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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$(1,1)d$ 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:

$$\begin{array}{ccc}(0\mid 1){\mathrm{TFTs}}^n(X)/\simeq & \leftarrow & H_{\mathrm{dR}}^n(X)\\ \downarrow & & \downarrow \\ (0\mid 1){\mathrm{TFT}}^0(X)/\simeq & \leftarrow & H_{\mathrm{dR}}^0(\mathrm{pt})\end{array}$$\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

$${\sigma }_1(M^n)=E:1-\mathrm{EB}\to \mathrm{tV}$$\sigma_1(M^n) = E : 1-EB \to tV

$$\mathrm{pt}\mapsto \Gamma M$$pt \mapsto \Gamma M

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

Example of $(1\mid 1)-\mathrm{EFT}$ associated to a spin manifold, there is the spinor bundle

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

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

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

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

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

$${\sigma }_{1\mid 1}(M):{\mathrm{Bord}}_{1\mid 1}\to \mathrm{TV}$$\sigma_{1|1}(M) : Bord_{1|1} \to TV

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

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

$$\mathrm{invol}\mapsto \mathrm{grading}\mathrm{involution}$$invol \mapsto grading involution

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

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

and these are mapped by the EFT as

$$I_{t,\theta }\mapsto e^{-t{D}^2+\theta D}$$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 $1\mid 1$ EFT a reduced non-susy field theory

$$\begin{array}{ccc}(1\mid 1)\mathrm{EBord}& \stackrel{E}{\to }& \mathrm{TV}\\ \uparrow & {\nearrow }_{E_{\mathrm{red}}}\\ {\mathrm{EBord}}_1^{\mathrm{spin}}\end{array}$$\array{ (1|1)EBord &\stackrel{E}{\to}& TV \\ \uparrow & \nearrow_{E_{red}} \\ EBord_1^{spin} }

Definition $E\in (1\mid 1)\mathrm{EFT}$, the partition function $Z_E$ of $E$ is the function

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

$$t\mapsto Z_{E_{\mathrm{red}}}(t)=E_{\mathrm{red}}(S_t^1)$$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

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

look at its reduced part

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

notice that by the above this assigns

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

$$S_t^1\mapsto \mathrm{str}(e^{-t{D}^2})=\mathrm{tr}(e^{-t{D}^2}){\mid }_{\mathrm{even}}-\mathrm{tr}(e^{-t{D}^2}){\mid }_{\mathrm{odd}}$$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

$$\mathrm{str}(e^{-t{D}^2})=\sum _{\lambda \in \mathrm{Spec}(D^2)}{e}^{-t\lambda }\mathrm{sdim}{E}_{\lambda }$$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

$$\dim E_{\lambda }^{+}-\dim E_{\lambda }^{-}$$dim E^+_\lambda - dim E^-_\lambda

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

is an isomorphism.

So further in the computation we have

$$\cdots =\dim \ker D_{+}-\dim \mathrm{coker}{D}_{+}=\hat{A}(M)$$\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 $t$,

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

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

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

given by the assignment

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

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

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

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

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

$${\mathrm{EBord}}_{(1\mid 1)}\to {\mathrm{EBord}}_1(X)\stackrel{E_{(V,\nabla )}}{\to }\mathrm{TV}$$EBord_{(1|1)} \to EBord_{1}(X) \stackrel{E_{(V,\nabla)}}{\to} TV

given by the assignment

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

${\mathrm{KO}}^0(X)=$ Grothendieck group of real vector bundles over $X$

$${\mathrm{KO}}^{-n}(\mathrm{pt})=\{\begin{array}{cc}\mathbb{Z}& n=0\mathrm{mod}4\\ {\mathbb{Z}}_2& n=1,2\mathrm{mod}8\\ 0& \mathrm{otherwise}\end{array}$$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 $\beta \in {\mathrm{KO}}^{-8}(\mathrm{pt})$

such that

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

$$\beta \mapsto u^2$$\beta \mapsto u^2

now the push-forward in topological K-theory

$$p:X^n\to \mathrm{pt}$$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

$${\int }_X:{\mathrm{KO}}^k(X)\to {\mathrm{KO}}^{k-n}(\mathrm{pt})$$\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

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

we get a map

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

involving the Thom isomorphism

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

then we set

$$\begin{array}{ccccc}{\mathrm{KO}}^k(X)& & \stackrel{{\int }_X}{\to }& & {\mathrm{KO}}^{k-n}(\mathrm{pt})\\ & {}_{\mathrm{Thom}\mathrm{iso}}\searrow & & & {\downarrow }_{\mathrm{suspension}}^{\simeq }\\ & & {\widetilde{\mathrm{KO}}}^{k+m}(T(\nu ))& \stackrel{C^{*}}{\to }& \end{array}$$\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:

$$\begin{array}{ccccccc}& & [E_{(V,\nabla )}]& & \stackrel{}{\leftarrow }& & [V^{+}-V^{-}]\\ 1\in & & (1\mid 1){\mathrm{EFT}}^0(X){/}_{\mathrm{conc}}& & \stackrel{\simeq }{\to }& & {\mathrm{KO}}^0(X)& & \ni 1\\ \downarrow & & {\downarrow }^{\mathrm{quantization}}& & & & {\downarrow }^{{\int }_X}& & \downarrow \\ {\sigma }_{(1\mid 1)}(X)& & {\mathrm{EFT}}^{-n}(\mathrm{pt}){/}_{\mathrm{conc}}& & \stackrel{\simeq }{\to }& & {\mathrm{KO}}^{-n}& & \alpha (X)\\ & \searrow & & {}_{\mathrm{partition}\mathrm{func}}\searrow & & {\swarrow }_{\simeq }& & {\swarrow }_{\mathrm{Atiyah}\prime s\alpha \mathrm{invariant}}\\ & & & & (\mathbb{Z}[u,u^{-1}]{)}^{-n}\\ & & & & \mathrm{index}D=\hat{A}(X)u^{n/4}\end{array}$$\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

• 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-spectral triple

References

• 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)