(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 (21)(2|1)-dimensional super-cobordism categories where cobordisms are Euclidean supermanifolds, and about the(21)the (2|1)-dimensional FQFTs given by functors on these.




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

As described at (2,1)-dimensional Euclidean field theories and tmf, the idea is that (21)(2|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

it will be shows that the claim is true at least for the cohomology ring over the point: the partition function of a (21)(2|1)-dimensional EFT is a modular form. Hence (21)(2|1)-dimensional EFTs do yield the correct cohomology ring of tmf over the point.


Let SDiff be the category of supermanifolds.

We will define a stack/fibered category on SDiffSDiff called EBord 21E Bord_{2|1} whose morphisms are smooth families of (2|1)-dimensional super-cobordisms, and a stack/fibered category sTV famsTV^{fam} of topological super vector bundles.

So recall

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 Σ\Sigma such that the “space” of maps Maps(Σ,X)Maps(\Sigma,X) naturally carries some measure that allows to perform a path integral.

This perspective suggests certain generalizations 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 generalization of Riemannian manifold with flat Riemannian metric.

notice that there is a canonical bijection between

  • flat Riemannian metrics on a dd-dimensional manifold XX

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

    Eucl( d):= dO( 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 (dδ)(d|\delta)-dimensional supermanifold is defined using the Euclidean super Lie group given by the semidirect product

Eucl( dδ):= dδSpin( d) Eucl(\mathbb{R}^{d|\delta}) := \mathbb{R}^{d|\delta} \rtimes Spin(\mathbb{R}^d)

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

first recall the notion of

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

This leads to the notion of

The morphisms of the category EBord (21)E Bord_{(2|1)} will be cobordisms that are Euclidean supermanifolds.

goal define the fibered category

EBord dδ sfam cSDiff \array{ E Bord_{d|\delta}^{sfam} \\ \downarrow \\ cSDiff }

where cSDiffcSDiff is the category of complex supermanifolds.

The objects of this fibered category are

Y ± Y Y c S \array{ Y^{\pm} &\to& Y &\leftarrow& Y^c \\ & \searrow & \downarrow & \swarrow \\ && S }

where YSY \to S is a family of Euclidean supermanifolds of dimension (dδ)(d|\delta).

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

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

S Y openU ϕ VS× cs dδ Y cUY c VS× d1δS× cs d1δ \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 0SY_0 \to S to Y 1SY_1 \to S is a diagram

Y 1 i 1 Σ i 0 Y 1 S \array{ Y_1 &\stackrel{i_1}{\to}& \Sigma &\stackrel{i_0}{\leftarrow}& Y_1 \\ & \searrow & \downarrow & \swarrow \\ && S }

where i 0,i 1i_0, i_1 are morphisms (of families of (X,G)(X,G)-spaces) satisfying the (+)-condition and the (c)-condition described at bordism categories following Stolz-Teichner.

Now a morphism in EBord dδ sfamE Bord^{sfam}_{d|\delta} from Y 0S 0Y_0 \to S_0 to Y 1S 1Y_1 \to S_1 is a bordism fitting into a diagram

Σ i 1 f *Y 1 Y 1 i 0 Y 1 S 0 f S1 \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 Σ,Σ\Sigma, \Sigma' if they are isometric – namely isomorphic in the category of Euclidean supermanifolds – “rel boundary”.

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

EFun csDiff (EBord (dδ) sfam,TV 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 cSDiffcSDiff of complex supermanifolds.

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

Recall that the ordinary category TVTV is the category of complete Hausdorff, locally convex topological vector space.

define the projective tensor product of two such V,WTVV, W \in TV. This is a certain completion of the algebraic tensor product V algWV \otimes_{alg} W with respect to the projective topology on V algWV \otimes_{alg} W.

This will be the coarsest topology (the one with the least open sets) making the following maps ff'

V algW Z f V×W \array{ V \otimes_{alg} W &\to& Z \\ & \nearrow_{f'} \\ V \times W }



C (M×N) C (M) algC (N) C (M)C (N) \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 TV sfamTV^{sfam} are pairs (S,V)(S,V) for SS a supermanifold and VV is a sheaf of locally complex 2\mathbb{Z}_2-graded topological vector space with the structure of a sheaf of modules of the structure sheaf O SO_S.

goal define the partition function of of a (21)(2|1)-dimensional Euclidean field theory.

definition Let EE be an EFT as above.

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

Z E: +×h Z_E : \mathbb{R}_+ \times h \to \mathbb {C}
(,τ)E(T ,τ) (\ell,\tau) \mapsto E(T_{\ell,\tau})

For the superversion we put

Z E:=Z E red Z_{E} := Z_{E_{red}}


EBord 21 sfam ρ E EBord 2,Spin fam E red TV fam TV sfam \array{ && E Bord_{2|1}^{sfam} \\ & {}^{\rho}\nearrow && \searrow^{E} \\ E Bord_{2, Spin}^{fam} &\stackrel{E_{red}}{\to}& TV^{fam} & \hookrightarrow & TV^{sfam} }


explicit description of EBord 1 famE Bord_{1}^{fam}

See bordism categories following Stolz-Teichner.

The category EBord 1 famE Bord_1^{fam} is generated from

  • the family of right elbows_

    1EBord + fam(,ptpt) R := +× S:= + \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

    L 0 S:=pt \array{ L_0 \\ \downarrow \\ S := pt }
  • the family of intervals in EBord + fam(pt,pt)E Bord^{fam}_{\mathbb{R}^+}(pt,pt)

    I 0 \array{ I \\ \downarrow \\ \mathbb{R}_{\geq 0} }


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

is determined by

E(pt)=:VTV E(pt) =: V \in TV
E(L 0)=:λ:VV E(L_0) =: \lambda : V \otimes V \to \mathbb{R}
E(R)=:ρTV +(,VV)C ( +,VV) E(R) =: \rho \in TV_{\mathbb{R}^+}(\mathbb{R}, V \otimes V) \simeq C^\infty(\mathbb{R}_+, V \otimes V)
E(I)=:e tHC ( ,End(V)) E(I) =: e^{-t H} \in C^\infty(\mathbb{R}_{\geq}, End(V))

forms a smooth semigroup under composition generated by

HEnd(V) H \in End(V)

(the Hamiltonian operator)

VV λ End(V) ρ e tH + \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,λ,H) (V, \lambda, H)

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

  • H:=ΔH:= \Delta the corresponding Laplace operator;

  • V:=C (M)L 2(M)V := C^\infty(M) \subset L^2(M);

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

where e tHe^{-t H} 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 EBord 2E Bord_{2}

The category EBord 2orientedfamE Bord_{2}_{oriented}^{fam} has the following generators:

objects are generated from

  • the circle K := 2/K_\ell := \mathbb{R}^2/\mathbb{Z}\cdot \ell of length >0\ell \gt 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 2SO(2)=Eucl or( 2)\mathbb{R}^2 \rtimes SO(2) = Eucl_{or}(\mathbb{R}^2)

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

morphisms are generated from

  • cylinders C ,τC_{\ell,\tau} which as a manifold is 2/\simeq \mathbb{R}^2/\mathbb{Z}\cdot \ell where τ\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 Re(τ)\ell \cdot Re(\tau) upwards and rotated by Im(τ)\ell \cdots 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 L_\ell, similar to the above, with arbitrary \ell but τ=0\tau = 0

  • the torus T τT_\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 R ,τ=T ,τ L_\ell \circ R_{\ell, \tau} = T_{\ell, \tau}

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

T ,τ=T ,τ T_{\ell', \tau'} = T_{\ell, \tau}

whenever =cτ+d\ell' = \ell \cdot|c \tau + d| and τ=aτ+bcτ+d\tau' = \frac{a \tau + b }{c \tau + d}

for (a b c d)SL 2()\left(\array{a & b \\ c & d }\right) \in SL_2(\mathbb{Z}).

Notice that SL 2()SL_2(\mathbb{Z}) is generated by

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

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

and now there is one more relation

T ,τ=T τ,1τ T_{\ell, \tau} = T_{\ell |\tau|, - \frac{1}{\tau}}

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

explicit description of EBord 2 famE Bord_{2}^{fam}

thwe category EBord 2,oriented famE Bord_{2, oriented}^{fam} is generated from


  • K S= +\array{K \\ \downarrow \\ S = \mathbb{R}_+}

  • morphisms

    L + \array{ L \\ \downarrow \\ \mathbb{R}_+ }
    R +×h/ \array{ R \\ \downarrow \\ \mathbb{R}_+ \times h/\mathbb{Z} }
    C +×(h)/ \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:


T α *T +×h α +×h (τ,1τ) < (,τ) \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

α *T=T. \alpha^* T = T \,.


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

Revised on August 14, 2012 17:47:45 by David Corfield (