Super J-holomorphic curves

Super J-holomorphic curves

Enno Keßler

Seminar on Algebra, Geometry and Physics October 26, 2021

Max-Planck-Institut für Mathematik, Bonn

J-holomorphic curves¹

• AJ-holomorphic curveφ: Σ→Nis a map from a Riemann surfaceΣto an almost Kähler manifold(N, ω,J)such that

∂_Jφ= 1

2(dφ+J dφI) = 0∈Γ T^∨Σ⊗φ^∗TN0,1

1McDuff and Salamon (2012).J-holomorphic curves and symplectic topology.

J-holomorphic curves

• J-holomorphic curves are absolute minimizers of the Dirichlet action.

• Under certain conditions the moduli spaceM^∗_p(A)of J-holomorphic curves and[imφ] =A∈H2(N,Z)is a manifold.

• There is a compactificationMp,k(A)via stable maps.

• Gromov–Witten invariants ofNcan be constructed as certain integrals overMp,k(A).

SuperJ-holomorphic curves²

• Asuper J-holomorphic curveΦ : M→Nis a map from a superRiemann surfaceMsuch that

DJΦ = 1

2 (dΦ +J dΦI)|_D= 0∈Γ D^∨⊗Φ^∗TN0,1

2Keßler, Sheshmani, and Yau (2021).“SuperJ-holomorphic Curves:

Construction of the Moduli Space.”

SuperJ-holomorphic curves of genus zero

• The differential equations of superJ-holomorphic curves couple the Cauchy–Riemann equations ofJ-holomorphic curves with a Dirac equation for spinors.

• SuperJ-holomorphic curves are critical points of the superconformal action or spinning string action.

• Under certain conditions the moduli spaceM(A)of J-holomorphic curves of genus0and[imφ] =A∈H2(N,Z) is asupermanifold.

• There is a compactificationM_0,k(A)viasuperstable maps.

• SuperGromov–Witten invariants?

Outline

Super Riemann Surfaces

Super J-holomorphic curves

Moduli Space of superJ-holomorphic curves

Super Stable Maps

Super Riemann Surfaces

Graßmann algebras

• Think of differential forms with exterior product∧.

• The exterior algebra of a vector spaceV is defined as V(V) =T(V)hv⊗vi.

• Z2-grading: V

(V) =V

0(V)⊕V

1(V).

• supercommutativeproduct: a·b= (−1)^p(a)p(b)b·a.

• ForV=Rⁿwe denote a basis byη^α and then any element a∈V

(Rⁿ)can be written

a=a0+η^α_αa+η^αη^β_αβa+. . .+η¹· · · · ·ηⁿ_1...na.

• Homomorphisms of Graßmann algebras preserve the Z2-grading.

Local theory of supermanifolds

Super geometry was developed in the 1980s to provide mathematical tools for supersymmetric field theories.³ The building block for supergeometry is the ringed space R^m|n= (R^m,O

R^m|n), where

OR^m|n =C^∞(R^m,R)⊗^ (Rⁿ).

• even coordinatesx¹, . . . ,x^m, odd coordinatesη¹, . . . , ηⁿ

• general function onR^2|2:

f(x, η) =₀f(x) +η^µ_µf(x) +η¹η²₁₂f(x)

• Supermanifolds are locally isomorphic toR^m|n.

• Maps of supermanifolds are maps of ringed spaces.

Super differential geometry

Let(x^a, η^α)be coordinates onR^m|n. Tangent vector fields on R^m|nare derivations on the functions onR^m|n. They can be written as a linear combination of the partial derivatives

∂_xa, ∂_η^α.

X=X^a∂_x^a+X^α∂η^α

Similarly: vector bundles, Lie groups, principal bundles, connections and (almost) complex structures…

Families of supermanifolds

Let(y^a, θ^α)be coordinates onR^p|qand(x^b, η^β)be coordinates onR^m|n. A mapΦ : R^p|q→R^m|nis completely determined by the image of the coordinate functions:

Φ^#x^b=₀f^b(y)

+θ^µ_µf^b(y)

+θ^µθ^ν_νµf^b(y) +· · · Φ^#η^β =

0f^β(y) +

θ^µ_µf^β(y)

+θ^µθ^ν_νµf^β(y)

+· · ·

For fullθ-expansion we need families of supermanifolds + base change. Here: Submersions. That is, we actually consider mapsΦ :R^p|q×B→R^m|n.

Families of supermanifolds

Let(y^a, θ^α)be coordinates onR^p|qand(x^b, η^β)be coordinates onR^m|n. A mapΦ : R^p|q→R^m|nis completely determined by the image of the coordinate functions:

Φ^#x^b=₀f^b(y) +θ^µ_µf^b(y) +θ^µθ^ν_νµf^b(y) +· · · Φ^#η^β = ₀f^β(y) +θ^µ_µf^β(y) +θ^µθ^ν_νµf^β(y) +· · · For fullθ-expansion we need families of supermanifolds + base change. Here: Submersions. That is, we actually consider mapsΦ :R^p|q×B→R^m|n.

Projective SuperspaceP^1|1_C

• The complex projective superspace of dimension1|1is a complex supermanifold given by two charts isomorphic to C^1|1 with coordinates(z1, θ1)and(z2, θ2)such that

z2 = 1 z1

, θ2 = θ1

z1

.

• AlternativelyP^1|1_C =Split_CS= P¹_C,V

C(H(S))

, where S→P¹_Cis the spinor line bundle, that isS⊗S=TP¹_C.

Super Riemann surfaces⁴

Definition

A super Riemann surface is a complex1|1-dimensional supermanifoldMwith an odd holomorphic distribution D ⊂TM, such that ¹₂[·,·] : D ⊗_CD 'TMD.

0 D TM TMD=D ⊗ D 0

4LeBrun and Rothstein (1988). “Moduli of super Riemann surfaces.”

Local structure of SRS

• Let(z, θ)be the standard coordinates onC^1|1 and define D ⊂TC^1|1 byD=h∂_θ+θ∂_zi. ThenD ⊗_CD 'TMDby

[∂θ+θ∂z, ∂θ+θ∂z] = 2∂z.

• Local uniformization: Every super Riemann surfaces is locally isomorphic toC^1|1with its standard super Riemann surface structure.

• A holomorphic mapΦ : C^1|1→Cgiven by

Φ(z, θ) =ϕ(z) +θψ(z)satisfiesDΦ =ψ(z) +θ∂zϕ.

Split Super Riemann Surfaces

• P^1|1_C is a super Riemann surface withDgenerated by

∂_θ1+θ1∂_z1 and∂_θ2−θ2∂_z2.

• By uniformization of super Riemann surfaces,P^1|1_C is the only super Riemann surface of genus zero.⁵

• More generally, for any Riemann surfaceΣand spinor bundleS→Σthe supermanifoldSplitScarries a canonical super Riemann surface structure.

5Crane and Rabin (1988). “Super Riemann surfaces: Uniformization and Teichmüller Theory.”

Odd deformations

LetM_red be the reduced manifold of a super Riemann surface M(overB) and set|M|=Mred×B. Pick a mapi:|M| →Mwhich is the identity on the topological spaces.

• The super Riemannn surfaceMis completely determined by a Riemannian metricg, a spinor bundleSand a gravitinoχ∈Γ (T^∨|M| ⊗S)on|M|.⁶

0 S=i^∗D i^∗TM i^∗TMD=T|M| 0

di

χ

6Keßler (2019).Supergeometry, Super Riemann Surfaces and the

Approaches to moduli spaces of SRS

• Deligne, 1987: Deformation Theory

• LeBrun–Rothstein, 1988: Moduli of marked SRS as

“canonical super orbifolds”

• Crane–Rabin, 1988: Uniformization of SRS

• Sachse, 2009: {MSRS}

Diff0M

• Donagi–Witten 2012: Super moduli space is not projected

• D’Hoker–Phong, 1988 /Keßler2019: Metrics and Gravitinos

Super J-holomorphic curves

SuperJ-holomorphic curves

Definition

LetMbe a super Riemann surface and(N, ω,J)an almost Kähler manifold. A mapΦ :M→Nis called a super J-holomorphic curve⁷if

DJΦ = 1

2 (dΦ +J dΦI)|_D= 0.

7Keßler, Sheshmani, and Yau (2021).“SuperJ-holomorphic Curves:

Construction of the Moduli Space.”

SuperJ-holomorphic curvesC^1|1×B→Cⁿ

Let(z, θ)be the superconformal coordinates onC^1|1,λ^σ coordinates ofBandZ^b complex coordinates onCⁿ. Any smooth mapΦ : C^1|1×B→Cⁿcan be written in coordinates as

Φ^b=ϕ^b(z,z, λ) +θψ^b(z,z, λ) +θψ^b(z,z, λ) +θθF^b(z,z, λ), The mapΦis superJ-holomorphic if

∂_θ+θ∂_z

Φ^b=ψ^b(z,z, λ)−θF^b(z,z, λ)

+θ∂_zϕ^b(z,z, λ)−θθ∂_zψ^b(z,z, λ)

= 0

• IfNis Kähler the mapΦis holomorphic.

SuperJ-holomorphic curves in component fields For a super Riemann surfaceM,i:|M| →Mand a map Φ :M→Ndefine

ϕ= Φ◦i:|M| →N

ψ= i^∗dΦ|_D∈Γ S^∨⊗ϕ^∗TN F=i^∗∆^DΦ∈Γ (ϕ^∗TN) Theorem

The mapΦis a super J-holomorphic curve if and only if

∂_Jϕ+hQχ, ψi= 0, (1 +I⊗J)ψ= 0 F= 0, Dψ/ −2h∨Qχ,dϕi+kQχk²ψ= 0 Here we have assumed for simplicity that N is Kähler.

SuperJ-holomorphic curves are super harmonic

Proposition

Any super J-holomorphic curveΦ :M→N is a critical point of the superconformal action

A(M,Φ) = Z

M

kdΦ|_Dk²[dvol]

= Z

|M|

kdϕk²_g^∨_⊗ϕ^∗_n+g^∨_S ⊗ϕ^∗n Dψ, ψ/

− kFk²_ϕ∗n

+ 4g^∨⊗ϕ^∗n(dϕ,hQχ, ψi) +kQχk²_g^∨_⊗gSkψk²_g∨ S⊗ϕ^∗n

− 1

6g^∨_S ⊗ϕ^∗n SR^N(ψ), ψ

dvolg

Moduli Space of super J-holomorphic

curves

B-points ofR^m|n

LetB=R^0|s. AB-point ofR^m|n is a mapp:B→R^m|n. p^#x^a=p^a p^#η^α=p^α The set ofB-points ofR^m|n is given by

R^m|n(B) = (OB)^m₀ ⊕(O_B)ⁿ₁ =^

s

m 0

⊕^

s

n 1

More generally for a supermanifoldMwe obtain a functor M:SPoint^op→Man

B7→M(B) that contains the full information onM⁸.

Space of all maps

LetHbe the infinite-dimensional supermanifold such that H(B) ={Φ :M×B→N}.

Charts can be constructed using the exponential map on the targetN.

Let furthermoreE → Hbe the vector bundle such that above Φwe have

E_Φ(B) = Γ D^∨⊗Φ^∗TN0,1

=codomainDJΦ

ThenS =DJ:H → E is a section ofEandS⁻¹(0)is the space of superJ-holomorphic curves.

Moduli Space

FixA∈H2(N,Z)and letj:M(A)→ Hbe the embedding of M(A)(B) ={Φ∈ H(B)| S(Φ) = 0and[imΦ] =A}.

j^∗kerdS j^∗TH j^∗S^∗E j^∗CokerdS

M(A)

j^∗dS

By the Theorem of Atiyah–Singer applied to the linearizations of∂_J andD:/

rk kerdS −rk CokerdS = 2n(1−p) + 2hc1(TN),Ai |2hc1(TN),Ai.

Moduli space of superJ-holomorphic curves

Theorem

Fix a closed compact super Riemann surface M overR^0|0of genus p, an almost Kähler manifold N and A∈H2(N). IfSis transversal to the zero section, i.e. dS is surjective,M(A)is a supermanifold of dimension

2n(1−p) + 2hc1(TN),Ai |2hc1(TN),Ai.

Sketch of Proof

Idea: Use implicit function theorem around a given J-holomorphic curve to obtain local charts forM(A). IfS is transversal to the zero section atΦ:

• Complete locally aroundΦto Sobolev spacesE^k,p→ H^k,p

• Apply Banach space implicit function theorem to obtain a local chart for S^k,p−1

(0).

• Show by elliptic regularity that preimages of zero are smooth, that is, inS⁻¹(0).

Sketch of Proof: Manifold structures

Assume thatΦ : M→Nhas component fieldsϕ:M_red →N, ψ= 0andF= 0. LetΦ_B= Φ×idB:M×B→N.

E(B) U_Φ(B)×Γ (D^∨⊗Φ^∗_BTN)^0,1₀

H(B) U_Φ(B)⊂Γ (Φ^∗_BTN)₀

S(B) (id_UΦ(B),F_Φ(B))

idD⊗P_exp^∇

ΦB

exp_ΦB

Γ (Φ^∗_BTN)₀∼= Γ (ϕ^∗TN)⊗(O_B)₀⊕Γ S^∨⊗ϕ^∗TN

⊗(O_B)₁

⊕Γ (ϕ^∗TN)⊗(O_B)₀ Γ D^∨⊗Φ^∗_BTN0,1

0 ∼= Γ S^∨⊗ϕ^∗TN0,1

⊗(O_B)₁

⊕

Γ (ϕ^∗TN)⊕Γ T^∨Mred⊗ϕ^∗TN0,1

⊗(O_B)₀

⊕Γ S^∨⊗ϕ^∗TN0,1

⊗(O_B) ²⁵

Transversality

Sis transversal to the zero section if the differential of the map F_Φ(C) : Γ (Φ^∗_CTN)₀ →Γ D^∨⊗Φ^∗_CTN0,1

0

X 7→

idD^∨⊗P^∇_exp

ΦCtX

−1

DJexp_ΦCX is surjective. In component fields the differential is given by

dF_Φ(C) : Γ (Φ^∗_CTN)₀→Γ D^∨⊗Φ^∗_CTN0,1 0

(ξ, ζ, σ)7→(ζ^0,1, σ,(1 +I⊗J)∇ξ,(1 +I⊗J) /Dζ^1,0) Note that(1 +I⊗J)∇ξand(1 +I⊗J) /Dζ^1,0 are(O_C)_a-linear.

Transversality

ByO_C-linearity of the differential operators it suffices to look at the reduced operators.

• (1 +I⊗J)∇ξis surjective for genericJifϕ_redis simple.

• (1 +I⊗J) /Dζ^1,0can be shown to be surjective ifMis of genus zero andNhas positive holomorphic sectional curvature.

• A particularly good example are superJ-holomorphic curvesΦ :P^1|1_C →Pⁿ_C.

Geometry of the moduli space

• Suppose that the targetNis a Kähler manifold and the domain super Riemann surface has vanishing gravitino, and the moduli spaceM(A)exists. ThenM(A) =SplitK whereK →M(A)is the bundle over the moduli space of (non-super)J-holomorphic curves such thatKφ=kerD/^1,0.

• In that case the moduli space carries an almost complex structure induced fromN.

Super Stable Maps

Compactification via stable maps

• The moduli spaceM₀(A)of classicalJ-holomorphic curves is in general not compact because sequences of

J-holomorphic spheres might “bubble”. That is they might converge to trees ofJ-holomorphic curves.

• Certain bubbles of a bubble tree might be constant.

Precomposing on a constant bubble with Möbius transformations leads to different descriptions of the same bubble tree.

Compactification via stable maps

Hence one generalizes to trees of bubbles with marked points.

This leads to so called stableJ-holomorphic curves.

M_0,k(A) = [

k-marked treesT

[

A=PAα

M0,T({Aα})

Superconformal automorphisms ofP^1|1_C

Automorphisms of the super Riemann surfaceP^1|1_C are of the form

l^#z1= az1+b

cz1+d ±θ₁ γz1+δ (cz1+d)² l^#θ₁= γz1+δ

cz1+d ±θ₁ 1 cz1+d withad−bc−γδ= 1.

• Any threeB-points ofP^1|1_C can be mapped by a unique superconformal automorphism to0(z1 = 0,θ1 = 0),1

(z1 = 1,θ₁ =) and∞(_z¹

1 = 0,θ₁= 0).

• Any superconformal automorphism mapping07→0, 1 7→1⁰ and∞ 7→ ∞implies that=±⁰ and the map is

Nodal supercurves Definition

LetT be akmarked tree, represented by vertices T={α, β, . . .}, the edge matrixEαβ and the markings {1, . . . ,k} →T. A nodal supercurve of genus zero overB, modeled onT is a tuple

z=

{zαβ}_E

αβ,{zi}_1≤i≤k

consisting ofB-pointszαβ:B→P^1|1_C andz_i:B→P^1|1_C such that for everyα∈Tthe reduction of the pointszαβ andz_i for p(i) =αare disjoint. Thezαβ are called nodal points andziare marked points.

The nodal curve is stable if at every node the number of special points, that is nodal points and marked points, is at least three.

Moduli space of stable supercurves

The moduli space of stable supercurves of fixed tree typeT is a superorbifold of dimension

2k−6−2e|2k−4 and singularities of typeZ^e+1₂ .

It can be realized as the quotient by automorphisms of an open subsupermanifold of

P^1|1_C

2e+k

.

The hard part of the proof is the definition of superorbifold

Stable superJ-holomorphic curves⁹

Definition

A super stable map of genus zero overBand modeled onT is a tuple

(z,Φ) = {zαβ}_E

αβ,{z_i}_1≤i≤k

,{Φ_α}_α∈T

given by a nodal supercurvezand superJ-holomorphic curves Φα:P^1|1_C ×B→Nsuch that

• Φα◦zαβ = Φ_β◦zβα,

• the number of special points on nodes with constantΦαis at least three.

9Keßler, Sheshmani, and Yau (2020).Super quantum cohomology I: Super stable maps of genus zero with Neveu-Schwarz punctures.

Stable superJ-holomorphic curves¹⁰

10Keßler, Sheshmani, and Yau (2020).Super quantum cohomology I: Super

Moduli space of super stable maps of fixed tree type The moduli spaceM_0,T({Aα})of stable of stable super

J-holomorphic curves of fixed tree typeT and partition{Aα}_α∈T of the homology classAis a superorbifold of dimension

2n+ 2hA,c1(TN)i −2e+ 2k−6|2hA,c1(TN)i+ 2k−4 and singularities of typeZ^e+1₂ .

It can be realized as the quotient by automorphisms of a subsupermanifold of

Y

α∈T

M^∗₀(Aα)× P^1|1_C

2e+k

.

Moduli space of super stable maps

The moduli space of super stable maps withkmarked points M_0,k(A)(B) = [

k-marked treesT

[

A=P Aα

M_0,T({Aα})(B) is not a superorbifold. Instead:

• We have proposed a generalization of Gromov topology.

• The reduced points form the compact space of classical stable maps.

• The reduction to fixed tree typeT is a superorbifold.

Conclusions

• We have a definition of superJ-holomorphic curve from a super Riemann surface to an almost Kähler manifold that mirrors and extends many of the properties of classical J-holomorphic curves.

• Under certain conditions onNwe have constructed the moduli space of superJ-holomorphic curvesΦ :M→N and its compactification in genus zero.

• We are working to understand the moduli space of super J-holomorphic curves better and search for super

analogues of Gromov–Witten invariants.

References

Crane, Louis and Jeffrey M. Rabin (1988). “Super Riemann surfaces: Uniformization and Teichmüller Theory.”In:

Communications in Mathematical Physics113.4, pp. 601–623.

Deligne, Pierre (1987). “Lettre à Manin.”Princeton.

Donagi, Ron and Edward Witten (2015). “Supermoduli Space Is Not Projected.”In:String-Math 2012. String-Math 2012 (Universität Bonn, Bonn, Germany, July 16–June 21, 2012).

Ed. by Ron Donagi et al. Proceedings of Symposia in Pure Mathematics 90. Providence: American Mathematical Society, pp. 19–71. arXiv:1304.7798 [hep-th].

Keßler, Enno (2019).Supergeometry, Super Riemann

Surfaces and the Superconformal Action Functional.Lecture Notes in Mathematics 2230. Berlin: Springer.

Keßler, Enno, Artan Sheshmani, and Shing-Tung Yau (2020).

Super quantum cohomology I: Super stable maps of genus zero with Neveu-Schwarz punctures.arXiv:2010.15634 [math.DG].

— (2021). “SuperJ-holomorphic Curves: Construction of the Moduli Space.”In:Mathematische Annalen. arXiv:

1911.05607 [math.DG].

LeBrun, Claude and Mitchell Rothstein (1988). “Moduli of super Riemann surfaces.”In:Communications in

Mathematical Physics117.1, pp. 159–176.

Leites, D. A. (1980). “Introduction to the theory of supermanifolds.”In:Russian Mathematical Surveys35.1, pp. 1–64.

McDuff, Dusa and Dietmar Salamon (2012).J-holomorphic curves and symplectic topology.2nd ed. American

Mathematical Society Colloquium Publications 52.

Providence, RI: American Mathematical Society. 726 pp.

Molotkov, Vladimir (2010). “Infinite Dimensional and Colored Supermanifolds.”In:Journal of Nonlinear Mathematical Physics17 (Special Issue in Honor of F. A. Berezin), pp. 375–446.

Sachse, Christoph (2009). “Global Analytic Approach to Super Teichmüller Spaces.”PhD thesis. Universität Leipzig.

arXiv:0902.3289 [math.AG].