Super-harmonic functions

Proposition A.3.4 The mapsΦχ,εhave a well-defined,Isom(_C^1|1)-invariant limitΦforε→0, which is independent ofχand satisfies

Φ(D∂(f)µ_E) = f,D∂(_Φ(σ))µ_E =σ.

Proof By definition,P_χ,ε is invariant under simultaneous left translation of both argu-ments and theU(1)-action. SinceΦχ,εcommutes with left invariant differential operators, it suffices that the components ofΦχ,ε(σ)have continuous limits whose supremum on a compact set is bounded by the supremum of a derivative of a component ofσ. To that end, we decomposeΦχ,ε(σ_od+ζσ_ev) = _Φ^ev_χ,ε(σ_ev) +ζΦ^od_χ,ε(σ_od); explicitly, using polar coordinates

(z⁰,z⁰) = (re^it,re^−it) with(r,t,θ)∈_R>0×_R/2πZ×_C^0|1, the integrals become

Φ^ev_χ,ε(g)(z,z) =−² π

Z

R>0×_R/2πZχ(ε⁻¹r)(logr)g(z+re^it,z+re^−it)rdrdt Φ^od_χ,ε(g)(z,z) =−²

π Z

R>0×_R/2πZ∂_z⁰(χ(ε⁻¹r)_logr)g(z+re^it,z+re^−it)rdrdt

= ² π

Z

R>0×_R/2πZ(χ(ε⁻¹r)logr)(∂_z⁰g(z+re^it,z+re^−it))rdrdt

= ² π

Z

R>0×_R/2πZ

(χ(ε⁻¹r)_logr)(∂_zg(z+re^it,z+re^−it))rdrdt

The limit then exists and is independent ofχsincerlogris bounded in a neighborhood of0.

In polar coordinates, we have∂_z∂_zf^idz₂^dz = ¹₄((r∂_r)²f+∂²_φ)^dr_r^dφ. We now prove that lim_ε→0Φ^ev

−∂_z∂_z(f)^idz₂^dz = f. By translation invariance, it suffices to prove this statement at0, where the left side is

lim

ε→0Φ^ev

−∂_z∂_z(f)^idzdz 2

= ² π

Z

R≥0×R/2πZlogr(∂_z∂_zf)(re^it,re^−it)rdrdt

= ¹ 2π

Z

R≥0×_R/2πZlogr(∂r(r∂rf) +r⁻¹∂²_tef)drdt

=

Z

R≥0

logr∂_r(r∂_ref)dr

= (rlogr)|_r=0f(0)−

Z

R≥0

r r∂refdr

= f(0)

where we integrated out the angular coordinatet, with ef(r) = _2π¹ R

R/2πZ f(re^it,re^−it)_dt, and then integrated by parts twice, using that ef extends continuously by ef(0) = f(0)_.

The proof of the corresponding statement forΦ^od, and the formula(D∂µ_E)◦_{Φ, works}

completely analogously.

We summarize the properties of the functionlimε→0P_χ,ε which we used in this proof in the following definition:

Definition A.3.5 LetE→Bbe a family of Euclidean supermanifolds. Anexact propagator is a functionP∈ O(E×_BEr∆)which satisfies the following properties:

• Pis even, i.e.,P◦σ =_{P, where}σswaps the two factors

• D₁∂₁P= D2∂2P=0

• In a local Euclidean chartB×U⊂ B×_C^1|1→_{E, we have} P(Z₁,Z₂) =−¹

π log|Z₂⁻¹Z₁|²+ f(Z₂⁻¹Z₁)

where f ∈ O(B×U)is a super-harmonic function which extends over0

Proposition A.3.6 Given an exact propagatorP ∈ O(E×_BEr∆), there is aparametrix ΦP : Ber_prop(E/B)→ O(E)which satisfies the following properties:

• Given a morphism of families (ef, id_B) : (E⁰ → B) → (E → B)_covering id_B, the diagram

Ber_prop,/B(E⁰) Ber_prop,/B(E)

O(E⁰) O(E)

ef∗

ΦP

ef^∗ Φef∗_P

commutes, while for a cocartesian morphism of families(ef,f):(E⁰ → B⁰)→(E→ B)_, the diagram

Ber_prop,/B⁰(E⁰) _Berprop,/B(E)

O(E⁰) O(E)

fe^∗

ΦP

fe^∗ Φ_ef∗_P

commutes

• The restriction ofΦP(σ)to the complement of the support ofσisR

E/BP(−,z)σ(z)

• For f ∈ O_prop(E)_{, we haveΦP}(D∂(f)µ_E) = f, and forσ ∈ Ber_prop(E/B)_{, we have} D∂(_ΦP(σ))µE =σ

Furthermore, these properties characterize the assignmentP7→_ΦPuniquely.

Proof We have seen that the standard parametrix satisfies these conditions. It is imme-diate that one may add an arbitrary left-invariant function which is super-harmonic in both variables to the standard propagator and parametrix which also satisfy them. This shows well-definedness and uniqueness for subsets ofC^1|1equipped with the standard Euclidean structure. In general, cover the familyE→Bby Euclidean charts, and use a

partition of unity argument to reduce to the charts.

Example A.3.7 Let Z_l = _C^1|1/2πilZ be the infinite-length cylinder of radiusl. We choose coordinatesx= ^z+_2l^z ∈_R,y= ^z_2il^−z ∈_R/2πZ,ξ =l^−1/2ζ ∈_C^0|1_{and define}

p_l(x,y,ξ) =−¹

πlog|e^x+iy−1|²+ ^x π

=−¹ πlog

e^(x+iy)/2−e^−(x+iy)/2

2

P_l(x₁,y₁,ξ₁;x₂,y₂,ξ₂) =p_l((x₂,y₂,ξ₂)⁻¹(x₁,y₁,ξ₁))

=−¹

πlog|e^{x2−x1+i(y2−y1)}−1|²+ ^x2−x₁ π

− ^ξ1ξ2 π

e^{x2−x1−i(y2−y1)} e^{x2−x1−i(y2−y1)}−₁ −¹

2

!

Note that even though it is not immediately obvious thatP_l is symmetric, the second line shows that p_l is even, which is equivalent.

One easily checks thatp_lis super-harmonic with a logarithmic singularity atx=y = ξ =_{0, so that}P_l defines a left translation-invariant propagator.

The observables of the free2|1-sigma model are given by functions on the classical solutions, i.e., super-harmonic functions. While these already form a convenient vector space, in the BV formalism they arise instead as the cohomology of a complex, i.e., as the quotient of compactly supported sections by the image of the Laplace–Dolbeault operator. For an open subset of a split cylinder orC^1|1, we first explain how to find explicit representatives in this quotient by theirsuper-harmonic dualand then prove that this complex has a contraction onto its cohomology.

We are interested in characterizing the cokernel of the operatorD∂µ_Eon compactly supported sections. To simplify the exposition, we restrict to open subsets of the cylinder Z_l = _C^1|1/2πilZ; the standard Euclidean supermanifoldC^1|1can be treated similarly using the standard propagator, and a non-split cylinder can be treated by iteratively modifying the exact propagator for split cylinders. For convenience, we fixl ∈ _R>0, noting that all results and proofs generalize to families in a straightforward manner.

Definition A.3.8 LetU⊂ Z_lbe an open subsupermanifold with compact closure. The

super-harmonic dualis the map

Φe :Γcs(U; Ber(U))→ lim−→

U∪V=Z_l

O(V)

σ7→ {_ΦPl(σ)|_V}_{{V|U∪V=Zl,σ|V=0}}

We say that a super-harmonic functiong ∈ O(V), defined on the complement of a compact subset of|C_l|_{, isregular at∞}if there are real numbersa,b,csuch that

g(x,y,ξ)−a|x| −bsgn(x)−csgn(x)ξ

and all of its derivatives go to zero forx→ ±_{∞. For}V⊂ |Z_l|_{open, let}O_D∂=0,reg(X) = colim_U∪V=Zl

Ucompact

O_D∂=0,reg(V)denote the LF space of germs of super-harmonic functions on Xwhich are regular at∞, where the Fréchet norms on the terms of the colimit are given by the sum decomposition of the asymptotic expansion, with the norms on harmonic functions which go to0at infinity given by the supremum of some derivative on some

cocompact subset.

In other words, the super-harmonic dual is the value of the parametrixΦP_l(σ)_{, but we} only remember its germ on a small neighborhood ofZ_lr^U.

Proposition A.3.9 The mapΦe takes values in germs of super-harmonic functions which are regular at∞.

Furthermore, it vanishes on sections of the formD∂(f)µ_E with f compactly supported, and defines an isomorphism of LF spaces

Γcs(U; Ber(U))/ imD∂µ_E ∼=O^reg_D∂=0(Z_lr^U).

Proof Letx₀ =max_x∈|U|x. The termlog|e^x+iy−1|of the one-variable propagator p_l and all of its derivatives are bounded by a constant factor ofe^x0−x, whereas the linear term gives forx1> _x0

Z

U

x₂−x₁

π + ^ξ1ξ2 2π

σ(x₂,y₂,ξ₂)µ_E(x₂) which is affine-linear inx₁,ξ₁.

For x → −∞, the same reasoning applies, and we get an asymptotic affine linear estimate with negated coefficients. By the propertyD∂ΦP_l(σ)µE =σ, we see thatΦe(σ) is super-harmonic outside of the support ofσ, and we have shown that it is regular at∞.

Let[f] ∈ O_D∂=0,reg(Z_lr^U), and choose a representative f which is defined on an open subsupermanifoldVwithU∪V =Z_l. Choose a bump functionχ∈ O(Z_l)_with χ|_ZlrV =0,χ|_ZlrU =_{1. Since} fχextends by0to all ofZ_l, and is super-harmonic outside U, the sectionσ= D∂(fχ)µ_Ehas compact support. CalculatingΦe(σ), we can integrate by parts as above to get the sum of f and a boundary term at infinity. If f and all of its

derivatives go to zero at infinity, this boundary term disappears in the limit, and we obtain f = Φe(σ). On the other hand, for each of the functions|x|, sgn(x)_andsgn(x)ζ the boundary terms at±_∞cancel, so that we obtainΦe(σ) = f as long as f is regular at

∞.

Letσ∈_Γcs(U; Ber(U))_{. Then}(1−χ)Φe(σ)is compactly supported, and we have D∂((1−χ)Φe(σ))µ_E = D∂(Φe(σ))µ_E−D∂(χΦe(σ))µ_E

=σ−D∂(χΦe(σ))µ_E

Thus the two sections are equal in the cokernel, and we have produced an inverse toΦ.e The explicit description shows that it is a morphism of LF-spaces.

Remark A.3.10 Essentially the same proof, using the standard propagator, identifies the cokernel ofD∂µ_Eon compactly supported sections on an open subsupermanifold U⊂ _C^1|1with germs of super-harmonic functions on the complement which have an asymptotic expansion inlog|z|and a super-harmonic function whose derivatives go to

0at∞.

By uniqueness of analytic continuation, the cohomology of the two-term complex of compactly supported sections f 7→ D∂(f)µ_E is concentrated in one degree for a con-nected, non-proper family. Formally applying the Künneth theorem, one expects the same to hold for its tensor powers. Since we work with the completed tensor powers, this argument can not be applied directly, however its conclusion still holds:

Proposition A.3.11 LetE→Bbe a fiberwise connected and non-compact family of Euclidean supermanifolds. Exterior tensor powers of the Laplace–Dolbeault operator define for each integer nthe complex

O_cs(E^×Bn) ^Ln_i=1Γcs(E^×Bn;p^∗_i Ber_/B(E)) . . .

[−n] [1−n]

. . . ^Ln_i=1Γcs(E^×Bn;^N_j6=ip^∗_j Ber_/B(E)) _Γcs(E^×Bn; Ber_B(E^×Bn))

[−1] [0]

Its cohomology is concentrated in degree0, where it is given by the convenient vector space O_D∂=0(E^×Bn)^∨obtained as the continuous dual of the super-harmonic functions on fiberwise products ofE→B, with the symmetric group acting by permuting the factors.

In fact, this complex is a filtered colimit along inclusions of complexes of convenient vector spaces with a contraction onto their cohomology.

Proof By the usual filtration argument, it suffices to prove this statement forB=_{pt, so} that we may assume thatE=_ΠK^−1/2 is a split Euclidean supermanifold obtained from a flat Riemann surfaceSequipped with a spin structureK^1/2. The linear complex is a contraction of

Ω^0,0(_Σ) _Ω^0,1(_Σ)

Ω^1,0(_Σ) _Ω^1,1(_Σ)∼=Ω^1,1(_Σ)

Ω^0,0(_Σ;K^1/2) _Ω^0,1(_Σ;K^1/2)

∂

∼=

∂

∂

as one easily sees using the homological perturbation lemma to remove the two new 1-form components to reproduce the Laplace operator∂∂(note that the middle row is obtained from the Dolbeault operator on(1, 0)-forms by complex conjugation, using the real structure onΩ^1,1(_Σ)∼=_Ω²(_Σ)⊗C). This complex is equipped with a filtration whose associated graded is the Dolbeault complex of the complex vector bundleC⊕ Ω^1,0⊕K^1/2, albeit with an interesting complex structure. Taking projective tensor powers yields a quasi-isomorphism to the Dolbeault complex of the bundle(_C⊕_Ω^1,0⊕K^1/2)ⁿ_, but equipped with a differential that is the sum of the operator∂and one that sends (p,q)_{-forms to}(p−_1,q+₁)-forms, i.e., lowers the Hodge filtration. Since S is non-compact, it is Stein [For81, Corollary 26.8], and therefore so are its powers. By [Pal73], on a relatively compact, Stein submanifold ofCⁿ the Dolbeault complex possesses a continuous retraction onto the holomorphic sections in degree0, and its construction via the Hodge decomposition shows that it defines a map of convenient vector spaces.

Using the integral pairing with the Dolbeault complex on distributional, compactly supported sections, we see that the latter has a contraction onto its cohomology, which is concentrated in the top degree. Therefore the same is true on theE²-page of the spectral sequence coming from the Hodge filtration on our complex, so that it degenerates on this page in the sense of Lemma 0.2. Finiteness of the Hodge filtration then shows that the cohomology is indeed concentrated in degree0.

The same argument applies using not necessarily compactly supported sections, show-ing that the cohomology of the resultshow-ing complex is concentrated in degree−n, where it is by definition given by the super-harmonic functions. By elliptic regularity, the integration pairing between the two complexes exhibits either one as the continuous

dual of the other.

In contrast, on a closed Euclidean supermanifold, i.e., a torus, the cohomology of this complex is finite-dimensional, but not concentrated in a single degree. In fact, there is an explicit deformation retraction onto the subspace of (even) translation-invariant sections.

Proposition A.3.12 LetMbe the moduli functor of pinned bounding super-tori from Propo-sition A.2.2. There is a unique functionalΦ_∞ : Ber_/M(_T²_M^|1)→ O(_T²_M^|1)with the following

properties:

• Φ∞isO(M)-linear andT²-equivariant

• The maps(D∂µ_E)◦_ΦandΦ◦(D∂µ_E)_{are the (T}²-equivariant) projections whose kernel are exactly theT²-invariant sections

• Φis self-adjoint, in the sense that given two sectionsσ_i ∈_Ber/M(_T²_M^|1)_{we have}

Z

T^2|1σ₁Φ_∞(σ₂) = (−1)^|σ1||σ2|

Z

T^2|1σ₂Φ_∞(σ₁).

Furthermore,Pis invariant under symmetries of the family, i.e.,U(1)×SL(2,Z)nT² -equiva-riant and commutes withD⁺_M.

Proof First, note that in our coordinates the section of the Berezinian defined by the Euclidean structure isµ_E =vol^1/2(1−vζ)dζdxdy; independence ofvis clear since

∂−∂|_v=0lies in the subbundle generated byDandD², whereas the change of basis from {∂,−D|²_v=0,D|_v=0}_to{∂,−D²,D}is given by the matrix with diagonal term{1,(1+ vζ)², 1+vζ}and only one off-diagonal entry, coming fromD² = D|²

v=0+vD(ζ)_D.

TheT²-action onT^2|1is diagonalized by the functionse_m,n = exp(2πi(mx+ny))_and em,nζ, with the action on sections of the Berezinian similarly diagonalized byem,nµ₀and e_m,nζµ₀, whereµ₀ = ^µE

vol^1/2(1−vζ) = dζdxdy. The operator f 7→ D∂(f)µ_E is therefore given by

em,n7→vol⁻¹

πv(−mw2+nw₁) + ^π

2|mw2−nw₁|²

vol ζ

em,nµ0

em,nζ 7→vol⁻¹π m(w₂−vζw₂) +n(vζw₁−w₁)em,nµ₀

(the fact that this operator is independent of the variablevin our presentation reflects the super-conformal invariance of the Laplace–Dolbeault operator). Thus the operator is2×2-block diagonal for the bases{em,n,em,nζ}_and{em,nζµ₀,em,nµ₀}with nilpotent off-diagonal terms and diagonal terms invertible for(m,n)6= (0, 0). An easy calculation then shows that the assignment

e_0,0µ₀7→ 0 e0,0ζµ07→ 0

em,nµ07→ ^vol

2·v

π²(mw₂−nw₁)² + ^vol·ζ π(mw₂−nw₁)

!

em,n for(m,n)6= (0, 0) em,nζµ07→ ^vol

2

π²|mw₂−nw₁|² + ^vol

2vζ π²(mw₂−nw₁)²

!

em,n for(m,n)6= (0, 0) satisfies[D∂µ_E,Φ] = id−p_H, withp_Hthe projection onto the1|1-dimensional subspace ofT²-invariant functions (the notation evokes that these are theharmonic functions)

Im Dokument Perturbative quantization of the two-dimensional supersymmetric sigma model (Seite 192-200)