J-holomorphic curves and Dirac-harmonic maps

TL;DR

This paper explores the relationship between J-holomorphic curves and Dirac-harmonic maps, showing how certain maps and spinors form Dirac-harmonic maps on Riemann surfaces with implications for topological string theory.

Contribution

It identifies conditions under which J-holomorphic curves on Riemann surfaces generate Dirac-harmonic maps into Kähler manifolds, linking geometric analysis with string theory models.

Findings

Construction of Dirac-harmonic maps from J-holomorphic curves.

Description of the moduli space tangent bundle as Dirac-harmonic maps.

Connection to the A-model in topological string theory.

Abstract

Dirac-harmonic maps are critical points of a fermionic action functional, generalizing the Dirichlet energy for harmonic maps. We consider the case where the source manifold is a closed Riemann surface with the canonical Spin^c-structure determined by the complex structure and the target space is a Kaehler manifold. If the underlying map f is a J-holomorphic curve, we determine a space of spinors on the Riemann surface which form Dirac-harmonic maps together with f. For suitable complex structures on the target manifold the tangent bundle to the moduli space of J-holomorphic curves consists of Dirac-harmonic maps. We also discuss the relation to the A-model of topological string theory.

Equations

$$D^f:\Gamma (S{\otimes }_{\mathbb{R}}{f}^{\ast }TM)⟶\Gamma (S{\otimes }_{\mathbb{R}}{f}^{\ast }TM)$$

$$\begin{array}{rl}\tau (f)& =\mathcal{R}(f,\psi )\\ D^f\psi & =0.\end{array}$$

$$D^f:\Gamma (S^c{\otimes }_{\mathbb{R}}{f}^{\ast }TM)⟶\Gamma (S^c{\otimes }_{\mathbb{R}}{f}^{\ast }TM)$$

$$S^{c+}$$

$$S^{c-}$$

$$S^{c+}$$

$$S^{c-}$$

$$D:\Gamma (S^{c\pm })⟶\Gamma (S^{c\mp })$$

$$\sqrt{2}(\bar{\partial }+{\bar{\partial }}^{\ast }).$$

$$D^f:\Gamma (S^{c\pm }{\otimes }_{\mathbb{R}}{f}^{\ast }TM)⟶\Gamma (S^{c\mp }{\otimes }_{\mathbb{R}}{f}^{\ast }TM).$$

$$S^c{\otimes }_{\mathbb{R}}{f}^{\ast }TM=(S^c{\otimes }_{\mathbb{C}}{f}^{\ast }{T}^{1,0}M)\oplus (S^c{\otimes }_{\mathbb{C}}{f}^{\ast }{T}^{0,1}M).$$

$$D^{f\prime }$$

$$D^{f\prime \prime }$$

$${\mathrm{ind}}_{\mathbb{C}}{D}^{f\prime }$$

$${\mathrm{ind}}_{\mathbb{C}}{D}^{f\prime \prime }$$

$$\ker D^{f\prime }$$

$$\cong {\mathrm{Def}}_J(f)\oplus {\mathrm{Obs}}_J(f).$$

$$\begin{array}[]{ll}\mathrm{ker}\,\bar{\partial}^{{\scriptscriptstyle f}\prime}\oplus\mathrm{ker}\,\bar{\partial}^{{\scriptscriptstyle f}\prime\prime},&\mathrm{ker}\,\bar{\partial}^{{\scriptscriptstyle f}\prime*}\oplus\mathrm{ker}\,\bar{\partial}^{{\scriptscriptstyle f}\prime\prime*}\\ \mathrm{ker}\,\bar{\partial}^{{\scriptscriptstyle f}\prime}\oplus\mathrm{ker}\,\bar{\partial}^{{\scriptscriptstyle f}\prime\prime*},&\mathrm{ker}\,\bar{\partial}^{{\scriptscriptstyle f}\prime\prime}\oplus\mathrm{ker}\,\bar{\partial}^{{\scriptscriptstyle f}\prime*}.\end{array}$$

$${\dim }_{\mathbb{R}}\mathcal{M}(A,J)=2n(1-g_{\Sigma })+2c_1(A).$$

$$L[f]=\frac{1}{2}{\int }_{\Sigma }|df{|}^2{\mathrm{dvol}}_h,$$

$$\tau (f)=0,$$

$$\tau (f)={\mathrm{tr}}_h({\nabla }^{f}df)=\sum _{\alpha }({\nabla }_{e_{\alpha }}^{f}df)(e_{\alpha }).$$

$$S{\otimes }_{\mathbb{R}}{f}^{\ast }TM\cong S{\otimes }_{\mathbb{C}}{f}^{\ast }{T}^{\mathbb{C}}M,$$

$$D^f:\Gamma (S{\otimes }_{\mathbb{R}}{f}^{\ast }TM)⟶\Gamma (S{\otimes }_{\mathbb{R}}{f}^{\ast }TM).$$

$$L[f,\psi ]=\frac{1}{2}{\int }_{\Sigma }\left(|df{|}^2+⟨\psi ,D^f\psi ⟩\right){\mathrm{dvol}}_h.$$

$${\psi }_0=\psi ,{\frac{d{\psi }_t}{dt}|}_{t=0}=\eta \in \Gamma (S^c{\otimes }_{\mathbb{R}}{f}^{\ast }TM),$$

$${\frac{d}{dt}|}_{t=0}{\int }_{\Sigma }⟨{\psi }_t,D^f{\psi }_t⟩{\mathrm{dvol}}_h=2{\int }_{\Sigma }⟨\eta ,D^f\psi ⟩{\mathrm{dvol}}_h.$$

$$f_0=f,{\frac{d{f}_t}{dt}|}_{t=0}=f^{\ast }X\in \Gamma (f^{\ast }TM),$$

$${\frac{d}{dt}|}_{t=0}{\int }_{\Sigma }|d{f}_t{|}^2{\mathrm{dvol}}_h=-2{\int }_{\Sigma }g(\tau (f),f^{\ast }X){\mathrm{dvol}}_h.$$

$${\frac{d}{dt}|}_{t=0}{\int }_{\Sigma }⟨{\psi }_t,D^{f_t}{\psi }_t⟩{\mathrm{dvol}}_h=2{\int }_{\Sigma }g(\mathcal{R}(f,\psi ),f^{\ast }X){\mathrm{dvol}}_h.$$

$$\tau (f)=0,$$

$$\psi =\sum _{\mu }{\psi }_{\mu }\otimes f^{\ast }{\partial }_{\mu },\text{with}{\psi }_{\mu }\in \Gamma (\tilde{U},S).$$

$$S^{c+}$$

Full text

1. Introduction

We briefly recall the definition of Dirac-harmonic maps (see [8, 9] and Section 2 for more details). Let $(\Sigma,h)$ and $(M,g)$ be Riemannian manifolds, where $\Sigma$ is closed and oriented, and $f\colon\Sigma\rightarrow M$ a smooth map. We assume that $\Sigma$ is a spin manifold and choose a spin structure $\mathfrak{s}$ with associated complex spinor bundle $S$. We can then form the twisted spinor bundle $S\otimes_{\mathbb{R}}f^{*}TM$ of spinors on $\Sigma$ with values in the pullback $f^{*}TM$ (also called spinors along the map $f$). The Dirac operator

[TABLE]

is determined by the Levi–Civita connections on $\Sigma$ and $M$.

Dirac-harmonic maps $(f,\psi)$, where $\psi\in\Gamma(S\otimes_{\mathbb{R}}f^{*}TM)$, are solutions of the following system of coupled equations [8, 9]:

[TABLE]

Here $\tau(f)\in\Gamma(f^{*}TM)$ is the so-called tension field of $f$ (cf. [12] and equation (2.2)). The curvature term $\mathcal{R}(f,\psi)$ is determined by the curvature tensor $R$ of the Riemannian metric $g$ on $M$ and is an algebraic expression in the differential $df$ and the spinor $\psi$ (linear in $df$ and quadratic in $\psi$); see Appendix B for a definition.

The system of equations (1.1) for Dirac-harmonic maps makes sense more generally if we replace the spin structure $\mathfrak{s}$ by a $\mathrm{Spin}^{c}$-structure $\mathfrak{s}^{c}$ and consider twisted spinors $\psi\in\Gamma(S^{c}\otimes_{\mathbb{R}}f^{*}TM)$, where $S^{c}$ is the complex spinor bundle associated to $\mathfrak{s}^{c}$. For the definition of the Dirac operator

[TABLE]

one has to choose (in addition to the Riemannian metrics $h$ and $g$) a Hermitian connection on the characteristic complex line bundle of $\mathfrak{s}^{c}$. We assume throughout that such a choice has been made and fixed (in the case we discuss there is a canonical choice of such a connection, determined by the Riemannian metric $h$).

Every almost Hermitian manifold $(\Sigma,j,h)$ has a canonical $\mathrm{Spin}^{c}$-structure $\mathfrak{s}^{c}$ whose associated spinor bundle $S^{c}=S^{c+}\oplus S^{c-}$ is the direct sum of the positive and negative Weyl spinor bundles

[TABLE]

We focus on the special case where $(\Sigma,j,h)$ is a closed Riemann surface, so that

[TABLE]

The Levi–Civita connection $\nabla^{h}$ induces a connection on $S^{c}$ with Dirac operator

[TABLE]

equal to the classical Dolbeault–Dirac operator

[TABLE]

Suppose that the target space $(M^{2n},J,g,\omega)$ is an almost Hermitian manifold with a Hermitian connection $\nabla^{M}$ and $f\colon\Sigma\rightarrow M$ a smooth map. We first derive a formula for the twisted Dirac operator

[TABLE]

Proposition 1.1**.**

The spinor bundle $S^{c}\otimes_{\mathbb{R}}f^{*}TM$ decomposes into two twisted complex spinor bundles

[TABLE]

There is a corresponding decomposition $D^{\scriptscriptstyle f}=D^{{\scriptscriptstyle f}\prime}+D^{{\scriptscriptstyle f}\prime\prime}$ of the Dirac operator into two twisted Dolbeault–Dirac operators

[TABLE]

The Hirzebruch–Riemann–Roch Theorem implies for the indices

[TABLE]

where $g_{\Sigma}$ is the genus of $\Sigma$, $A=f_{*}[\Sigma]\in H_{2}(M;\mathbb{Z})$ is the integral homology class represented by $\Sigma$ under $f$ and $c_{1}(A)=\langle c_{1}(TM,J),A\rangle$.

We then restrict to the case where $(M,J,g,\omega)$ is Kähler, $\nabla^{M}=\nabla^{g}$ the Levi–Civita connection and $f\colon\Sigma\rightarrow M$ a $J$-holomorphic curve. In this case, the Dolbeault operator $\bar{\partial}^{{\scriptscriptstyle f}\prime}$ is equal to the linearization $L_{f}\bar{\partial}_{J}$ of the non-linear Cauchy–Riemann operator $\bar{\partial}_{J}$ in $f$. In particular, the kernel of $D^{{\scriptscriptstyle f}\prime}$ is given by the direct sum of the deformation and obstruction space for the $J$-holomorphic curve $f$ (cf.  Remark 4.4):

[TABLE]

The pair $(f,J)$ is called regular if $\mathrm{Obs}_{J}(f)=0$.

Theorem 1.2**.**

Suppose that $(M,J,g,\omega)$ is a Kähler manifold of complex dimension $n>0$ and $f\colon\Sigma\rightarrow M$ a $J$-holomorphic curve. If $\psi\in\Gamma(S^{c}\otimes_{\mathbb{C}}f^{*}T^{\mathbb{C}}M)$ is an element of one of the following vector spaces, then $(f,\psi)$ is Dirac-harmonic:

[TABLE]

At least one of these vector spaces is non-zero, except possibly in the case that $g_{\Sigma}=1$ and $c_{1}(A)=0$.

Corollary 1.3**.**

Let $(\Sigma,j)$ be a Riemann surface, $A\in H_{2}(M;\mathbb{Z})$ and denote by $\mathcal{M}(A,J)$ the moduli space of all $J$-holomorphic curves $f\colon\Sigma\rightarrow M$ with $f_{*}[\Sigma]=A$. Suppose that $(f,J)$ is regular for all $f\in\mathcal{M}(A,J)$. Then $\mathcal{M}(A,J)$ is a smooth manifold (possibly empty) of dimension

[TABLE]

Every element $(f,\psi)\in T\mathcal{M}(A,J)$ of the tangent bundle of the moduli space is a Dirac-harmonic map.

Remark 1.4*.*

For the case of spin structures the vector spaces corresponding to the ones in (1.2) appear in the proof of [25, Theorem 1.1].

Remark 1.5*.*

Dirac-harmonic maps for the canonical $\mathrm{Spin}^{c}$-structure on Riemann surfaces are closely related to the A-model of topological string theory [27, 28] (with a fixed metric $h$, i.e. without worldsheet gravity); see Section 5 for a short discussion. In particular, in the A-model path integrals of certain operators localize to integrals over the finite-dimensional moduli spaces $\mathcal{M}(A,J)$ and the tangent bundle $T\mathcal{M}(A,J)$ can be identified with the space of $\chi$-zero modes (in our notation $\chi=\psi\in\mathrm{ker}\,\bar{\partial}^{{\scriptscriptstyle f}\prime}$).

In the last section we consider a generalization of Theorem 1.2 to twisted $\mathrm{Spin}^{c}$-structures $S^{c}\otimes_{\mathbb{C}}L$ with a holomorphic line bundle $L\rightarrow\Sigma$; see Corollary 7.2. For $L=K^{\scriptscriptstyle\frac{1}{2}}$ this includes the case of the spinor bundle $S=S^{c}\otimes_{\mathbb{C}}K^{\scriptscriptstyle\frac{1}{2}}$ of a spin structure $\mathfrak{s}$.

Dirac-harmonic maps $(f,\psi)$ from surfaces $\Sigma$ with a spin structure to Riemannian target manifolds $M$ have been studied before. We summarize some of the results in [2, 3, 9, 10, 20, 24, 25, 29].

Examples of Dirac-harmonic maps for $\Sigma=M=S^{2}$ were constructed in [9] where $f$ is a conformal map and $\psi$ is defined using a twistor spinor on $S^{2}$. This method was generalized in [20] to arbitrary Riemann surfaces $\Sigma$ admitting twistor spinors and arbitrary Riemannian manifolds $M$, where the map $f$ is harmonic (among closed surfaces only $S^{2}$ and $T^{2}$ admit non-zero twistor spinors [14, A.2.2]). In [29] and [10] it was shown that all Dirac-harmonic maps with source $\Sigma$ of genus $g_{\Sigma}$ and target $M=S^{2}$, so that $|\deg(f)|+1>g_{\Sigma}$, can be obtained using the constructions from [9, 20], where $f$ is holomorphic or antiholomorphic and $\psi$ is defined using a twistor spinor on $\Sigma$, possibly with isolated singularities (see also [24]).

Dirac-harmonic maps $(f,\psi)$ from spin Kähler manifolds to arbitrary Kähler manifolds were studied in [25]. In Example 7.5 below we consider the case where the source is a Riemann surface $\Sigma$ with a spin structure and the map $f$ is $J$-holomorphic.

Existence results for Dirac-harmonic maps related to the $\alpha$-genus $\alpha(\Sigma,\mathfrak{s},f)$ for a spin structure $\mathfrak{s}$ on $\Sigma$ were discussed in [2]. Section 10.1 in [2] contains several results for Dirac-harmonic maps from surfaces to Riemannian manifolds $M$ of dimension $\geq 3$. In [3] Dirac-harmonic maps from surfaces to Riemannian manifolds were constructed with methods related to an ansatz in [20].

In [15, 16, 17] another fermionic generalization of $J$-holomorphic curves was studied (see Remark 7.3 for a brief discussion of the relation to Dirac-harmonic maps).

Conventions**.**

In the following, all Riemann surfaces $\Sigma$ are closed (compact and without boundary), connected and oriented by the complex structure. For Riemannian metrics $h$ on $\Sigma$ and $g$ on $M$ we denote by $\nabla^{h}$ and $\nabla^{g}$ the Levi–Civita connections. Tensor products of vector spaces and vector bundles are over the complex numbers $\mathbb{C}$, unless indicated otherwise.

2. Some background on Dirac-harmonic maps

Recall that harmonic maps $f\colon\Sigma\rightarrow M$ from a closed, oriented Riemannian manifold $(\Sigma,h)$ to a Riemannian manifold $(M,g)$ are smooth maps, defined as the critical points of the Dirichlet energy functional [12]

[TABLE]

where $df$ is the differential of $f$ and $|df|^{2}$ is the length-squared determined by the metrics $h$ and $g$. The Euler–Lagrange equation for stationary points of $L[f]$ under variations of $f$ is

[TABLE]

where $\tau(f)$ is the tension field

[TABLE]

Here $df$ is considered as an element of $\Omega^{1}(f^{*}TM)$ and the connection $\nabla^{\scriptscriptstyle f}$ on the vector bundle $f^{*}TM\rightarrow\Sigma$ is induced from the Levi–Civita connection $\nabla^{M}=\nabla^{g}$. The basis $\{e_{\alpha}\}$ is a local orthonormal frame on $\Sigma$.

Remark 2.1*.*

If the connection $\nabla^{M}$ on $M$ is compatible with $g$, but not torsion-free, then harmonic maps $f$ do not necessarily satisfy $\tau(f)=0$.

Suppose that $\Sigma$ is a spin manifold and let $\mathfrak{s}$ be a spin structure on $\Sigma$ with associated complex spinor bundle $S$ and twisted spinor bundle $S\otimes_{\mathbb{R}}f^{*}TM$. Note that if $V$ is a complex vector space and $W$ a real vector space, then $V\otimes_{\mathbb{R}}W$ is a complex vector space isomorphic to $V\otimes_{\mathbb{C}}W^{\mathbb{C}}$, where $W^{\mathbb{C}}$ is the complexification $W\otimes_{\mathbb{R}}\mathbb{C}$. It follows that there is a (canonical) isomorphism of complex vector bundles

[TABLE]

with $T^{\mathbb{C}}M=TM\otimes_{\mathbb{R}}\mathbb{C}$ (see [29, Section 2]).

The Levi–Civita connection on $\Sigma$ and the connection $\nabla^{\scriptscriptstyle f}$ on $f^{*}TM$ yield a Dirac operator

[TABLE]

Dirac-harmonic maps $(f,\psi)$ are defined as the critical points of the fermionic action functional [8, 9]

[TABLE]

A pair $(f,\psi)$ is Dirac-harmonic if and only if it is a solution of the system of coupled Euler–Lagrange equations (1.1) (see [9, Proposition 2.1] for a proof of the formulae below):

• •

If $f$ is fixed and $\psi_{t}$ a variation of $\psi$ with

[TABLE]

then

[TABLE]

• •

If $f_{t}$ is a variation of $f$ with

[TABLE]

then

[TABLE]

Suppose in addition that $\psi_{t}=\sum_{\mu}\psi_{\mu}\otimes f_{t}^{*}\partial_{\mu}$ is a twisted spinor with time-independent components $\psi_{\mu}$ with respect to local coordinates $\{x_{\mu}\}$ (or a local frame) of $M$. If $\psi=\psi_{0}$ satisfies $D^{\scriptscriptstyle f}\psi=0$, then

[TABLE]

More details on the calculation of this variation can be found in Appendix B.

Dirac-harmonic maps are generalizations of harmonic maps: For the trivial spinor $\psi\equiv 0$, the curvature term $\mathcal{R}(f,\psi)$ vanishes identically and the system of equations (1.1) reduces to the equation

[TABLE]

i.e. $(f,0)$ is Dirac-harmonic for any harmonic map $f$.

The fermionic action functional (2.3) is motivated by theoretical physics: Suppose that $\Sigma$ is $2$-dimensional and $h,g$ Lorentzian metrics. The Dirichlet energy $L[X]$ for smooth maps $X\colon\Sigma\rightarrow M$ is (up to a normalization constant) the non-linear $\sigma$-model (Polyakov) action for bosonic strings propagating in $(M,g)$, cf. [7].

The functional $L[X,\psi]$ for Dirac-harmonic maps is part of the supersymmetric non-linear $\sigma$-model action [1]: Choosing coordinates $\{x_{\mu}\}$ on an open subset $U\subset M$ we can write every spinor $\psi\in\Gamma(S\otimes_{\mathbb{R}}f^{*}TM)$ on $\tilde{U}=f^{-1}(U)$ as

[TABLE]

The spinors $\psi_{\mu}$ are the fermionic superpartners of the scalar fields $X_{\mu}\in\mathcal{C}^{\infty}(\tilde{U},\mathbb{R})$, i.e. the coordinate fields of the map $X$ (in physics, the spinors $\psi_{\mu}$ take values in a Grassmann algebra).

In the supersymmetric non-linear $\sigma$-model action in [1] there is an additional curvature term which is determined by the curvature tensor $R$ of $g$ and of order $4$ in the spinor $\psi$ (cf. [11]). The full action for superstrings contains also a gravitino $\chi$, the superpartner of the metric $h$. This action was studied from a mathematical point of view in [19].

3. $\mathrm{Spin}^{c}$-structures on Riemann surfaces

We discuss some background material concerning $\mathrm{Spin}^{c}$-structures on Riemann surfaces (more details can be found e.g. in [18, 5, 13, 21]).

Let $(\Sigma,j,h)$ be a closed Riemann surface with complex structure $j$ and compatible Riemannian metric $h$. The canonical $\mathrm{Spin}^{c}$-structure $\mathfrak{s}^{c}$ on $\Sigma$ has spinor bundles

[TABLE]

where $\underline{\mathbb{C}}$ is the trivial complex line bundle and $K^{-1}=\bar{K}$ is the anticanonical line bundle. The spaces of smooth sections are

[TABLE]

Our notation for tangent vectors and $1$-forms of type $(1,0)$ and $(0,1)$ can be found in Appendix A. The Riemannian metric $h$ extends to Hermitian bundle metrics on $T^{1,0}\oplus T^{0,1}$ and $\Lambda^{1,0}\oplus\Lambda^{0,1}$ and the choice of a local $h$-orthonormal basis $(e_{1},e_{2})$ of $T\Sigma$ with $e_{2}=je_{1}$ determines local unit basis vectors

[TABLE]

and dual unit basis $1$-forms

[TABLE]

Any element $\beta\in\Lambda^{0,1}$ can be written as

[TABLE]

The spinor bundle $S^{c}$ has a Clifford multiplication

[TABLE]

that satisfies the Clifford relation

[TABLE]

Let $\alpha\in(T^{\mathbb{C}}\Sigma)^{*}$. For $\phi\in\underline{\mathbb{C}}=\Lambda^{0,0}$ Clifford multiplication is given by

[TABLE]

which implies

[TABLE]

For $\beta\in K^{-1}=\Lambda^{0,1}$ Clifford multiplication is given by contraction

[TABLE]

implying

[TABLE]

In particular, the volume form $\mathrm{dvol}_{h}=e_{1}^{*}\wedge e_{2}^{*}$ acts as

[TABLE]

The decomposition of the differential

[TABLE]

into $(1,0)$- and $(0,1)$-components is denoted by

[TABLE]

and the Dolbeault operator is given by

[TABLE]

with formal adjoint

[TABLE]

The Levi–Civita connection $\nabla^{h}$ of the Kähler metric $h$ satisfies $\nabla^{h}j=j\nabla^{h}$ and induces a connection on $K^{-1}$ and thus a Hermitian connection on $S^{c}$, compatible with Clifford multiplication. We consider the associated Dirac operator

[TABLE]

Lemma 3.1** (cf. [18]).**

The Dirac operator $D$ is equal to the Dolbeault–Dirac operator

[TABLE]

The Riemann–Roch theorem implies for the index

[TABLE]

where $g_{\Sigma}$ is the genus of $\Sigma$.

Proof.

Let $\phi\in\mathcal{C}^{\infty}(\Sigma,\mathbb{C})$ be a positive spinor. On $\mathcal{C}^{\infty}(\Sigma,\mathbb{C})$ the connection is just the differential $d$, hence

[TABLE]

where the last step follows from equation (3.1). Thus

[TABLE]

Since the Dirac operator is formally self-adjoint, the claim follows. ∎

Remark 3.2*.*

Riemann surfaces are spin, hence we can choose a spin structure $\mathfrak{s}$ on $\Sigma$, which is equivalent to the choice of a holomorphic square root $K^{\scriptscriptstyle\frac{1}{2}}$ of the canonical bundle $K$ (see [4, 18]). The spinor bundles of $\mathfrak{s}$ are

[TABLE]

and the spinor bundle of the canonical $\mathrm{Spin}^{c}$-structure is obtained by twisting

[TABLE]

There is another $\mathrm{Spin}^{c}$-structure with spinor bundle

[TABLE]

i.e.

[TABLE]

Remark 3.3*.*

Let $L\rightarrow\Sigma$ be a complex line bundle with a Hermitian bundle metric. Then there is a twisted $\mathrm{Spin}^{c}$-structure $\mathfrak{s}^{c}\otimes L$ with spinor bundles

[TABLE]

A connection $\nabla^{B}$ on $L$, compatible with the Hermitian bundle metric, together with the Levi–Civita connection $\nabla^{h}$ yields a Hermitian connection on $S^{c}\otimes L$ and a Dirac operator

[TABLE]

With the Dolbeault operator

[TABLE]

the Dirac operator $D_{B}$ is equal to the Dolbeault–Dirac operator

[TABLE]

4. Dirac operator along maps and $J$-holomorphic curves

Let $(\Sigma,j,h)$ be a Riemann surface and $(M,J,g,\omega)$ an almost Hermitian manifold of real dimension $2n$ with almost complex structure $J$, Riemannian metric $g$ and non-degenerate $2$-form $\omega$, related by

[TABLE]

We fix a Hermitian connection $\nabla^{M}$ on $TM$, i.e. an affine connection such that $\nabla^{M}g=0$ and $\nabla^{M}J=0$. For a general almost Hermitian manifold the connection $\nabla^{M}$ has non-zero torsion. The Hermitian connection $\nabla^{M}$ can be chosen torsion-free, hence equal to the Levi–Civita connection $\nabla^{g}$ of $g$, if and only if $(M,J,g,\omega)$ is Kähler.

Let $f\colon\Sigma\rightarrow M$ be a smooth map and consider the pullback $f^{*}TM\rightarrow\Sigma$ of the tangent bundle $TM$. If $X$ is vector field on $M$, then the pullback

[TABLE]

is a section of $f^{*}TM$. There is a unique Hermitian connection $\nabla^{\scriptscriptstyle f}$ on $f^{*}TM$ so that

[TABLE]

We consider the twisted spinor bundle

[TABLE]

on $\Sigma$. The Riemannian metric $g$ extends to a Hermitian bundle metric $\langle\cdot\,,\cdot\rangle$ on $T^{\mathbb{C}}M$. There is a decomposition into orthogonal $\pm i$-eigenspaces of the complex linear extension of $J$,

[TABLE]

and a corresponding decomposition of $S^{c}\otimes_{\mathbb{R}}f^{*}TM$ into two twisted complex spinor bundles (cf. [29, Section 3])

[TABLE]

(the tensor products on the right are over $\mathbb{C}$). The connection $\nabla^{M}$ extends to a Hermitian connection on $T^{\mathbb{C}}M$ which preserves both complex subbundles $T^{1,0}M$ and $T^{0,1}M$. The connections $\nabla^{h}$ and $\nabla^{\scriptscriptstyle f}$ thus define a Hermitian connection on $S^{c}\otimes_{\mathbb{R}}f^{*}TM$, also denoted by $\nabla^{\scriptscriptstyle f}$, which preserves both complex spinor bundles on the right hand side of equation (4.1).

Definition 4.1** (cf. [9]).**

The associated twisted Dirac operator

[TABLE]

is called the Dirac operator along the map $f$. Under the splitting in equation (4.1) the Dirac operator $D^{\scriptscriptstyle f}$ decomposes into two twisted Dirac operators

[TABLE]

Since the connection $\nabla^{\scriptscriptstyle f}$ on the twisted spinor bundle is obtained from the Levi–Civita connection $\nabla^{h}$ on $\Sigma$, the Dirac operator $D^{\scriptscriptstyle f}$ is formally self-adjoint. We consider the Dolbeault operators for the complex vector bundles $f^{*}T^{1,0}M$ and $f^{*}T^{0,1}M$,

[TABLE]

defined by

[TABLE]

The formal adjoints are denoted by $\bar{\partial}^{{\scriptscriptstyle f}\prime*}$ and $\bar{\partial}^{{\scriptscriptstyle f}\prime\prime*}$.

Proof of Proposition 1.1.

Let $\psi\in\Gamma(f^{*}T^{1,0}M)$. Then

[TABLE]

This implies the claim for the Dirac operator $D^{{\scriptscriptstyle f}\prime}$, because it is self-adjoint. The claim for $D^{{\scriptscriptstyle f}\prime\prime}$ follows similarly. ∎

Recall that a $J$-holomorphic curve is a smooth map $f\colon\Sigma\rightarrow M$ such that

[TABLE]

where

[TABLE]

is the differential. With the non-linear Cauchy–Riemann operator

[TABLE]

the map $f$ is a $J$-holomorphic curve if and only if

[TABLE]

Corollary 4.2**.**

Suppose that $(M,J,g,\omega)$ is Kähler, $\nabla^{M}=\nabla^{g}$ the Levi–Civita connection and $f\colon\Sigma\rightarrow M$ a $J$-holomorphic curve.

1. (1)

$T^{1,0}M\cong(TM,J)$* and $f^{*}T^{1,0}M$ is a holomorphic vector bundle over $\Sigma$.* 2. (2)

$\bar{\partial}^{{\scriptscriptstyle f}\prime}$* is equal to the linearization $L_{f}\bar{\partial}_{J}$ of the non-linear Cauchy–Riemann operator $\bar{\partial}_{J}$ in $f$.* 3. (3)

The kernel of $D^{{\scriptscriptstyle f}\prime}$ is given by

[TABLE] 4. (4)

The kernel of $D^{{\scriptscriptstyle f}\prime\prime}$ is given by

[TABLE]

Proof.

The claim in (2) follows from [22, p. 28]. For the formula in (3), note that

[TABLE]

The claim in (4) follows with Serre duality. ∎

Remark 4.3*.*

For a non-integrable almost complex structure $J$, the operators $\bar{\partial}^{{\scriptscriptstyle f}\prime}$ and $L_{f}\bar{\partial}_{J}$ differ by an operator of order [math], cf. [22, p. 28].

Remark 4.4* (cf. [22, 23, 26]).*

For an arbitrary smooth map $f\colon\Sigma\rightarrow M$, smooth sections of $f^{*}TM$ correspond to infinitesimal deformations of $f$. Suppose that $f$ is $J$-holomorphic. Then elements of

[TABLE]

correspond to infinitesimal deformations of $f$ through $J$-holomorphic curves. The vector space

[TABLE]

is called the obstruction space and the pair $(f,J)$ is called regular if $\mathrm{Obs}_{J}(f)=0$, i.e. $L_{f}\bar{\partial}_{J}$ is surjective. If $(f,J)$ is regular, then $(f^{\prime},J)$ is regular for all $J$-holomorphic curves $f^{\prime}\colon\Sigma\rightarrow M$ in a small neighbourhood of $f$ (inside the space of all smooth maps $\Sigma\rightarrow M$). In this case, it follows that the local moduli space, i.e. the set of all $J$-holomorphic curves $f^{\prime}$ near $f$, is a smooth manifold of real dimension $2\mathrm{ind}_{\mathbb{C}}D^{\scriptscriptstyle f}$ with tangent space in $f$ given by $\mathrm{Def}_{J}(f)$.

Remark 4.5*.*

For a twisted $\mathrm{Spin}^{c}$-structure $S^{c}\otimes L$ with complex line bundle $L\rightarrow\Sigma$, as in Remark 3.3, we can consider the spinor bundle $S^{c}\otimes L\otimes_{\mathbb{R}}f^{*}TM$. The choice of a Hermitian connection $B$ on $L$ then defines a connection $\nabla^{\scriptscriptstyle f\otimes B}$ on $S^{c}\otimes L\otimes_{\mathbb{R}}f^{*}TM$ with Dirac operator

[TABLE]

given by a generalization of Proposition 1.1.

5. Relation to topological string theory

Dirac-harmonic maps on Riemann surfaces $\Sigma$ with the canonical $\mathrm{Spin}^{c}$-structure are related to topological string theory, introduced by Edward Witten [27, 28]. We combine the $\mathrm{Spin}^{c}$ spinor bundles

[TABLE]

on the Riemann surface to a twisted complex spinor bundle

[TABLE]

with Weyl spinor bundles

[TABLE]

Here $(M,J,g,\omega)$ is a Kähler manifold of complex dimension $n$ and the pullback $f^{*}$ of $T^{1,0}M$ and $T^{0,1}M$ is abbreviated by an index $f$.

Definition 5.1**.**

We define the following subbundles111We follow the conventions in [28].:

$\bm{+}$** twist: **

[TABLE]

$\bm{-}$** twist: **

[TABLE]

We also define the following spinor bundles:

**A-model: **

[TABLE]

**B-model: **

[TABLE]

To explain these definitions we consider the action functional (2.3)

[TABLE]

The complete supersymmetric $\sigma$-model action functional also contains the quartic spinor term involving the Riemann curvature tensor of $g$, mentioned at the end of Section 2. We ignore this term in the following discussion.

We first consider the case where $(M,g)$ is a Riemannian manifold and the spinor a section $\psi\in\Gamma(S\otimes_{\mathbb{R}}f^{*}TM)$ for the spinor bundle $S$ of a spin structure on $\Sigma$. One allows a slightly more general situation where the Weyl spinor bundles come from different spin structures: Let $K^{\scriptscriptstyle\frac{1}{2}}$ and $\bar{K}^{\scriptscriptstyle\frac{1}{2}}$ be holomorphic square roots of $K$ and $\bar{K}$, not necessarily related by $\bar{K}^{\scriptscriptstyle\frac{1}{2}}=\overline{K^{\scriptscriptstyle\frac{1}{2}}}$. Then

[TABLE]

The non-linear $\sigma$-model has $N=2$ supersymmetry generated by spinors

[TABLE]

which are holomorphic and antiholomorphic sections of $K^{-\scriptscriptstyle\frac{1}{2}}$ and $\bar{K}^{-\scriptscriptstyle\frac{1}{2}}$, respectively.

Suppose that $(M,J,g,\omega)$ is a Kähler manifold of complex dimension $n$. We can decompose $T^{\mathbb{C}}M$ into the $(1,0)$- and $(0,1)$-part and denote the Weyl spinors by

[TABLE]

The non-linear $\sigma$-model now has $N=(2,2)$ supersymmetry generated by (anti)-holomorphic sections

[TABLE]

For a Riemann surface of genus $g_{\Sigma}\neq 1$ the canonical and anticanonical bundle are non-trivial, hence the sections in (5.1) have zeroes. In particular, the only covariantly constant sections, corresponding to global (rigid) supersymmetries, are identically zero.

This can be remedied with the topological $+$ and $-$ twists, i.e. using the $\mathrm{Spin}^{c}$-spinor bundle $S^{c}$ instead of the spinor bundle $S$. In the A-model the sections

[TABLE]

and in the B-model the sections

[TABLE]

can be chosen covariantly constant. These sections yield a global fermionic symmetry $Q$ of the non-linear $\sigma$-model for arbitrary genus $g_{\Sigma}$, which implies that the A-model and B-model (for suitable target spaces) define topological quantum field theories (TQFTs).

We consider the A-model spinor bundle in more detail. The vector bundle $\Delta_{A}$ can be decomposed as

[TABLE]

with sections

[TABLE]

The fermionic action (2.3) for the spinor bundle $\Delta_{A}$ can then be written as

[TABLE]

There is a complex antilinear bundle isomorphism

[TABLE]

given by complex conjugation and exchanging positive and negative Weyl spinors, which induces a corresponding isomorphism between $\mathrm{ker}\,D^{{\scriptscriptstyle f}\prime}$ and $\mathrm{ker}\,\bar{D}^{{\scriptscriptstyle f}\prime\prime}$. Defining the numbers of zero modes

[TABLE]

the index of the Dirac operator $D^{{\scriptscriptstyle f}\prime}$ is related to the so-called ghost number or $\mathrm{U}(1)_{A}$-anomaly by

[TABLE]

6. Dirac-harmonic maps to Kähler manifolds

Let $(\Sigma,j,h)$ be a Riemann surface and $(M,J,g,\omega)$ a Kähler manifold of complex dimension $n$ with Levi–Civita connection $\nabla^{M}=\nabla^{g}$.

Let $f\colon\Sigma\rightarrow M$ be a smooth map and $\psi\in\Gamma(S^{c}\otimes_{\mathbb{R}}f^{*}TM)$ a twisted spinor. Then $(f,\psi)$ is called a Dirac-harmonic map if it is a critical point of the fermionic action functional (2.3) (with the spinor bundle $S$ replaced by $S^{c}$). The same proof as in [9, Proposition 2.1] for spin structures shows that a pair $(f,\psi)$ is a Dirac-harmonic map if and only if it satisfies the Euler–Lagrange equations (1.1).

Definition 6.1**.**

For $A\in H_{2}(M;\mathbb{Z})$ let

[TABLE]

be the set of all smooth maps $f\colon\Sigma\rightarrow M$ with $f_{*}[\Sigma]=A$, where $[\Sigma]\in H_{2}(\Sigma;\mathbb{Z})$ is the generator determined by the complex orientation of $\Sigma$.

Proposition 6.2**.**

If $f\colon\Sigma\rightarrow M$ is $J$-holomorphic, then $f$ is harmonic and satisfies $\tau(f)=0$. More precisely, the absolute minima of the Dirichlet energy $L[f]$ on $\mathcal{X}_{A}$ are given by the $J$-holomorphic curves $f$ with $f_{*}[\Sigma]=A$. The Dirichlet energy of a $J$-holomorphic curve $f$ has value

[TABLE]

where $\omega$ is the Kähler form on $M$.

Proof.

The vanishing of the tension field $\tau(f)$ for $J$-holomorphic curves $f$ is well-known, cf. an example on [12, p. 118], and can be derived directly from formula (2.2) with respect to a local orthonormal frame $\{e_{1},e_{2}=je_{1}\}$, using that $df(e_{2})=Jdf(e_{1})$ and that the connection $\nabla^{M}=\nabla^{g}$ is torsion-free and Hermitian. The second part is proved in [23, Lemma 2.2.1] (note that deformations of $f$ do not change the integral homology class $f_{*}[\Sigma]$). ∎

Remark 6.3*.*

More generally, if the target manifold is only almost Kähler, [23, Lemma 2.2.1] shows that $J$-holomorphic maps from closed Riemann surfaces are still absolute minima of the Dirichlet energy functional, hence harmonic maps. However, if $\nabla^{M}$ has torsion, the equation $\tau(f)=0$ does not necessarily follow. Dirac-harmonic maps for connections $\nabla^{M}$ with torsion have been studied in [6].

The following statement appears in the proof of [25, Theorem 1.1] (more details on the definition of the curvature term $\mathcal{R}(f,\psi)$ can be found in Appendix B).

Proposition 6.4**.**

Let $f\colon\Sigma\rightarrow M$ be smooth map. Then

[TABLE]

for all twisted spinors $\psi$ which are sections of one of the following subbundles of $S^{c}\otimes f^{*}T^{\mathbb{C}}M$ (using the notation of Section 5):

[TABLE]

Proof.

This can be proved as in [25] by considering the expression (using the notation from Appendix B)

[TABLE]

Alternatively, consider a smooth map $f\colon\Sigma\rightarrow M$ with variation $f_{t}$ given by a vector field $X\in\Gamma(f^{*}TM)$. Any spinor $\psi\in\Gamma(S^{c}\otimes f^{*}T^{\mathbb{C}}M$) defines a spinor $\psi_{t}=\sum_{\mu}\psi_{\mu}\otimes f_{t}^{*}\partial_{\mu}$ with time-independent components $\psi_{\mu}$ with respect to local coordinates on $M$. By equation (2.4)

[TABLE]

For any variation $f_{t}$ the Dirac operator $D^{\scriptscriptstyle f_{t}}$ maps positive (negative) to negative (positive) Weyl spinors and preserves the $(1,0)$- and $(0,1)$-type of twisted spinors. Furthermore, the bundles $S^{c+}$ and $S^{c-}$ as well as $T^{1,0}M$ and $T^{0,1}M$ are orthogonal with respect to the Hermitian bundle metric.

This implies for every section $\psi$ of the bundles in (6.1) that the corresponding spinor $\psi_{t}$ satisfies

[TABLE]

∎

Remark 6.5*.*

The first two bundles in (6.1) can be described as the $(\mp i)$-eigenspaces of the bundle automorphism $\mathrm{dvol}_{h}=\mathrm{dvol}_{h}\otimes\mathrm{Id}$ on $S^{c}\otimes f^{*}T^{\mathbb{C}}M$ with $\mathrm{dvol}_{h}^{2}=-\mathrm{Id}$ (cf. equation (3.2)). The other two bundles are the $(\pm 1)$-eigenspaces of the bundle automorphism $I=\mathrm{dvol}_{h}\otimes J$ on $S^{c}\otimes f^{*}T^{\mathbb{C}}M$ with $I^{2}=\mathrm{Id}$.

Proof of Theorem 1.2.

The first claim is a direct consequence of the Euler–Lagrange equations (1.1) and Propositions 6.2, 6.4 and 1.1. The second claim follows because if all of the vector spaces are zero, then

[TABLE]

∎

Remark 6.6*.*

A Dirac-harmonic map $(f,\psi)$ as in Theorem 1.2, whose underlying map $f$ is harmonic, is called uncoupled in [2]. The Dirac-harmonic maps $(f,\psi)$ in Theorem 1.2 have minimal bosonic action $L[f]$ in their homology class $A$.

Example 6.7**.**

Suppose that $(M,J,g,\omega)$ is a Calabi–Yau manifold of complex dimension $n$, hence $c_{1}(TM)=0$, and $f\colon\mathbb{C}\mathrm{P}^{1}\rightarrow M$ is a $J$-holomorphic sphere. If $(f,J)$ is regular, then the vector space $\mathrm{ker}\,\bar{\partial}^{{\scriptscriptstyle f}\prime}$ has complex dimension $n$ and is the tangent space $\mathrm{Def}_{J}(f)$ in $f$ of the local moduli space of $J$-holomorphic spheres (compare with [17, Remark 2.4]). For every $\psi\in\mathrm{ker}\,\bar{\partial}^{{\scriptscriptstyle f}\prime}$, the pair $(f,\psi)$ is Dirac-harmonic.

Definition 6.8**.**

Let $(\Sigma,j)$ be a fixed Riemann surface. For a class $A\in H_{2}(M;\mathbb{Z})$ we denote by $\mathcal{M}(A,J)$ the space of all $J$-holomorphic curves $f\colon\Sigma\rightarrow M$ with $f_{*}[\Sigma]=A$.

Proof of Corollary 1.3.

This follows, because under the assumptions $T_{f}\mathcal{M}(A,J)=\mathrm{ker}\,\bar{\partial}^{{\scriptscriptstyle f}\prime}$ for all $f\in\mathcal{M}(A,J)$ (cf. Remark 4.4). ∎

Example 6.9**.**

Suppose that $(M,J,g,\omega)$ is a Kähler surface and $f\colon\mathbb{C}\mathrm{P}^{1}\rightarrow M$ an embedded $J$-holomorphic sphere representing a class $A$ of self-intersection $A^{2}=A\cdot A\geq-1$. Then every $f^{\prime}\in\mathcal{M}(A,J)$ is an embedding and $(f^{\prime},J)$ is regular (see [22, Corollary 3.5.4]). By the adjunction formula

[TABLE]

hence $\mathcal{M}(A,J)$ is a smooth manifold of real dimension $8+2A^{2}\geq 6$. The tangent bundle $T\mathcal{M}(A,J)$ is a complex vector bundle and consists of Dirac-harmonic maps.

7. Generalization to twisted $\mathrm{Spin}^{c}$-structures on $\Sigma$

We consider the following generalization for the same setup as in Section 6: Let $L\rightarrow\Sigma$ be a holomorphic Hermitian line bundle with Chern connection $\nabla$ and Dolbeault operator

[TABLE]

Then $\mathfrak{s}^{c}\otimes L$ is a $\mathrm{Spin}^{c}$-structure with holomorphic spinor bundles

[TABLE]

and Dolbeault–Dirac operator

[TABLE]

Lemma 7.1**.**

Let $f\colon\Sigma\rightarrow M$ be a smooth map. The twisted Dirac operator

[TABLE]

decomposes into the sum $D^{\scriptscriptstyle f}=D^{{\scriptscriptstyle f}\prime}+D^{{\scriptscriptstyle f}\prime\prime}$ of two twisted Dolbeault–Dirac operators

[TABLE]

In this situation we can define Dirac-harmonic maps $(f,\psi)$ as solutions of the analogue of the system of equations (1.1).

Corollary 7.2**.**

Let $f\colon\Sigma\rightarrow M$ be a $J$-holomorphic curve with $A=f_{*}[\Sigma]$. If $\psi\in\Gamma(S^{c}\otimes L\otimes f^{*}T^{\mathbb{C}}M)$ is an element of one of the following vector spaces, then $(f,\psi)$ is Dirac-harmonic:

[TABLE]

By the Hirzebruch–Riemann–Roch Theorem

[TABLE]

where we write $c_{1}(L)$ for $\langle c_{1}(L),[\Sigma]\rangle$.

Remark 7.3*.*

A Dirac-harmonic map $(f,\psi)$, where $f$ is a $J$-holomorphic curve and $\psi\in\mathrm{ker}\,\bar{\partial}^{{\scriptscriptstyle f}\prime}$, is a $(\nabla^{g},J)$-holomorphic supercurve as studied in [17], cf. also [15].

Example 7.4**.**

Consider again the situation in Example 6.9 of a Kähler surface $(M,J,g,\omega)$ with an embedded $J$-holomorphic sphere $f\colon\mathbb{C}\mathrm{P}^{1}\rightarrow M$ of self-intersection $A^{2}=A\cdot A\geq-1$ and smooth moduli space $\mathcal{M}(A,J)$. Let $L\rightarrow\Sigma$ be a holomorphic line bundle with $c_{1}(L)>0$. Then

[TABLE]

and the arguments in [22, Section 3.5] using the Kodaira vanishing theorem show that $\mathrm{coker}\,\bar{\partial}^{{\scriptscriptstyle f}\prime}=0$. Hence the complex vector space $\mathrm{ker}\,\bar{\partial}^{{\scriptscriptstyle f}\prime}$ has constant dimension

[TABLE]

for all $f\in\mathcal{M}(A,J)$. There is a complex vector bundle over the infinite-dimensional manifold $\mathcal{X}_{A}$ from Definition 6.1 with fibre $\Gamma(L\otimes f^{*}T^{1,0}M)$ over $f\in\mathcal{X}_{A}$. Since $\mathcal{M}(A,J)$ is a submanifold of $\mathcal{X}_{A}$, it follows that the subset of Dirac-harmonic maps $(f,\psi)$ with

[TABLE]

is a smooth complex vector bundle $E$ over $\mathcal{M}(A,J)$ of rank

[TABLE]

In particular, for $L=K^{\otimes(-q)}$ with integers $q\geq 1$, we have $c_{1}(L)=2q$ and the complex vector bundle $E$ over $\mathcal{M}(A,J)$ of Dirac-harmonic maps has rank

[TABLE]

which becomes arbitrarily large for $q\gg 1$.

Example 7.5**.**

Let $\mathfrak{s}$ be a spin structure on $\Sigma$ and $L=K^{\scriptscriptstyle\frac{1}{2}}$ the associated holomorphic square root of the canonical bundle $K$. Then $S\cong S^{c}\otimes K^{\scriptscriptstyle\frac{1}{2}}$ is the spinor bundle of $\mathfrak{s}$ with spin Dirac operator $D=\sqrt{2}(\bar{\partial}+\bar{\partial}^{*})$ (cf. [18]) and

[TABLE]

The vector spaces in (7.1) are called $V^{\pm}_{even}$ and $V^{\pm}_{odd}$ in the proof of [25, Theorem 1.1].

Acknowledgements

I would like to thank Uwe Semmelmann for discussions on a previous version of this paper and the referee for helpful remarks.

Appendix A

Let $(\Sigma,j,h)$ be a closed Riemann surface with complex structure $j$ and compatible Riemannian metric $h$. We fix some notation for the decomposition of tangent vectors and $1$-forms into those of type $(1,0)$ and $(0,1)$.

The almost complex structure $j$ on $T\Sigma$ extends canonically to a complex linear isomorphism on $T^{\mathbb{C}}\Sigma=T\Sigma\otimes_{\mathbb{R}}\mathbb{C}$ and we decompose

[TABLE]

into the complex $(+i)$- and $(-i)$-eigenspaces of $j$. The Riemannian metric $h$ extends to a Hermitian bundle metric on $T^{\mathbb{C}}\Sigma$ and the decomposition in (A.1) is orthogonal.

The dual space $(T^{\mathbb{C}}\Sigma)^{*}$ of complex linear $1$-forms on $T^{\mathbb{C}}\Sigma$ decomposes into

[TABLE]

where $\Lambda^{1,0}=K$ and $\Lambda^{0,1}=K^{-1}$ are the bundles of complex linear $1$-forms on $T^{1,0}$ and $T^{0,1}$. We have

[TABLE]

If $\tau\in(T^{\mathbb{C}}\Sigma)^{*}$ is a $1$-form, then its decomposition into $(1,0)$- and $(0,1)$-components is given by

[TABLE]

with

[TABLE]

Let $(e_{1},e_{2})$ with $e_{2}=je_{1}$ be a local $h$-orthonormal basis of $T\Sigma$. Then

[TABLE]

are local unit basis vectors of $T^{1,0}$ and $T^{0,1}$. We extend the dual real basis $(e_{1}^{*},e_{2}^{*})$ of $T^{*}\Sigma$ to a basis of complex linear $1$-forms of $(T^{\mathbb{C}}\Sigma)^{*}$. Then

[TABLE]

are the dual local unit basis vectors of $K$ and $K^{-1}$.

Appendix B

We summarize the definition of the curvature term $\mathcal{R}(f,\psi)$ that appears in the Euler–Lagrange equations (1.1) for Dirac-harmonic maps. Let $(\Sigma,j,h)$ be a Riemann surface, $(M^{n},g)$ a Riemannian manifold and $f\colon\Sigma\rightarrow M$ a smooth map. We denote by

[TABLE]

the curvature tensor, where we use the sign convention

[TABLE]

There is an induced map

[TABLE]

Composing with Clifford multiplication

[TABLE]

we get the map

[TABLE]

Definition B.1**.**

We define

[TABLE]

by

[TABLE]

With respect to a local orthonormal frame $e_{1},e_{2}$ for $T\Sigma$ we can write

[TABLE]

With the components of the curvature tensor $R$ with respect to a local frame $\{y_{k}\}_{k=1}^{n}$

[TABLE]

we obtain the original formula for the definition of the curvature term $\mathcal{R}$ in [9]:

[TABLE]

The symmetries

[TABLE]

imply that $\mathcal{R}(f,\psi)$ is indeed a real vector in $f^{*}TM$.

Suppose that $f_{t}$ a variation of the smooth map $f\colon\Sigma\rightarrow M$ with

[TABLE]

Let $\phi,\phi^{\prime}\in\Gamma(S^{c})$ be time-independent spinors on $\Sigma$, $Z,Z^{\prime}$ time-independent vector fields on $M$ and define spinors

[TABLE]

Definition B.2**.**

We set $df_{-}(e_{\alpha})$ for the vector field $df_{t}(e_{\alpha})$ along $f_{t}$ and

[TABLE]

In the last line we used that $[X,df_{-}(e_{\alpha})]=0$, since $f_{t}$ is generated (to first order) by the flow of $X$.

We calculate (cf. the proof of [9, Proposition 2.1])

[TABLE]

In particular, for $\psi^{\prime}=\psi$ and $D^{\scriptscriptstyle f}\psi=0$ we get formula (2.4), using that $D^{\scriptscriptstyle f}$ is formally self-adjoint.