# The heat flow for the full bosonic string

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## Abstract

We study harmonic maps from surfaces coupled to a scalar and a two-form potential, which arise as critical points of the action of the full bosonic string. We investigate several analytic and geometric properties of these maps and prove an existence result by the heat-flow method.

### Keywords

Harmonic maps with scalar and two-form potential Full bosonic string Heat flow### Mathematics Subject Classification

58E20 35K55 53C08 53C80## 1 Introduction and results

Harmonic maps between Riemannian manifolds are one of the most studied variational problems in differential geometry. Under the assumption that the target manifold has non-positive curvature Eells and Sampson established their famous existence result for harmonic maps making use of the heat-flow method [10]. Moreover, in the case that the domain is two-dimensional harmonic maps belong to the class of conformally invariant variational problems yielding a rich structure.

However, harmonic maps from surfaces also have a dual life in theoretical physics. More precisely, they arise as the *Polyakov action* in bosonic string theory. The full action for the bosonic string contains two additional terms, one of them being the pullback of a two-form from the target and the other one being a scalar potential. It is the aim of this article to study the full action of the bosonic string as a geometric variational problem.

There are already several mathematical results available for parts of the energy functional of the full bosonic string: in [11] the authors consider the harmonic map energy together with a scalar potential. The critical points of this energy functional are called *harmonic maps with potential*. One of the main results in that reference is that depending on the choice of potential, the qualitative behavior of harmonic maps with potential differs from the one of harmonic maps. There are several results available that characterize the properties of harmonic maps with potential: this includes gradient estimates [6] and Liouville theorems [7] for harmonic maps with potential from complete manifolds. In [8] an existence result for harmonic maps with potential from compact Riemannian manifolds with boundary is obtained, where it is assumed that the image of the map lies inside a convex ball. An existence result for harmonic maps with potential to a target with negative curvature was obtained in [12] by the heat-flow method. This result has been extended to the case of a domain manifold with boundary in [9]. Harmonic maps from surfaces coupled to a two-form potential have also been studied in the mathematical literature since they give rise to the *prescribed mean curvature equation*. Existence results via the heat flow for the prescribed mean curvature equation have been obtained for a flat target in [23] and for a three-dimensional target with negative curvature in [27].

In theoretical physics the two-form potential is interpreted as giving rise to an external magnetic field.

Harmonic maps coupled to a two-form potential and spinor fields instead of a scalar potential have been studied in [4].

We will call the critical points of the full bosonic string action *harmonic maps with scalar and two-form potential* and we will generalize several results already obtained for harmonic maps and harmonic maps with potential. Moreover, we will point out new phenomena that arise from the two-form potential.

This article is organized as follows. In Sect. 2, we analyze the energy functional of the full bosonic string and derive its critical points. In Sect. 3, we study analytic and geometric aspects of the critical points. In the last section, we derive an existence result via the heat-flow method for both compact and non-compact target manifolds.

## 2 The full bosonic string action

Throughout this article (*M*, *h*) is a Riemannian surface without boundary, we will mostly assume that *M* is compact, and (*N*, *g*) a closed, oriented Riemannian manifold of dimension \(\dim N\ge 3\). For a map \(\phi :M\rightarrow N\) we consider the square of its differential giving rise to the usual harmonic energy. Let *B* be a two-form on *N*, which we pull back by the map \(\phi \). In addition, let \(V:N\rightarrow \mathbb {R}\) be a scalar function, which we mostly assume to be smooth. By *R* we denote the scalar curvature of the domain *M*.

*R*on the domain

*M*is constant. In string theory, the potential \(V(\phi )\) is usually referred to as

*dilaton field*.

*U*(1)-valued functional

*B-field action*in string theory, which is defined as

*B-field action*as

*U*(1)-valued functional (2.2) is more difficult, and we will restrict ourselves to the functional (2.1).

Let us derive the critical points of (2.1).

### Proposition 2.1

*N*and \(e_1,e_2\) an orthonormal basis of

*TM*.

### Proof

*B*, we choose an orthonormal basis \(e_1,e_2\) of

*TM*and calculate

*M*we thus obtain

We call solutions of (2.3) *harmonic maps with scalar and two-form potential*.

### Remark 2.2

*M*and Latin indices on the target

*N*. Moreover, we will make use of the Einstein summation convention, that is, we sum over repeated indices. In terms of local coordinates \(x_\alpha \) on

*M*and \(y^i\) on

*N*the Euler–Lagrange equation reads

### Remark 2.3

*Z*can also be interpreted as arising from a metric connection with totally antisymmetric torsion. In this case one has

*X*,

*Y*are vector fields and

*A*(

*X*,

*Y*) is a skew-adjoint endomorphism. The endomorphism

*A*(

*X*,

*Y*) satisfies

### Remark 2.4

In principle one could also study the functional (2.1) for a higher-dimensional domain *M* with \(m=\dim M\ge 2\). However, then one needs to pull back an *m*-form from the target leading to an Euler–Lagrange equation with a higher nonlinearity on the right hand side, see [17, Chapter 2], for a detailed analysis.

### Lemma 2.5

### Proof

This follows by a direct calculation. \(\square \)

### Example 2.6

*M*,

*h*) into an oriented Riemannian three manifold (

*N*,

*g*). In this case the tension field \(\phi \) is related to the mean curvature vector \(H(\phi )\) by

*N*must be a multiple of the volume form \({\text {vol}}_g\), that is

*f*is constant and \(\nabla V\sim \nu \) the equation (2.7) has some similarity with the equation for

*linear Weingarten surfaces*. These are surfaces immersed in a three-dimensional manifold satisfying

*H*denotes the mean curvature,

*K*the Gauss curvature and

*a*,

*b*,

*c*are non-zero real numbers.

## 3 Properties of harmonic maps with scalar and two-form potential

In this section we study several properties of solutions of (2.3).

### Lemma 3.1

*M*and \(R^N\) the curvature tensor on

*N*.

### Proof

This follows by a direct calculation. \(\square \)

### Definition 3.2

*stress-energy tensor*.

### Lemma 3.3

The stress-energy tensor \(S_{\alpha \beta }\) is divergence free and symmetric if \(\phi \) is a harmonic map with scalar and two-form potential.

### Proof

Note that \(S_{\alpha \beta }\) is no longer trace-free, which corresponds to the fact that the scalar potential \(V(\phi )\) in the action functional does not respect the conformal symmetry.

With the help of (3.4) we now derive a monotonicity formula for \(e(\phi )+RV(\phi )\). A similar calculation for harmonic maps with scalar potential has been carried out in [20]. For the monotonicity formula for harmonic maps we refer to the book [28], for a general treatment of stress-energy tensors with applications to harmonic maps we refer to [2]. Note that we do not have to assume that *M* is compact to derive the monotonicity formula.

Let \((M,h_0)\) be a complete Riemannian surface with a pole \(x_0\). Let *r*(*x*) be the Riemannian distance function relative to the point \(x_0\). By \(\lambda _i,i=1,2\) we denote the eigenvalues of \({\text {Hess}}_{h_0}(r^2)\).

### Proposition 3.4

*M*is a complete Riemannian surface and

*N*a Riemannian manifold. Suppose that

*x*.

The proof is similar to the proof of Theorem 4.1 in [20].

### Proof

*S*the following identity holds

*X*. Integrating the divergence of (3.4) over the ball \(B_\rho (x_0)\) with radius \(\rho \) around the point \(x_0\), using (3.7) and the fact that the stress-energy-tensor is divergence free, we obtain

*M*is given by \(h=f^2h_0\). Hence, we find

*M*with respect to \(h_0\) such that \({\text {Hess}}_{h_0}(r^2)\) becomes a diagonal matrix with respect to \(e_1,e_2\). Then \(\tilde{e}_\alpha :=f^{-1}e_\alpha ,\alpha =1,2\) is an orthonormal basis with respect to

*h*. Thus, we obtain

As for harmonic maps ([25, Theorem 2]) and harmonic maps with potential ([11, Proposition 2]) we can prove a unique continuation theorem for harmonic maps with scalar and two-form potential. To obtain this result we recall the following ([1, p. 248])

### Theorem 3.5

*A*be a linear elliptic second-order differential operator defined on a domain

*D*of \(\mathbb {R}^n\). Let \(u=(u^1,\ldots ,u^n)\) be functions in

*D*satisfying the inequality

*D*.

Making use of this result we can prove the following

### Proposition 3.6

Let \(\phi ,\phi '\in C^2(M,N)\) be two harmonic maps with scalar and two-form potential. Moreover, assume that \(V:N\rightarrow \mathbb {R}\) is a \(C^{1,1}\) function. If \(\phi \) and \(\phi '\) are equal on a connected open set *W* of *M* then they coincide on the whole connected component of *M* which contains *W*.

### Proof

*U*be a coordinate ball on

*M*such that \(\phi =\phi '\) in some open subset. By shrinking

*U*if necessary we can assume that both \(\phi \) and \(\phi '\) map

*U*into a single coordinate chart in

*N*. We write \(y^i(x)\) for \(y^i(\phi (x))\) and \(z^i(x)\) for \(z^i(\phi '(x))\). We consider the function \(u^i:=y^i-z^i\). Using the local form of the Euler–Lagrange equation (2.5) we find

### Remark 3.7

In the case that the scalar potential \(V(\phi )\) vanishes identically, the energy functional (2.1) is conformally invariant. In this case one can exploit the conformal invariance to prove that critical points cannot have isolated singularities, whenever a certain energy is finite. This follows from the main theorem in [14], where a removable singularity theorem for critical points of a large class of conformally invariant energy functionals is established.

As a next step, we use the embedding theorem of Nash to isometrically embed *N* into some \(\mathbb {R}^q\). We denote the isometric embedding by \(\iota \) and consider the composite map \(u:=\iota \circ \phi :M\rightarrow \mathbb {R}^q\). By \(\tilde{N}\) we denote the tubular neighborhood of \(\iota (N)\subset \mathbb {R}^q\). Let \(\pi :\tilde{N}\rightarrow \iota (N)\) be the canonical projection which assigns to each \(z\in \tilde{N}\) the closest point in \(\iota (N)\) from *z*.

### Lemma 3.8

*Z*and \(\nabla V\) to the ambient space \(\mathbb {R}^q\).

### Proof

*Z*and \(\nabla V\) can be extended to the ambient space by projecting to a tubular neighborhood, for more details see [18, p. 463] and [12, p. 557]. \(\square \)

## 4 Existence results via the heat flow

In this section, we derive an existence result for critical points of (2.1) by the heat-flow method. In order to achieve this result, we will assume that \(\Omega \) is exact such that we have a variational structure that enables us to derive the necessary estimates for convergence of the gradient flow. A similar approach for geodesics coupled to a magnetic field was performed in [5].

In order to control the non-linearities arising from the two-form, we will have to restrict to target spaces with negative sectional curvature. A similar idea has been used in [27] for the heat flow of the prescribed mean curvature equation. More precisely, we will use the negative curvature of the target to control the non-linearities arising from the two-form potential.

For a general introduction to harmonic maps and their heat flows see the book [19].

*N*into \(\mathbb {R}^q\).

### Lemma 4.1

*N*is isometrically embedded into \(\mathbb {R}^q\). Then (4.1) acquires the form

### Proof

This follows from Lemma 3.8, we will omit the tildes in order not to blow up the notation. \(\square \)

As a first step, we establish the existence of a short-time solution.

### Lemma 4.2

For \(\phi _0\in C^{2+\alpha }(M,N)\) and \(V\in C^{1,1}(N)\) there exists a unique, smooth solution to (4.1) for \(t\in [0,T_{max})\).

### Proof

This can be proven using the Banach fixed point theorem, which requires that the potential \(V\in C^{1,1}(N)\). For more details, see [19, Chapter 5]. \(\square \)

To extend the solution beyond \(T_\mathrm{max}\) we will make use of the following Bochner formulae:

### Lemma 4.3

### Proof

The Bochner formulae will be the key-tool to achieve long-time existence and convergence of the evolution equation (4.1). However, we have to distinguish between the cases of a compact and a non-compact target.

### 4.1 Compact target

For a compact target manifold *N* we will prove the following:

### Theorem 4.4

*M*,

*h*) be a closed Riemannian surface and (

*N*,

*g*) be a closed, oriented Riemannian manifold with negative sectional curvature. Moreover, suppose that \(\Omega \) is exact, \(|B|_{L^\infty }<1/2\), \(V\in C^{2,1}(N)\) and that the homomorphism

*Z*satisfies

*N*. Then \(\phi _t:M\times [0,\infty )\rightarrow N\) subconverges as \(t\rightarrow \infty \) to a harmonic map with scalar and two-form potential \(\phi _\infty \), which is homotopic to \(\phi _0\).

We will divide the proof of Theorem 4.4 into several steps.

### Lemma 4.5

*N*.

### Proof

*N*is compact and we can estimate the Hessian of the potential \(V(\phi )\) by its maximum yielding the first claim.

Via the maximum principle we thus obtain the following:

### Corollary 4.6

### Proof

Using the assumptions the first statement follows by applying the maximum principle to (4.5). For the second statement, we apply the maximum principle to (4.6) using the bound (4.7). \(\square \)

### Lemma 4.7

*C*depends on \(M,N,Z,\nabla Z,V,|{\mathrm{d}}\phi _0|\) and \(T_\mathrm{max}\).

### Proof

Interpreting (4.2) as an elliptic equation we may apply elliptic Schauder theory ([16, p. 79]) and get that \(u\in C^{1+\alpha }(M,N)\) by the bounds (4.7) and (4.8). Making use of the regularity gained from elliptic Schauder theory we interpret (4.2) as a parabolic equation. The result then follows by application of parabolic Schauder theory ([16, p. 79]). \(\square \)

### Lemma 4.8

*C*. In particular, if \(u_0=v_0\) then \(u_t=v_t\) for all \(t\in [0,T_\mathrm{max})\).

### Proof

*C*will denote a universal constant that may change from line to line. We set \(h:=\frac{1}{2}|u-v|^2\). By projecting to a tubular neighborhood \(\mathrm {I\!I}(\mathrm{d}u,\mathrm{d}u), Z(\mathrm{d}u(e_1)\wedge \mathrm{d}u(e_2))\) and \(\nabla V(u)\) can be thought of as vector-valued functions in \(\mathbb {R}^q\), for more details see [21, p. 132] and also [12, p. 557]. Exploiting this fact a direct computation yields

*C*. The result then follows by application of the maximum principle. \(\square \)

### Proposition 4.9

Let \(\phi _t:M\times [0,T_\mathrm{max})\rightarrow N\) be a smooth solution of (4.1). If \(\frac{1}{2}|Z|_{L^\infty }^2\le \kappa ^N\) then there exists a unique smooth solution for all \(t\in [0,\infty )\).

### Proof

This follows from the continuation principle for parabolic partial differential equations. Suppose that there would be a maximal time of existence \(T_{fin}\), then using the estimates (4.7) and (4.8) one can show that we can continue the solution for some small \(\delta >0\) up to \(T_{fin}+\delta \) yielding a contradiction. The uniqueness follows from Lemma 4.8. \(\square \)

To achieve convergence of the evolution equation (4.1) we will make use of the following:

### Lemma 4.10

*M*,

*h*) is a compact Riemannian manifold. If a function \(u(x,t)\ge 0\) satisfies

*K*depending on

*M*.

### Proof

A proof can for example be found in [26, p. 284]. For more details on how the constant *C* in the estimate depends on geometric data, see [16, Lemma 2.3.1]. \(\square \)

### Proposition 4.11

(Convergence) Let \(\phi _t:M\times [0,\infty )\rightarrow N\) be a smooth solution of (4.1). If \(\Omega \) is exact, \(\frac{1}{2}|Z|_{L^\infty }^2\le \kappa ^N\) and \(|B|_{L^\infty }<1/2\) then the evolution equation (4.1) subconverges in \(C^2(M,N)\) to a harmonic map with scalar and two-form potential \(\phi _\infty \), which is homotopic to \(\phi _0\).

### Proof

*C*depends on \(|B|_{L^\infty },|R|_{L^\infty }|V|_{L^\infty }\) and \(E(\phi _0)\). Applying Lemma 4.10 to (4.5) and (4.9), we thus obtain a uniform bound on \(|{\mathrm{d}}\phi _t|^2\). Inserting this bound into (4.6) we find (for some positive constant

*C*)

*M*and

*t*from 0 to \(\infty \) we get

*t*, the limit \(\phi _\infty \) is homotopic to \(\phi _0\). \(\square \)

### 4.2 Non-compact target

In the case that the target manifold *N* is complete, but non-compact we have to make additional assumptions to control the image of *M* under the evolution of \(\phi \). However, we can use the potential \(V(\phi )\) to constrain \(\phi _t(M)\) to a compact set in *N*. This is similar to the case of the heat flow for harmonic maps with potential [12]. To this end, let \(\mathrm{d}_N(y)\) denote the Riemannian distance in *N* from some fixed point \(y_0\).

Finally, we will prove the following:

### Theorem 4.13

*M*,

*h*) be a closed Riemann surface and (

*N*,

*g*) a complete, oriented Riemannian manifold with negative sectional curvature. Moreover, suppose that \(\Omega \) is exact, \(|B|_{L^\infty }<1/2\), \(V\in C^{2,1}(N)\) and \(\frac{1}{2}|Z|_{L^\infty }^2\le \kappa _N\), where \(\kappa _N\) denotes and upper bound on the sectional curvature on

*N*. In addition, assume that the potential \(V(\phi )\) satisfies

*M*. Let \(\phi _t:M\times [0,\infty )\rightarrow N\) be a smooth solution of (4.1). Then \(\phi _t\) subconverges in \(C^2(M,N)\) to a harmonic map with scalar and two-form potential, which is homotopic to \(\phi _0\). Moreover, \(\phi _\infty \) is minimizing the energy in its homotopy class.

First of all, we make the following observation:

### Lemma 4.14

If the three-form \(\Omega \) is exact, then \(Z \in \Gamma ({\text {Hom}}(\Lambda ^2T^*N,TN))\) defined via (2.4) is parallel, that is \(\nabla Z=0\).

### Proof

*synchronous*in the point

*p*, meaning that

As a next step we show how we can use the Hessian of the scalar potential \(V(\phi )\) to constrain \(\phi _t(M)\) to a compact set. This idea has already been applied in the study of the heat flow for harmonic maps with potential, see [12, Proposition 2].

### Lemma 4.15

*M*,

*h*) be a closed Riemann surface and (

*N*,

*g*) a complete Riemannian manifold. Suppose that \(\Omega \) is exact and that \(\frac{1}{2}|Z|_{L^\infty }^2\le \kappa _N\). Moreover, assume that the potential \(V(\phi )\) satisfies

### Proof

*N*. We set

*Z*and the sectional curvature \(K^N\), we obtain from (4.6) by an argument similar to the compact case that

As noted in [9, Proposition 2.4], one can also constrain \(\phi _t(M)\) to a compact set if the maximum of the Hessian of the potential \(V(\phi )\) is bounded by the first eigenvalue of the Laplacian on *M*. This idea can also be applied here:

### Lemma 4.16

*M*,

*h*) be a closed Riemann surface and (

*N*,

*g*) a complete, oriented Riemannian manifold. Suppose that \(\Omega \) is exact and that \(\frac{1}{2}|Z|_{L^\infty }^2\le \kappa _N\). Moreover, assume that the potential \(V(\phi )\) satisfies

*M*. Then there exists a compact set \(K\subset N\) such that \(\phi _t(M)\subset K\) as long as the solution of (4.1) exists.

### Proof

*Z*and the sectional curvature \(K^N\) in the last step. By the Kato inequality \(\big |\nabla f\big |^2\ge \big |\nabla |f|\big |^2\) for a function \(f:M\rightarrow \mathbb {R}\) and the Poincaré inequality on

*M*we obtain

By making use of the previous Lemmata we thus find

### Lemma 4.17

*M*,

*h*) be a closed Riemann surface and (

*N*,

*g*) a complete, oriented Riemannian manifold. Moreover, suppose that \(\Omega \) is exact, \(|B|_{L^\infty }<1/2\), \(V\in C^{2,1}(N)\) and \(\frac{1}{2}|Z|_{L^\infty }^2\le \kappa _N\). In addition, assume that the potential \(V(\phi )\) satisfies

*M*. Let \(\phi _t:M\times [0,\infty )\rightarrow N\) be a smooth solution of (4.1). Then \(\phi _t\) converges in \(C^2(M,N)\) to a harmonic map with scalar and two-form potential, which is homotopic to \(\phi _0\).

### Proof

*K*. Thus, by Theorem 4.4 there exists a sequence \(t_k\) such that \(\phi _{t_k}\) converges to a harmonic map with scalar and two-form potential. Moreover, we have

### Remark 4.18

In the case of the heat flow for harmonic maps with potential to a non-compact target with a concave potential the limit \(\phi _\infty \) is trivial, see [12, p. 564]. This statement relies on the Bochner formula (3.2), due to the presence of the two-form potential we cannot draw the same conclusion here.

### Remark 4.19

If we would only require an upper bound on \(R{\text {Hess}}V\) in *N* then this would be enough to establish long-time existence of (4.1). Consequently, to achieve convergence of (4.1) we need the potential \(RV(\phi )\) to constrain \(\phi _t(M)\) to a compact set.

### Remark 4.20

Using the extrinsic version of the evolution equation (4.2) we can apply the maximum principle to bound the image of \(\phi _t(M)\). This idea was already used for related geometric flows: a similar criterion as below for the heat flow of harmonic maps with potential is given in [12, Proposition 2]. For the heat flow of the prescribed mean curvature equation in \(\mathbb {R}^3\) the same idea is used in Proposition 3.2 in [23].

*V*(

*u*) is hard to ensure.

### 4.3 Minimizing the energy

In this section, we briefly discuss if the limiting map constructed in Theorem 4.13 is minimizing energy in its homotopy class.

### Lemma 4.21

Under the assumptions of Theorem 4.13 the limiting map \(\phi _\infty \) is minimizing energy in its homotopy class.

### Proof

### Remark 4.22

Let \(\phi _t:M\times [0,\infty )\rightarrow N\) be a smooth solution of (4.1). Under the assumptions of Theorem 4.13 the energy \(E(\phi (t))\) is a convex function of *t*, which follows by a direct calculation.

Exploiting the convexity of the energy \(E(\phi (t))\) with respect to *t* we can prove the following uniqueness Theorem, which is very similar to the case of Hartman’s theorem for harmonic maps [15].

### Proposition 4.23

Under the assumptions of Theorem 4.13 the limit \(\phi _\infty \) is independent of the chosen subsequence \(t_k\).

### Proof

The proof is the same as in the case of harmonic maps and we omit it here. \(\square \)

## Notes

### Acknowledgments

Open access funding provided by [TU Wien (TUW)].

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