the two-dimensional supersymmetric sigma model
Dissertation zur
Erlangung des Doktorgrades (Dr. rer. nat.) der
Mathematisch-Naturwissenschaftlichen Fakultät der
Rheinischen Friedrich-Wilhelms-Universität Bonn
vorgelegt von Bertram Niklas Arnold
aus Rostock
Bonn, Oktober 2021
Rheinischen Friedrich-Wilhelms-Universität Bonn
Erstgutachter: Prof. Dr. Peter Teichner Zweitgutachter: Prof. Dr. Stephan Stolz Tag der Promotion: 28.01.2022
Erscheinungsjahr: 2022
Summary i
Introduction iii
1. Historical overview . . . iv
2. Statement of results . . . ix
3. Outline . . . xii
4. Future research directions . . . xiv
5. Acknowledgements . . . xvii
Conventions xix 1. Formal normal coordinates 1 1.1. Differential forms in a normal coordinate chart . . . 2
1.2. Secondary invariants forp1=0 . . . 17
1.3. Cohomology calculations and the basic subalgebra . . . 22
1.4. Globalization . . . 25
2. The classical field theory 37 2.1. Left-invariant local functionals on a super Lie group . . . 39
2.2. The free theory and conformally-invariant deformations . . . 46
2.3. The chain complex of classical observables . . . 53
2.4. The classical point observables . . . 57
3. The quantum field theory 63 3.1. Quantum field theories as effective interaction functionals . . . 63
3.2. Quantization and the one-loop obstruction . . . 77
3.3. Globalization on the source . . . 93
3.4. The quantum observables . . . 95
3.5. The quantum point observables . . . 111
4. The partition function 123
4.1. The factorization property and the partition function . . . 124
4.2. BV integration on closed supermanifolds . . . 128
4.3. Equivariant Localization . . . 137
4.4. Calculation of the partition function . . . 139
A. Euclidean supermanifolds 149 A.1. Euclidean2|1-supermanifolds . . . 149
A.2. The moduli stack of super tori . . . 155
A.3. Super-harmonic functions . . . 166
References 175
The two-dimensional nonlinear sigma model is a classical field theory whose fields are maps from a Riemann surfaceΣto a Riemannian manifoldX, and whose classical solutions are minimal surfaces. In this thesis, we study a supersymmetric extension, which has an additional fermionic field. Using a mathematical formulation of the Batalin–
Vilkovisky formalism developed by Costello and Gwilliam, we show that a perturbative quantization of this sigma model on flat surfaces exists if and only if the first Pontryagin classp1(TX)∈ H4(X;C)vanishes. IfXis in addition closed and oriented, we rigorously define the partition function of the resulting quantum field theory and show that it defines a weak modular form of weightdim2X. We calculate it exactly as the Witten genus R
Xexp
∑k≥2 1
4k(2πi)2kE2kpk(TX). The partition function is determined from local data onΣthrough the factorization algebra structure on quantum observables, and we show that it is a deformation of a family of free quantum field theories.
We prove existence of a quantization and calculate the partition function using a generalization of Gelfand–Kazhdan formal geometry to Riemannian manifolds, which reduces them to algebraic statements and Feynman diagram calculations. Our results are a first step in the Stolz–Teichner program for constructing geometric cocycles for elliptic cohomology.
In [Wit88], Witten proposed the following, at first surprising, result: LetXbe a smooth manifold, setSqn(TX) =Lk≥0qnkSymk(TX)for a formal parameterq, and define vec- tor bundles Rn over X by expanding Ln≥0qnRn = Nn≥1Sqn(TX). Then if the first Pontryagin classp1(TX)vanishes, the sum of indices of twisted Dirac operators
WitX(q) =
∑
n
qnind(D/⊗Rn)
is theq-expansion of amodular formof weightdimX/2, later called theWitten genus ofX. While a mathematical proof is not long [Zag88], Witten’s proposal came from interpreting the right side as theS1-equivariant index of the Dirac operator on loop space and formally writing it as an integral over the double loop space
WitX(q) =
Z
γ:T2→XPf(D/γ∗∇)exp
(2i¯h)−1
Z
T2h∂γ,∂γi
Dγ (†) whereq=e2πiτandτ∈His the moduli parameter for a complex structure on the genus one surfaceT2∼=C/(Z⊕τZ). The integrand is thePfaffianof the Dirac operator twisted with the pullbackγ∗TX, formally defined as the square root of the infinite product of its eigenvalues. A rigorous definition viazeta regularizationleads to the conditionp1(X) =0.
This lead him to conjecture various additional properties of this expression, such as rigidity, which is the statement that the equivariant index of each of the operatorsD/⊗Rn
on a manifold withS1-action is a trivialS1-representation. A proof was given using a formal perturbative expansion of the field integral [BT89].
Recently, the perturbative approach to quantum field theory has been axiomatized in works of Costello and Gwilliam using the language offactorization algebras[Cos11b;
CG16; CG21]. An early success was a geometric construction of the Witten genus of a holomorphic manifold [Cos11a] as the partition function of the holomorphic sigma model, for which the Feynman diagram expansion can be handled explicitly.
In this thesis, we expand their methods to a general Riemannian targetX. We show that a perturbative quantization exists if and only if p1(TX)vanishes rationally, and in this case, we can rigorously define and prove modularity ofWitX(q)as thepartition function. Furthermore, we compute it exactly usingsupersymmetric localization, obtaining Witten’s formula
WitX(e2πiτ) =
Z
Xexp
∑
k≥2
1
4k(2πi)2kE2k(τ)pk(TX)
!
whereE2k(τ) =∑ m,n∈Z2
(m,n)6=(0,0)
(mτ+n)−2kis anEisenstein series, a modular form of weight 2k. We also analyze the resultingfactorization algebra of quantum observables, which captures the locality of the quantized sigma model, and allows an interpretation of this calculation as an equivariant algebraic index theorem on the loop space.
1. Historical overview
Sigma models are classical field theories whose fields are mapsγ:Σ→Xfrom a source manifoldΣto a target manifoldX, both equipped with Riemannian metrics, defined by the action functional
S(γ) = 1 2
Z
Σkdγk2γ∗gX,gΣµΣ.
They have been studied in physics and mathematics alike for a long time; to be precise, in physics one is usually interested inLorentzianmetrics on the sourceΣ, but we will restrict ourselves to the Wick rotation to Euclidean signature. For instance,dimΣ=1 describes a particle constrained to move onX, and the solutions of the equations of motion are geodesics. FordimΣ = 2, classical solutions areminimal surfaces, and the theory isconformal, i.e., invariant under local rescaling of the source metric.
The physical approach to quantization, pioneered by Feynman, is to study the ill- defined integral over the fieldsR
γ:Σ→Xe−h¯−1S(γ)Dγby formal manipulations motivated by finite-dimensional toy models. For instance, theperturbative expansionin the parameter
¯
hcan be calculated as a sum over graphs by expanding around a classical solution as in the evaluation of Gaussian integrals using Wick’s lemma. A naive definition of the resulting Feynman diagram expansionleads to divergent integrals stemming from non-invertibility of the quadratic part of the action, the Laplace operator∆gΣ. While there are systematic ways to cancel these singularities by introduction of local counterterms, the result depends on many choices, which might not respect symmetries of the target manifold, and an exact computation of all terms in the perturbative expansion is essentially impossible.
In the 1980s, it was realized that the sigma model can often be enlarged to asuper- symmetricfield theory, in which the fields have even or odd parity and there are odd symmetries which square to translations along the source. The number of supersym- metries depends on the dimension ofΣand the geometry ofX; compare [Ket00] for a physical and [DF99b] for a mathematical review. Often these supersymmetric extensions have asuperspace description, obtained by replacing the sourceΣby asupermanifoldand the action functional by itssupersymmetric extensionsuch that the supersymmetry transforma- tions are implemented by odd symmetries of the source. Using the odd symmetries, the partition function can be exactly computed usingsupersymmetric localization, for which the toy model is the localization formula in equivariant cohomology [BV82]. Further- more, given enough supersymmetries one can find odd symmetries that square to zero, allowing the definition oftwistsof the field theory, which are often considerably easier
to analyze; compare [Wit91] for the original definition, and [ES19] for a mathematical formulation.
Mathematical approaches to quantum field theory have sought to axiomatize the expected properties of the Feynman integral, avoiding the divergences that abound when one attempts to define it naively by analogy to the Lebesgue integral. One candidate is thefunctorial approachpioneered by [Seg04] (for chiral conformal field theory) and [Ati88]
(for topological quantum field theory), which encodes the expected factorization of the field integral under a decomposition ofΣinto manifolds glued along their boundary as a symmetric monoidal functor from a category of cobordisms into vector spaces and has been instrumental in the development of higher category theory via thecobordism hypothesis[BD95; Lur09b].
For the non-topological sigma model, the cobordism hypothesis is not applicable, and a functorial description must take the geometry on the source into account. In particular, the linear map associated to a cobordism of Riemannian manifolds must depend smoothly on the metric of the cobordism. For supersymmetric quantum field theories, it is easier to use the functor of points formalism, so that objects and morphisms are families of supermanifolds that form a submersion over a test supermanifold B, which the field theory sends to vector bundles on Band linear maps between them.
Smoothness is captured by functoriality in the baseB. A key insight of Stolz and Teichner [ST04; ST11] was that this allows the definition of thespace of all field theoriesfor a certain dimension and number of supersymmetries, and by working with more generaltwisted field theoriesthey were able to deloop these spaces to aspectrum. Furthermore, these spectra can be explicitly described in low dimensions (withN=1supersymmetries):
in dimension0, one obtains the Eilenberg–MacLane spectrum HR[HKST11], and in dimension 1the real K-theory spectrum KO [HST10], giving rigorous mathematical meaning to the interpretation of the Euler characteristic andA-genus of a manifold asb partition functions of supersymmetric sigma models.
This raises the question what happens in the casedimΣ = 2,N = (0, 1), for which Witten’s supersymmetric sigma model is the prime example. A candidate is given by the spectrumTMFoftopological modular forms, anE∞-ring spectrum whose complexified homotopy groups are the classical ring of modular forms, for which the only known constructions so far use the methods of stable homotopy theory [Goe10; Lur18]. As a hint in this direction, if the target manifoldXhas aString structure, i.e., it has a Spin structure such that the integral lift of the classp1(TX)/2vanishes, the Witten genus admits a lift to πdimX(TMF)[AHR10; Lur09a], which contains interesting torsion information not seen at the level of the partition function. For instance, it gives an obstruction to the existence of Stein fillings of exotic23-spheres [BS20], and rigidity of the Witten genus has been proven using equivariant Thom classes for its complexification [AB02]. Furthermore, this conjectural relation has been used to exclude spontaneous breaking of supersymmetry for the sigma model with target S3 equipped with a non-bounding String structure [GJW21] and to argue for the absence of global anomalies for heterotic string theory
[TY21]. However, while there are partial results such as modularity of the partition function [ST11, Theorem 1.15], so far there is no example of a fully extended field theory in this formalism, as even the construction of the free fermion requires intricate analysis of gluing formulas for Pfaffian lines [LR20].
A different axiomatization is provided by the language offactorization algebras. Whereas the functorial perspective highlights the importance ofstates and their evolution, in this formalism the focus is onobservables and their algebraic structure. We refer to [CG16, Section 1.3] for a detailed exposition of this viewpoint, which in short says that locality of quantum field theory means that we can speak about measurements of the fields supported in some open subsetU ⊂ Σof the source manifold, and that two measurements can be performed simultaneously if their supports are disjoint. For instance, if the source is one-dimensional, this requires us to choose a time ordering for measurements, explaining the noncommutativity of observables in quantum mechanics.
The notion of factorization algebras originated in [BD04] as an algebraic axiomatization of the operator product expansion in chiral conformal field theory. A higher-categorical variant was used to define fully extended topological field theories without relying on the cobordism hypothesis [Sch14; Lur17]. Following [CG16, Definition 3.1.1], aprefactorization algebrais a precosheafU 7→ Obs(U)with values in a category of topological vector spaces, together withfactorization productmapsObs(U1)⊗ · · · ⊗Obs(Un) → Obs(V) for any pairwise disjoint inclusionU1t · · · tUn ⊂ V, satisfying natural associativity, commutativity, and unitality conditions. In a factorization algebra, the value on a setUis determined by the values on small open subsets and the maps between them. Crucially, this isnotthe usual cosheaf condition, which would assign the coproduct to the disjoint union of two subsets and make it impossible to define the factorization product. Instead, one uses the coarserWeiss topology, for whichSi∈IUi = Vis a cover if any finite subset S⊂Vis contained in a single open setUiin the cover.
While classical observables in the Lagrangian formalism are by definition functions on the classical solutions, i.e., the critical locus of the Lagrangian action functional, the definition for quantum observables is more involved. From finite-dimensional toy models, one expects the volume forme−h¯−1S(γ)Dγof the field integral to define a divergence operator¯hdivS :X7→ ¯hdiv0(X) +X(S)from vector fields to functionals of the fields such that the integral of a divergence against the volume form defined by the field integral vanishes. Turning this theorem into a definition, the quantum observables Obs(U) should be thought of as the quotient of all functionals by total divergences, allowing the use of partial integration in the Feynman graph expansion to derive the Dyson–Schwinger equations. In the classical limit, the volume form concentrates on the classical solutions, and the quotient by formal divergences amounts to enforcing the Euler–Lagrange equations dγ(S) =0. More generally, one can consider thedivergence complexdefined by the volume form, which Batalin and Vilkovisky used to study the existence of perturbative quantizations of gauge theories [BV81]. The key ingredient is theQuantum Master Equationdiv2S = 0, where we now consider this operation on
functionals on theextended fields, the (−1)-shifted cotangent bundle to the space of fields. The resulting chain complex has better formal properties than the quotient of gauge-invariant functionals by formal divergences, which form its zeroth cohomology.
Mathematically, it can be understood using the language ofderived differential geometry, which can handle degenerate critical loci of the classical action functional [AKSZ97;
PTVV13]. For sigma models, it is used to formulate the consistency conditions for the perturbative expansion around each point of the target.
Since the space of fields is infinite-dimensional, a naive definition ofdiv0 requires the multiplication of distributions and is therefore ill-defined. The divergence operator needs to be regularized, and it is not immediately clear how the result depends on the necessary choices. In [Cos11b], Costello achieved a rigorous formulation of this procedure by making all choices at once: To write down a perturbative quantum field theory in the BV formalism, one should give a regularized divergence operatordivS,Φ for any parametrixΦof the elliptic operator defining the free part of the (gauge-fixed) classical action via the datum of ascaleΦeffective interactionI[Φ], and the different choices should be related by theexact RG flow, which has an operadic interpretation as a generalization of the homotopy transfer theorem [DSV16]. Locality of the field theory can then be expressed via support conditions on I[Φ]in terms of the support ofΦ, and the choices of local counterterms needed for the perturbative expansion are all encapsulated in a (non-canonical) bijection between families of effective actions and local action functionals.
The Quantum Master Equation can be formulated completely in terms of the effective actions, and its space of solutions can be analyzed by obstruction theory. In particular, there is an iteratively defined sequence of obstructions to the existence of a quantization where the Quantum Master Equation holds modulo a fixed power of¯h.
The idea of making all choices at once was also used in [CG21, Chapter 8] to define the factorization algebra of quantum observables: The scaleΦdifferentialdivS,Φ turns compactly supported functionals on the extended fields into a cochain complex, and the exact RG flow defines isomorphisms between the complexes defined at different scales. The asymptotic locality of the scaleΦeffective action then allows the definition of thesupport of such an observable, and sending a subset U to the subcomplex of quantum observables supported on it defines the underlying precosheaf. AsdivS,Φis a second-order differential operator, it doesnotdefine a commutative differential graded algebra (cdga) structure on this complex. However, its failure to preserve the product is asymptotically local, so that the product of observables with disjoint support is a chain map, provided that one chooses the parametrixΦto have sufficiently small support.
Furthermore, for a compact manifold one can find parametrices such that the complex of quantum observables is isomorphic to a finite-dimensional one, which corresponds to the physical procedure of “integrating out the nonzero modes”.
These ideas have been applied with great success to certain sigma models. For in- stance, in the topological infinite-volume limit of the one-dimensional sigma model with target a symplectic manifoldX, the quantum observables on intervals give Fedosov’s
construction of a deformation quantization ofC∞(X), and the map to observables on a circle gives the unique normalized trace, which can be used to prove the algebraic index theorem [GG14; GLL17]. In two dimensions, the holomorphic twist of theN = (0, 1) sigma model can be quantized if the second Chern class of the target Kähler manifold vanishes [Cos11a], and in this case the quantum observables on disks inCcan be used to define the vertex algebra ofchiral differential operators, whose conformal blocks on an elliptic curve can be identified with the de Rham forms on the target, and the Witten genus arises as the character [GGW20, Corollary 19.2, Proposition 5.14]. To apply per- turbation theory, which requires an affine space of fields, to sigma models with curved targets, one uses theformal geometrypioneered by [GK71], and the derived viewpoint of the BV formalism allows the restatement of these constructions as theorems about the cohomology of the infinite-dimensional Lie algebra of (Hamiltonian) formal vector fields.
There are algebraic models for locally constant and holomorphic, translation-invariant factorization algebras, respectively, defined in terms of their values on small disks and the factorization products between them. Since finite disjoint unions of disks are a basis for the Weiss topology, the factorization property defines the quantum observables completely in terms of this local data. In the locally constant case, this is achieved by thefactorization homologyofEn-algebras, and in the chiral conformal case there is a construction of aC×nC-invariant, holomorphic factorization algebra onCfrom a vertex algebra [Brü21]. Such an algebraic description is not known for general factorization algebras, and previous work on the bosonic two-dimensional sigma model has focused on showing the existence of a quantization to all orders and expressing quantities of physical interest such as theβ-function in terms of the effective action [Ngu16; GW18].
We close this section by giving an overview how this general formalism works for the2|1-sigma model. As a first step, note that the Pfaffian in Eq. (†), which arises by formally integrating out the fermionic fields, is not a function, but rather a section of thePfaffian line bundleof the twisted Dirac operator over the double loop space [BF86].
To obtain a sensible expression, this bundle must be trivialized, for which there is both a global and local obstruction, given by its first Chern class and the curvature of the Bismut–Freed–Quillen connection, respectively. Both can be calculated by the family Atiyah–Singer index theorem, giving the transgression of the degree4cohomology class p1(TX)/2and its Chern–Weil representative to the double loop space. In perturbation theory around the constant maps, we only see the local obstruction, which needs to be canceled by aH-flux term, a deformation of the classical action that is defined in terms of a3-formηwith dη = hp¯ 1/2. In the BV formalism, this corresponds to a one-loop obstruction to quantization.
The Costello–Gwilliam formalism is most powerful if the vanishing of higher obstruc- tions can be derived algebraically, by showing that either the obstruction group is zero or quantization only involves one-loop diagrams. Neither is the case here, and after translating the above informal argument into the BV formalism, one a priori has to
calculate an infinite amount of obstructions, checking if each of them defines a vanishing degree4cohomology class. In this case, the partition function can be defined by inte- grating out the nonzero modes, which also gives an infinite expansion as a sum over graphs. The factorization algebra structure on quantum observables then exhibits the calculation of the partition function as the correct generalization of the index theorems we discussed above. In the absence of an algebraic description for factorization algebras on2|1-dimensional Euclidean supermanifolds, the quantum observables are arguably the correctdefinitionof the local operators in this sigma model, allowing us tostatethe equivariant algebraic index theorem on the loop space. Furthermore, unpublished work of Dwyer–Stolz–Teichner builds a twisted field theory from a factorization algebra, which could serve as an interesting testing ground for the proposed relation to topological modular forms. The integral over the non-zero modes identifies the restriction of this twist to compact Euclidean supermanifolds with that of the free fermion, and allows a rigorous definition of the partition function of the supersymmetric sigma model in the language of functorial field theory.
2. Statement of results
We apply the Costello–Gwilliam formalism to a formal neighborhood of the constant maps in the two-dimensional sigma model withN= (0, 1)supersymmetry, producing Witten’s calculation of the anomaly and partition function.
The first main result is the complete identification of obstructions to the quantization:
If the one-loop obstruction vanishes, a perturbative quantization exists.
Theorem A (Corollary 3.2.14) A quantization of the classical sigma model defined by a Rie- mannian manifold(X,g)andH-fluxηexists if and only if the first Pontryagin class of the target manifoldXvanishes rationally. In this case, the space of quantizations is given by pairs(g¯h,η¯h) of ah-dependent Riemannian metric and a¯ 3-formη¯hwith dη¯h= ¯hp1(gh¯)∈Ω4cl(X)[[h¯]]which reduce to(g,η)for¯h=0, modulo the action of¯h-dependent diffeomorphismsΦh¯ which reduce to idXfor¯h=0.
Our next result concerns the quantum observables on a certain family of closed Eu- clidean supermanifolds which provides an explicit model for the moduli stack of Eu- clidean super tori such that the underlying spin structure is non-bounding. We can formally integrate over the space of fields after inverting¯h, mapping these observables to a top-dimensional cohomology class which can be integrated over the targetX.
Theorem B (Corollary 4.2.8) After inverting¯h, the zeroth cohomology of the quantum observ- ables on a non-bounding super torus can be identified with the top cohomologyHdimX(X;C((¯h))) viaBV integration. This defines a vector bundle over the moduli stack of non-bounding super tori, and the global section defined by the unit observable defines aHdimX(X;C((h¯)))-valued modular form of weight dim2X, called thepartition functionof the quantum field theory.
So far, this treatment of the quantized field theory has not used supersymmetry, and the general formalism produces a formula for the partition function as a sum over graphs, where each term requires the careful inductive regularization of the divergent integrals that abound in the physical treatment of quantum field theory. The odd symmetry of the theory allows theexactcalculation of the partition function by supersymmetric localization, which can be carried out rigorously in the BV formalism, yielding the formula for the partition function that was predicted by Witten:
Theorem C (Corollary 4.4.5) The partition function of the quantized sigma modelΣX,g¯h,η¯h on a split super torusC1|1/(Zw1+Zw2)is given by
Witg¯h,ηh¯(X) =
"
exp
∑
∞ k=21
4k(2πi)2kE2k(w1,w2)pk(TX)
!#
dimX
where the square brackets denote the homogeneous component of degreedimX,E2k(w1,w2) =
∑ m,n∈Z (m,n)6=(0,0)
(mw1+nw2)−2k is an Eisenstein series of the latticeZw1⊕Zw2, andpk(TX) = Tr(R2k)is the suitably rescaled component of the Pontryagin character of form degree4k. In particular, it is independent of the parametersg¯h,ηh¯, andh.¯
Let us now explain in more detail how these theorems are proven. The general for- malism allows the study of existence of quantizations of a given classical field theory via obstruction theory. In our case, the scale invariance of the classical action can be used to reduce the space of quantizations, and thereby also of possible obstructions, which land in the (complexified) fourth cohomology of the target manifoldX: locally, the regularization procedure requires the choice of a3-form, with the difference of any two choices a closed form. The first cohomology group of the sheaf of closed3-forms then carries the obstruction for patching these local choices together consistently.
While the first obstruction can be computed relatively easily, at higher loop orders the number of contributing graphs grows exponentially, and the resulting obstruction depends on all previous choices. The main insight is that the obstruction will always be represented by a diffeomorphism-invariant polynomial in the derivatives of the metric tensor and a3-form, valued in closed4-forms, and the existence of normal coordinates shows that any such polynomial must be given by contracting covariant derivatives of the Riemann curvature tensor and the3-form using the metric tensor. This approach was used by Gilkey to identify the cohomology classes defined by coefficients in the asymptotic expansion of the heat kernel with polynomials in the Pontryagin classes, leading to an analytic proof of the Atiyah–Singer index theorem [Gil75; ABP73]. A slight modification, introducing the deformation parameter h¯, yields the following theorem, which is a differential-geometric rephrasing of Proposition 1.3.6 as explained in Remark 1.0.1:
Theorem D LetPh¯(g¯h,η¯h)be a diffeomorphism-invariant expression for a closed4-form, de- pending on a Riemannian metricgh¯ and a3-formηh¯ with dη¯h= ¯hp1(gh¯), which is polynomial in the respective derivatives and the inverse metric tensor(g¯h)−1, with all terms given by formal
power series with parameterh. Then there is a complex number¯ Cand a diffeomorphism-invariant expressionQh¯(gh¯,η¯h)for a3-form such thatPh¯(gh¯,η¯h) =Cp1(g¯h) + dQh¯(gh¯,η¯h).
The vanishing of higher-loop obstructions follows from this statement by first consid- ering the sigma model with target a formal disk equipped with a Riemannian metric, for which there is an additionalweight grading. TheN-loop obstruction lies in weight degree 2−2N, whereas we compute that the relevant obstruction group is concentrated in nonnegative weight degree, leading to the vanishing of theN-loop obstruction forN >1.
This result can then beglobalizedto a non-formal target by a version of Gelfand–Kazhdan formal geometry that incorporates the metric, which allows us to consistently perform the perturbative quantization in normal coordinate charts at every point.
Similar techniques can be used to show that the partition function is given by the integral of a top-dimensional form which is polynomial in the inverse metric tensor and covariant derivatives of the Riemann curvature tensor and the3-formη¯h. From this perspective, independence of(g¯h,ηh¯)and modularity are quite surprising. The latter is a feature of the supersymmetric nature of our field theory, which allows us to study our theory onnon-splitsuper tori. The covariantly constant spinor on the non-bounding torus gives rise to an odd vector field annihilating the partition function, implying holomorphic dependence on the lattice generatorswi, and rotation invariance shows that the partition function is a holomorphic function ofτ = ww1
2. It can be used to define a commuting differential on the quantum observables on a super torus, similar to Connes’sB-operator on Hochschild homology. Heuristically, the observables on a torus are the integral forms on the double loop space, and the odd vector field defines the Cartan differential d+ι∂z which depends on a complex structure on the torus; compare Example A.2.6. Together with the cohomology calculation of Chapter 1, it is used to calculate the partition function exactly:
Theorem E (Corollary 4.4.5) The partition function is the integral of the top-degree component ofυdimXWitTX,g2h¯,ηh¯ ∈ Ω∗(X)[υ], where
WitTX,g2 ¯h,η¯h(w1,w2) =exp
∑
k≥2
υ−2k
4k(2πi)2kE2k(w1,w2)pk(TX)
!
(mod dΩ∗(X)[υ−1]). The calculation of the partition function is the main mathematical result of this thesis.
To state and prove it, we only need to consider the quantum observables on closed surfaces. However, the quantum observables on flat open surfaces are very interesting as they determine the value on tori via thefactorization property. As argued above, they are the correct axiomatization of the locality of the sigma model, and we therefore also analyze them in some detail.
We determine the cohomology of the quantum observables on open supermanifolds via a (non-canonically) split filtration, and identify the map induced by the factorization product on the associated graded. As a variation on this calculation, we also analyze the cohomology of thepoint observablesand parts of theiroperator product expansionvia this
filtration. Along the way, we check that a quantization on the model geometryC1|1which is equivariant with respect to the action of isometries defines compatible quantizations on all Euclidean supermanifolds.
Theorem F (Proposition 3.4.13) The cohomology of the quantum observables on an open su- permanifold is concentrated in degree0, and the associated graded of the splitweight filtrationis isomorphic to functionals on super-harmonic functions which vanish on constants, tensored with polyvector fields on the targetX. On the associated graded, the factorization product induces a bilinear operation, which we identify with the factorization product of a family of free quantum field theories.
Similarly, the cohomology of the point observables is concentrated in degree0, and the scale- invariance of the classical field theory allows us to define an additional grading byapproximate conformal bidegree. The two-point operator product expansion can be calculated explicitly on the associated graded of the weight filtration.
3. Outline
Conventions collects the various conventions for Koszul signs, topological vector spaces, and homological algebra that we use.
Chapter 1 defines the algebra over which the universal quantization of the2|1-sigma model is defined, and calculates cohomology groups of various modules over it.
Section 1.1 introduces a commutative dg algebra which is given by differential forms that can be built out of covariant derivatives of the metric tensor of a Rieman- nian manifold, and computes the cohomology of various natural modules over it, showing that below the top dimension the cohomology is obtained from invariant polynomials ononvia the Chern–Weil construction.
Section 1.2 extends this computation to the situation where we also have a3-form whose de Rham differential is the Chern–Weil representative of the first Pontryagin class, showing that this does not give rise to new cohomology classes.
Section 1.3 introduces subcomplexes of basic and invariant elements, calculating their cohomology.
Section 1.4 explains how these subcomplexes can be globalized to a Riemannian manifold using a Koszul dual version of Gelfand–Kazhdan formal geometry.
Chapter 2 studies the classical2|1-sigma model (dimension2withN = (0, 1)super- symmetry) in the BV formalism.
Section 2.1 gives a straightforward, if technical, description of left-invariant local action functionals on a super Lie group in terms of its super Lie algebra.
Section 2.2 introduces the free sigma model defined by a finite-dimensional vector space with a non-degenerate bilinear form and computes its obstruction-deformation
complex, showing how the globalization procedure can be used to formulate the classical sigma model with curved target in the BV formalism.
Section 2.3 introduces the observables of this classical field theory, which are the measurements of a solution of the Euler–Lagrange equations on a given family of Euclidean supermanifolds, and relates them to harmonic distributions, which is the rigorous formulation of the perturbative solution of the equations of motion.
Section 2.4 introduces a similar notion of point observables, which are measurements that are localized at a single point, and have a more algebraic description.
Chapter 3 studies the existence of perturbative quantizations of the classical field theory and the resulting factorization algebra of quantum observables.
Section 3.1 recalls the treatment of quantum field theories in terms of effective in- teraction functionals from [CG21, Chapter 7] and defines prequantizations of the classical2|1-sigma model.
Section 3.2 refines this definition to quantizations, which are solutions of the regular- ized Quantum Master Equation, and studies their existence via obstruction theory, proving Theorem A by reduction to the cohomology calculations in Section 1.3.
Section 3.3 discusses how anIsom(C1|1)-invariant quantization on the model geom- etryC1|1defines compatible quantizations on all Euclidean supermanifolds.
Section 3.4 introduces the quantum observables and the factorization product fol- lowing [CG21, Chapter 8], which allows the simultaneous performance of two measurements in disjoint regions of the source supermanifold, and computes the induced maps on the associated graded of a filtration on quantum observables.
Section 3.5 introduces another notion of quantum observables following [CG21, Chapter 10], which are concentrated at a single point on the source and easier to describe algebraically, computes the two-point operator product expansion on the associated graded of a filtration, and uses the scale invariance of the classical field theory to formulate a conjecture for its general form.
Chapter 4 studies the quantum observables on non-bounding super tori and gives a rigorous identification of the partition function with the Witten genus.
Section 4.1 shows how the cohomology of the quantum observables defines a vector bundle with a non-integrable holomorphic structure over the moduli space of lattices inC, defines the partition function as the holomorphic section given by the unit observable, and explains how the factorization property defines it uniquely up to an overall normalization factor.
Section 4.2 identifies the quantum observables on a non-bounding torus with the top-dimensional de Rham cohomology of the target twisted by a line bundle on the moduli stack of super tori via an integration over the space of fields made rigorous
by the BV formalism, showing modularity of the partition function of a closed, oriented target.
Section 4.3 extends this quasi-isomorphism to a different complex incorporating the odd symmetries of non-bounding super tori, which is the rigorous version of the equivariant differential forms on the double loop space.
Section 4.4 uses the calculations of Chapter 1 to reduce the calculation of theequivari- ant Witten genusto one-loop diagrams, proving Theorem C.
Appendix A collects various definitions and results from the literature on Euclidean supermanifolds to set up consistent notation for the body of this thesis.
Appendix A.1 defines2|1-dimensional Euclidean supermanifolds as families of su- permanifolds modeled on the rigid geometryIsom(C1|1) C1|1.
Appendix A.2 studies one component of the moduli stack of these manifolds, relating it to the moduli stackH// Mp(2,Z)of elliptic curves equipped with a non-bounding spin structure.
Appendix A.3 studies a natural second order differential operator on Euclidean su- permanifolds, showing existence of closed contractions for the resulting elliptic complex, and explicitly describing them for the universal family of non-bounding tori.
4. Future research directions
OPEs in strictly renormalizable 2|1-EFTs. We describe the quantum observables of the sigma model as a factorization algebra on Euclidean supermanifolds, for which no algebraic description similar toEn-algebras (for topological field theories) or vertex algebras (for chiral conformal field theories) is known. In both cases, the physical interpretation of this algebraic structure is theOperator Product Expansion(OPE), which we study in Section 3.5. We conjecture an algebraic description of the two-point OPE in Conjecture 3.5.14, but forn≥3then-point OPE seems to be inherently analytical in nature. Nevertheless, associativity of the two-point OPE should be possible to formulate algebraically, and might give an interesting mathematical interpretation of the observables on a super torus and the partition function similar to conformal blocks of a vertex algebra on an elliptic curve. We also leave the analysis of factorization algebras on2|1-manifolds, with an eye towards an index-theoretic interpretation of the partition function, to future work.
Relation to the holomorphic sigma model. If the targetXis equipped with a Kähler structure, there are two ways to simplify the supersymmetric sigma model: On the one hand, we can use the splitting of the tangent bundle into (anti)holomorphic vector fields to redefine the classical action in thefirst-order formalism; compare [GGW20, Section 21] for a discussion of this procedure in the classical bosonic sigma model,
which immediately generalizes to the supersymmetric one by replacing the differen- tial operator∂with its odd square rootD. One can then pass to theinfinite volume limit, which splits as the sum of a super-holomorphic and an antiholomorphic field theory, i.e., where the translation operatorsD+and∂are homotopically trivialized, respectively. These are cotangent theories to the elliptic formal moduli problems of super-holomorphic and antiholomorphic maps to the target, respectively, and thus their quantization only involves one-loop diagrams.
On the other hand, the splitting of the odd fields allows us to decompose the odd symmetry D+ = D+1 +D2+ as the sum of two odd symmetries squaring to zero;
compare [ES19, Section 4.2] for a mathematical exposition of the twisting procedure, which amounts to adding one of these symmetries to the differential after regrading the fields so that it is even and of cohomological degree 1. This gives again the cotangent theory to holomorphic maps which was studied in [Cos11a; GGW20]: Its point observables form the vertex algebra ofchiral differential operators, and the partition function is its character, which evaluates to the Witten genus.
Thus for a Kähler target one can reach the holomorphic sigma model either as a limit (by rescaling the metric) or as a deformation (by twisting). In each case, the partition function is unchanged by a variation of the supersymmetric localization argument.
These physical arguments, and more generally holomorphic limits and twists of fac- torization algebras on Euclidean supermanifolds, should be studied rigorously using the BV formalism.
Dimensional reduction. As suggested by the title of Witten’s seminal paper [Wit88], one way to study the two-dimensional sigma model is via itsdimensional reduction, where we compactify one of the two coordinates to a circle of lengthL. The result is the equivariant one-dimensional sigma model with target the (infinite-dimensional) free loop space equipped with itsS1-action by reparametrization. We explain in Re- marks 3.4.16 and 4.1.5 how this procedure can be understood using the language of factorization algebras. Formally taking the limitL→∞, the action functional blows up away from the (infinite-dimensional) submanifold of maps from a cylinder that are constant along the compactified direction, for which the action is that of super- symmetric quantum mechanics. For the partition function, we obtain the value of the Witten genus at the cuspτ=i∞, which is theA-genus ofb X. It would be interesting to formulate this procedure in the Costello–Gwilliam language, recovering the treatment of supersymmetric quantum mechanics whose topological infinite volume limit has been treated in [GG14; GLL17] and fixing the normalization of the partition function.
Non-perturbative quantization and non-constant maps In the non-perturbative treat- ment, we have to cancel the global as well as the local anomaly. This requires a trivial- ization of the lift of the first Pontryagin class todifferential cohomology, and similarly restrictsH-flux terms to degree3differential cohomology classes to unambiguously define the exponentiated classical action [Bun11]. However, this integrality condition
is invisible in perturbation theory, as it implicitly gets multiplied with the parameterh.¯ We can more generally ask how our results are affected by working over the whole double loop space. For the partition function, this should not make a difference, as a formal application of supersymmetric localization suggests that its value can be calculated on theT2-fixed points, i.e., constant maps [Ber19, Corollary 4.11]. For the holomorphic twist, there are suggestions that nonperturbative effects can trivialize the algebra of observables [Wit07; TY08]. The methods of this thesis do not generalize easily, as we crucially used translation invariance of the classical solution around which we perturb.
From a physical perspective, the perturbative quantization only makes sense if the theory is asymptotically free, meaning that its coupling constants have nonnegative scaling degree under the action of the renormalization group. This condition can be formulated in the BV formalism via theβ-function, an obstruction to scale-invariant quantization, and we sketch a computation of its leading term in Remark 3.2.17.
However, we want to emphasize that renormalizability is not needed for the rigorous perturbative quantization which was the main goal of this thesis.
Manifolds with boundary. The Costello–Gwilliam formalism defines a projective vol- ume form on the fields, encoded in a divergence operator on compactly supported functionals. On a manifold with boundary, one can consider more general ellip- tic boundary conditions, leading to theBatalin–Fradkin–Vilkovisky formalismstudied mathematically in [CMR18; Rab20a]. This requires the choice of a Lagrangian of the (infinite-dimensional) symplectic manifold of boundary values of classical solutions, which conjecturally should identify spin structures on the loop space with string structures on the target manifold [Wal16]. As a result, we would obtain a rigorous perturbative approximation of the Hilbert space associated to the1|1-dimensional boundary manifold, which could be a first step towards investigation of the Stolz conjecture about a Lichnerowicz formula for the Dirac operator on loop space [Sto96].
Stolz–Teichner program. While related in spirit, our results are not directly applicable to the Stolz–Teichner program, as they are not formulated in terms of functorial field theories but rather the different axiomatization of factorization algebras. We close by highlighting how they should fit into their picture.
An extension to manifolds with boundary via the BFV formalism would allow us to build a functorial field theory by the gluing formulas in [CMR18, Section 3.6]. Without it, we can still define a twisted field theory from the factorization algebra of quantum observables, which might shed light on the periodicity of the spectrum defined from 2|1-Euclidean field theories.
From another perspective, the interpretation of field theories as geometric cocycles for generalized cohomology theories suggests that we should consider2|1-EFTs which are parametrized by maps from the source to a certain base manifoldY. The field integral
can then be seen as a wrong-way map: Given a submersionY→ Z, by integrating the maps defined by a field theory overYover all lifts of a map from the source to Z, we obtain a field theory over Z. Our field theory is the result of applying this procedure to the submersionX → pt and the trivial field theory over X, and the computation of the partition function aligns with the identification of the difference of the two orientations of rationalizedTMFwith the Witten genus [AHR10]. Because our perturbative expansion is local on the target, this procedure can be easily carried out in our setting, with the caveat that the resulting theory is only parametrized by a formal neighborhood of constant maps toZ. We suggest the study of theadiabatic limit, where the metric is rescaled to make the directions orthogonal to the fiber infinitely long, and the study of more general bases such as the stack of principalG-bundles as interesting questions for future work. The latter should generalize our constructions to the equivariant Witten genus, but would require new obstruction group calculations.
5. Acknowledgements
First and foremost, I want to thank my advisor Prof. Peter Teichner for introducing me to the mathematical aspects of quantum field theory and shaping my mathematical development from my undergraduate studies up to this thesis. I am especially happy to have Prof. Stephan Stolz serve as the second reviewer, as their joint program has been a constant source of inspiration. I want to express my gratitude to Prof. Margherita Disertori and Prof. Albrecht Klemm for agreeing to serve on the doctoral committee.
During my graduate studies, I was financially supported by the IMPRS on Moduli Spaces, and I want to thank the staff at the Max Planck Institute for Mathematics for excellent working conditions. I want to thank the other guests of the institute, especially my office mates in B25 and B27, for creating an inspiring mathematical atmosphere. In particular, I am grateful to Daniel Berwick-Evans, Daniel Brügmann, Robert Bryant, Owen Gwilliam, Enno Keßler, Achim Krause, Malte Lackmann, Si Li, Matthias Ludewig, Lennart Meier, Eugene Rabinovich, Jens Reinhold, David Reutter, Luuk Stehouwer, and Augusto Stoffel for enlightening discussions. Special thanks go to Lukas Müller for proofreading and offering helpful comments on a draft of this document.
My parents have always encouraged my mathematical interests, and I am grateful for their constant support throughout my studies.
Lastly, I want to thank Louisa for her love and support.
Supermanifolds. We work with cs supermanifolds, whose definition is recalled in Ap- pendix A. In particular, we will always work over the ground fieldC. For calculations, it is easiest to work in the functor of points approach, where every construction im- plicitly depends on a base supermanifoldB. The functor of points of a subsetU⊂Cn, considered first as a real manifold and then as a purely even cs supermanifold, is efficiently described by pairs of even functionsu,u∈ Oev(B)⊕nwhich are complex conjugate to each other modulo the nilpotent augmentation ideal. This condition will be implicitly assumed whenever we use a letter and its complex conjugate in a formula.
We denote even coordinates by Latin and odd coordinates by Greek letters, with the exception of the even coordinate(τ,τ)on the upper half plane in Appendix A.
Any work on supermanifolds would be remiss without a discussion of thecauchemar de signes[DF99b, p. 230] introduced by the Koszul sign rule. We follow the conventions in [DF99a]. Concretely, we consider vector spaces which are equipped with aZ×Z/2- grading, with the first factor giving the cohomological degree and the second factor the fermion degree. When two termsx,yswitch place, we obtain a sign of−1if both are of odd cohomological degree, and another sign if both are of even cohomological degree. We then take the product of both as the resulting Koszul sign, and slightly abuse notation by denoting this sign by(−1)|x||y|. We note that the functor of points formalism allows us to mostly absorb these signs in the choice of base supermanifold, and a careful analysis of fermionic signs is only needed in Appendix A.
Convenient vector spaces. Our exposition closely follows [CG16; CG21], and in particu- lar we need to consider topological vector spaces of functions and distributions on manifolds. We consider these asconvenient vector spaces[KM97, Theorem 2.14] which form a symmetric monoidal category under theprojective tensor product⊗π; compare [CG16, Definition B.5.0.1]. Since all spaces we consider arise as spaces of sections or distributions valued in vector bundles, we can always compute it explicitly via the formulaΓ(X;E)⊗πΓ(Y;F)∼=Γ(X×Y;EF), withEF= p∗1E⊗p2∗Fthe exterior tensor product. A dual formula holds for spaces of distributions. The extension to supermanifolds only uses finite-dimensional linear algebra and poses no additional analytical difficulties.
We often consider spaces of sections over total spaces of fiber bundlesE →B, with the baseBgiven by the background fields of the field theory or the test supermanifold of the functor of points formalism. Tensor products and duals should then be taken relative to the base, e.g.,D(E)∼=HomO(B)(O(E),O(B)).
Homological algebra and filtrations. We work throughout in cohomological grading.
This means that the suspensionC[n]of a cochain complex is given by shiftingdown, i.e., (C[n])k =Cn+k. IfVis a vector bundle without any cohomological grading, we letV[n] denote the cohomologically graded vector bundle withV[n]k =
(V fork =−n
0 else .
For(C, dC)a cochain complex, we denote byC\the underlying graded vector space.
Thesymmetric algebraSym∗V on aZ×Z/2-graded convenient vector space is the sumLn≥0(V⊗πn)Sn of orbits under the action of the symmetric group, and we denote the product of two of its elements by . We denote the product ∏n≥0(V⊗πn)Sn by Symd∗V.
The additive category of convenient vector spaces is not abelian, so that the construction of its derived∞-category is not straightforward. In particular, a chain map which induces an isomorphism on cohomology need not admit a chain homotopy inverse.
However, in this thesis we will always be able to produce a chain homotopy inverse via thehomological perturbation lemma[Mar01, Theorem 3], which we will only need in the simplest case ofsplit (complete) filtrations. Due to their central importance throughout this thesis, we recall these notions in this preparatory section:
Definition 0.1 Asplit filtered convenient vector space is a family V = {V(d)}d∈Z of convenient vector spaces, with morphisms f :V→Wgiven by maps of convenient vector spaces f = {fd,d0 :V(d) → W(d0)}d0≥d. Theassociated gradedis also the family GrV= {V(d)}d∈Z, but now considered in the category of graded vector spaces where morphisms are only defined ford=d0. Theinduced map on the associated gradedis given by the diagonal componentsGrf ={fd,d}d∈Z. A morphism of split filtered convenient vector spacesincreases filtrationif the induced map on the associated graded is zero.
Lemma 0.2 (Homological Perturbation Lemma) Let (C, dC)and (D, dD) be cochain complexes of filteredconvenient vector spaces andr:C→D,i:D→C,h:C[1]→C,δ : C→C[1]be maps of convenient vector spaces such that
Chain maps. r,iare chain maps, and(dC−δ)2=0; set p:= idC−[dC,h]
Strong deformation retraction. We haver◦i= idD andp=i◦r, in particular p2= p Side conditions. h2=h◦i=r◦h=0
Filtration. The mapsr,i,hare morphisms of filtered convenient vector spaces, andδincreases filtration
We call the mapsr,i,hcontraction data.
Then the followingperturbed contraction datasatisfy the same conditions:
d0C= dC−δ d0D = dD+r
∑
n≥0
(δh)nδ
! i
n
∑
≥0i0 =i+h
∑
n≥0
(δh)nδ
! i
h0 =h+h
∑
n≥0
(δh)nδ
! h
Here only a finite number of terms in the infinite sums are nonzero in each component, so that they converge to morphisms of filtered vector spaces.
In particular, let(C, dC)be a chain complex of split filtered convenient vector spaces, and let (r : GrC GrH : i,h)satisfy the above conditions on the associated graded, whereHis concentrated in a single cohomological degree. Then there is a contraction(r0 :C H:i0,h0), and in particular the cohomology ofCis concentrated in a single cohomological degree.
In the last situation, we slightly abuse language by saying thatthe spectral sequence defined by the filtration degenerates on theE2-page. We also note that the conclusion of the above lemma holds if we pass to the filtered colimit of a sequence of degreewise embeddings of complexesCi, each of which is equipped with contraction data, but where the obvious diagram doesnotneed to commute, as one can iteratively update the map between the subcomplexes Hi and use that cohomology commutes with filtered colimits.
We will employ several split filtrations, which we summarize here:
Nilpotent filtration. As explained in [CG21, Section 7.3], we work relative to a base dg manifold(B,B). HereBis an augmented sheaf ofC∞(B)-algebras, and we demand that the augmentation idealI is nilpotent. We then have a finite filtration by its powers.
Weight filtration. In Chapter 1, we first consider algebras equipped with aweight grading; the name was chosen by analogy to [LV12, Subsection 1.5.11]. We stress that this new grading is purely an algebraic bookkeeping device, and does not come with corresponding Koszul signs. Once we globalize our constructions to manifolds in Section 1.4, the grading becomes replaced by aweight filtration, which mixes the nilpotent filtration with the polynomial degree and the degree in the formal variableh. The key point is that the quantum field theory we consider is free modulo¯ terms of positive weight filtration, allowing for explicit calculations, which then serve as inputs to the homological perturbation lemma. For this reason, the name
“free-to-interacting filtration” chosen in [EGW21] might be more fitting.
¯
h-adic filtration. In Chapters 3 and 4, we always work with flat objects over the power series ringC[[h¯]]and the Laurent series ringC((h¯)) =C[[h¯]][¯h−1]. This defines a split
filtration, where the terms in the associated graded are all isomorphic, with the prefactorhN giving the filtration degree.
We warn the reader that this is not quite the same setup as in our main reference, which instead works with pro-convenient vector spaces [CG16, Section C.4]. Any split filtered vector spaceVdefines a pro-vector spacelim←−N
L
d≤NV(d), but this functor is not full (in pro-vector spaces, the mapsfd,d0only need to vanish ford0 ≥d00(d)withd00(d)−−−→d→∞ ∞).
We have not found the additional flexibility of pro-vector spaces necessary, and the weight grading and filtration are crucially used in the calculation of obstruction groups and description of the factorization product, respectively. In any case, the translation of their results to filtered vector spaces is completely straightforward.
