The free theory and conformally-invariant deformations

In the Costello–Gwilliam formalism, a free classical field theory is defined as an elliptic complex equipped with a(−1)-shifted symplectic structure [CG16, Definition 2.3.1]. In our case, the space of fields is concentrated in degrees0and1, so that the differential squares to zero automatically. The differential is therefore the Hamiltonian vector field of a quadratic function of cohomological degree0. The only scale-invariant such function is given by a symmetric bilinear tensorg∈Sym²V^∨, and the resulting differential operator

j

i i j i j

k

i j i

j i

i A

W

Figure 2.1.: The two types ofAffConf(_C^1|1)-invariant local action functionals. On the left, they are presented in the superspace formalism, and on the right, they are expanded into components. In the top row, we illustrate the “interaction” term R

C^1|1 A(v;Dv,∂v)µ_E for a bilinear form of weight degree4. The two decorated legs correspond to the inputs to which the differential operatorsDand∂are applied; in the Einstein sum convention, this is written asR

C^1|1 A_ij;Iv^I(Dvⁱ)(∂v^j)µ_E. Expanding the fieldv=x+ζψ, we obtain the three summands on the right, the last of which is pro-portional toR

C(∇A)_ijk;Ix^Iψⁱ(∂x^j)ψ^k. In the bottom row, we have similarly illustrated the “symmetry” termR

C^1|1W(v;V)for a vector fieldW ∈Vectd(V)of weight degree3.

is elliptic iff it defines a non-degenerate bilinear form. We therefore arrive at the following definition:

Definition 2.2.1 Afree,AffConf(_C^1|1)-invariant classical sigma modelis defined by a finite-dimensional vector spaceVand a non-degenerate symmetric bilinear formgonV. The corresponding elliptic complex is

O(_C^1|1)⊗V Ber(_C^1|1)⊗V^∨

[0] [1]

Q

whereQ= D∂µ⊗[is the tensor product ofD∂µ_E : f 7→(D∂(f))µ_E, theLaplace–Dolbeault operatordefined in Definition A.3.1, and[:V ∼=V^∨, the isomorphism defined byg.

Thegauge fixing operatorQ^GFis the tensor product of the operator Ber(_C^1|1)⊗V^∨ → O(_C^1|1)⊗V

fµ_E 7→D(f)

and the inverse isomorphism]= ([)⁻¹ :V^∨ ∼=V.

Remark 2.2.2 Unpacking this into the components, the elliptic complex splits as the sum C^∞(_C)⊗V Dens(_C)⊗V

Γ(K^1/2)⊗_{ΠV Ω}^1,0(K^1/2)⊗_ΠV^∨

[0] [1]

∂⊗[

−∂∂^idz₂^dz⊗[

while the gauge fixing operator is the trivialization ofDens(_C)_by ^idz₂^dz on the even component and the composition

Ω^1,0(K^1/2)⊗V^∨ ∼= _Ω^0,0(K^1/2)⊗V^{∨ −}−−−→^∂⊗] _Ω^0,1(K^1/2)⊗V ∼=_Γ(K^1/2)⊗V on the odd component.

While the line bundle K^1/2 and its conjugate are trivial, the notation captures the behavior under the action ofAffConf(_C^1|1). In particular, whileQis invariant under the action of this group, the gauge fixing operatorQ^GFis only invariant under the smaller groupIsom(_C^1|1)as it uses the isomorphismK^−1/2 ∼= K^1/2 and the trivialization of

Dens(_C)defined by the metric.

Proposition 2.2.3 The above data defines a free classical field theory and gauge fixing operator in the sense of [CG21, Definition 4.4.0.4-5]. Furthermore, we can replaceC^1|1with an arbitrary family of Euclidean supermanifolds.

Proof Both the operator defining the free theory and the gauge fixing operator square to zero by degree reasons. Their anticommutator is the operator∂D², which is the standard Laplace operator on the respective trivial bundle (and in particular becomes the Laplace operator on the even and odd constituent fields, to which the given definition applies.) Both operators are local and Isom(_C^1|1)-invariant, so that they make sense on an arbitrary family of Euclidean supermanifolds. Their anticommutator will then be the sum of the standard Laplacian and a nilpotent operator of lower degree, which depends on a splitting of the fields into even and odd constituents. This is still a generalized

Laplace operator in the sense of the given definition.

Proposition 2.2.4 The dglaMetd(V)nΩ³_cl(V)is isomorphic toAffConf(_C^1|1)-invariant de-formations of the free scalar field theory defined by(V,g).

Proof This is essentially formal: The bilinear formgdefines a classical action functional whose Hamiltonian vector fieldQsatisfies the Classical Master Equation, i.e., squares to zero, by degree reasons. Its scale-invariant deformations are then given by the twist of the dgla of scale-invariant local action functionalsVectd(V)n(_VB(_Sym²V^∨)⊕_Ω³_cl(V))_{by the} corresponding Maurer–Cartan element, which was the definition ofMetd(V)nΩ³_cl(V)_.

We now consider general conformal supersymmetric classical field theories. The philosophy of perturbation theory postulates that we should split the classical action into a quadratic part defining a free theory and an interacting part, which is of cubic or higher degree in the fields up to nilpotent terms. On the algebraic side, this corresponds to twisting the shifted Lie algebra by the Maurer–Cartan element defining the free theory, which (up to a shift) produces the dglaMetd(V)_.

To describe the formal neighborhood of constant maps in the2|1-dimensional sigma model with curved target, we need to enlarge the ground fieldCto a general derived manifold. The precise definition follows [CG21, Definition 7.3.0.1]; our definition is slightly more general by allowing a restricted class of pro-nilpotent structure sheaves, as suggested in the remark following the given reference.

Definition 2.2.5 Adg ringed manifold(X,A)is a pair of a smooth manifoldXand a sheaf of commutative dgΩ^∗(X)_-algebras A, together with an augmentationA → C^∞(X)_, whose underlying C^∞(X)-module is given by sheaves of sections of a graded vector bundleE→ M. The bundleEis concentrated in finitely many ranks, each of which is finite-dimensional, and the kernelI of the augmentationA →C^∞(X)_satisfiesI^N =0 forN 0. A morphism between dg ringed manifolds(X,A)_and(Y,B)is a smooth map f :X→Yand a map of augmented dgΩ^∗(X)_-algebrasα: f^∗B ⊗_f∗Ω^∗(Y)Ω^∗(X)→ A_.

Aweight-graded dg ringed manifoldis defined in the same manner, where nowA = {A^(d)}_d∈Zis a sheaf of augmentedΩ^∗(X)^\-cdgas equipped with an additional weight grading which is preserved by the differential and multiplication, where Ω^∗(X)^\ _is considered as a weight-graded (by cohomological degree) algebra with vanishing dif-ferential. We demand that each homogeneous component is given by sections of a finite-dimensional vector bundle and that the sum of the components of positive weight degree and the augmentation ideal in weight degree0is nilpotent.

Example 2.2.6 The main examples areA(X) = C^∞(X)_and A(X) = _Ω^∗(M)_{, which} model all and locally constant functions onM, respectively. We obtain formal moduli problems by consideringA(X) = _Ω^∗(X)⊗Afor an Artinian algebra A. The added generality captures the local nature of the formal moduli problem overX.

The cdga Ω^∗ X;O(NCd(TX)), dO

and its variants considered in Section 1.4 are examples of weight-graded dg ringed manifolds. Although neither definition is a special case of the other, we slightly abuse notation by referring to both notions as dg ringed manifolds, adding the assumption that the weight grading is preserved in the second case.

Note, however, that the cdga Ω^∗ X;O(NCd(TX)), d_∇+d_O−ι_R

which was studied in Section 1.4 is not finite-dimensional, and the differential does not preserve the weight grading. This issue can be sidestepped by careful analysis of the subspaceo(V)⊂NCd(V) of linear vector fields, by which the other components d_∇−ι_R enter, similarly to the

globalization of Harish-Chandra modules and morphisms.

We now give the definition of a classical sigma model with curved target in the BV

for-malism by a standard construction in derived differential geometry. While we formulate it for a general dg ringed manifold, the example to keep in mind isA=_Ω^∗_{, in which} case we use a connection to identify infinitesimally close points.

Definition 2.2.7 Aclassical2|1-sigma modelis a tuple(X,A,E_Σ,θ)_{, where}

• (X,A)is a dg ringed manifold in the sense of Definition 2.2.5

• E_Σ →Xis a finite-dimensional vector bundle, considered as a (cohomologically) graded vector bundle in degree0, equipped with a map d_∇ : Γ(X;A ⊗E_Σ) → Γ(X;A ⊗E_Σ[1]) which satisfies the Leibniz rule [d∇,a·] = (dAa)·, and a non-degenerate symmetric bilinear formg_E ∈ _Γ(X;A₀⊗Sym²E_Σ^∨)annihilated by d∇

• θis a Maurer–Cartan element of the curved dglaΓ X;A ⊗(Metd(E_Σ)nΩ³_cl(E_Σ)) with differential d∇ and curvature R∇ ∈ _Γ(X;A ⊗o(E_Σ)[2]), considered as a subalgebra ofΓ(X;A ⊗Metd(E_Σ)[2])_{, so that} d²_∇ = [R_∇,−]

IfAis weight-graded, we can complete the tensor product and demand thatθhas total weight-grading0. To distinguish this from the case whereAis nilpotent, we call such a sigma modelweight-graded.

In either case, the deformation-obstruction complexis the sheaf of dglas obtained by twistingA ⊗(Metd(E_Σ)nΩ³_cl(E_Σ))_byθ. We denote the resultingAffConf(_C^1|1)_-invariant action functional on the extended field spaceF_Σ =E_Σ[0]⊕E^∨_Σ[−1]⊗Ber(_C^1|1)_byS_θ = S_free+I_θ, where the first term denotes the quadratic term of the action. The curved Maurer–Cartan equation forIθ is equivalent to thecurved Classical Master Equation

I_R+ (d∇+Q)I_θ+ ¹

2{I_θ,I_θ}=0

whereI_Ris the degree1quadratic local action functional such that d²_∇ ={I_R,−}_{, which} in turn is equivalent to(d_∇+Q+{I_θ,−})² =0.

More generally, by covering a family of Euclidean supermanifoldsE→Bwith coordi-nate charts and using the Euclidean invariance ofS_θ, we obtain a solution of the curved Classical Master Equation onE→B, which we denote with the same symbolS_θ. Example 2.2.8 The definition of Metd(E_Σ) as a square-zero extension shows that we can decompose θ = τ+ _{A, where}τis a Maurer–Cartan element of the curved dgla A ⊗Vectd(E_Σ)_andAis a degree0closed cochain ofCE^∗ A ⊗Vectd(E_Σ); VB(Sym²E_Σ^∨⊕ Ω³_cl(E_Σ)). In other words,τgives a consistent choice of local coordinate charts modeled on E_Σ, allowing for a perturbative expansion of the classical action, and A gives a consistent Taylor expansion of the interaction term in these coordinate charts. The corresponding local action functionals, corresponding to the decomposition of I_θ = Iτ+I_Ainto “symmetry” and “interaction” terms, were illustrated in Illustration 2.1.7.

By definition, the Maurer–Cartan elementθfrom Proposition 1.1.8 defines a weight-graded classical2|1-sigma modelΣV = (pt,O(dNC(V)),V,θ). The obstruction-defor-mation complex is identified withO(NCd(V);Metd(V)⊕_Ω³_cl(V)), and in particular, the

recipient group of the one-loop obstruction isH¹ O(dNC(V);Ω³_cl(V)), which is spanned by the lift of the first Pontryagin class by Proposition 1.1.20. For globalization, it will be more convenient to work with the modifiedbasic invariantfunctional I_bi = I_θ−Io, where Io is the quadratic “symmetry” functional adjoint to the identity of o(V)_{, and} to consider the differential d∇,O = dO+ [τo,−]_onO(dNC(V))⊗V . It satisfies the curved Classical Master Equation withI_Rthe quadratic action functional corresponding toF(τo) = dOτo+ ¹₂[τo,τo]_{, is}O(V)-invariant and vanishes on the ideal generated by

o(V)^∨_.

We have seen that the formal moduli problem represented by the curved, filtered dgla Ω^∗(X;Metd(TX))is pointed and represented by the dglaVect(X)ngΓ(X; Sym²T^∨X)_{. On} the other hand, it is also equivalent to the formal moduli problem of formal exponentials.

We now show how this equivalence works explicitly. In particular, without a background Artinian algebraA, we obtain an essentially unique classical field theory describing the 2|1-sigma model on a formal neighborhood of the constant maps.

Proposition 2.2.9 The assignmentα7→ α∗θ_bidefines a natural equivalence from the formal moduli problem of formal exponentials to Maurer–Cartan elements ofΩ^∗(X;Metd(TX))_{. Given} a formal exponential, extensions of the resulting Maurer–Cartan element toΩ^∗(X;Metd(TX)n Ω³_cl(TX))are represented by the abelian dglaΩ^≥3(X). In particular, for any tuple(X,g,η)of a Riemannian manifold(X,g)_{and closed3-form}ηwe obtain a classical sigma model, which we denote byΣX,g,η.

Proof ByO(V)-equivariance, for any vector bundleE_Σ →Xequipped with a Rieman-nian metric, there is a canonical elementθ_bi,EΣ of the dgla O(NCd(E_Σ))⊗(Metd(E_Σ)n Ω³_cl(V)). Furthermore, if ∇^EΣ is any metric connection, this Maurer–Cartan element is a flat section, and it satisfies the curved Maurer–Cartan equationF(τ_o(EΣ)) + (_d∇+ d∇,O)θ_bi,EΣ+ ¹₂[θ_bi,EΣ,θ_bi,EΣ] =_0.

For a formal exponentialαon a Riemannian manifold(X,g), the elementα∗θ_bi,TXis a degree1element ofΩ^∗(X;Metd(TX))satisfying the curved Maurer–Cartan equation R∇^α+ d∇^α(α∗θ_bi,TX) +¹₂[α∗θ_bi,TX,α∗θ_bi,TX] =_{0, where} d∇^α = d∇+α∗τ_o(TX)is obtained by adding the globalization of the quadratic symmetry part ofτas a connection one-form to the Levi-Civita connection, andR_∇^α = α∗(F(τ_o(TX)))is its curvature. Since the splitting of∇^αinto the connection differential and linear orthogonal symmetries of the fields is arbitrary, we obtain thatα∗θ_TXis a Maurer–Cartan element. We have seen that this natural transformation is an equivalence in Proposition 1.4.12.

In particular, takingA=C, in weight degree−1the formal exponentialαis fixed to the isomorphismΩ⁰(X;T^∨X)∼= Ω¹(X), and since the Levi-Civita connection is torsion-free, it annihilates the generatorso(TX)^∨[−1]. For a generalA, we can think of a Maurer–

Cartan element as a family of metrics parametrized bySpec(A), and it is more natural to use the corresponding Levi-Civita connection∇^α. For now, we are simply choosing different basepoints of the affine space of orthogonal connections, but this viewpoint will be more useful in the quantization we study in the next chapter.

The obstruction-deformation complex of the resulting classical field theory is equiva-lent to the shifted dgla

Γ(X;TX)[−1]ng(_Γ(X; Sym²T^∨X)⊕_Ω^≥3(X)).

In particular, obstructions to quantization lie in H¹(_Ω^≥3(X)) ∼= H⁴(X;C)_{. A closed} degree0elementη ∈_Ω^≥3(X)⊗Agives a deformationηeof the classical action which satisfies the Maurer–Cartan equation, defining a classical2|1-sigma model ΣX,g,η =

(X,Ω^∗(X),TX,α∗τ+η_e)_.

Remark 2.2.10 From the construction, we see that the classical2|1-sigma model defines a sheaf of classical field theories on the site of Riemannian manifolds. The one-loop obstruction defines a section of the sheafΩ^≥3(−)on this site, i.e., is locally defined and invariant under diffeomorphisms (a priori, this is to be interpreted in a suitably derived sense). We will not explore this viewpoint further, as it is a straightforward consequence of our investigation of the universal deformationΣVof the free2|1-sigma model and the methods of Section 1.4.

In this thesis, we will only investigate the classical sigma modelsΣVandΣX,g,η. How-ever, since the former is the universal deformation of the free sigma model, our methods generalize to more general ground rings, as long as the vector bundleE_Σis concentrated in cohomological degree0. More generally, we could for instance allowghostsin coho-mological degree−1, which would allow the study of the sigma model with target an Artin stack, e.g., the quotient stack of an isometric action of a Lie group on a Riemannian

manifold.

Remark 2.2.11 In the language of [Cos11a, Definition 2.1.1], the degree−1part of the classical action defines the structure of a curved shifted L_∞-algebra on V, while the degree0part defines a non-degenerate, invariant, symmetric bilinear pairing and a closed3-form. The BV Poisson bracket defined by the symmetry component{Iτ,−}_of the classical action corresponds to the Chevalley–Eilenberg differential, and the Maurer–

Cartan equation to the (curved)L_∞-relations. In particular, the globalized version on a Riemannian manifold recovers Costello’s description of smooth manifolds via curved L_∞-algebras.

The BV Poisson bracketQ+{I_A,−}of the interaction component can also be computed explicitly. To this end, we have to identify the Hamiltonian vector field corresponding to the local action functionalI_A, which is a straightforward calculation using integration by parts. We perform this calculation for the universal sigma modelΣV, using the flat, torsion-free connection∇^V to calculate partial derivatives of tensors onV. Here dv is a compactly supported infinitesimal, even change in the fields, so that there are no

boundary terms and fermionic signs are handled by the functor of points formalism.

dI_A(v) =−¹ 2

Z

C^1|1

(∇^VA)(v; dv,Dv,∂v) +A(v;Ddv,∂v) +A(v;Dv,∂dv)µ_E

= −¹ 2 Z

C^1|1 D(A(v; dv,∂v)) +∂(A;Dv, dv)µ_E

| {z }

=0

+ ¹ 2

Z

C^1|1 A(v;D∂v, dv) +A(v; dv,D∂v)

+ (∇^VA)(v;Dv, dv,∂v) + (∇^VA)(v;∂v,Dv, dv)

−(∇^VA)(v; dv,Dv,∂v)

! µ_E

The last expression is given by evaluating dvon a smooth section and therefore extends to distributional sections.

Let g(v;X,Y) = g₀+ ¹₂(A(v;X,Y) +A(v;Y,X)) be the symmetrization of g₀+_A, which is a non-degenerate symmetric bilinear form (non-degeneracy can be checked forv =0, where it is equivalent to non-degeneracy ofg₀). We can then define a tensor Γ∈VB((V^∨)^⊗2⊗V)_by

g(_Γ(v;X,Y),Z) = ¹ 2

(∇^VA)(v;X,Y,Z) + (∇^VA)(v;Z,X,Y)−(∇^VA)(v;Y,X,Z) IfAis symmetric, this defines the connection1-form of the Levi-Civita connection ofg in the trivialization of the tangent bundle determined by the affine structure onV. As expected from the classification of deformations of the classical action, the antisymmetric component of Aonly contributes through its de Rham derivative, and it is given by raising one index of the three-formηwith the metric. Together, this defines a connection

∇^η =∇^V+Γ. Note that∇^ηg=0, whereas the torsion of∇^η is given by raising an index ofηusingg.

The BV Poisson bracket then becomes the Koszul differential computing the derived zero locus of the Euler–Lagrange equations∇^η_∂Dv =∇⁻_D^η∂v=0. Concretely, applying it to a linear functionalX7→ hω,Xi(0)of the antifields shows thathω,∇^η

∂Dvi(0)_{is exact} under the BV bracket with the classical action, which is a differential by the curved

Classical Master Equation.

Im Dokument Perturbative quantization of the two-dimensional supersymmetric sigma model (Seite 72-79)