nLab (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

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• Axiomatic field theories and their motivation from topology.

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• (2,1)-dimensional Euclidean field theories and tmf

$(1,1)d$ 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:

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

Example of 1-EFT

$$\sigma_1(M^n) = E : 1-EB \to tV$$

$$pt \mapsto \Gamma M$$

$$(pt \stackrel{[0,t]}{\to}) \mapsto e^{- t \Delta}$$

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

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

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

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

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

$$D = \left( \array{ 0 & D_- \\ D+ & 0 } \right)$$

$$\sigma_{1|1}(M) : Bord_{1|1} \to TV$$

$$\mathbb{R}^{0|1} \mapsto E(\mathbb{R}^{0|1}) = \Gamma(S)$$

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

$$invol \mapsto grading involution$$

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

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

and these are mapped by the EFT as

$$I_{t,\theta} \mapsto e^{-t D^2 + \theta D}$$

(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 $E$ a $1|1$ EFT a reduced non-susy field theory

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

Definition $E \in (1|1)EFT$, the partition function $Z_E$ of $E$ is the function

$$Z_E : \mathbb{R}_+ \to \mathbb{C}$$

$$t \mapsto Z_{E_{red}}(t) = E_{red}(S^1_t)$$

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

Example Consider from above the EFT

$$E = \sigma_{1|1}(M)$$

look at its reduced part

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

notice that by the above this assigns

$$[0,t] \stackrel{E_{red}}{\mapsto} e^{-t D^2}$$

$$S^1_t \mapsto str(e^{-t D^2}) = tr(e^{-t D^2})|_{even} - tr(e^{-t D^2})|_{odd}$$

where on the right we have the super trace.

This evaluates to

$$str(e^{-t D^2}) = \sum_{\lambda \in Spec(D^2)} e^{-t \lambda} sdim E_{\lambda}$$

where the super dimension? of the eigenspace $E_\lambda$ is

$$dim E^+_\lambda - dim E^-_\lambda$$

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

is an isomorphism.

So further in the computation we have

$$\cdots = dim ker D_+ - dim coker D_+ = \hat A(M)$$

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 $t$,

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

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

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

given by the assignment

$$E_{(V,\nabla)} : 1s EB(X) \to TV$$

$$(x : pt \to X) \mapsto V_x = fiber of V over x$$

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

$$\gamma \mapsto (V_{\gamma_x} \to V_{\gamma_y})$$

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

example of a $(1|1)$-EFT over $X$ consider

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

given by the assignment

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

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

now to K-theory:

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

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

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

such that

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

$$\beta \mapsto u^2$$

now the push-forward in topological K-theory

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

for $X$ a closed spin structure manifold

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

then we define

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

as follows:

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

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

we get a map

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

involving the Thom isomorphism

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

then we set

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

---

now start with $X^n$ again a spin manifold

then

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

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

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

answer presumeably, but not worked out yet

Related concepts

• supersymmetric quantum mechanics

• Euclidean quantum field theory

• spectral triple

• spectral action

• higher category theory and physics: Spectral standard model and gravity

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

• (2,1)-dimensional Euclidean field theories and tmf

• 2-spectral triple

References

• Stephan Stolz, Peter Teichner, What is an elliptic object? in Topology, geometry and quantum field theory , London Math. Soc. LNS 308, Cambridge Univ. Press (2004), 247-343. (pdf)

• 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)

• Stefan Stolz, Peter Teichner, Supersymmetric Euclidean field theories and generalized cohomology , in Hisham Sati, Urs Schreiber (eds.), Mathematical Foundations of Quantum Field and Perturbative String Theory Proceedings of Symposia in Pure Mathematics, AMS (2011)