(1,1)-dimensional Euclidean field theories and K-theory

This is a sub-entry of geometric models for elliptic cohomology and A Survey of Elliptic Cohomology

See there for background and context.

This entry here indicates how 1-dimensional FQFTs (the superparticle) may be related to topological K-theory.

raw material: this are notes taken more or less verbatim in a seminar – needs polishing




(1,1)d(1,1)d EFTs

recall the commercial for supergeometry with which we ended last time: the grading introduced by supergeometry makes it possible to have push-forward diagrams of the kind:

(01)TFTs n(X)/ H dR n(X) (01)TFT 0(X)/ H dR 0(pt) \array{ (0|1)TFTs^n(X)/\simeq &\leftarrow& H^n_{dR}(X) \\ \downarrow && \downarrow \\ (0|1)TFT^0(X)/\simeq &\leftarrow& H^0_{dR}(pt) }

Example of 1-EFT

σ 1(M n)=E:1EBtV \sigma_1(M^n) = E : 1-EB \to tV
ptΓM pt \mapsto \Gamma M
(pt[0,t])e tΔ (pt \stackrel{[0,t]}{\to}) \mapsto e^{- t \Delta}

Example of (11)EFT(1|1)-EFT associated to a spin manifold, there is the spinor bundle

S=S +S S = S^+ \oplus S^-

a /2\mathbb{Z}/2-graded vector bundle and on this there is the Dirac operator

D:Γ(S)Γ(S) D : \Gamma(S) \to \Gamma(S)

where Γ(S)=Γ(S +)Γ(S )\Gamma(S) = \Gamma(S^+) \oplus \Gamma(S^-). So we can write

D=(0 D D+ 0) D = \left( \array{ 0 & D_- \\ D+ & 0 } \right)
σ 11(M):Bord 11TV \sigma_{1|1}(M) : Bord_{1|1} \to TV
01E( 01)=Γ(S) \mathbb{R}^{0|1} \mapsto E(\mathbb{R}^{0|1}) = \Gamma(S)

there is an involution invol: 01 01invol : \mathbb{R}^{0|1} \to \mathbb{R}^{0|1}. It maps to

involgradinginvolution invol \mapsto grading involution

we have the following moduli space of super intervals (super 1d-bordisms)

+ 11{superintervalsI t,θ}/ \mathbb{R}^{1|1}_+ \simeq \{super intervals I_{t,\theta}\}/\sim

and these are mapped by the EFT as

I t,θe tD 2+θD I_{t,\theta} \mapsto e^{-t D^2 + \theta D}

(here we are implicitly working in the topos of sheaves on the category of supermanifolds and these equations have to be interpreted in that topos-logic, mapping generalized elements to generalized elements).

So we have for EE a 111|1 EFT a reduced non-susy field theory

(11)EBord E TV E red EBord 1 spin \array{ (1|1)EBord &\stackrel{E}{\to}& TV \\ \uparrow & \nearrow_{E_{red}} \\ EBord_1^{spin} }

Definition E(11)EFTE \in (1|1)EFT, the partition function Z EZ_E of EE is the function

Z E: + Z_E : \mathbb{R}_+ \to \mathbb{C}
tZ E red(t)=E red(S t 1) t \mapsto Z_{E_{red}}(t) = E_{red}(S^1_t)

that sends a length to the value of the EFT on the circle of that circumferene.

Example Consider from above the EFT

E=σ 11(M) E = \sigma_{1|1}(M)

look at its reduced part

z E(t)=E red(S t 1) z_E(t) = E_{red}(S^1_t)

notice that by the above this assigns

[0,t]E rede tD 2 [0,t] \stackrel{E_{red}}{\mapsto} e^{-t D^2}
S t 1str(e tD 2)=tr(e tD 2) eventr(e tD 2) odd S^1_t \mapsto str(e^{-t D^2}) = tr(e^{-t D^2})|_{even} - tr(e^{-t D^2})|_{odd}

where on the right we have the super trace.

This evaluates to

str(e tD 2)= λSpec(D 2)e tλsdimE λ str(e^{-t D^2}) = \sum_{\lambda \in Spec(D^2)} e^{-t \lambda} sdim E_{\lambda}

where the super dimension? of the eigenspace E λE_\lambda is

dimE λ +dimE λ dim E^+_\lambda - dim E^-_\lambda

and this vanishes for λ0\lambda \neq 0 since there D:E λ +E λ D : E_\lambda^+ \stackrel{\simeq}{\to} E_\lambda^-

is an isomorphism.

So further in the computation we have

=dimkerD +dimcokerD +=A^(M) \cdots = dim ker D_+ - dim coker D_+ = \hat A(M)

where the last step is the Atiyah-Singer index theorem.

So due to supersymmetry , the partition function has two very special properties:

  • it is constant – in that it does not depend on tt,

  • it takes integer values \in \mathbb{N} \subset \mathbb{R}.

recall from VXV \to X a vector bundle with connection \nabla we get a 1d EFT

E (V,)1dEFT(X) E_{(V,\nabla)} \in 1d EFT(X)

given by the assignment

E (V,):1sEB(X)TV E_{(V,\nabla)} : 1s EB(X) \to TV
(x:ptX)V x=fiberofVoverx (x : pt \to X) \mapsto V_x = fiber of V over x

a morphism is an interval [0,t][0,t] of length tt equipped with a map γ:[0,t]X\gamma : [0,t] \to X, this is sent to the parallel transport associated with the connection on a bundle

γ(V γ xV γ y) \gamma \mapsto (V_{\gamma_x} \to V_{\gamma_y})

Now refine this example to super-dimension (11)(1|1):

example of a (11)(1|1)-EFT over XX consider

EBord (11)EBord 1(X)E (V,)TV EBord_{(1|1)} \to EBord_{1}(X) \stackrel{E_{(V,\nabla)}}{\to} TV

given by the assignment

(Σ (11)X)((Σ red (11)X)paralleltransportasbefore (\Sigma^{(1|1)} \to X)( \mapsto (\Sigma^{(1|1)}_{red} \to X) \mapsto parallel transport as before

so we just forget the super-part and consider the same parallel transport as before.

now to K-theory:

KO 0(X)= KO^0(X) = Grothendieck group of real vector bundles over XX

KO n(pt)={ n=0mod4 2 n=1,2mod8 0 otherwise KO^{-n}(pt) = \left\{ \array{ \mathbb{Z} & n = 0 mod 4 \\ \mathbb{Z}_2 & n = 1,2 mod 8 \\ 0 & otherwise } \right.

there is a Bott element βKO 8(pt)\beta \in KO^{-8}(pt)

such that

KO *(pt) [u,u 1] KO^*(pt) \stackrel{\simeq_{\mathbb{Q}}}{\to} \mathbb{Z}[u,u^{-1}]
βu 2 \beta \mapsto u^2

now the push-forward in topological K-theory

p:X npt p : X^n \to pt

for XX a closed spin structure manifold

then there exists an embedding XS n+mX \hookrightarrow S^{n+m}. Let ν\nu be the normal bundle to this embedding.

then we define

X:KO k(X)KO kn(pt) \int_X : KO^k(X) \to KO^{k-n}(pt)

as follows:

let D(ν)D(\nu) be the disk bundle? and S(ν)S(\nu) be the sphere bundle of ν\nu. Then the Thom bundle? is

T(ν):=D(ν)/S(ν) T(\nu) := D(\nu)/S(\nu)

we get a map

S n+mCT(ν):=D(ν)/S(ν) S^{n+m} \stackrel{C}{\to} T(\nu) := D(\nu)/S(\nu)

involving the Thom isomorphism

C(X)={X ifxD(ν) * otherwise C(X) = \left\{ \array{ X & if x \in D(\nu) \\ * & otherwise } \right.

then we set

KO k(X) X KO kn(pt) Thomiso suspension KO˜ k+m(T(ν)) C * \array{ KO^k(X) && \stackrel{\int_X}{\to}&& KO^{k-n}(pt) \\ & {}_{Thom iso}\searrow &&& \downarrow^{\simeq}_{suspension} \\ && \tilde KO^{k+m}(T(\nu)) &\stackrel{C^*}{\to}& }

now start with X nX^n again a spin manifold


theorem (Stolz-Teichner): we have the horizontal isomorphism in the following diagram:

[E (V,)] [V +V ] 1 (11)EFT 0(X)/ conc KO 0(X) 1 quantization X σ (11)(X) EFT n(pt)/ conc KO n α(X) partitionfunc Atiyahsαinvariant ([u,u 1]) n indexD=A^(X)u n/4 \array{ && [E_{(V,\nabla)}]&& \stackrel{}{\leftarrow} && [V^+ - V^-] \\ 1 \in &&(1|1)EFT^0(X)/_{conc} &&\stackrel{\simeq}{\to}&& KO^0(X) && \ni 1 \\ \downarrow &&\downarrow^{quantization} &&&& \downarrow^{\int_X} && \downarrow \\ \sigma_{(1|1)}(X) &&EFT^{-n}(pt)/_{conc} &&\stackrel{\simeq}{\to}&& KO^{-n} && \alpha(X) \\ &\searrow&&{}_{partition func}\searrow&& \swarrow_{\simeq} && \swarrow_{Atiyah's \alpha invariant} \\ &&&& (\mathbb{Z}[u,u^{-1}])^{-n} \\ &&&& index D = \hat A(X) u^{n/4} }

question if we don’t divide out concordance, do we get differential K-theory on the right?

answer presumeably, but not worked out yet


  • Pokman Cheung, Supersymmetric field theories and cohomology (arXiv:0811.2267)

  • Stefan Stolz (notes by Arlo Caine), Supersymmetric Euclidean field theories and generalized cohomology Lecture notes (2009) (pdf)

Revised on March 21, 2014 09:25:03 by Urs Schreiber (