preparation) there is a hopefully canonical
trivialization of $L_\Psi \otimes L_B \to S$

action being section of bundle instead of function: annomaly:

sources

* 1) integrals over fermions $L_\Psi$: spin structure need not be equivariant under $\mathbb{Z}_2$-action

* 2) simultaneous electric and magnetic current or alternatively self-dual current

interply between 1 and 2 leads to anomaly cancellation

* 3) boundary of topological terms, like WZW, Chern-Simons

what about $L_B$: exotic orientation

## Kevin Costello;  part I ##

### deformation quantization ###

* classical mechanics: $A^{cl}$ commutative algebra with
Poisson brackets $\{-,-\}$

* this is the classical observable algebra

* to quantize this we need to find some associative algebra
  $A^q$ over the ring $\mathbb{R}[\![\hbar]\!]$

* such that

  * 1) $A^q/\hbar A^q = A^cl$

  * 2) if $a,b \in A^c{l}$, $\tilde a, \tilde b$ are lifts in $A^q$ then
    $$
      \{a,b\} = \frac{1}{\hbar} [\tilde a, \tilde b] mod \hbar
    $$

**goal** of these lectures: want to give an analog of ths picture for QFT

* 1) need to explain wha plays the role of commutative, Poisson and
  associative algebras

* 2) explain how classical field theory is encoded in commutative and Poisson

* 3) explain how to quantize

structure that play the role of associative algebras is a
[[factorization algebra]]

this is a $C^\infty$-analog (i.e. differential geometric analog)
of a **chiral algebra** in the sense of
Beilinson and Drinfeld

* let $M$ be a manifold (on which we do QFT)

* Let $B(M) = \{smooth balls in M\}$; this is an $\infty$-dim manifold

>audience: would open balls here form a manifold? does it matter?
  >answer: well, really we don't think of manifolds but of [[diffeological space]]s, of course (sheaves on manifold)

* Let $B_n(M) = \{n disjoint smooth balls embedded in larger ball in M\}$

there are obvious projection maps

$$
  B(M) \stackrel{q}{\leftarrow} B_n(M) \stackrel{p}{\to} B(M)^n
$$

everything now joint work with O. Gwilliam

**def** A **factorization algebra** is

* a [[vector bundle]] $F$ on $B(M)$

  * equipped with maps $p^*(F^{\times n}) \to q^*(F)$

    * satisfying some evident compatibility condition

    * such that everything is invariant under the obvious $S_n$-action on
      $B_n(M)$ that exchanges the order of the balls
* concretely:

  * $F$ assigns a vector space to every ball $B \subset M$

  * if we have some configuration of balls, like 2 balls $B_1, B_2$ inside a big one $B_3$
    we get a map $F(B_1)\otimes F(B_2) \to F(B_3)$

  * these must vary smoothly as the configuration of the balls varies

This is an algebra over the embedded little disk [[operad]], which is a
"colored operad" (i.e. a [[multicategory]] with more than one object)

* where the colors are $B(M)$

* $n$-ary operations are $B_n(M)$

* with extra conditions such that the vector space we assign to each color forms a smooth vector bundle

  * and all operad maps are compatible with this

notice that it happenss that for given in and out colors, there is at least one
morphism in the operad.

* there are several different reductions of this structure that are more familiar

* notion of [[vector bundle]] comes in three natural flavors

  * 1) $C^\infty$

  * 2) holomorphic

  * 3) locally constant sheaves

* definition of factorization algebra can be modified to the case of 2) and 3)

**def** a locally constant factorization algebra is like a factorization algebra,
except that instead of being a vector bundle $F$ is a locally constant sheaf on $B(M)$
of co[[chain complex]]es
and the structure maps of locally constant sheaves

>question: cohomologically locally constant or really locally constant

  > answer: I think cohomologically locally constant

let $F$ be a locally constant factorization algebra in $\mathbb{R}^n$, then
since $B(\mathbb{R}^n)$ is contractible $F$ is quasi-isomorphic to a
trivial sheaf with fiber $V$ a cochain complex

so then for instance the map $V__{B_1} \otimes V_{B_2}  \to V_{B_3}$
depends only on homotopies of the configuration,

so, a locally constant factorization algebra on $\mathbb{R}^n$ is an
$E_n$-[[algebra over an operad|algebra]]

next specialization: **holomorphic factorization algebras**

Let $\Sigma$ be a Riemannian surface.

We know what it means for a map from a complex manifold to $B(M)$ to be
holomorphic; so we can talk about holomorphic algebras on $B(\Sigma)$

a homomorphic map $U \to B(M)$ is a bundle $M \to U$ all of whose fibers are balls
and a map
$$
  \array{
    M &\to& \Sigma
    \\
    \downarrow
    \\
    U
  }
$$

actually $B(M)$, too, is also not a complex manifold but a sheaf on complex manifold

>Andre Henriquez: but with this definition, won't every map $U \to B(M)$ be holomorphic:
  > answer: oops, right

??

let's consider a holomorphic factorization algebra on $\mathcal{C}$, which is
translation invariant and dilation invariant

let $V = F$ (any round disk)

if we have a configuratoin of disks with $B_1$ and $B_2$ in $B_3$ with radii
$\epsilon_i$ with one disk in the center of the big one and the other at
complex parameter $z$, the map
$$
  m_z : V \otimes V \to V
$$
must vary holomorphically with $z$

so $z \mapsto m_z$ is a holomorphic map $Annulus \to Hom(V \otimes V, V)$, so it has
a Laurent expansion

$$
  m_z \sim \sum_{k \in \mathbb{Z}} Z^k a_k
$$

with $a_k$ in some completion of $Hom(V \otimes V , V)$

>question by Ulrich Bunke: algebraic tensor product or not?
  > answer: no, in examples tensor product will be projective tensor product

reminiscent of vertex operator algebra

notice that Beilinson-Drinfeld make the same def in the algebraic setting, in their
case the axioms are equivalent to that of a vertex operator algebra;

they show axioms for chiral algebra on $\mathcal{C}$ are essentially equivalent
to those of a vertex operator algebra

**claim**: structure of factorization algebra: good to encode quantum field theory

notice that factorizations algebras on real line tend to be associativ algebras,
so that fits in with the expectation from quantum mechanics.

***

No previous day --- [[Oberwolfach Workshop, June 2009 -- Strings, Fields, Topology|Main workshop page]] --- [[Oberwolfach Workshop, June 2009 -- Tuesday, June 9|Next day]]