Here are notes by Urs Schreiber for Tuesday, June 9, from Oberwolfach.
smooth cohomology
idea:
main diagram
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.
$I$ projects a differential cohomology to its underlying ordinary cohomology class;
$R$ send the differential cohomology class to its curvature_ differential form data
we want an exact sequence
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
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
that gives the above kind of exact sequence
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
definition
$\hat E$ has $S^1$-integration if there is a natural (in $M$) transformation
compatible with $\int$ of forms and for $E$ it is given by the suspension isomorphism
and
for $p : M \times S^1 \to M$ and
remark ordinary cohomology theories are supposed to be homotopy invariant, but differential forms are not, so in general the differential cohomology is not
Lemma Given $\hat E$ a smooth cohomology theory. The homotopy formula:
given $h : M \times [0,1] \stackrel{smooth}{\to} N$ a smooth homotopy we have
corollary $ker(R)$ (i.e. flat cohomology) is a homotopy invariant functor
def \hat _H{flat} := ker(R)
proof of lemma
suffices to show
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
a calculation: $\hat H^1_{flat}(pt) = \hat H^1(pt) = \mathbb{R}/\mathbb{Z} = \hat K^1(pt)$
Theorem (Hopkins-Singer)
For each generalized cohomology theory $E^*$ a differential version $\hat E^*$ as in the above definition does exist
Moreover $\hat E_{flat}^* = E \mathbb{R}/\mathbb{Z}^{\bullet -1}$
remark it’s not evident hot to obtain more structure like multiplication
theorem using geometric models, multiplicative smooth extensions with $S^1$-integration are constructed for
K-theory (Bunke-Schick)
MU-bordisms (unitary bordisms)
(Bunke-Schröder-Schick-Wiethaupts; and from there Landweber exact cohomology theories)
uniqueness theorem (Bunke-Schick) (Simons-Sullivan proved this for ordinary integral cohomology)
assume $E^*$ is rationally even, meaning that
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.
example if we don’t require compatibility with $S^1$-integration, then there are “exotic” abelian group structures on $\hat K^1$
(I gave up taking notes in that one, maybe somebody else has notes?)
alternative to yesterday’s axioms:
replace $B(M)$ by $Embeddings(\bar D^n, M)$
and replace $B_n(M)$ by
what is the classical analog of a factorization algebra?
and how do we get classical QFT?
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
def a 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 $E \to M$
is the classical action
$S$ is local: obtained by $int$ of a Lagrangian
if $B \subset M$ is a ball, let
Freed: notice that you are doing here classical QFT in Euclidean signature Costello: yes
rough idea
the classical factorization algebra $X_S$ associated to $S$ assigns to $B$, the algebra
of functions on the set of solutions to $EL$.
we want maps
for $B_i$ in $B_{n+1}$
we have a map
this yields a map
as desried
simple example
fields are $C^\infty$ functions on $M$
Euler-Lagrange equation is $\Delta \phi = 0$
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 derived space of EL solutions
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
$E_0(n) = \emptyset$ for $n \geq 1$
$E_0(0) = pt$
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
Beilinson and Drinfeld define an operad over (i.e. in the category of cochain complex moudles over) the ring of formal power series over $\hbar$
as follows:
generated by $\cdot$, a commutative product
$\{-,-\}$ a Poisson bracket of deg +1
with differential $d(-) = \hbar \{-,-\}$
call this the BD operad
so
rant on BV-theory termionology
framed $E_2$, which is often called the BV-operad has nothing to do with the BV-theory
instead: the BD-operad from above is the one related to BV-theory
def the $P_0$ (or $Poisson_0$) operad is the operad of commutative Poisson algebras with $\{-,-\} of deg 1$
so $P_0 = BD/\hbar$
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
here $\Lambda^k T M$ is in degree $-k$ with differential $\vee d f$
has Schouten bracket, which is of degree +1
This “wants to become” $E_0$
observation if $M$ has a measure, then $O(crit^n(f))$ has a canonical quantization to an algebra over $BD$.
then quantization is
here $\Delta$ arises whenever $M$ has a measure
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
Example $\phi \in C^\infty(M)$, $S(\phi) = \int \phi \Delta \phi$
derived space of solutions to EL is the complex
if $B \subset M$ is a ball, then
this is a commutative dga and defines a commutative factorizaton algebra
if we add an interaction term to the action functional
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
the algebra of functions iss
with diffeential including YM action
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
see Kevin Costello’s book linked to on hos website for details
theorem (joint with O. Gwilliam)
(“wave” version)
conssider the scalar field theory, with an action of the form
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
$S^{(k)}$ is a local function, but this is non-canonical
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
with maps $T^{(n)} \to Q^{(n)}(X_S)$
such that $T^{(n)}$ fits into a homotopy fiber diagram
so we get obstructions; for instance for $\phi^4$-theory the obstruction is the famous $\beta$-function
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 $\hbar \mathbb{R}[\![\hbar]\!]$
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
is defined by doing both observations
correlation functions should be cochain maps
if $B_1, \cdots, B_n$ are disjoint, if $O_i \in F(B_i)$
then
is a measurement of how to observe $O_i$ correlations
if $B_1, B_2 \subset \tilde B$ the diagram
should commute (operator product expansion)
we can consider correlation functions with coefficients in any cochain complex, we require they must satisfy this equation
def (Beilinson-Drinfeld)
(Kevin Walker (blob homology), Jacob Lurie (topological chiral homology))
lemma for a massive scalar field,
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)