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\section*{Oberwolfach Workshop, June 2009 -- Tuesday, June 9}

Here are notes by [[Urs Schreiber]] for Tuesday, June 9, from [[Oberwolfach Workshop, June 2009 -- Strings, Fields, Topology|Oberwolfach]].

\hypertarget{schick_differential_cohomology}{}\subsection*{{Schick: differential cohomology}}\label{schick_differential_cohomology}

smooth cohomology

\begin{itemize}%
\item idea:

\begin{itemize}%
\item combine cohomology + differential forms

\end{itemize}


\end{itemize}
main diagram

\begin{displaymath}
\itexarray{
    \hat H(M) &\stackrel{I}{\to}& H^\bullet(M)
    \\
    \downarrow^{R} && \downarrow
    \\
    \Omega^\bullet_{d=0}(M)
    &\stackrel{}{\to}&
    H^\bullet_{dR}(M)
    \simeq H^\bullet(M,\mathbb{R})
  }
\end{displaymath}
so differential cohomology $\hat H^\bullet(M)$ combines the ordinary cohomology $H^\bullet(M)$ with a differential form representative of its image in real cohomology.

\begin{itemize}%
\item $I$ projects a differential cohomology to its underlying ordinary cohomology class;


\item $R$ send the differential cohomology class to its \textbf{curvature\_ differential form data}



\end{itemize}
we want an exact sequence

\begin{displaymath}
\itexarray{
   H^{\bullet-1}(M)
   &\stackrel{ch}{\to}&
   \Omega^{\bullet-1}(M)/{im(d)}
   &\stackrel{d}{\to}&
   \hat H(M) 
   &\stackrel{I}{\to}& 
   H^\bullet(M) \to 0
   \\
   &&&
   {}_{d}\searrow
   &
   \downarrow^R
   \\
   &&&&
   \Omega^\bullet_{d=0}(M)
 }
\end{displaymath}
\textbf{definition}

Given cohomology theory $E^\bullet$, a smooth refinement $\hat E^\bullet$ is a functor $\hat E : Diff \to Grps$ with transformations $I, R$ such that

\begin{displaymath}
\itexarray{
    \hat E(M) &\stackrel{I}{\to}& E^\bullet(M)
    \\
    \downarrow^{R} && \downarrow
    \\
    \Omega^\bullet_{d=0}(M, V)
    &\stackrel{}{\to}&
    E^\bullet_{dR}(M)
    \simeq E^\bullet(M,\mathbb{R})
  }
\end{displaymath}
where

$V = E^\bullet(pt)\otimes \mathbb{R}$

is the graded non-torsion cohomology of $E$ on the point. So now all the gradings above denote total grading.

and such that there is a transformation

\begin{displaymath}
a : \Omega^{\bullet -1}(M)/{im(d)}
  \to
  \hat E^*(M)
\end{displaymath}
that gives the above kind of exact sequence

\textbf{definition}

if $E^*$ is multiplicative, we say $\hat E^*$ is multiplicative with product $\vee$ if $\hat E$ takes values in graded rings and the transformations are compatible with multiplicative structure, where

\begin{displaymath}
a(\omega) \vee x = a(\omega \wedge R(x))
\end{displaymath}
\textbf{definition}

$\hat E$ has $S^1$-integration if there is a natural (in $M$) transformation

\begin{displaymath}
\int : \hat E^*(M \times S^1) \to \hat E^{\bullet -1}(M)
\end{displaymath}
compatible with $\int$ of forms and for $E$ it is given by the suspension isomorphism

and

\begin{displaymath}
\int \circ p^* = 0
\end{displaymath}
for $p : M \times S^1 \to M$ and

\begin{displaymath}
\int \circ (  id \times (z \mapsto \bar z) )^* = - \int
\end{displaymath}
\textbf{remark} ordinary cohomology theories are supposed to be homotopy invariant, but differential forms are not, so in general the differential cohomology is not

\textbf{Lemma} Given $\hat E$ a smooth cohomology theory. The \textbf{homotopy formula}:

given $h : M \times [0,1] \stackrel{smooth}{\to} N$ a smooth homotopy we have

\begin{displaymath}
h^*_1(X) - h^*_0(X)
  =
  a( \int_{M \times [0,1]/M}  h^*(R(x)))
\end{displaymath}
\textbf{corollary} $ker(R)$ (i.e. flat cohomology) is a homotopy invariant functor

\textbf{def} $\hat _H{flat} := ker(R)$

\textbf{proof of lemma}

suffices to show

\begin{displaymath}
\iota_1^*(x) - \iota_0^*(x) = a(\int_{M\times [0,1]/M} R(x))
\end{displaymath}
for all $x \in \hat E(M \times [0,1])$

observe if $x = p^* y$ the left hand side vanishes, $\int R(p^* y) = 0$

for general $x$ $\exists y \in \jhat E(M)$; $x - p^*(y) = a (\omega)$ $\omega \in \Omega(M \times [0,1])$

Stokes' theorem gives $i^*_1 \omega - i^*_0 \omega = \int_{[0,1]} d \omega$ $= \int R(a(\omega)) = \int R(x-p^* \omega) = \int R(x)$

on the other hand

\begin{displaymath}
i^*_1(x) - i^*_0(x) = i^*_1(a(\omega))
    - i^*_0(a(\omega))
    = a(\int R(x))
\end{displaymath}
a calculation: $\hat H^1_{flat}(pt) = \hat H^1(pt) = \mathbb{R}/\mathbb{Z} = \hat K^1(pt)$

\textbf{Theorem (Hopkins-Singer)}

For each generalized cohomology theory $E^*$ a differential version $\hat E^*$ as in the above definition does exist

Moreover $\hat E_{flat}^* = E \mathbb{R}/\mathbb{Z}^{\bullet -1}$

\textbf{remark} it's not evident hot to obtain more structure like multiplication

\textbf{theorem} using geometric models, multiplicative smooth extensions with $S^1$-integration are constructed for

\begin{itemize}%
\item K-theory (Bunke-Schick)


\item MU-bordisms (unitary bordisms)\newline (Bunke-Schr\"o{}der-Schick-Wiethaupts; and from there Landweber exact cohomology theories)



\end{itemize}
\textbf{uniqueness theorem (Bunke-Schick)} (Simons-Sullivan proved this for ordinary integral cohomology)

assume $E^*$ is \emph{rationally even}, meaning that

\begin{displaymath}
E^k(pt)\otimes \mathbb{Q} = 0 \;\; for odd k
\end{displaymath}
plus one further technical assumption

then any two smooth extensions $\hat E^*$, $\tilde E^*$ are naturally isomorphic

such that if required to be compatible with integration the ismorphism is unique

if $\hat E, \tilde E$ are multiplicative, then this isomorphism is, as well.

\textbf{example} if we don't require compatibility with $S^1$-integration, then there are ``exotic'' abelian group structures on $\hat K^1$

\hypertarget{bunke_smooth_ktheory}{}\subsection*{{Bunke: smooth K-theory}}\label{bunke_smooth_ktheory}

(I gave up taking notes in that one, maybe somebody else has notes?)

\hypertarget{costello_part_ii}{}\subsection*{{Costello, part II}}\label{costello_part_ii}

alternative to yesterday's axioms:

replace $B(M)$ by $Embeddings(\bar D^n, M)$

and replace $B_n(M)$ by

\begin{displaymath}
Embed(D^n, M) \times Enb(\bar D^n \coprod \cdots \coprod \bar D^n, \bar D^n)
\end{displaymath}
\begin{itemize}%
\item what is the classical analog of a factorization algebra?


\item and how do we get classical QFT?



\end{itemize}
\textbf{basic idea}

factorization algebras form a symmetric monoidal category

so we can look at [[algebra over an operad]] in the category of factorization algebras

if $F, F'$ are factorization algebras, then

\begin{displaymath}
(F\otimes F')(B) = F(B) \otimes F'(B)
\end{displaymath}
\textbf{def} a \textbf{classical factorization algebra} is a commutative algebra in the category of factorization algebras

recall, an $E_\infty$ object in $E_n$-algebras is an $E_\infty$-algebra

idea of how to associate a classical factorization algebra to a classical field theory is as follows

suppose we have classical field theory, e.g. space of fields is section of a vector bundle $E \to M$

\begin{displaymath}
S : \Gamma(M,E) \to \mathbb{R}
\end{displaymath}
is the classical action

$S$ is local: obtained by $int$ of a Lagrangian

if $B  \subset M$ is a ball, let

\begin{displaymath}
EL(B) = 
  \left\{
    \phi \in \Gamma(interior(B), E)
    such that \phi satisfies Euler-Lagrange equations
  \right\}
\end{displaymath}
\begin{quote}%
Freed: notice that you are doing here classical QFT in Euclidean signature Costello: yes


\end{quote}
\textbf{rough idea}

the classical factorization algebra $X_S$ associated to $S$ assigns to $B$, the algebra

\begin{displaymath}
O(EL(B))
\end{displaymath}
of functions on the set of solutions to $EL$.

we want maps

\begin{displaymath}
X_S(B_1) \otimes \cdots \otimes X_S(B_n)
  \to
  X_S(B_{n+1})
\end{displaymath}
for $B_i$ in $B_{n+1}$

we have a map

\begin{displaymath}
EL(B_{n+1}) \to EL(B_1)\times \cdots \times EL(B_n)
\end{displaymath}
this yields a map

\begin{displaymath}
O(EL(B_1)) \otimes \cdots \otimes O(EL(B_{n+1}))
\end{displaymath}
as desried

\textbf{simple example}

fields are $C^\infty$ functions on $M$

\begin{displaymath}
S(\phi) := \int_M \phi \Delta \phi
\end{displaymath}
Euler-Lagrange equation is $\Delta \phi = 0$

\begin{displaymath}
EL(B) = \{Harmonic functions on the interior of B\}
\end{displaymath}
\begin{displaymath}
O(EL(B)) = \prod_{n \geq 0} Hom(EL(B)^{\otimes n}, \mathbb{R})^{S_n}
\end{displaymath}
where Hom means continuous linear maps, and where $\otimes$ is the completed tensor product

later, for more complex examples, what we really want to do is to take the \emph{derived} space of EL solutions

\textbf{question}

Why does this classical factorization algebra want to become just a factorization algebra?

recall that fact-algebras form a symmetric monoidal category

the $E_0$-[[operad]] is defined by

\begin{itemize}%
\item $E_0(n) = \emptyset$ for $n \geq 1$


\item $E_0(0) = pt$



\end{itemize}
so for instance an $E_0$-algebra in $Vect$ is a vector space with an element

forgot to mention that factorization algebras need to have a unit, a section $F$ on $B(M)$ which is a unit for the product

So: an $E_0$-algebra in factorization algebras is just a factorization algebra

\begin{displaymath}
\itexarray{
     graded commutative algebra with Poisson bracket of deg +1
     &\stackrel{quantize}{\to}& 
     E_0-algebras (in co[[chain complex]]es)
     \\
     Poisson &\stackrel{quantize}{\to}& E_1-algebras
     \\
     graded comm algebra with Poisson bracket of deg -1
    &\stackrel{quantize}{\to}& 
    E_2-algebras
     \\
     graded comm algebra with Poisson bracket of deg -2
    &\stackrel{quantize}{\to}& 
    E_3-algebras
  }
\end{displaymath}
Beilinson and Drinfeld define an operad over (i.e. in the category of cochain complex moudles over) the ring of formal power series over $\hbar$

\begin{displaymath}
\mathbb{R}[\![\hbar]\!]
\end{displaymath}
as follows:

\begin{itemize}%
\item generated by $\cdot$, a commutative product


\item $\{-,-\}$ a Poisson bracket of deg +1


\item with differential $d(-) = \hbar \{-,-\}$



\end{itemize}
call this the \textbf{BD operad}

\begin{displaymath}
BD/\hbar BD = operad of commutative algebras with \{-,-\} of deg +1
\end{displaymath}
\begin{displaymath}
H_\bullet(BD(n) \otimes_{\mathbb{R}[\![\hbar]\!]} \mathbb{R}(\hbar))  = 0
\end{displaymath}
so

\begin{displaymath}
BD \otimes_{\mathbb{R}[\![\hbar]\!]} \mathbb{R}[\![\hbar]\!]
  \simeq E_0
\end{displaymath}
\textbf{rant on [[BV-theory]] termionology}

framed $E_2$, which is often called the \emph{BV-operad} has \textbf{nothing} to do with the [[BV-theory]]

instead: the BD-operad from above is the one related to [[BV-theory]]

\textbf{def} the $P_0$ (or $Poisson_0$) operad is the operad of commutative Poisson algebras with $\{-,-\} of deg 1$

so $P_0 = BD/\hbar$

\textbf{general fact}

let $M$ be a manifold, and $f : M \to \mathbb{R}$ a function, then $O(derived critical locus of f)$ is a $P_0$-operad

derived critical locus has as functions the [[differential graded algebra]]

\begin{displaymath}
\cdots 
   \Gamma(M,\Lambda^2 T M)
  \stackrel{\vee d f}{\to}
   \Gamma(M,\Lambda^ T M)
  \stackrel{\vee d f}{\to}
   O(M)
  =
  \Gamma(M, \Lambda^\bullet T M)
\end{displaymath}
here $\Lambda^k T M$ is in degree $-k$ with differential $\vee d f$

\begin{displaymath}
\Gamma(M, \Lambda^\bullet T M)
\end{displaymath}
has Schouten bracket, which is of degree +1

This ``wants to become'' $E_0$

\textbf{observation} if $M$ has a [[measure]], then $O(crit^n(f))$ has a canonical quantization to an algebra over $BD$.

then quantization is

\begin{displaymath}
\Gamma(M, \Lambda^\bullet T M), \vee d f + \hbar \Delta
\end{displaymath}
here $\Delta$ arises whenever $M$ has a measure

\begin{displaymath}
\Delta X = Div X
\end{displaymath}
\begin{displaymath}
X \in \Gamma(M, T M)
\end{displaymath}
\hypertarget{costello_part_iii}{}\subsection*{{Costello, part III}}\label{costello_part_iii}

so recall that the derived critical locis if a function is a $P_0$-algebra, so it wants to quantize to $E_0$

if we have a classical field theory, the derived space of solutions to EL yields a $P_0$ algebra in factorizatoin algebra

so it wants to become a factorization algebra

\textbf{Example} $\phi \in C^\infty(M)$, $S(\phi) = \int \phi \Delta \phi$

derived space of solutions to EL is the complex

\begin{displaymath}
\itexarray{
  C^\infty(M) &\stackrel{\Delta}{\to}&
    C^\infty(M)
  \\
    0 && 1
 }
\end{displaymath}
if $B \subset M$ is a ball, then

\begin{displaymath}
O(EL^n(B))
  =
  \Pi_{n \geq 0}(
     Hom( C^\infty(int B) \stackrel{\Delta}{\to} C^\infty(int B))^{\otimes n},
     \mathbb{R}
  )^{S_n}
\end{displaymath}
this is a commutative dga and defines a commutative factorizaton algebra

if we add an interaction term to the action functional

\begin{displaymath}
S(\phi)
  =
  \int \phi \Delta \phi + \phi^3
\end{displaymath}
then we get the same algebra of functions but the differential changes

in Yang-Mills theory with gauge Lie algebra $g$: first, we consider the derived quotient of $\Omega^1(M)\otimes g$ by $\Omega^0(M)\otimes g$, then, take derived critical locus of YM action

What we get, when linearized looks like

\begin{displaymath}
(
  \itexarray{
     \Omega^0(M) &\to& \Omega^1(M) &\stackrel{d \star d}{\to}& \Omega^3(M)
       &\stackrel{}{\to}& \Omega^4(M)
     \\
     -1 && 0 && 1 && 2
  }) \otimes g
\end{displaymath}
the algebra of functions iss

\begin{displaymath}
\Pi_{n \geq 0} Hom(E^{\otimes n}; \mathbb{R})^{S_n}
\end{displaymath}
with diffeential including YM action

\textbf{theorem} if we take the derived space of solutions to the EL equations, looking infinitesimally near a fixed solution, then we find a $P_0$-algebra internal to factorization algebras on $M$

this amounts to quantizing the action $S$ into a solution of the quantum master equation

this requires machinery of counter-terms, Wilsonian effective actions, to even define the quantum master equation

\begin{quote}%
see Kevin Costello's book linked to on hos website for details


\end{quote}
\textbf{theorem} (joint with O. Gwilliam)

(``wave'' version)

conssider the scalar field theory, with an action of the form

\begin{displaymath}
S(\phi) = \int \phi (\Delta \phi + m^2 \phi)
  + arbitrary local higher terms
\end{displaymath}
use the above theorem, around the 0-solution

Let $X_S$ be the classical factorization algebra associated to it (it is a $P_0$-algebra)

Let $Q^{(n)}(X_S)$ be the set of quantizations

= $\{lifts of$X\_S$to an algebra over BD/{\hbar^{n+1}}\}$

there is a sequence $T^{(n)} \to T^{(n-1)} \to \cdots \to T^{(1)} \to pt$

where $T^{(n)}$ maps to $Q^{(n)}(X_S)$, so that the obvious diagram commutes

where $T^{(n)} \to T^{(n-1)}$ is a torsor for the abelian group of local functions of the field $\phi$

so $T^{(\infty)} = lim_n T^n$ then

\begin{displaymath}
T^{(\infty)} \simeq \{\sum_{k \geq 1} \hbar^k S^{(k)}\}
\end{displaymath}
$S^{(k)}$ is a local function, but this is non-canonical

\textbf{more sophisticated version}

consider any reasonable classical theory, with its classical factorization algebra $X_S$

let $Q^{(n)}(X_S) = simplicial set of possible quantizations defined mod \hbar^{n+1}$

$Der_{loc}(X_S)$ is the cochain complex of derivations of $X_S$, preserving $P_0$-structure (is, in fact, local functions on an extended space)

theorem

there exists a sequence of simplicial sets

\begin{displaymath}
T^{(n)} \to T^{(n-1)}
  \to 
  \cdots \to 
  T^{(1)}
  \to pt
\end{displaymath}
with maps $T^{(n)} \to Q^{(n)}(X_S)$

such that $T^{(n)}$ fits into a [[homotopy limit|homotopy fiber diagram]]

\begin{displaymath}
\itexarray{
    T^{(n)}
    &\to&
    0
    \\
    \downarrow
    &&
    \downarrow
    \\
    T^{n-1} &\stackrel{obstruction}{\to}& Der_{loc}(X_S)[2]
  }
\end{displaymath}
so we get obstructions; for instance for $\phi^4$-theory the obstruction is the famous $\beta$-function

\textbf{theorem} Let $g$ be a simple Lie algebra. Then there is a quantization of Yang-Mills theory on $\mathbb{R}^4$ which is ``renormalizable'' (behaves well with respect to scaling)

the set of quantizations is 1-dimensional term by term

The set of such quantizations is $\hbar \mathbb{R}[\![\hbar]\!]$

\textbf{correlation functions}

Where do correlation functions appear?

If $F$ is a factorization algebra on $M_S$, corresponding to some QFT, then $F(B) = \{measurements we can make on the ball B\}$

if $B_1, B_2 \subset B$ are disjoint, the maps

\begin{displaymath}
F(B_1)\otimes F(B_2) \to F(B)
\end{displaymath}
is defined by doing both observations

correlation functions should be cochain maps

\begin{displaymath}
\langle 
    F(B_1) \otimes \cdots \otimes F(B_n) \to \mathbb{R}
  \rangle
\end{displaymath}
if $B_1, \cdots, B_n$ are disjoint, if $O_i \in F(B_i)$

then

\begin{displaymath}
\langle O_1, \cdots, O_n\rangle
\end{displaymath}
is a measurement of how to observe $O_i$ correlations

if $B_1, B_2 \subset \tilde B$ the diagram

\begin{displaymath}
\itexarray{
    F(B1) \otimes \cdots \otimes F(B_n)
    &\stackrel{\langle \cdots \rangle}{\to}&
    \mathbb{R}
    \\
    \downarrow && \uparrow^{\langle \cdots \rangle}
    \\
    F(\tilde B) \otimes F(B_3) \otimes \cdots \otimes F(B_n)
  }
\end{displaymath}
should commute (operator product expansion)

we can consider correlation functions with coefficients in any cochain complex, we require they must satisfy this equation

\textbf{def} (Beilinson-Drinfeld)

\begin{displaymath}
CH_\bullet(M,F)
  =
  homotopy universal recipient of correlation functions
\end{displaymath}
\begin{displaymath}
= colim_{B_1, \cdots B_n \subset M disjoint}
F(B_1) \otimes \cdots F(B_n)
\end{displaymath}
(Kevin Walker (blob homology), Jacob Lurie (topological chiral homology))

\textbf{lemma} for a massive scalar field,

\begin{displaymath}
CH_\bullet(M,F) \simeq \mathbb{R}[\![\hbar]\!]
\end{displaymath}
in general

$CH_\bullet(M,F)$ looks like measures on the space of critical points of the classical action

if we perturb around isolated critical points, $CH_\bullet(M,F)
  =
  \mathbb{R}[\![\hbar]\!]$

in this situation correlation functions exist and are unique

general program: correlation functions define a measure on the space of classical solutions

(Feynman graphs appear here as homotopies between operads, or something, see his book)


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