Super J-holomorphic curves

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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 TΣ⊗φTN0,1

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 spaceMp(A)of J-holomorphic curves and[imφ] =AH2(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 : MNis a map from a superRiemann surfaceMsuch that

DJΦ = 1

2 (dΦ +J dΦI)|D= 0∈Γ D⊗ΦTN0,1

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] =AH2(N,Z) is asupermanifold.

• There is a compactificationM0,k(A)viasuperstable maps.

SuperGromov–Witten invariants?



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)hvvi.

• Z2-grading: V

(V) =V



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

• ForV=Rnwe denote a basis byηα and then any element a∈V

(Rn)can be written

a=a0ααaαηβαβa+. . .+η1· · · · ·η

• 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 Rm|n= (Rm,O

Rm|n), where

ORm|n =C(Rm,R)⊗^ (Rn).

• even coordinatesx1, . . . ,xm, odd coordinatesη1, . . . , ηn

• general function onR2|2:

f(x, η) =0f(x) +ηµµf(x) +η1η212f(x)

• Supermanifolds are locally isomorphic toRm|n.

• Maps of supermanifolds are maps of ringed spaces.


Super differential geometry

Let(xa, ηα)be coordinates onRm|n. Tangent vector fields on Rm|nare derivations on the functions onRm|n. They can be written as a linear combination of the partial derivatives

xa, ∂ηα.


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


Families of supermanifolds

Let(ya, θα)be coordinates onRp|qand(xb, ηβ)be coordinates onRm|n. A mapΦ : Rp|q→Rm|nis completely determined by the image of the coordinate functions:



µθννµfb(y) +· · · Φ#ηβ =

0fβ(y) +



+· · ·

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


Families of supermanifolds

Let(ya, θα)be coordinates onRp|qand(xb, ηβ)be coordinates onRm|n. A mapΦ : Rp|q→Rm|nis completely determined by the image of the coordinate functions:

Φ#xb=0fb(y) +θµµfb(y) +θµθννµfb(y) +· · · Φ#ηβ = 0fβ(y) +θµµfβ(y) +θµθννµfβ(y) +· · · For fullθ-expansion we need families of supermanifolds + base change. Here: Submersions. That is, we actually consider mapsΦ :Rp|q×B→Rm|n.


Projective SuperspaceP1|1C

• The complex projective superspace of dimension1|1is a complex supermanifold given by two charts isomorphic to C1|1 with coordinates(z1, θ1)and(z2, θ2)such that

z2 = 1 z1

, θ2 = θ1



• AlternativelyP1|1C =SplitCS= P1C,V


, where S→P1Cis the spinor line bundle, that isSS=TP1C.


Super Riemann surfaces4


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 onC1|1 and define D ⊂TC1|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 toC1|1with its standard super Riemann surface structure.

• A holomorphic mapΦ : C1|1→Cgiven by

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


Split Super Riemann Surfaces

• P1|1C is a super Riemann surface withDgenerated by

θ11z1 and∂θ2−θ2z2.

• By uniformization of super Riemann surfaces,P1|1C 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

LetMred 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=iD iTM iTMD=T|M| 0



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}


Donagi–Witten 2012: Super moduli space is not projected

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


Super J-holomorphic curves


SuperJ-holomorphic curves


LetMbe a super Riemann surface and(N, ω,J)an almost Kähler manifold. A mapΦ :MNis 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 curvesC1|1×B→Cn

Let(z, θ)be the superconformal coordinates onC1|1σ coordinates ofBandZb complex coordinates onCn. Any smooth mapΦ : C1|1×B→Cncan be written in coordinates as

Φbb(z,z, λ) +θψb(z,z, λ) +θψb(z,z, λ) +θθFb(z,z, λ), The mapΦis superJ-holomorphic if


Φbb(z,z, λ)−θFb(z,z, λ)

+θ∂zϕb(z,z, λ)−θθ∂zψb(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 :MNdefine

ϕ= Φ◦i:|M| →N

ψ= idΦ|D∈Γ S⊗ϕTN F=iDΦ∈Γ (ϕTN) Theorem

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

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


SuperJ-holomorphic curves are super harmonic


Any super J-holomorphic curveΦ :MN is a critical point of the superconformal action

A(M,Φ) = Z



= Z


kdϕk2g⊗ϕn+gS ⊗ϕn Dψ, ψ/

− kFk2ϕn

+ 4g⊗ϕn(dϕ,hQχ, ψi) +kQχk2g⊗gSkψk2g S⊗ϕn

− 1

6gS ⊗ϕn SRN(ψ), ψ



Moduli Space of super J-holomorphic



B-points ofRm|n

LetB=R0|s. AB-point ofRm|n is a mapp:B→Rm|n. p#xa=pa p#ηα=pα The set ofB-points ofRm|n is given by

Rm|n(B) = (OB)m0 ⊕(OB)n1 =^


m 0



n 1

More generally for a supermanifoldMwe obtain a functor M:SPointop→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×BN}.

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

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

EΦ(B) = Γ D⊗ΦTN0,1


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


Moduli Space

FixAH2(N,Z)and letj:M(A)→ Hbe the embedding of M(A)(B) ={Φ∈ H(B)| S(Φ) = 0and[imΦ] =A}.

jkerdS jTH jSE jCokerdS



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


Fix a closed compact super Riemann surface M overR0|0of genus p, an almost Kähler manifold N and AH2(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 spacesEk,p→ Hk,p

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


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


Sketch of Proof: Manifold structures

Assume thatΦ : MNhas component fieldsϕ:MredN, ψ= 0andF= 0. LetΦB= Φ×idB:M×BN.

E(B) UΦ(B)×Γ (D⊗ΦBTN)0,10

H(B) UΦ(B)⊂Γ (ΦBTN)0

S(B) (idUΦ(B),FΦ(B))




Γ (ΦBTN)0∼= Γ (ϕTN)⊗(OB)0⊕Γ S⊗ϕTN


⊕Γ (ϕTN)⊗(OB)0 Γ D⊗ΦBTN0,1

0 ∼= Γ S⊗ϕTN0,1


Γ (ϕTN)⊕Γ TMred⊗ϕTN0,1


⊕Γ S⊗ϕTN0,1

⊗(OB) 25



Sis transversal to the zero section if the differential of the map FΦ(C) : Γ (ΦCTN)0 →Γ D⊗ΦCTN0,1


X 7→




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

dFΦ(C) : Γ (ΦCTN)0→Γ D⊗ΦCTN0,1 0

(ξ, ζ, σ)7→(ζ0,1, σ,(1 +I⊗J)∇ξ,(1 +I⊗J) /Dζ1,0) Note that(1 +I⊗J)∇ξand(1 +I⊗J) /Dζ1,0 are(OC)a-linear.



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

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

• (1 +I⊗J) /Dζ1,0can be shown to be surjective ifMis of genus zero andNhas positive holomorphic sectional curvature.

• A particularly good example are superJ-holomorphic curvesΦ :P1|1C →PnC.


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 whereKM(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.

M0,k(A) = [

k-marked treesT





Superconformal automorphisms ofP1|1C

Automorphisms of the super Riemann surfaceP1|1C are of the form

l#z1= az1+b

cz1+d ±θ1 γz1+δ (cz1+d)2 l#θ1= γz1

cz1+d ±θ1 1 cz1+d withadbc−γδ= 1.

• Any threeB-points ofP1|1C 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




consisting ofB-pointszαβ:B→P1|1C andzi:B→P1|1C such that for everyα∈Tthe reduction of the pointszαβ andzi 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 typeZe+12 .

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




The hard part of the proof is the definition of superorbifold


Stable superJ-holomorphic curves9


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

(z,Φ) = {zαβ}E



given by a nodal supercurvezand superJ-holomorphic curves Φα:P1|1C ×BNsuch that

• Φαzαβ = Φβzβα,

• 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 spaceM0,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 typeZe+12 .

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



M0(Aα)× P1|1C




Moduli space of super stable maps

The moduli space of super stable maps withkmarked points M0,k(A)(B) = [

k-marked treesT


A=P Aα

M0,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.



• 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 :MN 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.



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