A. Euclidean supermanifolds 149
A.2. The moduli stack of super tori
In this section, we describe a subcategory ofEucl2|1in more detail. Consider first the full subcategory where the fibers of the map E → Bare compact and connected. By Proposition A.1.15, its restriction to the category of ordinary manifolds is the fibered category of closed connected Riemann surfaces equipped with a flat metric and a spin structure. The underlying topological space of such a surface must be a torus, and there are two orbits of spin structures under the action of the diffeomorphism group, distinguished by theZ/2-valued A-genus [Ati71]. We restrict our attention to the fullb subcategory where the spin structure is non-bounding, so that the Riemann surface, together with its spin structure, is the quotient ofCby a (|B|-dependent) latticeΛ. The resulting Euclidean supermanifold is then the quotient ofC1|1under left multiplication byΛ× {0} ⊂C×C0|1.
For a general base supermanifoldB, the latticeΛhas interesting deformations to a commutative subgroup ofC1|1, which will give non-split elements ofEucl2|1(B). This viewpoint was taken in [Ber20] to give an explicit model for the full subcategory ofEucl2|1 such that the fiber is a torus equipped with the non-bounding spin structure, which is a categoryEuclnb2|1fibered in groupoids oversMan. We reproduce this computation in a slightly different parametrization, which is more amenable to the computations in Chapter 4. Most importantly, we realize the odd symmetry of the universal family such that it squares to a translation alongT2, allowing the definition of the equivariant differential in Definition 4.3.1.
We start by introducing the universal family of Euclidean supermanifolds whose fibers are tori equipped with the non-bounding spin structure, together with a trivialization of the localZ2-system of the first homology of the fiber, a choice of trivialization of the U(1)-torsorP, and a section. We call this additional data apinning. The base space of this family is a generalized supermanifold, i.e., a sheaf on the site of supermanifolds.
Definition A.2.1 LetM →sManbe the category with objects
• a supermanifoldB
• two even functions(w1,w1),(w2,w2) ∈ sMan(B,C)such thatvol := 2i1(w2w1− w1w2)>0
• two odd functionsv,v∈ Ood(B)such thatvv=0
and morphisms given by mapsB → B0which pull back the functions in the obvious way.
Theuniversal family of pinned, non-bounding super toriis the functorM →Eucl2|1, which sends an object(B,wi,wi,v,v)to
• The productE=T2|1×B→B, whereT2:= (R/Z)2andT2|1:=T2×C0|1
• with the trivialU(1)-torsor, equipped with its canonical flat connection
• and the mapτ : C1+1|1 → Vect/B(E)whose value on the basis elements are the two vector fields
Dwi,v =vol−1/2(1−vζ)∂ζ−vol1/2ζ ∂z+v∂ζ
=vol−1/2
(1−vζ)∂ζ −ζ
−w2∂x+w1∂y
2i
∂wi,v =∂z+v vol−1∂ζ+ζ∂z
=vol−1
w2∂x−w1∂y
2i +v
∂ζ+ζ
−w2∂x+w1∂y 2i
where∂z = 1
w2w1−w1w2 w2∂x−w1∂y)
∂z = 1
w2w1−w1w2 −w2∂x+w1∂y)
and morphisms to the product of the mapB→B0with the identity ofT2|1. Note that−D2=∂z+v vol−1∂ζ+ζ∂z
agrees with the complex conjugate of∂modulo nilpotents, and that[D,∂] =0by virtue of the conditionvv=0, so that this data indeed defines a Euclidean structure.
Proposition A.2.2 The functor M → Eucl2|1 identifies the source with the Grothendieck construction of the sheafsMan → Setwhich sends a supermanifold B to the isomorphism classes of families(p:E→B∈ Eucl2|1(B),P, . . .)of Euclidean supermanifolds overBwhose underlying spin Riemann surface is a non-bounding torus, together with a section B → E, a trivialization of the flat U(1)-torsor P → E, and a trivialization of the local Z2-system
|b| 7→ H1(|p−1(b)|) on|B|which preserves the orientation determined by the Riemannian structure on the fiber.
Proof This statement was proven in a slightly different parametrization in [Ber20, Lemma 3.9]. Call the data of a section and trivialization ofPand the localZ-system apinning, and denote the category of families of non-bounding super tori equipped with a pinning byEucl2p|1. Given a pinned family of non-bounding super toriE → B, there is a unique local diffeomorphism B×C1|1 → E which sends0 to the chosen section, is the universal covering|B| ×C/Z2 → |E|defined by the local Z2-system on the underlying topological spaces, and pulls the Euclidean structure onEback to the standard Euclidean structure onC1|1 (with the isomorphism between flatU(1) -torsors given by the respective trivializations). The preimage of the zero-section defines a subgroupΛ ⊂ sMan(B,C1|1) isomorphic tosMan(B,Z2)(to be precise, a priori it is a subgroup ofIsom(C1|1)that acts properly discontinuously, and the resulting spin structure is non-bounding iff theU(1)-component is the identity), and the preimages of the two generators of Z2 give B-points (wi,wi,vi) ofC1|1, which have to satisfy v1v2 = 0. Settingv = −vol−1/2(w1v1+w2v2),v = −vol−1/2(w1v1+w2v2)then
gives an inverse equivalenceEucl2p|1→ M, as we can see by finding a Euclidean chart centered on(0, 0, 0):
For this, we need to solve the equations
∂z =1 Dζe=1 Dz=−ζe Dz=∂ζe=∂z =0 (z,z,ζe)(0, 0, 0) =0
Starting from the casev=v =0, it is easy to see that the unique solution in a neighbor-hood of(0, 0, 0)is
z= w1x+w2y
z= w1x+w2y−2ζ v(w1x+w2y) +v(w1x+w2y) ζe=vol1/2ζ−vol−1/2 v(w1x+w2y) +v(w1x+w2y)
These functions in fact define an isomorphism between the universal cover, whose underlying supermanifold is canonically identified withM ×C1|1, with the constant familyM ×C1|1equipped with the standard Euclidean structure. This isomorphism sends the preimageZ2of the pointZ= (0, 0, 0)to the subgroup ofC∞(B,C1|1)given by elements
mw1+nw2,mw1+nw2,−(mw1+nw2)v−(mw1+nw2)v vol1/2
(m,n)∈C∞(B,Z)2
.
We now investigate symmetries of the universal pinned family, for which it is advanta-geous to adapt the language ofstacks; compare [Hei05] for an introduction to stacks on the site of manifolds, which applies verbatim to supermanifolds.
Proposition A.2.3 The odd vector field
DM+ =∂ζ+ ζ(−w2∂x+w1∂y)
2i +
∑
i
2(wiv+wiv)∂wi
is a Euclidean symmetry ofT2M|1 → Msquaring to the fiber-preserving symmetryvol·∂z =
−w2∂x+w1∂y
2i .
The (super) Lie groupU(1)×SL(2,Z)nT2acts by Euclidean symmetries onT2M|1→ M, withSL(2,Z)×T2preserving the fibers and commuting withDM+. The weights of the various variables under theU(1)-action are listed in Table A.1.
ξ kξ
wi 2 v,DM+ 1 x,y 0
ζ −1 wi −2 v −3
Table A.1.: Transformation properties of various objects onT2M|1under theU(1)-action, where the variableξ on the left transforms asu◦ξ = ukξξ.
This gives a (homotopy) pullback diagram of stacks
(M ×C0|1)//(U(1)×SL(2,Z)nT2) M//(U(1)×SL(2,Z)nT2)
M//(U(1)×SL(2,Z)nT2) Euclnb2|1 where all maps are representable submersions with fiberC0|1.
Proof First note that DM+ is a lift of the vector field2∑i(wiv+wiv)∂wi on M and therefore defines a symmetry of the familyT2M|1 → M. To see thatDM+ is a Euclidean isometry, we have to check that it commutes with the two vector fields Dand∂, for which we first collect some auxiliary results and then use that any product of the odd parametersv,vvanishes:
DM+(vol) =2v·vol DM+, vol·∂z
=2vol·(v∂z+v∂z) D+M,∂z
=2v∂z
DM+, vol·∂z
=0 D+M,∂z
=−2v∂z D+M,∂ζ
=vol·∂z DM+,ζ∂ζ
=∂ζ −vol·∂z D+M, vol·ζ∂z
=vol·∂z
D+M,∂
=2v∂z−v[DM+, vol−1∂ζ+∂z]
=2v∂z−2v∂z
=0 D+M,D
=−vD+vol−1/2
D+M,∂ζ
−DM+, vol·ζ∂z +v
DM+,ζ∂ζ
=−vD+vol−1/2 vol·∂z−vol·∂z+v(∂ζ−vol∂z)
=0
For the square of this symmetry, we use[∂wi, vol·∂z] = [∂wi,∂wj] =0to get (D+M)2= ∂ζ+vol·ζ∂z2
= −w2∂x+w1∂y
2i .
The groupT2acts by translating the coordinatesx,y, and the groupSL(2,Z)acts by a b
c d
| {z }
=:A
·(wi,wi,v,v;x,y,ζ) = (Aw,Aw,v,v;dx−cy,ay−bx,ζ).
It is easy to verify that these actions, as well as theU(1)-action described in Table A.1, preserve the Euclidean structure and the vector fieldDM+, which has weight1under theU(1)-action. This defines commutingU(1)- andSL(2,Z)nT2-actions onT2M|1 → M, which on the corresponding family of pinned non-bounding tori correspond to changing the trivialization ofU(1)-torsor,SL(2,Z)-torsor, and base point, respectively.
The new base point is constrained to lie in a submanifold of codimension0|1, and in general, one also has to allow odd translations in the direction ofDM+. It follows that a general family of non-bounding tori over a base supermanifoldBis isomorphic to the category whose objects are pairs of aU(1)×SL(2,Z)nT2-torsor P → Band an equivariant map f :P→ M, where morphisms from(P,f)to(P0,f0)are given by an isomorphismΦ : P ∼= P0 and an odd functionυ ∈ Ood(P)which hasU(1)-degree−1 such that f = f0◦Φ+υD+M. The composition of two such morphisms(Φi,υi)is given by (Φ1◦Φ2+υ1υ2Xf1,f2(Φ1,Φ2),υ1+Φ∗1υ2), where the termXf(Φ1,Φ2)encodes the action of the Lie algebra element(D+M)2=vol·∂zofT2which depends on the parameters f.
The indicated pullback diagram is the straightforward translation of this explicit
statement into the language of stacks.
While the stackEuclnb2|1can not be (easily) represented as the quotient of a super Lie group action on a (generalized) supermanifold, the homotopy pullback we constructed tells us that the valueF(Euclnb2|1)for a stackF with respect to the submersion topology on it can be described iteratively in the following manner:
• Start withM, and form the stacky quotient with respect to the trivialT2-action, i.e., the product withpt //T2. ThenF(M//T2) ∼= F(M)T2 is the category of objects inF(M)together with a (smooth)T2-action.
• take the “central extension of Lie groupoids” defined by the odd symmetryD+M. The resulting stack classifies families of non-bounding tori with a trivialization of the resultingU(1)×SL(2,Z)-torsor, but no choice of basepoint. EvaluatingF on it yields the category ofT2-objects inF(M)together with a lift of the odd symmetry 2∑i(wiv+wiv)∂wi whose square is the infinitesimal generator−w2∂x2i+w1∂y of the T2-action (the required derivatives exist sinceF defines a stack on all supermani-folds).
• The Lie groupU(1)×SL(2,Z)acts on this category, andF(Euclnb2|1)are the ho-motopy fixed points, i.e., objects equipped with a (smooth) action of this group, twisted by the action onF(M)T2 andD+M.
We now introduce a smaller model forEuclnb2|1by breaking the symmetry between the two parametersw1,w2. The idea is that we can use theU(1)-action to rotate the parameterw2into the positive real line. However, this condition is not preserved by the action ofSL(2,Z)or the odd vector fieldD+M, and the choice of element ofU(1)is not unique since the involution−1preserves it. The consequence is that theSL(2,Z)-action on the upper half planeH ⊂ Chas to be lifted to a central extension, themetaplectic groupMp(2,Z), whose elements are pairs of a matrix
a b c d
and a choice of square root of the nowhere vanishing holomorphic functionτ 7→ cτ+d on the upper half plane. This choice fixes the element
cτ+d cτ+d
1/4
∈U(1), which means that a parameter of U(1)-weightkwill now transform with the factor
cτ+d cτ+d
k/4
. We can change this factor by multiplying the parameter with the factor(=τ)k/4, using the identity=cτaτ++db = | =τ
cτ+d|2. This leads to a convenient parametrization ofEuclnb2|1, whose proof is however slightly technical since the resulting stack is not obtained as a global quotient; it is essentially carried out in [Ber20, Section 3.3].
Proposition A.2.4 Let M>0 be the subfunctor of M for which w2 = w2 > 0, which we parametrize by
v=vol= w1w2−w1w2 2i τ= w1
w2 τ= w1 w2
λ= (=τ)−1/4v τ= (=τ)−3/4v ξ = (=τ)1/4ζ
The resulting Euclidean structure is defined by the vector fields
∂= (v=τ)−1/2
∂z+ θ
2(τ−τ)(∂ξ +ξ∂z)
D= (v=τ)−1/4
1− λξ 2
∂ξ−ξ∂z where∂z = −∂x+τ∂y
∂z = −∂x+τ∂y
The odd vector field
D+M>0 =∂ξ+ξ∂z+λv∂v+θ∂τ
defines an odd symmetry, which squares to the generator∂zof the torus action.
The metaplectic group acts by Euclidean symmetries, with the action of an element A = a b
c d
,τ7→ (cτ+d)1/2
given by the fiber-preserving diffeomorphismΦAdefined by Φ∗A(x,y) = (dx−cy,−bx+ay)
Φ∗Aξ = (cτ+d)1/2ξ Φ∗Aτ= aτ+b
cτ+d Φ∗Aτ= aτ+b cτ+d
Φ∗Aλ= (cτ+d)−1/2λ
Φ∗Aθ = (cτ+d)−1/2(cτ+d)2θ This defines a homotopy pullback square of stacks
(M>0×C0|1)//(Mp(2,Z)nT2) M>0//(Mp(2,Z)nT2)
M>0//(Mp(2,Z)nT2) Euclnb2|1
Proof In principle, this proposition can be proven by plugging the given formulas into Proposition A.2.3. The idea is that one can use suitable powers ofw1/22 ,w1/22 to replace all parameters except(w2,w2)by ones which transform trivially under theU(1)-action, which leads to a modification of theSL(2,Z)-action. Although the odd vector fieldDM+ does not preserve the subfunctorM>0, its sum with a suitable element ofU(1)does, leading to the formula forD+M
>0. Similarly, theSL(2,Z)-action does not preserveM>0, and after passing to Mp(2,Z)we can use theU(1)-action to move w2 back into the positive real axis.
It is arguably easier to verify directly that the indicated formulas define a Euclidean structure with compatible symmetries. The only difference is that now the Euclidean sym-metryD+M
>0and theMp(2,Z)-action have to be combined with a change of trivialization of theU(1)-torsor, as we have
[D+M
>0,∂] =−2
λ
4 + θ
4(τ−τ)
∂
[D+M
>0,D] =
λ
4 + θ
4(τ−τ)
D Φ∗A∂=
cτ+d cτ+d
1/2
∂
Φ∗AD=
cτ+d cτ+d
−1/4
D Φ∗ADM+
>0 = (cτ+d)−1/2DM+
>0
The result follows as before by noting thatR>0×H//(Mp(2,Z)nT2)is a presentation of the moduli stack of tori equipped with a flat metric and a non-bounding spin structure, for which the coefficients ofλandθare the universal odd deformations. In other words, we can locally choose a trivialization of theMp(2,Z)-torsor of isomorphisms of the underlying spin Riemann surface to a fixed non-bounding torus and choose a section to produce an isomorphism to the model geometry described above, and different choices of trivialization and section are related by theMp(2,Z)-action and theT2-action and odd vector field, respectively. Again, the expression of this statement in the language of
stacks yields the indicated pullback diagram.
We now unpack what this explicit description means for (convenient) vector bundles on the stack of non-bounding super tori, i.e., natural transformations between the corre-sponding sheaves of categories, which form the recipients for the partition function of twisted Euclidean field theories. A reference for vector bundles over stacks on manifolds is [Hei05, Definition 2.11]. We can use the submersionM → Euclnb2|1 to identify them with vector bundles over the underlying manifoldLatanddescent data. Such descent data are naturally produced from the quantum field theories that we consider in the body of this thesis.
Furthermore, by working with the smaller presentationM>0×Mp(2,Z)×T2|1 ⇒ M>0, we can recover a presentation of vector bundles in terms of vector bundles on R>0×H//(Mp(2,Z)nT2). We can formulate our main result, namely modularity of the partition function of the quantized2|1-sigma model, directly in terms of descent data, avoiding the use of stacks altogether, at the expense of notation and loss of knowledge that this completely describes the partition function on all families of non-bounding super tori.
Corollary A.2.5 The category of vector bundles on the stackEuclnb2|1is equivalent to the category ofU(1)×SL(2,Z)nT2-equivariant vector bundlesE→Lat, together with aSL(2,Z)nT2 -equivariant, odd self map∇D+
M ofΓ(Lat;E)h1,v,visatisfying
∇D+
M(Φ∗uσ) =u−1Φ∗u(∇D+
Mσ)
∇D+
M(fσ) =2
∑
i
(vwi+vwi)∂wif
!
σ+ f∇D+
Mσ for f ∈C∞(Lat)
∇D+
M(vσ) =−v∇D+
Mσ
∇D+
M(vσ) =−v∇D+
Mσ (∇D+
M)2= 1
2iL−w2∂x+w1∂y
whereΦu:(w1,w2)7→(u2w1,u2w2)is the action ofU(1)onLatandLXis the action of the Lie algebra elementX∈C∞(Lat;C2).
This category is equivalent to the category ofZ/2-gradedMp(2,Z)-equivariant vector bundles onR>0×H, where the action ofMp(2,Z)on the first factor and the action ofT2on both factors is trivial, together with an odd endomorphismDE ∈Γ(R>0×H; Endod(E))and even self-maps v∇v,∇τofΓ(R>0×H;E)satisfying the following properties:
• For f ∈C∞(R>0×H), we have theLeibniz rules
[v∇v,f·] = (v∂vf)·,[∇τ,f·] = (∂τf)·
In other words,v∇vdefines a partial connection along the first coordinate, and∂E :σ7→
(∇τσ)dτdefines a holomorphic structure onEfor a fixedv
• v∇visMp(2,Z)nT2-equivariant, whereas we have a b
c d
◦DE = (cτ+d)1/2DE◦ a b
c d
, a b
c d
◦ ∇τ = (cτ+d)2∇τ◦ a b
c d
i.e.,∂EisMp(2,Z)nT2-equivariant
• We have[v∇v,DE] =0,[∇τ,DE] =0, andD2Eis the action of the Lie algebra element
−∂x+τ∂yofT2
Global sections of(E,DE,v∇v,∂τ)are tuples(σev,σλ,σθ)whereσev ∈Γ(R>0×H;ETev2)Mp(2,Z) is even, invariant, and annihilated by the operatorDE,σλ ∈ Γ(R>0×H;EodT2)Mp(2,Z)is odd and invariant withDEσλ =v∇vσev, andσθdτ∈Γ(R>0×H;Ω0,1(H)⊗EodT2)Mp(2,Z)is odd and invariant withDEσθ =∇τσev.
Proof Since vector bundles satisfy descent for submersions of supermanifolds, we can use the representable submersionsM//(U(1)×SL(2,Z)nT2),M>0//(Mp(2,Z)n
T2)→Euclnb2|1to identify the category of vector bundles onEuclnb2|1with descent data for equivariant vector bundles onMandM>0, respectively. The additional descent datum is an extension of the equivariant structure overC0|1, i.e., an equivariant lift of the vector fieldD+MorD+M
>0, which must square to the chosen lift of its square.
Restriction defines a functorVect(M>0)→Vect(R>0×H), which one easily checks to be an isomorphism by induction over the filtration by powers of the ideal of nilpotent elements. Furthermore, this restriction isMp(2,Z)nT2-equivariant, identifying descent data for the substack where the odd parameter is set to zero with equivariant vector bundles onR>0×H. An extension of the isomorphismΦ|M>0×Mp(2,Z)×T2 is given by a covariant derivative∇D+
M,>0 in the direction of the projection of the odd symmetry DM+,>0 =vλ∂v+θ∂τ+. . . to the base, and decomposing its effect as∇D+
M,>0σ= DEσ+ λ(v∇vσ) +ω(∇τσ)gives the three maps from the statement. The cocycle condition then enforcesMp(2,Z)nT2-equivariance, with a factor of(cτ+d)1/2 coming from the fact that the vector fieldDM+
>0 transforms nontrivially, and the identification of(∇D+
M,>0)2 with the action of the indicated function to the Lie algebra ofT2.
Global sections are given by morphisms from the trivial vector bundle. In terms of descent data, they are sections ofEwhose pullbacks are sent to each other by the isomorphismΦ. Equivariant, even sections of a split equivariant vector bundle over M>0 are given by Γ(R>0×H;ETev2 ⊕ETod2hλ,θi)Mp(2,Z), and spelling out the descent condition in this decomposition leads to the three indicated conditions.
Example A.2.6 Let(C, dC)be a cohomologically graded chain complex, considered as aZ/2-graded vector space with aU(1)-action u·x = u|x|xencoding the grading.
Furthermore, suppose that C is equipped with a smooth SL(2,Z)nT2-action such that the action ofT2isSL(2,Z)-equivariantly trivialized, i.e., there are self-mapsιXof cohomological degree−1ofCsuch that
LX= [dC,ιX],[ιX,ιY] =0,[LX,ιY] =0,[(A·),ιX] =ιAX.
TheU(1)×SL(2,Z)nT2-equivariant structure then lifts the trivial vector bundleC→ M toVect(M//(U(1)×SL(2,Z)nT2)). The additional descent datum needed to obtain a vector bundle onEuclnb2|1 is an equivariant lift ofDM+, which one obtains by adding dC+2i1ι−w2∂x+w1∂y to the lift provided by the canonical flat connection on a trivial vector bundle.
We now want to identify the resulting vector bundle overR>0×Hwith the sum of the trivial vector bundles with fiberCk, wherekranges over the integers. For this, we trivialize theU(1)-equivariant structure by multiplying withwk/22 on the componentCk; ifkis odd, the result is to be interpreted as a section of a nontrivial vector bundle over Lat. In terms of this trivialization, the equivariant structure is given by identifying the fibers overτand aτcτ++dbby multiplication with(cτ+d)k/2, with the choice of square root depending on the lift to the metaplectic group. (In particular, the centerZ/4⊂Mp(2,Z) stabilizes every point, and it acts on the fibers by multiplication withik.) In the new
trivialization, the operatorDE is given by dC+ι−∂x+τ∂Y, whereasv∇vand∂E are the canonical connection and holomorphic structure on a trivial bundle, respectively.
It follows that global sections areC-valued functions
{xkev,xkλ ∈C∞(R>0×H,Ck),xkθdτ∈Ω0,1(R>0×H;Ck)}k∈Z that satisfy the modular transformation properties
xevk
aτ+b cτ+d
= (cτ+d)k/2xkev xλk
aτ+b cτ+d
= (cτ+d)k/2xkλ xkθ
aτ+b cτ+d
= (cτ+d)k/2(cτ+d)2xθk and the conditions
dCxkev−1=ι∂x−τ∂yxkev+1 dCxkλ−1+ι−∂x+τ∂yxkλ+1= xkev
dCxkθ−1+ι−∂x+τ∂yxkθ+1
dτ=∂xevk
Explicitly, the elementxev(u) =∑kxkevukdefines a closed element of the equivariant com-plex(C[u±1], dC+uι−∂x+τ∂y), whereuis a formal parameter of degree2, andxλ(u),xθ(u) are homotopical trivializations of its dependence onvand failure to be holomorphic, respectively. In particular, forE=C[−k],DE =0the global sections are modular forms of weightk/2(albeit without the condition of meromorphicity at the cuspτ→∞).
Remark A.2.7 Observe that∇D+
M>0 squares to zero on the subspace ofT2-equivariant sectionsC∞(R>0×H,ET2)h1,v,vi. For the vector bundle we described above, it can be interpreted as the Cartan model for a chiral half ofT2-equivariant cohomology; compare [Ber19, Definition 4.2] and [Ber20, Remark 3.26], where the case C = (Ω∗(M), d)is identified with functions on the moduli stack of non-bounding tori equipped with a map to a target manifold Mwhose underlying smooth function is constant along the fibers. We obtain almost the same complex as the BV model for equivariant forms on the double loop space in Section 4.3.
Specific cocycles representing a given cohomology class lead to interesting secondary invariants; compare [GJW21], where the partition function of different (non-perturbative) quantizations of the2|1-sigma model with targetS3 were distinguished by a Z/24-valued invariant which can be obtained from the nullhomotopyσθ of the failure of the partition functionσevto be holomorphic (albeit not at the level of rigor accessible in the
perturbative quantization).