Dirac-harmonic maps with torsion

Dirac-harmonic maps with torsion

Volker Branding TU Wien
Institut für diskrete Mathematik und Geometrie
Wiedner Hauptstraße 8–10, A-1040 Wien
[
July 26, 2019
Abstract.

We study Dirac-harmonic maps from surfaces to manifolds with torsion, which is motivated from the superstring action considered in theoretical physics. We discuss analytic and geometric properties of such maps and outline an existence result for uncoupled solutions.

Key words and phrases:
Dirac-harmonic Maps with torsion, Regularity, Removal of Singularities, Existence of uncoupled solutions
2010 Mathematics Subject Classification:
53C27, 58E20, 58J20, 53C43

]volker@geometrie.tuwien.ac.at

1. Introduction and Results

Dirac-harmonic maps arise as the mathematical version of the simplest supersymmetric non-linear sigma model studied in quantum field theory. They are critical points of an energy functional that couples the equation for harmonic maps to so-called vector spinors [CJLW06]. If the domain is two-dimensional, Dirac-harmonic maps belong to the class of conformally invariant variational problems.

Many results for Dirac-harmonic maps have already been obtained. This includes the regularity of solutions [CJLW05], [Zhu09], [WX09] and the energy identity [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 flow approach 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 functional one is led to Dirac-harmonic maps with curvature term, see [CJW07]. From an analytical point of view the latter are more difficult and not much is known about solutions of these equations. Dirac-harmonic maps coupled to a two-form potential, called Magnetic Dirac-harmonic maps, are studied in [Bra14].

The full supersymmetric nonlinear -model considered in theoretical physics involves additional terms that are not captured by the previous analysis. Some of these additional terms can be interpreted as considering both Dirac-harmonic maps and Dirac-harmonic maps with curvature term into manifolds having a connection with torsion.

In this note we want to extend the framework of Dirac-harmonic maps to target spaces with torsion. It turns out that most of the known results for Dirac-harmonic maps still hold, in particular the regularity of weak solutions and the removable singularity theorem. Moreover, we outline an approach to the existence question for Dirac-harmonic maps with torsion using index theory.

This paper is organized as follows. In the second section we provide some background material on the superspace formalism used in theoretical physics and briefly review orthogonal connections with torsion dating back to Cartan. Section three then introduces Dirac-harmonic maps with torsion and afterwards we discuss geometric (Section 4) and analytic aspects (Section 5) of these maps. In the last section we comment on Dirac-harmonic maps with curvature term to target manifolds with torsion.

Acknowledgements: The author would like to thank Christoph Stephan and Florian Hanisch for several discussions about torsion and supergeometry.

2. Some Background Material

2.1. The (1,1) supersymmetric nonlinear -model in superspace

In this section we want to give a short overview on how physicists formulate supersymmetric sigma models as field theories in superspace. For a detailed discussion we refer to the books [Fre99] and [Del99], for more specific details of the supersymmetric -model one may consult [Pol05], p.106, [CT89], Chapter 5 and references therein.

In two-dimensional superspace we have the usual commuting coordinates and in addition anti-commuting coordinates . The central objects are the so-called superfields , whose components are given in terms of local coordinates by

(2.1)

Here, denotes a usual map and are certain spinors taking values in a Grassmann algebra. We have neglected any auxiliary fields. To obtain an action functional, we need the supercovariant derivatives

Using the metric on the target manifold we obtain a conformal invariant action for the supersymmetric -model in superspace by setting

(2.2)

Expanding the superfield and interpreting the terms from a geometric point of view yields (the precise definition of all terms is given in Section 3)

(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 maps with curvature term.

But there is another way to write down a conformal invariant action for a supersymmetric nonlinear -model in superspace using a two-form on the target manifold. More precisely, one studies the action

(2.4)

Again, we may expand this action in terms of ordinary fields, which gives

(2.5)

with a two-form , a three-form and some quantity . The geometric version of the full supersymmetric nonlinear -model is then governed by the action

We want to analyze this action from the point of view of differential geometry.

2.2. A Shortcut to Torsion

Orthogonal connections with torsion have already been classified by Cartan, see [Car23], [Car24] and [Car25]. However, here we mostly follow the presentation from [PS12], Section 2.

Consider a manifold with a Riemannian metric . By we denote the Levi-Civita connection. For any affine connection there exists a -tensor field such that

(2.6)

for all vector fields . We demand that the connection is orthogonal, that is for all vector fields one has

(2.7)

where denotes the scalar product of the metric . Combing (2.6) and (2.7) we follow that the endomorphism is skew-adjoint, that is

(2.8)

The curvature tensors of and satisfy the following relation

(2.9)

Regarding the symmetries of the curvature tensor of an orthogonal connection with torsion, we have (with )

However, in general the curvature tensor is not symmetric under swapping the first two entries with the last two entries, see [Agr06], Remark 2.3.

Any torsion tensor induces a tensor by setting

We define the space of all possible torsion tensors on by

For and one sets

where is a local basis of and we sum over . The following classification result is due to Cartan:

Theorem 2.1.

Assume that . Then the space has the following irreducible decomposition

which is orthogonal with respect to and is explicitly given by

Moreover, for we have

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 vectorial, with torsion tensor in totally anti-symmetric and with torsion tensor in of Cartan type.

In terms of local coordinates we have from (2.8)

(2.10)

and from (2.9)

(2.11)

For more details on the geometric/physical interpretation of manifolds with torsion 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 with torsion in detail. We assume that is a closed Riemannian spin surface with spinor bundle and is a compact Riemannian manifold. Let be a map. Together with the pull-back bundle we may consider the twisted bundle . Sections in this bundle are called vector spinors, in terms of local coordinates on they can be expressed as

(3.1)

Note that these spinors do not take values in a Grassmann algebra. We are using the Einstein summation convention, that is, we sum over repeated indices. Indices on will be denoted by Latin letters, whereas indices on are labeled by Greek letters. On we have a connection that is induced from the connections on and , we will denote this connection by . We will mostly be interested in connections with torsion on the target manifold , in this case we will write and then we have the following decomposition

(3.2)

On the spinor bundle we have the Clifford multiplication with tangent vectors, which is skew-symmetric

for all and .

We now consider the twisted Dirac operator on , namely

where denotes a local orthonormal basis of . This operator is elliptic and self-adjoint with respect to the norm, since the connection on is metric. In terms of local coordinates we may express it as

with the Christoffel symbols and the torsion coefficients on . Moreover, denotes the usual Dirac operator acting on sections of .

We may now study the energy functional

(3.3)

which is part of the full 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 hand this follows from the fact that the operator is elliptic and self-adjoint, on the other hand we note

where we used the skew-symmetry of the Clifford multiplication and the skew-adjointness of the endomorphism .

Remark 3.2.

We only consider a connection with torsion on the target manifold . Of course, we could also consider a connection with torsion on the domain . It is known that in this case the Dirac operator is still self-adjoint if

see [FS79], Satz 2 and also [PS12], Cor. 4.6. However, by Theorem 2.1 we know that in the case of a two-dimensional domain only the vectorial torsion contributes. Hence, we would get a twisted Dirac operator which is no 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

(3.4)
(3.5)

with the curvature term

and the torsion term defined by (3).

Proof.

We choose a local orthonormal basis on such that and also at a considered point. Consider a smooth variation of the pair satisfying . Using the skew-adjointness of the endomorphism , we find

Moreover, we calculate

Expanding the curvature tensor using (2.9), we find

Using the symmetries of the curvature tensor without torsion, we get

For the rest of the terms we define by

(3.6)

and evaluating at

(3.7)

gives the result. ∎

We call solutions of the system (3.4) and (3.5) Dirac-harmonic maps with torsion.

Expanding the connection on , we find

(3.8)
(3.9)

For a general torsion tensor the expression cannot be brought into a “nicer” form. However, for vectorial torsion we find

where is a vector field on .

In terms of local coordinates on , the equations for Dirac-harmonic maps with torsion (3.4) and (3.5) acquire the form

Remark 3.4.

We do not get a torsion contribution for the tension field when starting from a variational principle. However, if we just take the harmonic map equation and change to a connection with torsion, then we do get a contribution. In this case the torsion piece in the tension field vanishes for totally antisymmetric torsion due to symmetry reasons. This is the reason why physical models usually consider only skew-symmetric torsion.

We call a solution of the Euler-Lagrange equations (3.4) and (3.5) uncoupled, if 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 twisted Dirac-operator does not change when considering a connection with torsion on the arguments from [AG12] can also be applied in our case. Thus, let us briefly recall the following facts:

Let be a closed Riemannian spin manifold of dimension with spin structure and let be a vector bundle with metric connection. The twisted Dirac-operator has an index ([LM89], p.141, p.151), where

On the other hand, the index can be calculated from using [LM89], Thm. 7.13 (with being the Chern character of the bundle ):

These statements still hold in our case since we are assuming that we have a metric connection on . From the variational formula (3.7) it can be deduced that in order to obtain an existence result we have to do the following: For a given harmonic map and , where , we have to construct for any smooth variation of a smooth variation of satisfying . This is the same argument as Cor. 5.2 in [AG12]. Note that and have the same principal symbol and the same index. Hence, this smooth variation can be constructed by assuming that the index 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-harmonic maps with torsion from surfaces. Since the presence of torsion on the target manifold does not affect the conformal structure on the domain , Dirac-harmonic maps with torsion share many nice properties with “usual” Dirac-harmonic maps.

Lemma 4.1.

In two dimensions the functional is conformally invariant.

Proof.

It is well-known that the following terms are invariant under conformal transformations

and thus the energy functional is conformally invariant. For more details, the reader may take a look at Lemma 3.1 in [CJLW06]. ∎

For both harmonic and Dirac-harmonic map there exists a quadratic holomorphic differential, we can find something similar here. Thus, let be a Dirac-harmonic map with torsion. On a small domain of we choose a local isothermal parameter and set

(4.1)

with and .

By varying with respect to the metric of the domain , we obtain the energy-momentum tensor:

(4.2)

It is clear that is symmetric and traceless, when is a Dirac-harmonic map with torsion.

Proposition 4.2.

Let be a Dirac-harmonic map with torsion. Then the energy momentum tensor is covariantly conserved, that is

Proof.

We choose a local orthonormal basis of with at the considered point. By a direct calculation, using the skew-adjointness of the endomorphism , we obtain

Again, calculating directly, we get

where we used that is a solution of (3.5). On the other hand, we find

Adding up the different contributions then yields the assertion. ∎

Proposition 4.3.

The quadratic differential is holomorphic.

Proof.

This follows directly from the last Lemma. ∎

Lemma 4.4.

The square of the twisted Dirac operator satisfies the following Weitzenböck formula

(4.3)

where denotes the connection Laplacian on .

Proof.

This follows from a direct calculation or from the general Weitzenböck formula for twisted Dirac operators, see for example [LM89], p. 164, Theorem 8.17 and (2.9). ∎

As a next step we rewrite the Euler-Lagrange equations, for more details see [Zhu09]. By the Nash embedding theorem we can embed isometrically in some of sufficient high dimension . We then have that with . The vector spinor becomes a vector of untwisted spinors , more precisely . The condition that is along the map is now encoded as

If we think of the torsion tensor as an endomorphism on we can extend it to the ambient space by parallel transport.

Lemma 4.5.

Assume that . Moreover, assume that and . Then the Euler-Lagrange equations acquire the form

(4.4)
(4.5)

where denotes the second fundamental form in and the shape operator.

5. Analytic Aspects of Dirac-harmonic maps with torsion

In this section we study analytic aspects of Dirac-harmonic maps with torsion. This includes the regularity of solutions as well as the removal of isolated singularities.

5.1. Regularity of solutions

First of all, we need the notion of a weak solution of (3.4) and (3.5). Therefore, we define

Definition 5.1 (Weak Dirac-harmonic Map with torsion).

A pair is called weak Dirac-harmonic map with torsion from to if and only if the pair solves (4.4) and (4.5) in a distributional sense.

Note that the analytic structure of Dirac-harmonic maps with torsion is the same as the one of Dirac-harmonic maps

Thus, the regularity theory developed for Dirac-harmonic maps can easily be applied. More precisely, we may use the following (where denotes the unit disc)

Theorem 5.2.

Let be a weak Dirac-harmonic map with torsion. If is continuous, then the pair is smooth.

This was proved in [CJLW05], Theorem 2.3, for Dirac-harmonic maps and can easily be generalized to our case. Hence, we have to ensure the continuity of the map . Thus, we will apply the following result due to Rivière (see [Riv07]):

Theorem 5.3.

For every in (that is for all and ), every solving

(5.1)

is continuous. The notation should be understood as for all .

To apply Theorem 5.3 we further rewrite the Euler-Lagrange equations. We will follow the presentation in [Zhu09] for Dirac-harmonic maps. We denote coordinates in the ambient space by . Let be an orthonormal frame field for the normal bundle . In addition, let be a domain in and consider a weak Dirac-harmonic map with torsion . We choose local isothermal coordinates , set , and use the notation . The term involving the second fundamental form can be rewritten as

(5.2)

see for example [Riv07]. Following [CJWZ13], p.7, the term on the right hand side of (4.4) involving the shape operator can also be written in a skew-symmetric way, namely

(5.3)

Here, denotes the projection map .

After these preparations we may now state the following

Proposition 5.4.

Let be a closed Riemannian spin surface and let be a compact Riemannian manifold. Assume that is a weak solution of (4.4) and (4.5). Let be a simply connected domain of . Then there exists such that

(5.4)

holds.

Proof.

By assumption is compact, we denote its unit normal field by . Exploiting the skew-symmetry of (5.2), (5.3), we denote

with

and we get the same expression for with changed to . Thus, we can write (4.4) in the following form