aa r X i v : . [ m a t h . DG ] M a y DIRAC-HARMONIC MAPS WITH TORSION
VOLKER BRANDING
Abstract.
We study Dirac-harmonic maps from surfaces to manifoldswith torsion, which is motivated from the superstring action consideredin theoretical physics. We discuss analytic and geometric properties ofsuch maps and outline an existence result for uncoupled solutions. Introduction and Results
Dirac-harmonic maps arise as the mathematical version of the simplestsupersymmetric non-linear sigma model studied in quantum field theory.They are critical points of an energy functional that couples the equationfor harmonic maps to so-called vector spinors [CJLW06]. If the domain istwo-dimensional, Dirac-harmonic maps belong to the class of conformallyinvariant variational problems.Many results for Dirac-harmonic maps have already been obtained. This in-cludes the regularity of solutions [CJLW05], [Zhu09], [WX09] and the energyidentity [CJLW05]. In addition, an existence result for uncoupled solutions[AG12], for the boundary value problem [CJW], [CJWZ13] and for a non-linear version of Dirac-geodesics [Iso12] have been established. A heat flowapproach for Dirac-harmonic maps has been studied in [Bra13a], see also[Bra13b].However, in quantum field theory more complicated models are studied.Taking into account an additional curvature term in the energy functionalone is led to
Dirac-harmonic maps with curvature term , see [CJW07]. Froman analytical point of view the latter are more difficult and not much isknown about solutions of these equations. Dirac-harmonic maps coupled toa two-form potential, called
Magnetic Dirac-harmonic maps , are studied in[Bra14].The full (1 ,
1) supersymmetric nonlinear σ -model considered in theoreticalphysics involves additional terms that are not captured by the previousanalysis. Some of these additional terms can be interpreted as consideringboth Dirac-harmonic maps and Dirac-harmonic maps with curvature terminto manifolds having a connection with torsion.In this note we want to extend the framework of Dirac-harmonic maps totarget spaces with torsion. It turns out that most of the known results forDirac-harmonic maps still hold, in particular the regularity of weak solutionsand the removable singularity theorem. Moreover, we outline an approach Date : August 30, 2018.2010
Mathematics Subject Classification.
Key words and phrases.
Dirac-harmonic Maps with torsion, Regularity, Removal ofSingularities, Existence of uncoupled solutions. to the existence question for Dirac-harmonic maps with torsion using indextheory.This paper is organized as follows. In the second section we provide somebackground material on the superspace formalism used in theoretical physicsand briefly review orthogonal connections with torsion dating back to Car-tan. Section three then introduces Dirac-harmonic maps with torsion andafterwards we discuss geometric (Section 4) and analytic aspects (Section5) of these maps. In the last section we comment on Dirac-harmonic mapswith curvature term to target manifolds with torsion.
Acknowledgements:
The author would like to thank Christoph Stephanand Florian Hanisch for several discussions about torsion and supergeometry.2.
Some Background Material
The (1,1) supersymmetric nonlinear σ -model in superspace. In this section we want to give a short overview on how physicists formulatesupersymmetric sigma models as field theories in superspace. For a detaileddiscussion we refer to the books [Fre99] and [Del99], for more specific detailsof the (1 ,
1) supersymmetric σ -model one may consult [Pol05], p.106, [CT89],Chapter 5 and references therein.In two-dimensional superspace we have the usual commuting coordinates ξ + , ξ − and in addition anti-commuting coordinates θ + , θ − . The central ob-jects are the so-called superfields Φ( ξ, θ + , θ − ), whose components are givenin terms of local coordinates byΦ j ( ξ, θ + , θ − ) = φ j ( ξ ) − iθ − ψ j + ( ξ ) + iθ + ψ j − ( ξ ) . (2.1)Here, φ ( ξ ) denotes a usual map and ψ + ( ξ ) , ψ − ( ξ ) are certain spinors takingvalues in a Grassmann algebra. We have neglected any auxiliary fields. Toobtain an action functional, we need the supercovariant derivatives D ± = i ∂∂θ ∓ + θ ∓ ∂∂ξ ± . Using the metric g on the target manifold we obtain a conformal invariantaction for the (1 ,
1) supersymmetric σ -model in superspace by setting E (1 , SY M (Φ) = 12 Z g ( D + Φ , D − Φ) d ξdθ + dθ − . (2.2)Expanding the superfield and interpreting the terms from a geometric pointof view yields (the precise definition of all terms is given in Section 3) E ( φ, ψ ) SY M = 12 Z M | dφ | + h ψ, /Dψ i + 16 h R N ( ψ, ψ ) ψ, ψ i . (2.3)The first two terms in the functional give rise to the energy for Dirac-harmonic maps , including also the third terms leads to
Dirac-harmonic mapswith curvature term .But there is another way to write down a conformal invariant action for asupersymmetric nonlinear σ -model in superspace using a two-form B on thetarget manifold. More precisely, one studies the action E (1 , ASY M (Φ) = 12 Z B ( D + Φ , D − Φ) d ξdθ + dθ − . (2.4) IRAC-HARMONIC MAPS WITH TORSION 3
Again, we may expand this action in terms of ordinary fields, which gives E ( φ, ψ ) ASY M = 12 Z M φ − B + C ( e α · ψ, ψ, dφ ( e α )) + H ( ψ, ψ, ψ, ψ ) (2.5)with a two-form B , a three-form C and some quantity H . The geometricversion of the full (1 ,
1) supersymmetric nonlinear σ -model is then governedby the action E ( φ, ψ ) = E ( φ, ψ ) SY M + E ( φ, ψ ) ASY M . We want to analyze this action from the point of view of differential geom-etry.2.2.
A Shortcut to Torsion.
Orthogonal connections with torsion havealready been classified by Cartan, see [Car23], [Car24] and [Car25]. However,here we mostly follow the presentation from [PS12], Section 2.Consider a manifold N with a Riemannian metric g . By ∇ LC we denote theLevi-Civita connection. For any affine connection there exists a (2 , A such that ∇ Tor X Y = ∇ LC X Y + A ( X, Y ) (2.6)for all vector fields
X, Y ∈ Γ( T N ). We demand that the connection is orthogonal , that is for all vector fields
X, Y, Z one has ∂ X h Y, Z i = h∇ X Y, Z i + h Y, ∇ X Z i , (2.7)where h· , ·i denotes the scalar product of the metric g . Combing (2.6) and(2.7) we follow that the endomorphism A ( X, · ) is skew-adjoint, that is h A ( X, Y ) , Z i = −h Y, A ( X, Z ) i . (2.8)The curvature tensors of ∇ LC and ∇ Tor satisfy the following relation R Tor ( X, Y ) Z = R LC ( X, Y ) Z + ( ∇ LC X A )( Y, Z ) − ( ∇ LC Y A )( X, Z ) (2.9)+ A ( X, A ( Y, Z )) − A ( Y, A ( X, Z )) . Regarding the symmetries of the curvature tensor of an orthogonal connec-tion with torsion, we have (with
X, Y, Z, W ∈ Γ( T N )) h R Tor ( X, Y ) Z, W i = −h R Tor ( Y, X ) Z, W i , h R Tor ( X, Y ) Z, W i = −h R Tor ( X, Y ) W, Z i . However, in general the curvature tensor is not symmetric under swappingthe first two entries with the last two entries, see [Agr06], Remark 2.3.Any torsion tensor A induces a (3 ,
0) tensor by setting A XY Z = h A ( X, Y ) , Z i . We define the space of all possible torsion tensors on T p N by T ( T p N ) = (cid:8) A ∈ ⊗ T ∗ p N | A XY Z = − A XZY
X, Y, Z ∈ T p N (cid:9) . For A ∈ T ( T p M ) and Z ∈ T p M one sets c ( A )( Z ) = A ∂ yi ∂ yi Z , where ∂ y i is a local basis of T N and we sum over i . The following classifi-cation result is due to Cartan: VOLKER BRANDING
Theorem 2.1.
Assume that dim N ≥ . Then the space T ( T p N ) has thefollowing irreducible decomposition T ( T p N ) = T ( T p N ) ⊕ T ( T p N ) ⊕ T ( T p N ) , which is orthogonal with respect to h· , ·i and is explicitly given by T ( T p N ) = { A ∈ T ( T p N ) | ∃ V s.t. A XY Z = h X, Y ih V, Z i − h
X, Z ih V, Y i} , T ( T p N ) = { A ∈ T ( T p N ) | A XY Z = − A Y XZ ∀ X, Y, Z } , T ( T p N ) = { A ∈ T ( T p N ) | A XY Z + A Y ZX + A ZXY = 0 , c ( A )( Z ) = 0 } . Moreover, for dim N = 2 we have T ( T p N ) = T ( T p N ) . A proof of the above Theorem can be found in [TV83], Theorem 3.1.We call the torsion of a connection, whose torsion tensor is contained in T ( T p N ) vectorial , with torsion tensor in T ( T p N ) = Λ T ∗ p N totally anti-symmetric and with torsion tensor in T ( T p N ) of Cartan type .In terms of local coordinates we have from (2.8) A ijk = − A ikj (2.10)and from (2.9) R Tor ijkl = R LC ijkl + ∇ i A jkl − ∇ j A ikl + A irl A rjk − A jrl A rik . (2.11)For more details on the geometric/physical interpretation of manifolds withtorsion we refer to the lecture notes [Agr06] and the survey article [Sha02].3. Dirac-harmonic maps with torsion
Let us now describe the geometric framework for Dirac-harmonic maps withtorsion in detail. We assume that (
M, h ) is a closed Riemannian spin surfacewith spinor bundle Σ M and ( N, g ) is a compact Riemannian manifold. Let φ : M → N be a map. Together with the pull-back bundle φ − T N we mayconsider the twisted bundle Σ M ⊗ φ − T N . Sections in this bundle are called vector spinors , in terms of local coordinates y i on N they can be expressedas ψ = ψ i ⊗ ∂∂y i ( φ ( x )) . (3.1)Note that these spinors do not take values in a Grassmann algebra. Weare using the Einstein summation convention, that is, we sum over repeatedindices. Indices on N will be denoted by Latin letters, whereas indices on M are labeled by Greek letters. On Σ M ⊗ φ − T N we have a connection thatis induced from the connections on Σ M and φ − T N , we will denote thisconnection by ˜ ∇ . We will mostly be interested in connections with torsionon the target manifold N , in this case we will write ˜ ∇ Tor and then we havethe following decomposition˜ ∇ Tor = ∇ Σ M ⊗ φ − T N + Σ M ⊗ ∇ φ − T N + Σ M ⊗ A ( · , · ) . (3.2)On the spinor bundle Σ M we have the Clifford multiplication with tangentvectors, which is skew-symmetric h X · ψ, χ i Σ M = −h ψ, X · χ i Σ M for all X ∈ T M and ψ, χ ∈ Γ(Σ M ). IRAC-HARMONIC MAPS WITH TORSION 5
We now consider the twisted Dirac operator on Σ M ⊗ φ − T N , namely /D Tor := e α · ˜ ∇ Tor e α = /D + e α · A ( dφ ( e α ) , · ) , where { e α } denotes a local orthonormal basis of T M . This operator iselliptic and self-adjoint with respect to the L norm, since the connectionon φ − T N is metric. In terms of local coordinates we may express it as /D Tor ψ = /∂ψ i ⊗ ∂∂y i + e α · ψ k ⊗ Γ ijk ∂φ j ∂x α ∂∂y i + e α · ψ k ⊗ A ijk ∂φ j ∂x α ∂∂y i with the Christoffel symbols Γ ijk and the torsion coefficients A ijk on N .Moreover, /∂ denotes the usual Dirac operator acting on sections of Σ M .We may now study the energy functional E Tor ( φ, ψ ) = 12 Z M | dφ | + h ψ, /D Tor ψ i (3.3)= 12 Z M | dφ | + h ψ, /Dψ i + h ψ, A ( dφ ( e α ) , e α · ψ ) i , which is part of the full (1 ,
1) supersymmetric non-linear sigma model (2.5)as described in the introduction.
Remark 3.1.
The energy functional (3.3) is real-valued. One the one handthis follows from the fact that the operator /D Tor is elliptic and self-adjoint,on the other hand we note h ψ, e α · A ( dφ ( e α ) , ψ ) i = h e α · A ( dφ ( e α ) , ψ ) , ψ i = −h A ( dφ ( e α ) , ψ ) , e α · ψ i = h ψ, e α · A ( dφ ( e α ) , ψ ) i , where we used the skew-symmetry of the Clifford multiplication and theskew-adjointness of the endomorphism A . Remark 3.2.
We only consider a connection with torsion on the targetmanifold N . Of course, we could also consider a connection with torsionon the domain M . It is known that in this case the Dirac operator is stillself-adjoint if A ∈ T ( T M ) ⊕ T ( T M ) , see [FS79], Satz 2 and also [PS12], Cor. 4.6. However, by Theorem 2.1 weknow that in the case of a two-dimensional domain M only the vectorialtorsion contributes. Hence, we would get a twisted Dirac operator which isno longer self-adjoint.As for Dirac-harmonic maps we can compute the critical points of (3.3): Proposition 3.3.
The critical points of the functional (3.3) are given by τ ( φ ) = R ( φ, ψ ) + F Tor ( φ, ψ ) , (3.4) /D Tor ψ = 0 (3.5) with the curvature term R ( φ, ψ ) = 12 R N ( e α · ψ, ψ ) dφ ( e α ) and the torsion term F Tor ( φ, ψ ) ∈ Γ( φ − T N ) defined by (3.6) . VOLKER BRANDING
Proof.
We choose a local orthonormal basis { e α } on M such that [ e α , ∂ t ] = 0and also ∇ ∂ t e α = 0 at a considered point. Consider a smooth variation of thepair ( φ, ψ ) satisfying ( ∂φ t ∂t , ˜ ∇ Tor ψ t ∂t ) (cid:12)(cid:12) t =0 = ( η, ξ ). Using the skew-adjointnessof the endomorphism A , we find ∂∂t (cid:12)(cid:12) t =0 Z M | dφ t | = Z M h ∇ LC ∂t dφ t ( e α ) , dφ t ( e α ) i (cid:12)(cid:12) t =0 = Z M h∇ LC e α ∂φ t ∂t , dφ t ( e α ) i (cid:12)(cid:12) t =0 = − Z M h ∂φ t ∂t , ∇ LC e α dφ t ( e α ) i (cid:12)(cid:12) t =0 = − Z M h τ ( φ ) , η i . Moreover, we calculate ∂∂t Z M h ψ t , /D Tor ψ t i = 12 Z M h ˜ ∇ LC ψ t ∂t , /D Tor ψ t i + h ψ t , ˜ ∇ LC ∂t /D Tor ψ t i = 12 Z M h ˜ ∇ Tor ψ t ∂t , /D Tor ψ t i + h ψ t , ˜ ∇ Tor ∂t /D
Tor ψ t i = Z M Re h ˜ ∇ Tor ψ t ∂t , /D Tor ψ t i + 12 h ψ t , e α · R N Tor ( dφ t ( ∂ t ) , dφ t ( e α )) ψ t i . Expanding the curvature tensor using (2.9), we find h ψ t , e α · R N Tor ( dφ t ( ∂ t ) , dφ t ( e α )) ψ t i = h ψ t , e α · R N ( dφ t ( ∂ t ) , dφ t ( e α )) ψ t i + h ψ t , e α · ( ∇ dφ t ( ∂ t ) A )( dφ t ( e α ) , ψ t ) i − h ψ t , e α · ( ∇ dφ t ( e α ) A )( dφ t ( ∂ t ) , ψ t ) i + h ψ t , e α · A ( dφ t ( ∂ t ) , A ( dφ t ( e α ) , ψ t )) i − h ψ t , e α · A ( dφ t ( e α ) , A ( dφ t ( ∂ t ) , ψ t )) i . Using the symmetries of the curvature tensor without torsion, we get h ψ t , e α · R N ( dφ t ( ∂ t ) , dφ t ( e α )) ψ t i (cid:12)(cid:12) t =0 = h R N ( e α · ψ, ψ ) dφ ( e α ) , η i . For the rest of the terms we define F Tor ( φ, ψ ) ∈ Γ( φ − T N ) by h F Tor ( φ, ψ ) , η i := 12 (cid:0) h ψ, e α · ( ∇ η A )( dφ ( e α ) , ψ ) i − h ψ, e α · ( ∇ dφ ( e α ) A )( η, ψ ) i + h ψ, e α · A ( η, A ( dφ ( e α ) , ψ )) i − h ψ, e α · A ( dφ ( e α ) , A ( η, ψ )) i (cid:1) (3.6)and evaluating at t = 0 ddt E Tor ( φ t , ψ t ) (cid:12)(cid:12) t =0 = Z M Re h ξ, /D Tor ψ t i + h η, − τ ( φ ) + R ( φ, ψ ) + F Tor ( φ, ψ ) i (3.7)gives the result. (cid:3) We call solutions ( φ, ψ ) of the system (3.4) and (3.5)
Dirac-harmonic mapswith torsion .Expanding the connection on N , we find τ ( φ ) = 12 R N ( e α · ψ, ψ ) dφ ( e α ) + F Tor ( φ, ψ ) , (3.8) /Dψ = − A ( dφ ( e α ) , e α · ψ ) . (3.9) IRAC-HARMONIC MAPS WITH TORSION 7
For a general torsion tensor A the expression F Tor ( φ, ψ ) cannot be broughtinto a “nicer” form. However, for vectorial torsion we find F Tor ( φ, ψ ) = h V, dφ ( e α ) ih V, e α · ψ i ψ − | V | h dφ ( e α ) , e α · ψ i ψ + h V, ψ ih dφ ( e α ) , e α · ψ i V − h dφ ( e α ) , e α · ψ ih ψ, ( ∇ V ) ♯ i− h∇ dφ ( e α ) V, e α · ψ i ψ, where V is a vector field on N .In terms of local coordinates x α on M , the equations for Dirac-harmonicmaps with torsion (3.4) and (3.5) acquire the form τ m ( φ ) = 12 R mlij h ψ i , e α · ψ j i Σ M ∂φ l ∂x α + 12 (cid:0) ∇ m A lji − ∇ l A mji + A mri A rlj − A lri A m rj (cid:1) h ψ i , e α · ψ j i Σ M ∂φ l ∂x α ,/∂ψ i = − ( A ijk + Γ ijk ) e α · ψ j ∂φ k ∂x α . Remark 3.4.
We do not get a torsion contribution for the tension field τ ( φ ) when starting from a variational principle. However, if we just takethe harmonic map equation and change to a connection with torsion, thenwe do get a contribution. In this case the torsion piece in the tension fieldvanishes for totally antisymmetric torsion due to symmetry reasons. Thisis the reason why physical models usually consider only skew-symmetrictorsion.We call a solution of the Euler-Lagrange equations (3.4) and (3.5) uncoupled,if φ is a harmonic map.Using tools from index theory, a general existence result for uncoupled Dirac-harmonic maps could be derived in [AG12]. Since the index of the twistedDirac-operator does not change when considering a connection with torsionon φ − T N the arguments from [AG12] can also be applied in our case. Thus,let us briefly recall the following facts:Let (
M, h ) be a closed Riemannian spin manifold of dimension m with spinstructure σ and let E → M be a vector bundle with metric connection.The twisted Dirac-operator /D E : Γ(Σ M ⊗ E ) → Γ(Σ M ⊗ E ) has an index α ( M, σ, E ) ∈ KO m ( pt )([LM89], p.141, p.151), whereKO m ( pt ) ∼ = Z if m = 0(4) Z if m = 1 , . On the other hand, the index α ( M, σ, E ) can be calculated from ker /D E using [LM89], Thm. 7.13 (with ch(E) being the Chern character of thebundle E ): α ( M, σ, E ) = { ch(E) · b A ( T M ) } [ M ] if m = 0(8)[dim C (ker( /D E ))] Z if m = 1(8)[ dim C (ker( /D E ))2 ] Z if m = 2(8) { ch(E) · b A ( T M ) } [ M ] if m = 4(8) VOLKER BRANDING
These statements still hold in our case since we are assuming that we havea metric connection on E = φ − T N . From the variational formula (3.7)it can be deduced that in order to obtain an existence result we have todo the following: For a given harmonic map φ and ψ ∈ ker( /D Tor ), where /D Tor ψ ∈ Γ(Σ M ⊗ φ − T N ), we have to construct for any smooth variation φ t of φ a smooth variation of ψ t satisfying ddt R M h ψ t , /D Tor t ψ t i (cid:12)(cid:12) t =0 = 0. Thisis the same argument as Cor. 5.2 in [AG12]. Note that /D and /D Tor have thesame principal symbol and the same index. Hence, this smooth variationcan be constructed by assuming that the index α ( M, σ, E ) is non-trivial, see[AG12], Prop. 8.2 and Section 9.4.
Geometric Aspects of solutions
In this section we analyze some geometric properties of Dirac-harmonicmaps with torsion from surfaces. Since the presence of torsion on the tar-get manifold N does not affect the conformal structure on the domain M ,Dirac-harmonic maps with torsion share many nice properties with “usual”Dirac-harmonic maps. Lemma 4.1.
In two dimensions the functional E Tor ( φ, ψ ) is conformallyinvariant.Proof. It is well-known that the following terms are invariant under confor-mal transformations Z M | dφ | , Z M h ψ, /Dψ i , Z M | ψ | and thus the energy functional E Tor ( φ, ψ ) is conformally invariant. For moredetails, the reader may take a look at Lemma 3.1 in [CJLW06]. (cid:3) For both harmonic and Dirac-harmonic map there exists a quadratic holo-morphic differential, we can find something similar here. Thus, let ( φ, ψ ) bea Dirac-harmonic map with torsion. On a small domain ˜ M of M we choosea local isothermal parameter z = x + iy and set T ( z ) dz =( | φ x | − | φ y | − i h φ x , φ y i (4.1)+ h ψ, ∂ x · ˜ ∇ T or∂ x ψ i − i h ψ, ∂ x · ˜ ∇ T or∂ y ψ i ) dz with ∂ x = ∂∂x and ∂ y = ∂∂y .By varying E Tor ( φ, ψ ) with respect to the metric h αβ of the domain M , weobtain the energy-momentum tensor : T αβ = 2 h dφ ( e α ) , dφ ( e β ) i − δ αβ | dφ | + h ψ, e α · ˜ ∇ Tor e β ψ i . (4.2)It is clear that T αβ is symmetric and traceless, when ( φ, ψ ) is a Dirac-harmonic map with torsion. Proposition 4.2.
Let ( φ, ψ ) be a Dirac-harmonic map with torsion. Thenthe energy momentum tensor is covariantly conserved, that is ∇ e α T αβ = 0 . IRAC-HARMONIC MAPS WITH TORSION 9
Proof.
We choose a local orthonormal basis of
T M with ∇ e α e β = 0 at theconsidered point. By a direct calculation, using the skew-adjointness of theendomorphism A , we obtain ∇ e α (2 h dφ ( e α ) , dφ ( e β ) i − δ αβ | dφ | ) =2 h τ ( φ ) , dφ ( e β ) i =2 hR ( φ, ψ ) , dφ ( e β ) i + 2 h F Tor ( φ, ψ ) , dφ ( e β ) i . Again, calculating directly, we get ∇ e α h ψ, e α · ˜ ∇ Tor e β ψ i = h ˜ ∇ LC e α ψ, e α · ˜ ∇ Tor e β ψ i + h ψ, /D ( ˜ ∇ Tor e β ψ ) i = h A ( dφ ( e α ) , e α · ψ )) , ˜ ∇ Tor e β ψ i + h ψ, /D ( ˜ ∇ Tor e β ψ ) i = h ψ, /D Tor ( ˜ ∇ Tor e β ψ ) i , where we used that ψ is a solution of (3.5). On the other hand, we find h ψ, /D Tor ˜ ∇ Tor e β ψ i = h ψ, ˜ ∇ Tor e β /D Tor ψ | {z } =0 i + h ψ, e α · R Σ M ( e α , e β ) ψ i | {z } = h ψ, Ric( e β ) · ψ i =0 + h ψ, e α · R N Tor ( dφ ( e α ) , dφ ( e β )) ψ i = − hR ( φ, ψ ) , dφ ( e β ) i − h F Tor ( φ, ψ ) , dφ ( e β ) i . Adding up the different contributions then yields the assertion. (cid:3)
Proposition 4.3.
The quadratic differential T ( z ) dz is holomorphic.Proof. This follows directly from the last Lemma. (cid:3)
Lemma 4.4.
The square of the twisted Dirac operator /D Tor satisfies thefollowing Weitzenb¨ock formula ( /D Tor ) ψ = − ˜∆ Tor ψ + R ψ + 12 e α · e β · R N ( dφ ( e α ) , dφ ( e β )) ψ (4.3)+ e α · e β · (cid:0) ( ∇ dφ ( e α ) A )( dφ ( e β ) , ψ ) + A ( dφ ( e α ) , A ( dφ ( e β ) , ψ )) (cid:1) , where ˜∆ Tor denotes the connection Laplacian on Σ M ⊗ φ − T N .Proof.
This follows from a direct calculation or from the general Weitzenb¨ockformula for twisted Dirac operators, see for example [LM89], p. 164, Theo-rem 8.17 and (2.9). (cid:3)
As a next step we rewrite the Euler-Lagrange equations, for more details see[Zhu09]. By the Nash embedding theorem we can embed N isometricallyin some R q of sufficient high dimension q . We then have that φ : M → R q with φ ( x ) ∈ N . The vector spinor ψ becomes a vector of untwisted spinors ψ , ψ , . . . , ψ q , more precisely ψ ∈ Γ(Σ M ⊗ T R q ). The condition that ψ isalong the map φ is now encoded as q X i =1 ν i ψ i = 0 for any normal vector ν at φ ( x ) . If we think of the torsion tensor A ( · , · ) as an endomorphism on T N we canextend it to the ambient space R q by parallel transport. Lemma 4.5.
Assume that N ⊂ R q . Moreover, assume that φ : M → R q and ψ : M → Σ M ⊗ T R q . Then the Euler-Lagrange equations acquire theform − ∆ φ =II( dφ, dφ ) + P (II( e α · ψ, dφ ( e α )) , ψ ) + F Tor ( φ, ψ ) , (4.4) /∂ψ =II( dφ ( e α ) , e α · ψ ) + A ( dφ ( e α ) , e α · ψ ) , (4.5) where II denotes the second fundamental form in R q and P the shape oper-ator. Analytic Aspects of Dirac-harmonic maps with torsion
In this section we study analytic aspects of Dirac-harmonic maps with tor-sion. This includes the regularity of solutions as well as the removal ofisolated singularities.5.1.
Regularity of solutions.
First of all, we need the notion of a weaksolution of (3.4) and (3.5). Therefore, we define χ ( M, N ) := { ( φ, ψ ) ∈ W , ( M, N ) × W , ( M, Σ M ⊗ φ − T N )with (4 .
4) and (4 .
5) a.e. } . Definition 5.1 (Weak Dirac-harmonic Map with torsion) . A pair ( φ, ψ ) ∈ χ ( M, N ) is called weak Dirac-harmonic map with torsion from M to N ifand only if the pair ( φ, ψ ) solves (4.4) and (4.5) in a distributional sense.Note that the analytic structure of Dirac-harmonic maps with torsion is thesame as the one of Dirac-harmonic maps − ∆ φ ≤ C ( | dφ | + | dφ || ψ | ) ,/∂ψ ≤ C | ψ || dφ | . Thus, the regularity theory developed for Dirac-harmonic maps can easilybe applied. More precisely, we may use the following (where D denotes theunit disc) Theorem 5.2.
Let ( φ, ψ ) : D → N be a weak Dirac-harmonic map withtorsion. If φ is continuous, then the pair ( φ, ψ ) is smooth. This was proved in [CJLW05], Theorem 2.3, for Dirac-harmonic maps andcan easily be generalized to our case. Hence, we have to ensure the continuityof the map φ . Thus, we will apply the following result due to Rivi`ere (see[Riv07]): Theorem 5.3.
For every B = B ij , ≤ i, j ≤ q in L ( D, so ( q ) ⊗ R ) (thatis for all i, j ∈ , . . . q, B ij ∈ L ( D, R ) and B ij = − B ji ), every φ ∈ W , ( D, R q ) solving − ∆ φ = B · ∇ φ (5.1) is continuous. The notation should be understood as − ∆ φ i = P qj =1 B ij ·∇ φ j for all ≤ i ≤ q . IRAC-HARMONIC MAPS WITH TORSION 11
To apply Theorem 5.3 we further rewrite the Euler-Lagrange equations.We will follow the presentation in [Zhu09] for Dirac-harmonic maps. Wedenote coordinates in the ambient space R q by ( y , y , . . . , y q ). Let ν l , l = n + 1 , . . . , q be an orthonormal frame field for the normal bundle T ⊥ N .In addition, let D be a domain in M and consider a weak Dirac-harmonicmap with torsion ( φ, ψ ) ∈ χ ( M, N ). We choose local isothermal coordinates z = x + iy , set e = ∂ x , e = ∂ y and use the notation φ α = dφ ( e α ). Theterm involving the second fundamental form can be rewritten asII m ( φ α , φ α ) = φ iα φ jα (cid:18) ∂ν il ∂y j ν ml − ∂ν ml ∂y j ν il (cid:19) , m = 1 , , . . . , q, (5.2)see for example [Riv07]. Following [CJWZ13], p.7, the term on the righthand side of (4.4) involving the shape operator can also be written in askew-symmetric way, namelyRe P m (II( φ α , e α · ψ ) , ψ ) = (5.3) φ iα h ψ k , e α · ψ j i (cid:18) ∂ν l ∂y j (cid:19) ⊤ ,i (cid:18) ∂ν l ∂y k (cid:19) ⊤ ,m − (cid:18) ∂ν l ∂y k (cid:19) ⊤ ,i (cid:18) ∂ν l ∂y j (cid:19) ⊤ ,m ! . Here, ⊤ denotes the projection map ⊤ : R q → T y N .After these preparations we may now state the following Proposition 5.4.
Let ( M, h ) be a closed Riemannian spin surface and let N be a compact Riemannian manifold. Assume that ( φ, ψ ) ∈ χ ( M, N ) is aweak solution of (4.4) and (4.5). Let D be a simply connected domain of M . Then there exists B mi ∈ L ( D, so ( q ) ⊗ R ) such that − ∆ φ m = B mi · ∇ φ i (5.4) holds.Proof. By assumption N ⊂ R q is compact, we denote its unit normal fieldby ν l , l = n + 1 , . . . , q . Exploiting the skew-symmetry of (5.2), (5.3), wedenote B mi = (cid:18) f mi g mi (cid:19) , i, m = 1 , , . . . , q with f mi := (cid:0) ∂ν il ∂y j ν ml − ∂ν ml ∂y j ν il (cid:1) φ jx + h ψ k , ∂ x · ψ j i Σ M (cid:18)(cid:18) ∂ν l ∂y j (cid:19) ⊤ ,i (cid:18) ∂ν l ∂y k (cid:19) ⊤ ,m − (cid:18) ∂ν l ∂y k (cid:19) ⊤ ,i (cid:18) ∂ν l ∂y j (cid:19) ⊤ ,m (cid:19) + 12 h ψ k , ∂ x · ψ j i Σ M ( ∇ m A ijk − ∇ i A mjk + A mrk A rij − A irk A m rj )and we get the same expression for g mi with x changed to y . Thus, we canwrite (4.4) in the following form − ∆ φ m = B mi · ∇ φ i . It remains to show that B mi ∈ L ( D, so ( q ) ⊗ R ). This follows directly sincethe pair ( φ, ψ ) is a weak solution of (4.4), (4.5) and the Sobolev embedding | ψ | L ≤ C | ψ | W , . The skew-symmetry of B mi can be read of from itsdefinition. (cid:3) Corollary 5.5.
Let ( M, h ) be a closed Riemannian spin surface and more-over, let N ⊂ R q be a compact manifold. Suppose that ( φ, ψ ) ∈ χ ( M, N ) isa weak Dirac-harmonic map with torsion. Then by the last Proposition andTheorem 5.3, we may deduce that φ m is continuous, m = 1 , , . . . , q , hence φ ∈ C ( M, N ) . Remark 5.6.
If we would consider a torsion contribution in the tensionfield, we could still deduce the continuity of the map φ due to the skew-adjointness of the endomorphism A .We may summarize our considerations by the following Theorem 5.7.
Let ( φ, ψ ) : D → N be a weak Dirac-harmonic map withtorsion. Then the pair ( φ, ψ ) is smooth. Removable Singularity Theorem.
In this section we want to provea removable singularity theorem for Dirac-harmonic maps with torsion.More precisely, we want to show that solutions ( φ, ψ ) of (3.4) and (3.5)cannot have isolated singularities, whenever a certain energy is finite. It iswell known that such a theorem holds for both harmonic maps [SU81] andalso Dirac-harmonic maps [CJLW06]. Let us define the following “energy”:
Definition 5.8.
Let U be a domain on M . We define the energy of the pair( φ, ψ ) on U by E ( φ, ψ, U ) := Z U ( | dφ | + | ψ | ) . (5.5)This energy is conformally invariant and thus plays an important role.First of all, we need some local energy estimates (with the unit disc D ). Theorem 5.9.
Let ( M, h ) be a closed Riemannian spin surface and ( N, g ) a compact Riemannian manifold. Assume that the pair ( φ, ψ ) is a Dirac-harmonic map with torsion. There is a small constant ε > such that if thepair ( φ, ψ ) satisfies Z D ( | dφ | + | ψ | ) < ε, (5.6) then | dφ | C k ( D ) + | ψ | C k ( D ) ≤ C ( | dφ | L ( D ) + | ψ | L ( D ) ) , (5.7) where the constant C depends on N and k . Since the presence of torsion does not affect the analytic structure of theEuler-Lagrange equations the same proof as for Theorem 4.3 in [CJLW06]still holds.The behaviour of ( φ, ψ ) near a singularity can be described by the following
Corollary 5.10.
There is an ε > small enough such that if the pair ( φ, ψ ) is a smooth solution of (3.4) and (3.5) on D \ { } with finite energy E ( φ, ψ, D ) < ε , then for any x ∈ D we have | dφ ( x ) || x | ≤ C | dφ | L ( D | x | ) , (5.8) | ψ ( x ) | | x | + |∇ ψ ( x ) || x | ≤| ψ | L ( D | x | ) . (5.9) IRAC-HARMONIC MAPS WITH TORSION 13
Proof.
Fix any x ∈ D \ { } and define ( ˜ φ, ˜ ψ ) by˜ φ ( x ) := φ ( x + | x | x ) and ˜ ψ ( x ) := | x | ψ ( x + | x | x ) . It is easy to see that ( ˜ φ, ˜ ψ ) is a smooth solution of (3.4) and (3.5) on D with E ( ˜ φ, ˜ ψ, D ) < ε . By application of Theorem 5.9, we have | d ˜ φ | L ∞ ( D ) ≤ C | d ˜ φ | L ( D ) , | ˜ ψ | C ( D ) ≤ C | ˜ ψ | L ( D ) and scaling back yields the assertion. (cid:3) Proposition 5.11.
Let ( φ, ψ ) be a smooth Dirac-harmonic map with torsionon D \ { } satisfying E ( φ, ψ, D ) < ε . Then we have Z π r (cid:12)(cid:12) ∂φ∂θ (cid:12)(cid:12) dθ = Z π (cid:12)(cid:12) ∂φ∂r (cid:12)(cid:12) + h ψ, ∂ r · ˜ ∇ ψ∂r i + h ψ, A ( ∂φ∂r , ∂ r · ψ ) i = Z π (cid:12)(cid:12) ∂φ∂r (cid:12)(cid:12) − r h ψ, ∂ θ · ˜ ∇ ψ∂θ i − r h ψ, A ( ∂φ∂θ , ∂ θ · ψ ) i , where ( r, θ ) are polar coordinates on the disc D centered around the origin.Proof. By Proposition (4.3) we know that T = | φ x | − | φ y | − i h φ x , φ y i + h ψ, ∂ x · ˜ ∇ Tor ∂ x ψ i − i h ψ, ∂ x · ˜ ∇ Tor ∂ y ψ i is holomorphic on D \ { } with z = x + iy ∈ D . By the last Corollary wealso know that | ψ || ˜ ∇ Tor ψ | ≤ C ( | ψ ||∇ Σ M ψ | + | dφ || ψ | ) ≤ C | z | − , | dφ | ≤ C | z | − , which, altogether, gives | T ( z ) | ≤ Cz − . Moreover, it is easy to see that R D | T ( z ) | < ∞ . Hence, we may follow that zT ( z ) is holomorphic on the disc D and by Cauchy’s integral theorem we deduce0 = Im Z | z | = r zT ( z ) dz = Z π Re( z T ( z )) dθ. (5.10)Moreover, a direct calculation givesRe( z T ( z )) = r (cid:12)(cid:12) ∂φ∂r (cid:12)(cid:12) − (cid:12)(cid:12) ∂φ∂θ (cid:12)(cid:12) − h ψ, ∂ θ · ˜ ∇ Tor ψ∂θ i , (5.11)which finally proves the result. (cid:3) Now we are in the position to state the
Theorem 5.12 (Removable Singularity Theorem) . Let ( φ, ψ ) be a solutionof (3.4) and (3.5), which is smooth on U \ { p } for some p ∈ U . If ( φ, ψ ) hasfinite energy E ( φ, ψ, D ) , then ( φ, ψ ) extends to a smooth solution on U .Proof. With the help of the last Proposition the same proof as for Theorem4.6 in [CJLW06] can be applied. (cid:3) Dirac harmonic with curvature term and torsion
As we have seen in the introduction the full (1 ,
1) non-linear supersym-metric sigma model studied in quantum field theory involves an additionalcurvature term in the energy functional, namely E c ( φ, ψ ) = 12 Z M | dφ | + h ψ, /Dψ i + 16 h R N ( ψ, ψ ) ψ, ψ i . (6.1)Here, the indices are contracted as follows h R N ( ψ, ψ ) ψ, ψ i = R ijkl h ψ i , ψ k ih ψ j , ψ l i , which ensures that the action is real-valued. The factor in front of the cur-vature term is required by supersymmetry, see [Fre99], p.78. This functionalhas already been analyzed in [CJW07]. Proposition 6.1 ([CJW07]) . The critical points of the energy functional(6.1) are given by τ ( φ ) = R ( φ, ψ ) + ˜ R ( ψ ) , (6.2) /Dψ = 13 R N ( ψ, ψ ) ψ (6.3) with the curvature terms ˜ R ( ψ ) = 112 h ( ∇ R ) ♯ ( ψ, ψ ) ψ, ψ i , R ( φ, ψ ) = 12 R N ( e α · ψ, ψ ) dφ ( e α ) . Here, ♯ : T ∗ N → T N denotes the musical isomorphism.
Solutions ( φ, ψ ) of the system (6.2), (6.3) are called
Dirac-harmonic mapswith curvature term .At present very little is known about the properties of Dirac-harmonic mapswith curvature term. Compared to Dirac-harmonic maps the main differencearises in the fact that Dirac-harmonic maps with curvature term constitutea coupled system of two non-linear equations. This makes the analysis ofsolutions of (6.2) and (6.3) substantially harder. However, (6.3) has aninteresting limit. In the case that the map φ is trivial (6.3) gives rise to thespinorial Weierstrass representation of surfaces. The analytic aspects of thisequation are investigated in [CJW08] and [Wan10].The functional (6.1) also has a natural extension to target spaces with torsion(see again E ASY M (Φ) in the introduction). In this case, we have to specify H ( ψ, ψ, ψ, ψ ) in (2.5). If we replace the curvature tensor in (6.1) with thecurvature tensor of a connection with torsion and contract the indices thesame way, then in general the action will no longer be real-valued. This is dueto the fact that the curvature tensor is not symmetric under swapping thefirst with the second pair of indices, see Rem. 2.3. in [Agr06]. However, if westick to totally anti-symmetric torsion and impose the additional conditionthat the torsion is parallel, then the curvature tensor has the necessarysymmetries to obtain a real-valued action. When studying the critical pointsof this functional one obtains a set of equations that has the same analyticstructure as Dirac-harmonic maps with curvature term. IRAC-HARMONIC MAPS WITH TORSION 15
Remark 6.2.
Together with the analysis performed in [Bra14] one getsa full description of the full (1 ,
1) non-linear supersymmetric sigma modelfrom the perspective of differential geometry.
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