nLab Oberwolfach Workshop, June 2009 -- Tuesday, June 9

Here are notes by Urs Schreiber for Tuesday, June 9, from Oberwolfach.

Schick: differential cohomology

smooth cohomology

  • idea:

    • combine cohomology + differential forms

main diagram

H^(M) I H (M) R Ω d=0 (M) H dR (M)H (M,) \array{ \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}) }

so differential cohomology H^ (M)\hat H^\bullet(M) combines the ordinary cohomology H (M)H^\bullet(M) with a differential form representative of its image in real cohomology.

  • II projects a differential cohomology to its underlying ordinary cohomology class;

  • RR send the differential cohomology class to its curvature_ differential form data

we want an exact sequence

H 1(M) ch Ω 1(M)/im(d) d H^(M) I H (M)0 d R Ω d=0 (M) \array{ 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) }


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

E^(M) I E (M) R Ω d=0 (M,V) E dR (M)E (M,) \array{ \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}) }


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

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

and such that there is a transformation

a:Ω 1(M)/im(d)E^ *(M) a : \Omega^{\bullet -1}(M)/{im(d)} \to \hat E^*(M)

that gives the above kind of exact sequence


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

a(ω)x=a(ωR(x)) a(\omega) \vee x = a(\omega \wedge R(x))


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

:E^ *(M×S 1)E^ 1(M) \int : \hat E^*(M \times S^1) \to \hat E^{\bullet -1}(M)

compatible with \int of forms and for EE it is given by the suspension isomorphism


p *=0 \int \circ p^* = 0

for p:M×S 1Mp : M \times S^1 \to M and

(id×(zz¯)) *= \int \circ ( id \times (z \mapsto \bar z) )^* = - \int

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 E^\hat E a smooth cohomology theory. The homotopy formula:

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

h 1 *(X)h 0 *(X)=a( M×[0,1]/Mh *(R(x))) h^*_1(X) - h^*_0(X) = a( \int_{M \times [0,1]/M} h^*(R(x)))

corollary ker(R)ker(R) (i.e. flat cohomology) is a homotopy invariant functor

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

proof of lemma

suffices to show

ι 1 *(x)ι 0 *(x)=a( M×[0,1]/MR(x))\hat _ \iota_1^*(x) - \iota_0^*(x) = a(\int_{M\times [0,1]/M} R(x))

for all xE^(M×[0,1])x \in \hat E(M \times [0,1])

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

for general xx yjhatE(M)\exists y \in \jhat E(M); xp *(y)=a(ω)x - p^*(y) = a (\omega) ωΩ(M×[0,1])\omega \in \Omega(M \times [0,1])

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

on the other hand

i 1 *(x)i 0 *(x)=i 1 *(a(ω))i 0 *(a(ω))=a(R(x)) i^*_1(x) - i^*_0(x) = i^*_1(a(\omega)) - i^*_0(a(\omega)) = a(\int R(x))

a calculation: H^ flat 1(pt)=H^ 1(pt)=/=K^ 1(pt)\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 *E^* a differential version E^ *\hat E^* as in the above definition does exist

Moreover E^ flat *=E/ 1\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 1S^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 *E^* is rationally even, meaning that

E k(pt)=0foroddk E^k(pt)\otimes \mathbb{Q} = 0 \;\; for odd k

plus one further technical assumption

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

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

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

example if we don’t require compatibility with S 1S^1-integration, then there are “exotic” abelian group structures on K^ 1\hat K^1

Bunke: smooth K-theory

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

Costello, part II

alternative to yesterday’s axioms:

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

and replace B n(M)B_n(M) by

Embed(D n,M)×Enb(D¯ nD¯ n,D¯ n) Embed(D^n, M) \times Enb(\bar D^n \coprod \cdots \coprod \bar D^n, \bar D^n)
  • 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,FF, F' are factorization algebras, then

(FF)(B)=F(B)F(B) (F\otimes F')(B) = F(B) \otimes F'(B)

def a classical factorization algebra is a commutative algebra in the category of factorization algebras

recall, an E E_\infty object in E nE_n-algebras is an E 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 EME \to M

S:Γ(M,E) S : \Gamma(M,E) \to \mathbb{R}

is the classical action

SS is local: obtained by intint of a Lagrangian

if BMB \subset M is a ball, let

EL(B)={ϕΓ(interior(B),E)suchthatϕsatisfiesEulerLagrangeequations} EL(B) = \left\{ \phi \in \Gamma(interior(B), E) such that \phi satisfies Euler-Lagrange equations \right\}

Freed: notice that you are doing here classical QFT in Euclidean signature Costello: yes

rough idea

the classical factorization algebra X SX_S associated to SS assigns to BB, the algebra

O(EL(B)) O(EL(B))

of functions on the set of solutions to ELEL.

we want maps

X S(B 1)X S(B n)X S(B n+1) X_S(B_1) \otimes \cdots \otimes X_S(B_n) \to X_S(B_{n+1})

for B iB_i in B n+1B_{n+1}

we have a map

EL(B n+1)EL(B 1)××EL(B n) EL(B_{n+1}) \to EL(B_1)\times \cdots \times EL(B_n)

this yields a map

O(EL(B 1))O(EL(B n+1)) O(EL(B_1)) \otimes \cdots \otimes O(EL(B_{n+1}))

as desried

simple example

fields are C C^\infty functions on MM

S(ϕ):= MϕΔϕ S(\phi) := \int_M \phi \Delta \phi

Euler-Lagrange equation is Δϕ=0\Delta \phi = 0

EL(B)={HarmonicfunctionsontheinteriorofB} EL(B) = \{Harmonic functions on the interior of B\}
O(EL(B))= n0Hom(EL(B) n,) S n O(EL(B)) = \prod_{n \geq 0} Hom(EL(B)^{\otimes n}, \mathbb{R})^{S_n}

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


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

recall that fact-algebras form a symmetric monoidal category

the E 0E_0-operad is defined by

  • E 0(n)=E_0(n) = \emptyset for n1n \geq 1

  • E 0(0)=ptE_0(0) = pt

so for instance an E 0E_0-algebra in VectVect is a vector space with an element

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

So: an E 0E_0-algebra in factorization algebras is just a factorization algebra

gradedcommutativealgebrawithPoissonbracketofdeg+1 quantize E 0algebras(inco[[chaincomplex]]es) Poisson quantize E 1algebras gradedcommalgebrawithPoissonbracketofdeg1 quantize E 2algebras gradedcommalgebrawithPoissonbracketofdeg2 quantize E 3algebras \array{ 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 }

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

[[]] \mathbb{R}[\![\hbar]\!]

as follows:

  • generated by \cdot, a commutative product

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

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

call this the BD operad

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


BD [[]][[]]E 0 BD \otimes_{\mathbb{R}[\![\hbar]\!]} \mathbb{R}[\![\hbar]\!] \simeq E_0

rant on BV-theory termionology

framed E 2E_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 0P_0 (or Poisson 0Poisson_0) operad is the operad of commutative Poisson algebras with {,}ofdeg1\{-,-\} of deg 1

so P 0=BD/P_0 = BD/\hbar

general fact

let MM be a manifold, and f:Mf : M \to \mathbb{R} a function, then O(derivedcriticallocusoff)O(derived critical locus of f) is a P 0P_0-operad

derived critical locus has as functions the differential graded algebra

Γ(M,Λ 2TM)dfΓ(M,Λ TM)dfO(M)=Γ(M,Λ TM) \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)

here Λ kTM\Lambda^k T M is in degree k-k with differential df\vee d f

Γ(M,Λ TM) \Gamma(M, \Lambda^\bullet T M)

has Schouten bracket, which is of degree +1

This “wants to become” E 0E_0

observation if MM has a measure, then O(crit n(f))O(crit^n(f)) has a canonical quantization to an algebra over BDBD.

then quantization is

Γ(M,Λ TM),df+Δ \Gamma(M, \Lambda^\bullet T M), \vee d f + \hbar \Delta

here Δ\Delta arises whenever MM has a measure

ΔX=DivX \Delta X = Div X
XΓ(M,TM) X \in \Gamma(M, T M)

Costello, part III

so recall that the derived critical locis if a function is a P 0P_0-algebra, so it wants to quantize to E 0E_0

if we have a classical field theory, the derived space of solutions to EL yields a P 0P_0 algebra in factorizatoin algebra

so it wants to become a factorization algebra

Example ϕC (M)\phi \in C^\infty(M), S(ϕ)=ϕΔϕS(\phi) = \int \phi \Delta \phi

derived space of solutions to EL is the complex

C (M) Δ C (M) 0 1 \array{ C^\infty(M) &\stackrel{\Delta}{\to}& C^\infty(M) \\ 0 && 1 }

if BMB \subset M is a ball, then

O(EL n(B))=Π n0(Hom(C (intB)ΔC (intB)) n,) S n 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}

this is a commutative dga and defines a commutative factorizaton algebra

if we add an interaction term to the action functional

S(ϕ)=ϕΔϕ+ϕ 3 S(\phi) = \int \phi \Delta \phi + \phi^3

then we get the same algebra of functions but the differential changes

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

What we get, when linearized looks like

(Ω 0(M) Ω 1(M) dd Ω 3(M) Ω 4(M) 1 0 1 2)g ( \array{ \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

the algebra of functions iss

Π n0Hom(E n;) S n \Pi_{n \geq 0} Hom(E^{\otimes n}; \mathbb{R})^{S_n}

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 0P_0-algebra internal to factorization algebras on MM

this amounts to quantizing the action SS 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

S(ϕ)=ϕ(Δϕ+m 2ϕ)+arbitrarylocalhigherterms S(\phi) = \int \phi (\Delta \phi + m^2 \phi) + arbitrary local higher terms

use the above theorem, around the 0-solution

Let X SX_S be the classical factorization algebra associated to it (it is a P 0P_0-algebra)

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

= {liftsof\{lifts of X_StoanalgebraoverBD/ n+1} to an algebra over BD/{\hbar^{n+1}}\}

there is a sequence T (n)T (n1)T (1)ptT^{(n)} \to T^{(n-1)} \to \cdots \to T^{(1)} \to pt

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

where T (n)T (n1)T^{(n)} \to T^{(n-1)} is a torsor for the abelian group of local functions of the field ϕ\phi

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

T (){ k1 kS (k)} T^{(\infty)} \simeq \{\sum_{k \geq 1} \hbar^k S^{(k)}\}

S (k) 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 SX_S

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

Der loc(X S) Der_{loc}(X_S) is the cochain complex of derivations of X SX_S, preserving P 0P_0-structure (is, in fact, local functions on an extended space)


there exists a sequence of simplicial sets

T (n)T (n1)T (1)pt T^{(n)} \to T^{(n-1)} \to \cdots \to T^{(1)} \to pt

with maps T (n)Q (n)(X S)T^{(n)} \to Q^{(n)}(X_S)

such that T (n)T^{(n)} fits into a homotopy fiber diagram

T (n) 0 T n1 obstruction Der loc(X S)[2] \array{ T^{(n)} &\to& 0 \\ \downarrow && \downarrow \\ T^{n-1} &\stackrel{obstruction}{\to}& Der_{loc}(X_S)[2] }

so we get obstructions; for instance for ϕ 4\phi^4-theory the obstruction is the famous β\beta-function

theorem Let gg be a simple Lie algebra. Then there is a quantization of Yang-Mills theory on 4\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 FF is a factorization algebra on M SM_S, corresponding to some QFT, then F(B)={measurementswecanmakeontheballB}F(B) = \{measurements we can make on the ball B\}

if B 1,B 2BB_1, B_2 \subset B are disjoint, the maps

F(B 1)F(B 2)F(B) F(B_1)\otimes F(B_2) \to F(B)

is defined by doing both observations

correlation functions should be cochain maps

F(B 1)F(B n) \langle F(B_1) \otimes \cdots \otimes F(B_n) \to \mathbb{R} \rangle

if B 1,,B nB_1, \cdots, B_n are disjoint, if O iF(B i) O_i \in F(B_i)


O 1,,O n \langle O_1, \cdots, O_n\rangle

is a measurement of how to observe O iO_i correlations

if B 1,B 2B˜B_1, B_2 \subset \tilde B the diagram

F(B1)F(B n) F(B˜)F(B 3)F(B n) \array{ 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) }

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)

CH (M,F)=homotopyuniversalrecipientofcorrelationfunctions CH_\bullet(M,F) = homotopy universal recipient of correlation functions
=colim B 1,B nMdisjointF(B 1)F(B n) = colim_{B_1, \cdots B_n \subset M disjoint} F(B_1) \otimes \cdots F(B_n)

(Kevin Walker (blob homology), Jacob Lurie (topological chiral homology))

lemma for a massive scalar field,

CH (M,F)[[]] CH_\bullet(M,F) \simeq \mathbb{R}[\![\hbar]\!]

in general

CH (M,F) 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 (M,F)=[[]] 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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