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

superalgebra

and

supergeometry

## Applications

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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# Contents

## $\left(1,1\right)d$ 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:

$\begin{array}{ccc}\left(0\mid 1\right){\mathrm{TFTs}}^{n}\left(X\right)/\simeq & ←& {H}_{\mathrm{dR}}^{n}\left(X\right)\\ ↓& & ↓\\ \left(0\mid 1\right){\mathrm{TFT}}^{0}\left(X\right)/\simeq & ←& {H}_{\mathrm{dR}}^{0}\left(\mathrm{pt}\right)\end{array}$\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

${\sigma }_{1}\left({M}^{n}\right)=E:1-\mathrm{EB}\to \mathrm{tV}$\sigma_1(M^n) = E : 1-EB \to tV
$\mathrm{pt}↦\Gamma M$pt \mapsto \Gamma M
$\left(\mathrm{pt}\stackrel{\left[0,t\right]}{\to }\right)↦{e}^{-t\Delta }$(pt \stackrel{[0,t]}{\to}) \mapsto e^{- t \Delta}

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

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

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

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

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

$D=\left(\begin{array}{cc}0& {D}_{-}\\ D+& 0\end{array}\right)$D = \left( \array{ 0 & D_- \\ D+ & 0 } \right)
${\sigma }_{1\mid 1}\left(M\right):{\mathrm{Bord}}_{1\mid 1}\to \mathrm{TV}$\sigma_{1|1}(M) : Bord_{1|1} \to TV
${ℝ}^{0\mid 1}↦E\left({ℝ}^{0\mid 1}\right)=\Gamma \left(S\right)$\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

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

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

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

and these are mapped by the EFT as

${I}_{t,\theta }↦{e}^{-t{D}^{2}+\theta D}$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

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

Definition $E\in \left(1\mid 1\right)\mathrm{EFT}$, the partition function ${Z}_{E}$ of $E$ is the function

${Z}_{E}:{ℝ}_{+}\to ℂ$Z_E : \mathbb{R}_+ \to \mathbb{C}
$t↦{Z}_{{E}_{\mathrm{red}}}\left(t\right)={E}_{\mathrm{red}}\left({S}_{t}^{1}\right)$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

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

look at its reduced part

${z}_{E}\left(t\right)={E}_{\mathrm{red}}\left({S}_{t}^{1}\right)$z_E(t) = E_{red}(S^1_t)

notice that by the above this assigns

$\left[0,t\right]\stackrel{{E}_{\mathrm{red}}}{↦}{e}^{-t{D}^{2}}$[0,t] \stackrel{E_{red}}{\mapsto} e^{-t D^2}
${S}_{t}^{1}↦\mathrm{str}\left({e}^{-t{D}^{2}}\right)=\mathrm{tr}\left({e}^{-t{D}^{2}}\right){\mid }_{\mathrm{even}}-\mathrm{tr}\left({e}^{-t{D}^{2}}\right){\mid }_{\mathrm{odd}}$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

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

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

$\cdots =\mathrm{dim}\mathrm{ker}{D}_{+}-\mathrm{dim}\mathrm{coker}{D}_{+}=\stackrel{^}{A}\left(M\right)$\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 $\in ℕ\subset ℝ$.

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

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

given by the assignment

${E}_{\left(V,\nabla \right)}:1s\mathrm{EB}\left(X\right)\to \mathrm{TV}$E_{(V,\nabla)} : 1s EB(X) \to TV
$\left(x:\mathrm{pt}\to X\right)↦{V}_{x}=\mathrm{fiber}\mathrm{of}V\mathrm{over}x$(x : pt \to X) \mapsto V_x = fiber of V over x

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

$\gamma ↦\left({V}_{{\gamma }_{x}}\to {V}_{{\gamma }_{y}}\right)$\gamma \mapsto (V_{\gamma_x} \to V_{\gamma_y})

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

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

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

given by the assignment

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

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

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

such that

${\mathrm{KO}}^{*}\left(\mathrm{pt}\right)\stackrel{{\simeq }_{ℚ}}{\to }ℤ\left[u,{u}^{-1}\right]$KO^*(pt) \stackrel{\simeq_{\mathbb{Q}}}{\to} \mathbb{Z}[u,u^{-1}]
$\beta ↦{u}^{2}$\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 $X↪{S}^{n+m}$. Let $\nu$ be the normal bundle to this embedding.

then we define

${\int }_{X}:{\mathrm{KO}}^{k}\left(X\right)\to {\mathrm{KO}}^{k-n}\left(\mathrm{pt}\right)$\int_X : KO^k(X) \to KO^{k-n}(pt)

as follows:

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

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

we get a map

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

involving the Thom isomorphism

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

then we set

$\begin{array}{ccccc}{\mathrm{KO}}^{k}\left(X\right)& & \stackrel{{\int }_{X}}{\to }& & {\mathrm{KO}}^{k-n}\left(\mathrm{pt}\right)\\ & {}_{\mathrm{Thom}\mathrm{iso}}↘& & & {↓}_{\mathrm{suspension}}^{\simeq }\\ & & {\stackrel{˜}{\mathrm{KO}}}^{k+m}\left(T\left(\nu \right)\right)& \stackrel{{C}^{*}}{\to }& \end{array}$\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:

$\begin{array}{ccccccc}& & \left[{E}_{\left(V,\nabla \right)}\right]& & \stackrel{}{←}& & \left[{V}^{+}-{V}^{-}\right]\\ 1\in & & \left(1\mid 1\right){\mathrm{EFT}}^{0}\left(X\right){/}_{\mathrm{conc}}& & \stackrel{\simeq }{\to }& & {\mathrm{KO}}^{0}\left(X\right)& & \ni 1\\ ↓& & {↓}^{\mathrm{quantization}}& & & & {↓}^{{\int }_{X}}& & ↓\\ {\sigma }_{\left(1\mid 1\right)}\left(X\right)& & {\mathrm{EFT}}^{-n}\left(\mathrm{pt}\right){/}_{\mathrm{conc}}& & \stackrel{\simeq }{\to }& & {\mathrm{KO}}^{-n}& & \alpha \left(X\right)\\ & ↘& & {}_{\mathrm{partition}\mathrm{func}}↘& & {↙}_{\simeq }& & {↙}_{\mathrm{Atiyah}\prime s\alpha \mathrm{invariant}}\\ & & & & \left(ℤ\left[u,{u}^{-1}\right]{\right)}^{-n}\\ & & & & \mathrm{index}D=\stackrel{^}{A}\left(X\right){u}^{n/4}\end{array}$\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

## References

• Stefan Stolz (notes by Arlo Caine), Supersymmetric Euclidean field theories and generalized cohomology Lecture notes (2009) (pdf)
Revised on May 17, 2013 02:42:20 by Urs Schreiber (82.169.65.155)