Perturbative N=2 supersymmetric quantum mechanics and L-theory with complex coefficients

0.37 MB·English

by  Daniel Berwick-Evans

PERTURBATIVE N =2 SUPERSYMMETRIC QUANTUM MECHANICS
AND L-THEORY WITH COMPLEX COEFFICIENTS
DANIELBERWICK-EVANS
5
1
Abstract. WeconstructL-theorywithcomplexcoefficientsfromthegeometryof1|2-
0
dimensional perturbative mechanics. Methods of perturbative quantization lead to
2
wrong-way maps that we identify with those coming from the MSO-orientation of L-

n theorytensoredwiththecomplexnumbers.
a
J
Contents
3
1
1. Introduction 1

2. Background: 1|2-dimensional geometry and physics 3
]
T 3. Classical vacua 5
A 4. L-theory with complex coefficients 8
. 5. Perturbative quantization and the L-class 9
h
References 11
t
a
m
[

1. Introduction
1
v
Inthispaperwebeginaninvestigationofd|2-dimensionalsuperEuclideanfieldtheories
1
(EFTs) in the style of Stolz-Teichner, starting with d=1. Since 1|2-dimensional quantum
0
mechanics has twice the minimal number of fermions required for supersymmetry, it often
2
3 goes by the moniker N = 2 supersymmetric quantum mechanics. Concrete examples are
0 providedbytheHodge-deRhamcomplexofanorientedRiemannianmanifold,asexplained
1. by Witten in his study of Morse theory [Wit82] and reviewed briefly in Section 2 below.
0 The exact geometrical and topological information captured by 1|2-dimensional theo-
5 ries remains unclear to us, and we view the present paper as extracting the most easily
1
computable quantities. In Theorem 1.1 we show that the fiberwise partition function for
:
v a family of field theories over X defines a class in L(X)⊗C, where L is the cohomology
i theory defined by the symmetric L-theory spectrum. We note that—being a cohomology
X
theoryoverC—L⊗Cisnothingmorethan4-periodicdeRhamcohomology. Ourreasonfor
r
a viewingitinthismorecomplicatedlightisthenaturalappearanceofHirzebruch’sL-genus:
a perturbative quantization procedure constructs a pushforward to the point that agrees
withonecomingfromtheMSO-orientationofL-theorytensoredwithC(seeTheorem1.2).
In particular, π : L0(X)⊗C→L−n(pt)⊗C sends 1 to the L-genus (or signature) of X.
!
Our long-term goal is to leverage an understanding of the 1|2-dimensional case to gain
traction on the more complicated 2|2- and 3|2-dimensional theories. This is in analogy
with Stolz and Teichner’s approach to a geometric model for elliptic cohomology: they are
motivated in large part by the relation between 1|1-dimensional EFTs and Dirac operators
on Riemannian spin manifolds [HST10, ST11]. In the footsteps of G. Segal [Seg88] they
argue by analogy that 2|1-EFTs ought to capture structures related to Dirac operators
on loop spaces. Nuances of super Euclidean geometry rule out EFTs of dimension d|1
for d > 2 (e.g., see [Fre99]), so the next class of Euclidean field theories to consider are of
dimension d|2, which exist for d = 0,1,2,3. Unbridled optimism might lead one to hope
Date:January15,2015.
1
2 DANIELBERWICK-EVANS
for a relationship between 2|2-dimensional field theories, TMF with level structure, and
the Ochanine genus. However, as yet such connections are purely by analogy to the lower
dimensional case described below, coupled to Witten’s description of the Ochanine genus
as the signature of loop space [Wit88].
1.1. Statement of results. Drawing on methodology from [BE13b], we extract homo-
topical data from 1|2-Euclidean field theories associated to the moduli of 1|2-dimensional
super circles. We focus on a connected component of this moduli space characterized by
the holonomy of a 0|2-dimensional odd vector bundle over an ordinary circle which takes
the form diag(+1,−1); we call these periodic-antiperiodic (PA) circles. For a smooth mani-
fold X we define a stack Φ1|2 (X) that consists of periodic-antiperiodic circles with a map
0,PA
to X of zero classical action. There is a line bundle over Φ1|2 (X) denoted L. We call a
0,PA
sectionofL⊗k supersymmetric ifitextendstoasectionoverastackΦ1|2 (X),thatroughly
(cid:15),PA
consists of PA super circles mapping to X with nilpotent action. Tensor products of line
bundles yields a graded super commutative algebra of supersymmetric sections. Our first
result is the following.
Theorem 1.1. Concordance classes of supersymmetric sections of L• over the periodic-
antiperiodic vacua of X are naturally isomorphic to 4-periodic de Rham cohomology,
(cid:77)H•+4i(X)∼=Γ (Φ1|2 (X);L•)/concordance
dR susy 0,PA
i∈Z
where the cup product on the left coincides with the tensor product of sections on the right.
The symmetric L-theory spectrum defines a 4-periodic cohomology theory that after
tensoring with C is 4-periodic de Rham cohomology, Lk(X)⊗C∼=(cid:76) Hk+4i(X). Hence,
i∈Z dR
we can rephrase the above result in terms of an isomorphism with L-theory with complex
coefficients. The connection between 1|2-dimensional field theories and L-theory does not
end here: our second main result is a geometric construction of wrong-way maps
π
Γ (Φ1|2 (X);Lk) ! Γ (Φ1|2 (pt);Lk−n)
susy 0,PA susy 0,PA
(cid:8) dim(X)=n
π
Lk(X)⊗C ! Lk−n(pt)⊗C
compatible with those coming from the MSO-orientation of L-theory tensored with C.
This construction starts with the linearization of the fields of the classical 1|2-sigma
model over Φ1|2 (X), which defines an infinite dimensional vector bundle over Φ1|2 (X).
0,PA 0,PA
The linearization of the classical action defines a family of operators on the fibers denoted
by ∆1|2 , called the kinetic operators. We define the ζ-super determinant of the kinetic
X,PA
operators as
pf (∆1|2 | )
sdet (∆1|2 ):= ζ X,PA fer
ζ X,PA det (∆1|2 | )1/2
ζ X,PA bos
where the numerator is the ζ-regularized Pfaffian on odd (or fermionic) sections and the
denominator is the square root of the ζ-determinant on even (or bosonic) sections. Our
convention is to take the positive square root of the determinant. From our point of view,
perturbative quantization is the construction of a volume form on the fibers Φ1|2 (X) →
0,PA
Φ1|2 (pt) which in turn defines a wrong-way map. Explicitly, such a volume form is the
0,PA
product of the super determinant of the kinetic operator with the canonical volume form
that comes from integration of differential forms.
Theorem 1.2. The ζ-super determinant of ∆1|2 represents the L-class of X as a su-
X,PA
persymmetric function on Φ1|2 (X). This function modifies the canonical volume form
0,PA
PERTURBATIVE SUPERSYMMETRIC QUANTUM MECHANICS AND L-THEORY 3
on Φ1|2 (X) so that integration of supersymmetric sections of L• coincides with the wrong-
0,PA
way map associated with the MSO-orientation for L-theory with complex coefficients. In
particular, the total volume of Φ1|2 (X) is the L-genus of X.
0,PA
The ζ-determinant computation underlying Theorem 1.2 is of a fairly standard variety
inthephysicsliterature,sointhisregardourcontributionistoironoutsomemathematical
details, explainthegeometryleadingtotherelevantoperators, andcontextualizetheresult
via Theorem 1.1.
Remark 1.3. The analogous constructions for periodic-periodic super circles (i.e., those de-
fined by an R0|2-bundle over S1 with trivial holonomy) give quite different results. Owing
to a failure of excision, supersymmetric sections for tensor powers of any line bundle over
periodic-periodic vacua cannot be given the structure of a nontrivial cohomology theory.
However, the only proof we know of this fact is by brute force, and therefore rather un-
enlightening (see [BE13c], Chapter 4). A computation similar to the one constructing the
L-class shows that the ζ-super determinant of the relevant kinetic operator over periodic-
periodic circles 1. However, in this case the classical action defines an interesting volume
formonactionzerofieldsthatweanalyzedindepthin[BE13a]. Withrespecttothisvolume
form,thetotalvolumeofperiodic-periodicvacuaistheEulercharacteristicofX. Inphysics
parlance, the contribution to the path integral of the nonzero modes is 1, and the contri-
bution from the zero modes is the Euler characteristic. This result can be anticipated from
the physical interpretation of the index theorem [AG83]: N = 2 supersymmetric quantum
mechanics of an oriented Riemannian manifold X encodes two super traces of the operator
d+d∗ acting on Ω•(X), one corresponding to the signature, and the other corresponding
to the Euler characteristic, e.g., see [Wit82].
1.2. Notation and conventions. This paper follows the conventions for supermanifolds
andsuperstackssetforthin[BE13b],andwereferthereadertothesubsectionthereinwith
thesametitleasthepresentonefordetails. Insummary, oursupermanifoldshavecomplex
algebras of functions (called cs-manifolds in [DM99]), we work with the functor of points
throughout, and we frequently identify a stack with a groupoid presenting it as articulated
in [Blo08]. One important convention for group actions on mapping spaces (or stacks) is
that the action of φ∈Aut(Σ) on f ∈Map(Σ,X) is by f (cid:55)→f ◦φ−1.
1.3. Acknowledgements. Part of this work was completed while I was a Ph.D. student,
and I warmly thank my advisor Peter Teichner for his support and guidance. I also thank
Owen Gwilliam for several helpful conversations in the early stages of this work.
2. Background: 1|2-dimensional geometry and physics
2.1. 1|2-Euclidean manifolds. Let R1|2 denote the super group with multiplication
(t,θ ,θ )·(t(cid:48),θ(cid:48),θ(cid:48))=(t+t(cid:48)+iθ θ(cid:48)+iθ θ(cid:48),θ +θ(cid:48),θ +θ(cid:48)), (t,θ ,θ ),(t(cid:48),θ(cid:48),θ(cid:48))∈R1|2(S).
1 2 1 2 1 1 2 2 1 1 2 2 1 2 1 2
This defines the 1|2-dimensional super translation group. The Lie algebra of right-invariant
vectorfieldsonthissuperLiegrouphasgeneratorsD :=∂ −iθ ∂ andD :=∂ −iθ ∂
1 θ1 1 t 2 θ2 2 t
satisfying the relations
1 1
D2 = [D ,D ]=D2 = [D ,D ]=−i∂ , [D ,D ]=0.
1 2 1 1 2 2 2 2 t 1 2
The left-invariant vector fields differ by a sign, e.g., ∂ +iθ ∂ .
θ1 1 t
We will be interested in field theories that are invariant under the automorphism that
when restricted to R is the orientation reversing map t (cid:55)→ −t. Physically, this is the auto-
morphismoftime-reversal. Thereareessentiallytwoindependentliftsofthisautomorphism
to R1|2,
(t,θ ,θ )(cid:55)→r+ (t,iθ ,iθ ), (t,θ ,θ )(cid:55)→r− (t,−iθ ,iθ ).
1 2 1 2 1 2 1 2
Weobservethatr andr bothhaveorder4,commutewitheachother,andsatisfyr2 =r2.
+ − + −
They generate a discrete abelian group we denote by T which acts on R1|2 by automor-
phisms.
4 DANIELBERWICK-EVANS
Remark 2.1. Typically time-preserving and time-reversing automorphisms are of different
flavors: the former define linear operators on the Hilbert space of states, whereas the latter
defineanti-linearoperators,e.g.,seeFreedandMoore[FM13]foranextensivediscussion. In
thispaperweareonlyconsideringthevalueoftheoriesoncircles,whichrequiresinvariance
under symmetries rather than equivariance. This allows us to effectively ignore the issue.
We define the 1|2-Euclidean group as the semidirect product, R1|2(cid:111)T. The left action
of R1|2(cid:111)T on R1|2 defines a 1|2-Euclidean model space; see [HST10], Section 6.3 for detail.
SuchamodelspaceinturndefinesafiberedcategoryofsuperEuclideanmanifolds: forany
supermanifold S there is a category of 1|2-Euclidean manifolds over S, i.e., S-families of
supermanifolds that admit atlases where each chart is isomorphic to an open submanifold
of R1|2 and transition functions are given by restrictions of the action of R1|2(cid:111)T on R1|2.
Between such families one can define isometries over S, which locally are given by the
action of R1|2 (cid:111)T. Let T < T denote the subgroup that acts by orientation preserving
+
automorphisms of R ⊂ R1|2. The subgroup R1|2(cid:111)T < R1|2(cid:111)T is the subgroup of time
+
preserving isometries.
Our main examples of 1|2-Euclidean manifolds will be 1|2-super circles. Let R1|2 be
>0
the super semigroup given by restricting the structure sheaf and group structure of R1|2 to
R ⊂R. GivenageneratorR∈R1|2(S)andA∈T wedefineanS-familyofsupercircles
>0 >0 +
as the quotient, S × R1|2 := S ×R1|2/((R,A)·Z), by the right Z-action generated
(R,A)
by the composition of the R and A actions on S ×R1|2. We view R as a super radius
and A as a holonomy. When A = id (respectively, A = r r ) we call the associated circle
+ −
periodic-periodic or PP (respectively, periodic-antiperiodic or PA).
2.2. A brief tour of N =2 supersymmetric mechanics. Fields of the 1|2-dimensional
sigma model consist of maps φ: S × I1|2 → X for S × I1|2 a proper S-family of 1|2-
dimensional Euclidean intervals and X a fixed Riemannian manifold. These fields have
automorphisms coming from super Euclidean isometries S×I1|2 →S×I(cid:48)1|2 over S. There
is a function on fields called the classical action,
(cid:90)
S(φ):= (cid:104)D φ,D φ(cid:105),
1 2
I1|2×S/S
where the integral is fiberwise over S. Being built from right-invariant vector fields, S is
invariantundertheleftactionofsuperEuclideanisometries. Tounpackthisactioninterms
of more familiar geometric quantities, the map φ determines a map x: S ×I → X from
a family of ordinary 1-manifolds to X, and sections ψ ,ψ ∈ x∗πTX of the odd tangent
1 2
bundle pulled back to this family. Then (after discarding of terms without derivatives) we
obtain the action
(cid:90) (cid:16) (cid:17)
(1)S(x,ψ ,ψ )= (cid:104)x˙,x˙(cid:105)+(cid:104)ψ ,∇ ψ (cid:105)+(cid:104)ψ ,∇ ψ (cid:105)+R(ψ ,ψ ,ψ ,ψ ) dt
1 2 1 x˙ 1 2 x˙ 2 1 2 1 2
I×S/S
where x˙ denotes the derivative of the path determined by x, ∇ is the covariant derivative
x˙
along this path for the Levi-Civita connection, and R denotes the curvature of the Levi-
Civita connection.
One can quantize the classical theory using techniques of geometric quantization. The
result is a representation of R1|2 on the vector space Ω•(X) of differential forms on X,
>0
which can be viewed as a super time evolution. One mathematical perspective on this
timeevolutionistopackagethequantumtheoryasafunctorfromacategory1|2-Euclidean
bordismstothecategoryof(topological)vectorspaces;thenthesemigroupofsuperintervals
actsonthisvectorspace,whichisexactlytherepresentationofR1|2, whereasuperinterval
>0
hasalengthinR1|2. Explicitly,theactionisdeterminedbytheLiealgebraactionwhereD
>0 1
acts by d+d∗ and D acts by i(d−d∗) on forms, and it is important that both of these
2
operators square to the Hodge Laplacian. We also get an action of T on differential forms
+
that is determined by r2 = r2 acting as the mod 2 grading involution on Ω•(X), and the
+ −
action of r r is by the Hodge star, ∗, normalized such that ∗2 =+1.
+ −
PERTURBATIVE SUPERSYMMETRIC QUANTUM MECHANICS AND L-THEORY 5
As usual in the bordism-type approach, the value of a field theory on a circle is the
(super) trace of an operator. In the case above, the value on a periodic-periodic circle is
the Euler characteristic of X, and the value on periodic-antiperiodic circle is the signature
of X.
Remark 2.2. In this example, one can extend the action of T to one by O(2), where the
+
SO(2) < O(2) action encodes the Z-grading on forms. This leads to a more restrictive
notion of field theory, but might end up being the mathematically desirable one in the end.
We have chosen to stay with the more general class here, since the restricted versions will
still map to L⊗C by our result.
3. Classical vacua
3.1. Fields and vacua.
Definition 3.1. Define the stack of fields for the 1|2-dimensional sigma model with tar-
getX,denotedΦ1|2(X),asthestackassociatedtotheprestackwithobjectsoverS consist-
ing of triples (R,A,φ) where R ∈ R1|2(S) and A ∈ T determine a family of super circles
>0 +
and φ: S× R1|2 →X is a map. Morphisms over S consist of commuting triangles
RA
∼
=
S×RA R1|2 S×RA(cid:48)(cid:48) R1|2
(cid:8)
(2)
φ φ(cid:48)
X
wherethehorizontalarrowisanisomorphismofS-familiesofsuperEuclidean1|2-manifolds.
We will define a substack of Φ1|2(X) for families of periodic-antiperiodic circles whose
map to X has zero classical action. First, for r r ∈ T and R ∈ R (S) ⊂ R1|2(S),
+ − + >0
hereafterletS× R1|2denotethefamilyS× R1|2. Next,letproj: S× R1|2 →S×R0|1
R (R,r+r−) R
denote the r r -invariant projection R1|2 → R0|1 given by the formula (t,θ ,θ ) (cid:55)→ θ in
+ − 1 2 1
the notation of subsection 2.1.
Definition 3.2. Let Φ1|2 (X) be the full substack of Φ1|2(X) generated by objects over S
0,PA
definedby(R,φ)whereR∈R (S)⊂R1|2(S)andφ: S× R1|2 →X isamapthatfactors
>0 R
through proj.
Remark 3.3. A quick check against Equation 1 verifies that the maps Φ1|2 (X) do indeed
0,PA
have zero classical action. In the component description, for a field to have zero action
the map x: S ×S1 → X must be constant along the fibers, i.e., must factor through the
projection S ×S1 → S. Then, the odd sections ψ ,ψ must be parallel along the circle,
1 2
which (given the constancy of x) means ψ and ψ are constant sections. Finally, given the
1 2
antiperiodic boundary conditions on ψ , we require ψ =0.
2 2
3.2. Periodic-antiperiodic circles with nilpotent action. To define supersymmetric
sections in the statement of Theorem 1.1, we need to define a stack Φ1|2 (X)⊃Φ1|2 (X)
(cid:15),PA 0,PA
that consists of fields with nilpotent classical action.
Let R∈R1|1(S)(cid:44)→R1|2(S) for the inclusion (t,ρ )(cid:55)→(t,ρ ,0). Again, to simplify the
>0 >0 1 1
notationweletS× R1|2 :=S× R1|2. Weshalldefinemorphismsproj : S× R1|2 →
R (R,r+r−) R R
S×R0|1. WeobservethattheprojectionR1|2 →R0|1 oftheprevioussectionisnotinvariant
under the Z-action associated to the lattice R=(r,ρ ,0) for ρ (cid:54)=0, so will not define such
1 1
a map proj . Instead, for (r,ρ ,0)∈(R ×R0|2)(S), we claim the map
R 1 >0
t
proj : R1|2×S →R0|1×S, proj (t,θ ,θ ,s)=(θ −ρ ,s)
R R 1 2 1 1r
6 DANIELBERWICK-EVANS
is invariant under the left action of Z on the family R1|2×S. Denoting the action of the
generator by µ , we compute
R
t+r+iρ θ t
(proj ◦µ )(t,θ ,θ ,s)=proj (t+r+iθ ρ ,θ +ρ ,−θ ,s)=θ +ρ −ρ 1 1 =θ −ρ .
R R 1 2 R 1 1 1 1 2 1 1 1 r 1 1r
This shows the map proj is invariant, as claimed, so defines a map which we also denote
R
by proj
R
proj : S× R1|2 →S×R0|1.
R R
where the source circles S× R1|2 have R∈R1|1(S)⊂R1|2(S).
R >0 >0
Definition 3.4. Define the stack Φ1|2 (X) as the full substack of Φ1|2(X) consisting of
(cid:15),PA
pairs (R,φ) where R ∈ R1|1(S) (cid:44)→ R1|2(S) as above and φ: S× R1|2 → X is a map that
R
factors through proj .
R
3.3. Groupoid presentations. Thefactorizationconditiononthemapφinthedefinition
ofΦ1|2 (X)allowsustoidentifyitwithamorphismφ : S×R0|1 →X forφ=φ ◦proj . We
(cid:15),PA 0 0 R
can therefore find a groupoid that presents the stack Φ1|2 (X) whose objects are (R,φ )∈
(cid:15),PA 0
(R1|1×SMfld(R0|1,X))(S). Themorphisms(R,φ )→(R(cid:48),φ(cid:48)),aredeterminedbyS-families
>0 0 0
of super Euclidean isometries; lifting to the universal cover of the family of super circles,
these isometries are determined (non-uniquely) by an S-point (u,ν ,ν ,A)∈Euc(R1|2)(S).
1 2
However, not all super Euclidean isometries of R1|2 descend to ones on super circles, so the
first task is to compute those that do.
We define the group R1|1(cid:111)T by the multiplication
(u,ν)·(u(cid:48),ν(cid:48))=(u+u(cid:48)+νν(cid:48),ν+ν(cid:48)), (u,ν),(u(cid:48),ν(cid:48))∈R1|1(S),
and where generators of T act as r ·(u,ν)=(−u,iν) and r ·(u,ν)=(−u,−iν). There is
+ −
an evident inclusion,
(3) R1|1(cid:111)T (cid:44)→R1|2(cid:111)T, (u,ν,r)(cid:55)→(u,ν,0,r).
Lemma 3.5. An S-point of Euc(R1|2) descends to map between family of super circles
S× R1|2 if and only if it is in the image of the inclusion (3).
R
Proof. First we verify the assertion for super translations. Let (t,θ ,θ ) ∈ R1|2(S) denote
1 2
a lift of coordinates on a family of super circles, let (u,ν ,ν ) ∈ R1|2(S) be a family of
1 2
super translations, and let µPA denote the action of the generator (R,r r ) of the family
R + −
of lattices defining the family of super circles, where R = (r,ρ ). To obtain a well-defined
1
action by super translations, we require
µPA((u,ν ,ν )·(t,θ ,θ ))=(u,ν ,ν )·µPA(t,θ ,θ ).
R 1 2 1 2 1 2 R 1 2
Computing the left side we get
µPA(u+t+iν θ +iν θ ,ν +θ ,ν +θ )=(u+t+iν θ +iν θ +r+i(ν +θ )ρ ,ν +θ +ρ ,−(ν +θ )),
R 1 1 2 2 1 1 2 2 1 1 2 2 1 1 1 1 1 1 2 2
whereas computing the right we get
(u,ν ,ν )·(t+r+iθ ρ ,θ +ρ ,−θ )=(u+t+r+iθ ρ +iν (θ +ρ )−iν θ ,ν +θ +ρ ,ν −θ )
1 2 1 1 1 1 2 1 1 1 1 1 2 2 1 1 1 2 2
from which we conclude that the translation action of (u,ν ,ν ) ∈ R1|2(S) descends to
1 2
an isometry of an S-family of super circles if and only if ν = 0. This action by super
2
translations changes the base-points on supercircles so we obtain a map
(4) S× R1|2 →S× R1|2
R R(cid:48)
where R = (r,ρ ,0) and R(cid:48) = (r + 2ν ρ ,ρ ,0) ∈ R1|2(S), which is computed by the
1 1 1 1
conjugation action of a super translation on R∈R1|2(S).
>0
The action of r on R ∈ R1|1(S) is R = (r,ρ ) (cid:55)→ (−r,±iρ ). For R ∈ R1|1(S)
± 1 1 >0
a generator of a lattice action, the image of the lattice under this map is a lattice with
positive generator R(cid:48) =(r,∓iρ )∈R1|1(S). The map
1 >0
(r,ρ ,t,θ ,θ )(cid:55)→(r,∓ρ ,−r,±iθ ,iθ )
1 1 2 1 1 2
PERTURBATIVE SUPERSYMMETRIC QUANTUM MECHANICS AND L-THEORY 7
defines an isometry as in (4), and so the lemma is proved. (cid:3)
The above proof determines the target family of super circles for a fixed source, i.e.,
we have formulas for R(cid:48) given R and an isometry. It remains to understand how these
isometries affect the maps φ : S ×R0|1 → X. This amounts to finding the arrow that
0
makes the diagram commute
R1|2×S id×→projR R0|1×S
(5) (u,ν,r )·↓ (cid:57)
± (cid:57)
(cid:75)
R1|2×S pr→ojR(cid:48) R0|1×S,
where the left vertical arrow is the action of R1|1(cid:111)T on R1|2.
Lemma 3.6. The unique dotted arrow in diagram (5) is given by
(cid:18) (cid:19)
u+iν θ
θ (cid:55)→±i ν +θ−ρ 1 1
1 1 r
for θ a coordinate on R0|1.
Proof. The action by T on R0|1 is determined by the action of generators r , and it is easy
±
to verify that the action through ±i on θ leads to a commutative square for R and R(cid:48) as
1
in the proof of Lemma 3.5. For super translations, we compute
u+t+iν θ
proj ((u,ν )·(t,θ ,θ ))=proj (u+t+iν θ ,ν +θ ,θ )=(ν +θ −ρ 1 1)
R(cid:48) 1 1 2 R(cid:48) 1 1 1 1 2 1 1 1 r
and comparing with proj (t,θ ,θ ), we deduce the lemma. (cid:3)
R 1 2
Putting this all together, for an S-point of SMfld(R0|1,X)∼=πTX of the form
φ =x+θψ, x: C∞(X)→C∞(S),ψ ∈Der (C∞(X),C∞(S))
0 x
wherexisanalgebrahomomorphismandψ isanoddderivationwithrespecttox,wehave
an action on (r,ρ ,x,ψ)∈(R1|1×πTX)(S) by
1
(6) (u,ν )·(r,ρ ,x,ψ)(cid:55)→(cid:0)r+2iνρ ,ρ ,x+(ν − ρ1u)ψ,(1+ iρ1ν1)ψ(cid:1).
1 1 1 1 1 r r
Definition3.7. Asurjectivemapofstacks isamorphismofstacksthatonS-pointsinduces
an essentially surjective, full morphism of groupoids.
We have obtained the following groupoid description.
Proposition 3.8. There is a surjective map of stacks
(R1|1×πTX)//(R1|1(cid:111)T)→Φ1|2 (X),
>0 (cid:15),PA
where the surjection comes from lifting an isometry of super circles to the universal cover.
Restricting to R∈R (S)⊂R1|1(S) we obtain the following.
>0 >0
Proposition 3.9. There is an equivalence of stacks
(R ×πTX)//(T1|1(cid:111)T)∼=Φ1|2 (X).
>0 R 0,PA
Remark 3.10. The above is not quite a quotient stack since the groups T1|1 ∼= R1|1/R·Z
R
depend on R, and we use the subscript R as a reminder for this subtlety. However, all
groups T1|1 are isomorphic and in the computations below we will be primarily concerned
R
with invariance under the action of T1|1, so this subtlety gets lost in the wash.
R
8 DANIELBERWICK-EVANS
4. L-theory with complex coefficients
4.1. Line bundles over classical vacua. Giventhegroupoiddescriptionsoftheprevious
section, the homomorphisms
T ∼=Z/4×Z/2→p1 Z/4⊂C× ∼=Aut(C0|1)
define odd line bundles denoted L over Φ1|2 (X) that are natural in X, where the first
(cid:15),PA
isomorphism sends r to the generator of Z/4 and r (cid:55)→ (3,1) ∈ Z/4 × Z/2. By this
− +
description, L is 4-periodic: we have isomorphisms L• ∼=L•+4k for all k.
We require the following definition to make contact with L⊗C.
Definition 4.1. A section s∈Γ(Φ1|2 (X);Lk) is called supersymmetric when it is in the
0,PA
image of the restriction map i∗: Γ(Φ1|2 (X);Lk)→Γ(Φ1|2 (X);Lk). We denote the set of
(cid:15),PA 0,PA
supersymmetric sections by Γ (Φ1|2 (X);Lk)⊂Γ(Φ1|2 (X);Lk).
susy 0,PA 0,PA
Proof of Theorem 1.1. We will show there is a natural isomorphism of rings
Γ (Φ1|2 (X);L•)∼=(cid:77)Ω•+4i(X)
susy 0,PA cl
i∈Z
between supersymmetric sections and 4-periodic closed differential forms. The theorem
followssinceconcordanceclassesofclosedformsarepreciselydeRhamcohomologyclasses,
e.g., see the Appendix of [BE13b]. We compute supersymmetric sections as functions on
R ×πTX that are invariant under the R1|1-action, equivariant for the T-action, and in
>0
the image of the restriction map that defines supersymmetric sections.
First we understand the supersymmetric condition, by characterizing the R1|1-action
determined by (6) on C∞(R1|1×πTX).
>0
Lemma 4.2. The R1|1-action in (6) on C∞(R1|1 ×πTX) ∼= (C∞(R )[ρ ])⊗Ω•(X) is
>0 >0 1
determined by the formula exp(iuQ2+ν Q) for the infinitesimal generator
1
d ρ
Q:=2iρ ⊗id−id⊗d+i 1 ⊗deg
1dr r
where d is the de Rham d and deg is the degree endomorphism on differential forms.
This is proved as Lemma 2.11 in [BE13b], where we use the fact that the formula (6)
for the action is identical to the R1|1-action on R ×πTX in [BE13b].
>0
Returning to the proof of Theorem 1.1, it suffices to consider functions g ∈C∞(R ×
>0
πTX) ⊂ C∞(R1|1×πTX), i.e., those independent of ρ. We write express such a function
>0
as
(cid:88)
g(r,ρ,x,ψ)= g (r)⊗α g (r)∈C∞(R ), α ∈Ω•(X).
l l l >0 l
To be a section of Lk, by definition it must be equivariant for the action of T. This in
turndemandsanequivariancefortheZ/4-actiongeneratedbyr ,andinvarianceunderthe
−
Z/2-action generated by r r . This later Z/2-action is trivial, so invariance is automatic.
+ −
The Z/4 action on g is by (i)deg(αl), so being a section of Lk requires that deg(αl) = k
mod 4, i.e., we obtain 4-periodic differential forms.
The remaining condition for g to descend to a section of L on Γ (Φ1|2 (X);Lk) is
susy 0,PA
that it be Q-closed. So we compute
Qg =(cid:88)(2i·ρdgl ⊗α −g ⊗dα +i·deg(α )g ρ/r⊗α )
dr l l l l l l
where α ∈Ωk(X)⊂C∞(πTX). We first observe that Q-closed functions have dα =0 for
l l
all k (e.g., by restricting to the locus ρ=0). Furthermore,
dg deg(α )g
l = l l,
dr 2r
sogl(r)=c·rdeg(αl)/2 forsomeconstantc,andwithoutlossofgeneralitywemaytakec=1.
SincethedependenceonR isthereforecompletelydeterminedbythedegreeoftheformα,
>0
PERTURBATIVE SUPERSYMMETRIC QUANTUM MECHANICS AND L-THEORY 9
we may identify the function g with a sum of even or odd closed forms, depending on the
parity of k. This completes the proof. (cid:3)
The above proof shows that there are various choices of isomorphism between super-
symmetric sections and cocycles. In the next section we will use the fixed choice
(7) Γ (Φ1|2 (X);L•)→∼ (cid:77)Ω•+4i(X), rk/2⊗ω (cid:55)→ 1 ω
susy 0,PA cl (2π)k/2
i∈Z
for ω ∈Ωk(X).
cl
5. Perturbative quantization and the L-class
5.1. The normal bundles. Now let X be a Riemannian manifold that we equip with its
Levi-Civita connection, ∇. Given a map φ: S×R0|1 →X, the pullback bundle φ∗TX has
both a metric and connection.
Definition 5.1. Let NΦ1|2 (X) be stack whose S-points are triples (R,φ,δν) where
0,PA
(R,φ) ∈ Φ1|2 (X)(S) and δν ∈ Γ (S × R1|2,φ∗TX) where the zero subscript denotes
0,PA 0 R
thesectionsintheorthogonalcomplementtotheconstantsections(usingtheusualpairing
on functions on the circle and the Riemannian metric on X). Morphisms (R,φ,δν) →
(R(cid:48),φ(cid:48),δν(cid:48)) over S are determined by morphisms (R,φ) → (R(cid:48),φ(cid:48)) in Φ1|2 (X)(S), where
0,PA
δν(cid:48) is the pullback of δν along the map between families of super circles.
Remark 5.2. TheRiemannianexponentialmaponX definesanexponentialmaponsections
of NΦ1|2 (X) with values in Φ1|2(X); see [BE13b] Section 4.2 for details.
0,PA
We define the linearized classical action as
(cid:90)
(8) S (φ,δν)= (cid:104)φ∗∇ δν,φ∗∇ δν(cid:105),
lin D1 D2
S×RR1|2/S
and we will use the following lemma (proved in [BE13b], Section 4.4) to unravel this defi-
nition.
Lemma 5.3. Let φ : S ×R0|1 → X be a map with Taylor expansion φ = x +θψ and
0 0 0 0
i : S →S×R0|1 the inclusion at θ =0. The C∞(S)[θ]-linear map
0
v (cid:55)→i∗v+θi∗((φ∗∇ )v), Γ(S×R0|1,φ∗E)→Γ(S,x∗E)[θ]∼=Γ(S×R0|1,p∗x∗E)
0 0 0 ∂θ 0 0 1 0
gives an isomorphism of vector bundles over S × R0|1. The image of φ∗∇ along this
∂θ
inverse of this isomorphism is the operator ∂ +θF(ψ ,ψ ), where F(ψ ,ψ ) denotes the
θ 0 0 0 0
curvature of the connection ∇ viewed as an End(x∗E)-valued function on S.
0
After pulling back along proj, the above lemma gives
φ∗∇ =(proj∗φ∗∇) =∂ +θ R(ψ ,ψ ).
∂θ1 0 ∂θ1 θ1 1 1 1
The maps φ: S × R1|2 → X are independent of t and θ , by virtue of factoring through
R 2
S×R0|1. Hence,withrespecttotheisomorphisminLemma5.3pulledbacktoS× R1|2,we
R
mayidentifyφ∗∇ =∂ andφ∗∇ =∂ ,andweidentifyδν witha+θ η +θ η +θ θ G
∂t t ∂θ2 θ2 1 1 2 2 1 2
fora,GsectionsofTX pulledbacktoS andη ,η sectionsofπTX pulledbacktoS. Then
1 2
10 DANIELBERWICK-EVANS
we compute
(cid:90)
S(φ,δν) = (cid:104)φ∗∇ δν,φ∗∇ δφ(cid:105)
D1 D2
S×RR1|2/S
(cid:90)
= (cid:104)(∂ −iθ ∂ +θ R(ψ ,ψ ))(a+θ η +θ η +θ θ G),
θ1 1 t 1 1 1 1 1 2 2 1 2
S×RR1|2/S
(∂ −iθ ∂ )(a+θ η +θ η +θ θ G)(cid:105)
θ2 2 t 1 1 2 2 1 2
(cid:90)
= dt(cid:0)|a˙|2+i(cid:104)η˙ ,η (cid:105)−i(cid:104)R(ψ ,ψ )a,a˙(cid:105)+(cid:104)R(ψ ,ψ )η ,η (cid:105)
2 2 1 1 1 1 2 2
S1×S/S
(cid:1)
−i(cid:104)η ,η˙ (cid:105)+(cid:104)G,G(cid:105) .
1 1
(cid:90)
= dt(−(cid:104)a,D a(cid:105)−i(cid:104)η ,D η (cid:105)−i(cid:104)η ,D η (cid:105)+(cid:104)G,G(cid:105))
a 1 η2 1 2 η 2
S1×S/S
(cid:90)
=: dt(cid:104)δν,∆1|2 δν(cid:105).
X,PA
S1×S/S
where the sections a,η ,η ,G satisfy the boundary conditions:
1 2
a(t)=a(t+r), η (t)=η (t+r), η (t)=−η (t+r), G(t)=−G(t+r)
1 1 2 2
and we have separated the even and odd pieces of ∆1|2 as
X,PA
d2 d d d
D :=Id ⊗ −iR(ψ ,ψ )⊗ , D :=Id ⊗ , D :=Id ⊗ +iR(ψ ,ψ )⊗id,
a n dt2 1 1 dt η1 n dt η2 n dt 1 1
where Id denotes the identity operator on sections of the pullback of TX and πTX.
n
Following the standard rules of functional integration in this easy case, we define the ζ-
super determinant as
Pf (D )Pf (D )
sdet (∆1|2 ):= ζ η1 ζ η2 .
ζ X,PA det(D )1/2
a
5.2. The L-class. The L-class is associated to the characteristic series
x/2 xcosh(x/2)
=
tanh(x/2) 2sinh(x/2)
From the product formulas
(cid:89)∞ (cid:18) (cid:16) x (cid:17)2(cid:19) (cid:89)∞ (cid:32) (cid:18) x (cid:19)2(cid:33)
sinh(x/2)=(x/2) 1+ , cosh(x/2)= 1+
2πk 2π(k−1/2)
k=1 k=1
we derive
sinh(x/2) (cid:16) (cid:88)∞ x2k (cid:17) (cid:16) (cid:88)∞ (cid:16) x2k (cid:88)∞ 1 (cid:17)(cid:17)
=exp − 2ζ(2k) , cosh(x)=exp − 2 ,
x/2 2k(2πi)2k 2k(2πi)2k (n−1/2)2k
k=1 k=1 n=1
whereζ denotestheRiemannζ-function. ThiswillgiveusaconvenientformfortheL-class.
Proof of Theorem 1.2. Tocomputetheζ-superdeterminantwerequiretwofacts. First,the
ζ-regularized product associated to the sequence {(2πk)n}∞ is rn/2; e.g., see Example 2
rn k=1
of [QHS93]. Second, if A is determinant class, then det (AD) = det (A)·det (D) where
ζ Fr ζ
det denotestheFredholmdeterminant,andsimilarlyforζ-regularizedPfaffians. Weclaim
Fr
(cid:18)d2 (cid:19) (cid:32) (cid:18)d(cid:19)−1(cid:33)
det (D ) = det ⊗Id det Id−iR(ψ ,ψ )⊗
ζ a ζ dt2 TX Fr 1 1 dt
(cid:18)d (cid:19) (cid:32) (cid:18)d(cid:19)−1(cid:33)
pf (D ) = pf ⊗Id pf Id+iR(ψ ,ψ )⊗ .
ζ η2 ζ dt TX Fr 1 1 dt