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(2,1)-dimensional Euclidean field 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 is about the definition of $(2\mid 1)$-dimensional super-cobordism categories where cobordisms are Euclidean supermanifolds, and about $\mathrm{the}(2\mid 1)$-dimensional FQFTs given by functors on these.

Previous:

• Axiomatic field theories and their motivation from topology

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

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

• bordism categories following Stolz-Teichner

Idea

Previously we had defined smooth categories of Riemannian cobordisms. Now we pass from Riemannian manifolds to Euclidean supermanifolds and define the correponding smooth cobordism category. Then we define $(d\mid \delta )$-dimensional Euclidean field theories to be smooth respentations of these categories.

As described at (2,1)-dimensional Euclidean field theories and tmf, the idea is that $(2\mid 1)$-dimensional Euclidean field theories are a geometric model for tmf cohomology theory. While there is no complete proof of this so far, in the next and final session

• modular forms from partition functions

it will be shows that the claim is true at least for the cohomology ring over the point: the partition function of a $(2\mid 1)$-dimensional EFT is a modular form. Hence $(2\mid 1)$-dimensional EFTs do yield the correct cohomology ring of tmf over the point.

Details

Let SDiff be the category of supermanifolds.

We will define a stack/fibered category on $\mathrm{SDiff}$ called $E{\mathrm{Bord}}_{2\mid 1}$ whose morphisms are smooth families of (2|1)-dimensional super-cobordisms, and a stack/fibered category ${\mathrm{sTV}}^{\mathrm{fam}}$ of topological super vector bundles.

So recall

• supergeometry.

question: What is the right notion of Riemannian or Euclidean structure on super-cobordisms?

strategy: From the path integral perspective we need some structure on $\Sigma$ such that the “space” of maps $\mathrm{Maps}(\Sigma ,X)$ naturally carries some measure that allows to perform a path integral.

This perspective suggests certain genralizations of the notion of Riemannian manifold to supermanifolds which may be a little different than what one might have thought of naively.

We want to define Euclidean supermanifolds as a genralization of Riemannian manifold with flat Riemannian metric.

notice that there is a canonical bijection between

• flat Riemannian metrics on a $d$-dimensional manifold $X$

• a maximal atlas? on $X$ consisting of charts such that all transition functions belong the the Euclidean group or Galileo group

  $$\mathrm{Eucl}({ℝ}^d):={ℝ}^d⋊O({ℝ}^d)$$Eucl(\mathbb{R}^d) := \mathbb{R}^d \rtimes O(\mathbb{R}^d)

  (rigid translations and rotations)

In analogy to that we define:

Similarly a Euclidean structure on a $(d\mid \delta )$-dimensional supermanifold is defined using the Euclidean super Lie group given by the semidirect product

$$\mathrm{Eucl}({ℝ}^{d\mid \delta }):={ℝ}^{d\mid \delta }⋊\mathrm{Spin}({ℝ}^d)$$Eucl(\mathbb{R}^{d|\delta}) := \mathbb{R}^{d|\delta} \rtimes Spin(\mathbb{R}^d)

where the Spin group acts on the translations in ${ℝ}^{d\mid \delta }$ in a way to be specified.

first recall the notion of

• complex supermanifold

goal replace the standard Euclidean group $({ℝ}^d,\mathrm{Eucl}({ℝ}^d))$ by the super Euclidean group $(X,G)$ where $X$ is a suitable supermanifold and $G$ a suitable super Lie group.

This leads to the notion of

• Euclidean supermanifold.

The morphisms of the category $E{\mathrm{Bord}}_{(2\mid 1)}$ will be cobordisms that are Euclidean supermanifolds.

goal define the fibered category

$$\begin{array}{c}E{\mathrm{Bord}}_{d\mid \delta }^{\mathrm{sfam}}\\ \downarrow \\ \mathrm{cSDiff}\end{array}$$\array{ E Bord_{d|\delta}^{sfam} \\ \downarrow \\ cSDiff }

where $\mathrm{cSDiff}$ is the category of complex supermanifolds.

The objects of this fibered category are

$$\begin{array}{ccccc}{Y}^{\pm }& \to & Y& \leftarrow & Y^c\\ & \searrow & \downarrow & \swarrow \\ & & S\end{array}$$\array{ Y^{\pm} &\to& Y &\leftarrow& Y^c \\ & \searrow & \downarrow & \swarrow \\ && S }

where $Y\to S$ is a family of Euclidean supermanifolds of dimension $(d\mid \delta )$.

For the non-super, non-family version of Euclidean bordism we require that the core $Y^c$ is totally geodesic in $Y$.

now for the superversion we require that there exist charts (in the open atlas) of $Y\to S$ covering all of $Y^c$ such that

$$\begin{array}{ccc}& & S\\ & \nearrow & & \nwarrow \\ Y{\supset }_{\mathrm{open}}U& & \stackrel{\varphi }{\to }& & V\subset S\times {ℝ}_{\mathrm{cs}}^{d\mid \delta }\\ {\downarrow }^{\supset }& & & & {\downarrow }^{\supset }\\ Y^c\supset U\cap Y^c& & \stackrel{\simeq }{\to }& & V\cap S\times {ℝ}^{d-1\mid \delta }\subset S\times {ℝ}_{\mathrm{cs}}^{d-1\mid \delta }\end{array}$$\array{ && S \\ & \nearrow && \nwarrow \\ Y \supset_{open} U &&\stackrel{\phi}{\to}&& V \subset S \times \mathbb{R}^{d|\delta}_{cs} \\ \downarrow^{\supset} &&&& \downarrow^{\supset} \\ Y^c \supset U \cap Y^c &&\stackrel{\simeq}{\to}&& V \cap S \times \mathbb{R}^{d-1|\delta} \subset S \times \mathbb{R}^{d-1|\delta}_{cs} }

next, a Euclidean superbordism from $Y_0\to S$ to $Y_1\to S$ is a diagram

$$\begin{array}{ccccc}{Y}_1& \stackrel{i_1}{\to }& \Sigma & \stackrel{i_0}{\leftarrow }& Y_1\\ & \searrow & \downarrow & \swarrow \\ & & S\end{array}$$\array{ Y_1 &\stackrel{i_1}{\to}& \Sigma &\stackrel{i_0}{\leftarrow}& Y_1 \\ & \searrow & \downarrow & \swarrow \\ && S }

where $i_0,i_1$ are morphisms (of families of $(X,G)$-spaces) satisfying the (+)-condition and the (c)-condition described at bordism categories following Stolz-Teichner.

Now a morphism in $E{\mathrm{Bord}}_{d\mid \delta }^{\mathrm{sfam}}$ from $Y_0\to S_0$ to $Y_1\to S_1$ is a bordism fitting into a diagram

$$\begin{array}{ccccc}\Sigma & \stackrel{i_1}{\leftarrow }& f^{*}{Y}_1& \to & Y_1\\ {\uparrow }^{i_0}& \searrow & \downarrow & & \downarrow \\ Y_1& \to & S_0& \stackrel{f}{\to }& S1\end{array}$$\array{ \Sigma &\stackrel{i_1}{\leftarrow}& f^* Y_1 &\to& Y_1 \\ \uparrow^{i_0} &\searrow& \downarrow && \downarrow \\ Y_1 &\to & S_0 &\stackrel{f}{\to}& S1 }

and we identify bordisms $\Sigma ,\Sigma \prime$ if they are isometric – namely isomorphic in the category of Euclidean supermanifolds – “rel boundary”.

definition A $(d\mid \delta )$-dimensional Euclidean field theory is a symmetric monoidal functor

$$E\in {\mathrm{Fun}}_{\mathrm{csDiff}}^{\otimes }(E{\mathrm{Bord}}_{(d\mid \delta )}^{\mathrm{sfam}},{\mathrm{TV}}^{\mathrm{sfam}})$$E \in Fun_{csDiff}^\otimes(E Bord_{(d|\delta)}^{sfam}, TV^{sfam})

of symmetric monoidal fibered cateories (i. symmetric monoidal functor as well as cartesian functor? ) over the category $\mathrm{cSDiff}$ of complex supermanifolds.

Definition (roughly) ${\mathrm{TV}}^{\mathrm{sfam}}$ is the category of families of topological vector spaces parameterized by complex supermanifolds.

Recall that the ordinary category $\mathrm{TV}$ is the category of complete Hausdorff, locally convex topological vector space.

define the projective tensor product? of two such $V,W\in \mathrm{TV}$. This is a certain completion of the algebraic tensor product $V{\otimes }_{\mathrm{alg}}W$ with respect to the projective topology on $V{\otimes }_{\mathrm{alg}}W$.

This will be the coarsest topology (the one with the least open sets) making the following maps $f\prime$

$$\begin{array}{ccc}V{\otimes }_{\mathrm{alg}}W& \to & Z\\ & {\nearrow }_{f\prime }\\ V\times W\end{array}$$\array{ V \otimes_{alg} W &\to& Z \\ & \nearrow_{f'} \\ V \times W }

continuous.

Remark

$$\begin{array}{ccc}{C}^{\mathrm{\infty }}(M\times N)& \leftarrow & C^{\mathrm{\infty }}(M){\otimes }_{\mathrm{alg}}{C}^{\mathrm{\infty }}(N)\\ & {}_{\simeq }\nwarrow & {\downarrow }^{\subset }\\ & & C^{\mathrm{\infty }}(M)\otimes C^{\mathrm{\infty }}(N)\end{array}$$\array{ C^\infty(M \times N) &\leftarrow& C^\infty(M) \otimes_{alg} C^\infty(N) \\ & {}_{\simeq}\nwarrow & \downarrow^{\subset} \\ && C^\infty(M) \otimes C^\infty(N) }

Definition the objects of ${\mathrm{TV}}^{\mathrm{sfam}}$ are pairs $(S,V)$ for $S$ a supermanifold and $V$ is a sheaf of locally complex ${\mathbb{Z}}_2$-graded topological vector space with the structure of a sheaf of modules of the structure sheaf $O_S$.

goal define the partition function of of a $(2\mid 1)$-dimensional Euclidean field theory.

definition Let $E$ be an EFT as above.

We may think of ${ℝ}_{+}\times h$ (positive axis times upper half plane) as moduli space of Euclidean tori, where for $(\ell ,\tau )\in {ℝ}_{+}\times h$ we get a torus (regarded as a cobordism) denoted $T_{\ell ,\tau }$. This is the torus given by the lattice spanned by $(\mathrm{1,0})$ and $\ell \mathrm{Re}(\tau )+\mathrm{Im}(\tau )$ in the upper half plane. Then for the ordinary EFT we would define

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

$$(\ell ,\tau )\mapsto E(T_{\ell ,\tau })$$(\ell,\tau) \mapsto E(T_{\ell,\tau})

For the superversion we put

$$Z_E:=Z_{E_{\mathrm{red}}}$$Z_{E} := Z_{E_{red}}

where

$$\begin{array}{ccc}& & E{\mathrm{Bord}}_{2\mid 1}^{\mathrm{sfam}}\\ & {}^{\rho }\nearrow & & {\searrow }^E\\ E{\mathrm{Bord}}_{2,\mathrm{Spin}}^{\mathrm{fam}}& \stackrel{E_{\mathrm{red}}}{\to }& {\mathrm{TV}}^{\mathrm{fam}}& \hookrightarrow & {\mathrm{TV}}^{\mathrm{sfam}}\end{array}$$\array{ && E Bord_{2|1}^{sfam} \\ & {}^{\rho}\nearrow && \searrow^{E} \\ E Bord_{2, Spin}^{fam} &\stackrel{E_{red}}{\to}& TV^{fam} & \hookrightarrow & TV^{sfam} }

Examples

explicit description of $E{\mathrm{Bord}}_1^{\mathrm{fam}}$

See bordism categories following Stolz-Teichner.

The category $E{\mathrm{Bord}}_1^{\mathrm{fam}}$ is generated from

• the family of right elbows_

  $$\begin{array}{cccc}1-E{\mathrm{Bord}}_{{ℝ}_{+}}^{\mathrm{fam}}(\varnothing ,\mathrm{pt}\coprod \mathrm{pt})& \ni & R& :={ℝ}_{+}\times ℝ\\ & & \downarrow \\ & & S:={ℝ}_{+}\end{array}$$\array{ 1-E Bord_{\mathbb{R}_+}^{fam}(\emptyset, pt \coprod pt) & \ni& R & := \mathbb{R}_+ \times \mathbb{R} \\ && \downarrow \\ && S := \mathbb{R}_+ }

• the point-family of the left elbox

  $$\begin{array}{c}{L}_0\\ \downarrow \\ S:=\mathrm{pt}\end{array}$$\array{ L_0 \\ \downarrow \\ S := pt }

• the family of intervals in $E{\mathrm{Bord}}_{{ℝ}^{+}}^{\mathrm{fam}}(\mathrm{pt},\mathrm{pt})$

  $$\begin{array}{c}I\\ \downarrow \\ {ℝ}_{\ge 0}\end{array}$$\array{ I \\ \downarrow \\ \mathbb{R}_{\geq 0} }

Because:

$E\in {\mathrm{Fun}}_{\mathrm{Diff}}^{\otimes }(E{\mathrm{Bord}}^{\mathrm{fam}},{\mathrm{TV}}^{\mathrm{fam}})$ $

is determined by

$$E(\mathrm{pt})=:V\in \mathrm{TV}$$E(pt) =: V \in TV

$$E(L_0)=:\lambda :V\otimes V\to ℝ$$E(L_0) =: \lambda : V \otimes V \to \mathbb{R}

$$E(R)=:\rho \in {\mathrm{TV}}_{{ℝ}^{+}}(ℝ,V\otimes V)\simeq C^{\mathrm{\infty }}({ℝ}_{+},V\otimes V)$$E(R) =: \rho \in TV_{\mathbb{R}^+}(\mathbb{R}, V \otimes V) \simeq C^\infty(\mathbb{R}_+, V \otimes V)

$$E(I)=:e^{-tH}\in C^{\mathrm{\infty }}({ℝ}_{\ge },\mathrm{End}(V))$$E(I) =: e^{-t H} \in C^\infty(\mathbb{R}_{\geq}, End(V))

forms a smooth semigroup under composition generated by

$$H\in \mathrm{End}(V)$$H \in End(V)

(the Hamiltonian operator)

$$\begin{array}{ccccc}V\otimes V& & \stackrel{\lambda }{\hookrightarrow }& & \mathrm{End}(V)\\ & {}_{\rho }\nwarrow & & {\nearrow }_{e^{-tH}}\\ & & {ℝ}_{+}\end{array}$$\array{ V \otimes V &&\stackrel{\lambda}{\hookrightarrow}&& End(V) \\ & {}_{\rho}\nwarrow && \nearrow_{e^{- t H}} \\ && \mathbb{R}_+ }

so due to smoothness the data collapses to the infinitesimal data

$$(V,\lambda ,H)$$(V, \lambda, H)

example – ordinary quantum mechanics Let $M$ be a Riemannian manifold. Then set

• $H:=\Delta$ the corresponding Laplace operator?;

• $V:=C^{\mathrm{\infty }}(M)\subset L^2(M)$;

• $\lambda$ is the restriction of the $L^2(M)$ inner product to $V$

where $e^{-tH}$ is “trace class” in the non-standard sense described above in that it makes the above diagram commute.

So everything as known from standard quantum mechanics textbooks, except that we don’t use the full Hilbert space of states, but just the Frechet space of smooth functions.

explicit description of $E{\mathrm{Bord}}_2$

The category $E{\mathrm{Bord}}_2{}_{\mathrm{oriented}}{}^{\mathrm{fam}}$ has the following generators:

objects are generated from

• the circle $K_{\ell }:={ℝ}^2/\mathbb{Z}\cdot \ell$ of length $\ell >0$ (with collars!! that’s why it looks like a cylinder of circumference $\ell$)

  notice that we may think of $\ell$ as parameteriing translation by $\ell$ in ${ℝ}^2⋊\mathrm{SO}(2)={\mathrm{Eucl}}_{\mathrm{or}}({ℝ}^2)$

  and the circle with $(+)$/$(-)$-collars reversed

morphisms are generated from

• cylinders $C_{\ell ,\tau }$ which as a manifold is $\simeq {ℝ}^2/\mathbb{Z}\cdot \ell$ where $\tau$ parameterizes the embedding of the outgoing circle: the incoming circle is embedded in the canonical way (the identity map on the cylinder, really), while the outgoing circle is embedded by translating the cylinder by $\ell \cdot \mathrm{Re}(\tau )$ upwards and rotated by $\ell \cdots \mathrm{Im}(\tau )$.

• right elbows which are the same as the cylinder, except that now the second circle is embedded after reflection so that it becomes an ingoing circle.

• the thin left elbow $L_{\ell }$, similar to the above, with arbitrary $\ell$ but $\tau =0$

• the torus $T_{\tau }$ obtained from the cylinder by gluing incoming and outgoing

notice the pair of pants is not a morphism in the category at all! since, recall, we require all bordisms to be flat and all boundaries to be geodesics . There is no way to put such a flat metric on the trinion.

satisfying the relations

$$L_{\ell }\circ R_{\ell ,\tau }=T_{\ell ,\tau }$$L_\ell \circ R_{\ell, \tau} = T_{\ell, \tau}

as for the non-family version, but now also with the new relations

$$T_{\ell \prime ,\tau \prime }=T_{\ell ,\tau }$$T_{\ell', \tau'} = T_{\ell, \tau}

whenever $\ell \prime =\ell \cdot \mid c\tau +d\mid$ and $\tau \prime =\frac{a\tau +b}{c\tau +d}$

for $\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)\in {\mathrm{SL}}_2(\mathbb{Z})$.

Notice that ${\mathrm{SL}}_2(\mathbb{Z})$ is generated by

• translation $(\ell ,\tau )\mapsto (\ell ,\tau +1)$

• S-matrix $(\ell ,\tau )\mapsto (\ell \cdot \mid \tau \mid ,-\frac{1}{\tau })$

and now there is one more relation

$$T_{\ell ,\tau }=T_{\ell \mid \tau \mid ,-\frac{1}{\tau }}$$T_{\ell, \tau} = T_{\ell |\tau|, - \frac{1}{\tau}}

as usual write $q:=e^{2\pi i\tau }$ which is on the pointed unit disk since $\tau$ is half plane since $\tau$

explicit description of $E{\mathrm{Bord}}_2^{\mathrm{fam}}$

thwe category $E{\mathrm{Bord}}_{2,\mathrm{oriented}}^{\mathrm{fam}}$ is generated from

objects:

• $\begin{array}{c}K\\ \downarrow \\ S={ℝ}_{+}\end{array}$

• morphisms

  $$\begin{array}{c}L\\ \downarrow \\ {ℝ}_{+}\end{array}$$\array{ L \\ \downarrow \\ \mathbb{R}_+ }
  $$\begin{array}{c}R\\ \downarrow \\ {ℝ}_{+}\times h/\mathbb{Z}\end{array}$$\array{ R \\ \downarrow \\ \mathbb{R}_+ \times h/\mathbb{Z} }
  $$\begin{array}{c}C\\ \downarrow \\ {ℝ}_{+}\times (h\cup ℝ)/\mathbb{Z}\end{array}$$\array{ C \\ \downarrow \\ \mathbb{R}_+ \times (h \cup \mathbb{R})/\mathbb{Z} }

subject to the relations

… as before (homework 3, problem 4).. and the furhter one:

for

$$\begin{array}{ccc}T& & {\alpha }^{*}T\\ \downarrow & & \downarrow \\ {ℝ}_{+}\times h& \stackrel{\alpha }{\leftarrow }& {ℝ}_{+}\times h\\ \\ (\ell \cdot \mid \tau \mid ,-\frac{1}{\tau })& \stackrel{}{<\leftarrow }& (\ell ,\tau )\end{array}$$\array{ T && \alpha^* T \\ \downarrow && \downarrow \\ \mathbb{R}_+ \times h &\stackrel{\alpha}{\leftarrow}& \mathbb{R}_+ \times h \\ \\ (\ell\cdot |\tau|, -\frac{1}{\tau}) &\stackrel{}{\lt\leftarrow}& (\ell, \tau) }

the relation is

$${\alpha }^{*}T=T.$$\alpha^* T = T \,.