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(2,1)-dimensional Euclidean field theory

Context

Quantum field theory

Supergeometry

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

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Contents

Idea

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 (d|δ)(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 (2|1)(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 (2|1)(2|1)-dimensional EFT is a modular form. Hence (2|1)(2|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 SDiffSDiff called EBord 2|1E 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

In analogy to that we define:

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

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

where the Spin group acts on the translations in d|δ\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 (2|1)E Bord_{(2|1)} will be cobordisms that are Euclidean supermanifolds.

goal define the fibered category

EBord d|δ 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 (d|δ)(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 d|δ Y cUY c VS× d1|δS× cs d1|δ \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 d|δ 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 (d|δ)(d|\delta)-dimensional Euclidean field theory is a symmetric monoidal functor

EFun csDiff (EBord (d|δ) 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 }

continuous.

Remark

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 (2|1)(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}}

where

EBord 2|1 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} }

Examples

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

Because:

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

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

morphisms are generated from

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

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

objects:

subject to the relations

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

for

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 \,.

References