Coarse regularity of solutions to a nonlinear sigmamodel with \(L^p\) gravitino
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Abstract
The regularity of weak solutions of a twodimensional nonlinear sigma model with coarse gravitino is shown. Here the gravitino is only assumed to be in \(L^p\) for some \(p>4\). The precise regularity results depend on the value of p.
Mathematics Subject Classification
53C43 58E201 Introduction
The action functionals of the various models of quantum field theory yield many examples of beautiful variational problems. These problems are usually analytically very difficult, because they represent borderline cases, due to phenomena like conformal invariance. What makes them still tractable usually is their intricate algebraic structure resulting from the various symmetries of and the interactions between the various fields involved. Mathematically, often a geometric interpretation of these algebraic structures is possible. In any case, the analysis needs to use the special structure of the action functional. A well known instance is the theory of harmonic mappings from Riemann surfaces to Riemannian manifolds, which in the context of QFT arise from the action functional of the nonlinear sigma model, or the Polyakov action of string theory. Here, a particular skew symmetry of the nonlinear term in the Euler–Lagrange equations could be systematically exploited and generalized in the work of Hélein, Rivière and Struwe, see [14, 21, 22, 23]. This is also our starting point, both conceptually—because we generalize the harmonic map problem—and methodologically—because we shall use their techniques. In fact, the action functional of the nonlinear sigma model and the Polyakov action of string theory constitute only the simplest of their kind. In more sophisticated models, other fields enter, in particular a spinor field. Also, when one investigates the harmonic action functional mathematically, naturally also another object enters, the metric g or the conformal structure of the underlying Riemann surface, and for many purposes, not only the field, but also g should be varied. Again, however, in the advanced QFT models, there arises another object, a kind of partner of the metric g, the gravitino \(\chi \), also called the Rarita–Schwinger field. In harmonic map theory, or in related theories, like Teichmüller theory à la AhlforsBers, one often needs to consider metrics g that are not necessarily smooth, and this may lead to delicate regularity questions. Likewise, the gravitino is not necessarily smooth, and in this paper we address the related regularity questions.
In fact, this article is a part of our systematic study of an action functional motivated from super string theory. Let us now describe its ingredients in more precise terms. They are a map from an oriented Riemann surface to a compact Riemannian manifold and its super partner, a vector spinor, with the Riemannian metric of the domain and its super partner, the gravitino, as parameters. This action functional is the twodimensional nonlinear sigma model of quantum field theory, which has been studied for a long time both in physics and mathematics. Such models have been used in supersymmetric string theory since the 1970s, see e.g. [5, 12]. We refer to [11, 16, 17] for more details about the mathematical aspects.
In a recent work [19], a corresponding geometric model was set up and some analytical issues were studied. In contrast with the previous models which use anticommuting fields and which are therefore not directly amenable to the methods of geometric analysis, this model uses only commuting fields and thus is given within the context of Riemannian geometry. Though this approach makes the supersymmetries involved less transparent, it has the advantages that this model is closely related to mathematically longstudied models such as harmonic maps and Diracharmonic maps and their various variants. In [19], a detailed setup for this twodimensional nonlinear sigma model was developed. On this basis, now the regularity issues can be investigated. The smoothness of weak solutions of the Euler–Lagrange equations, with smooth Riemannian metric and gravitino, was obtained in [19].
The analysis of twodimensional harmonic maps, and even more so, of Diracharmonic maps is quite subtle, because they constitute borderline cases for the regularity theory, with phenomena like bubbling. While the harmonic map case by now can be considered as well understood, and much is known about Diracharmonic maps, it turns out that major new difficulties from the analytical perspective are caused by the gravitino, even if the gravitino is treated only as a parameter and not as a dependent variable in its own right. These difficulties arise from the way the gravitino is coupled with the spinor field in the action functional, see (1) below. These difficulties become even more severe if the gravitino in the model is not smooth. More precisely, we encounter the following question: what is the weakest possible assumption on the gravitino and under such an assumption how smooth will the critical points of the action functional be? Apparently in general we can no longer expect \(C^\infty \) differentiability, but one may still hope to improve the original regularity of the weak solutions. Here we explore this issue. We shall combine the regularity theory of [21, 22, 23] with Morrey space theory and a subtle iteration argument to achieve what should be the optimal regularity results in our setting.
We remark that the Lagrangian of the action appears in this form for reasons of supersymmetry. Note that in the particular case where the gravitino vanishes, this reduces to the Diracharmonic map functional with curvature term introduced in [8] and further studied in e.g. [3, 4, 18]. If in addition, the curvature terms in the Lagrangian also vanish, this reduces to the Diracharmonic map functional introduced in [6, 7], which is studied to a great extent in e.g. [9, 15, 25, 27, 28].
Definition 1.1
Let \(1<p\le \infty \). We say that a measurable function \(u:(X,\mu )\rightarrow \mathbb {R}\) is an almost \(L^p\) function, denoted by \(u\in L^{po}(X,\mu )\), if \(u\in L^q(X,\mu )\) for any \(1\le q<p\).
Theorem 1.1
 (1)If \(p>p_0\), then \(\psi \in W^{1,p/2}_{loc}(B_1)\) and \(\phi \in W^{1,p}_{loc}(B_1)\). Furthermore, there exists an \(\varepsilon =\varepsilon (p)>0\) such that whenever \(\Vert \phi \Vert _{W^{1,2}(B_1)}+\Vert \psi \Vert _{L^4(B_1)}\le \varepsilon \), then for any \(U\Subset B_1\),for some constant \(C=C(p,U,\Vert Q\chi \Vert _{L^p(B_1)})>0\).$$\begin{aligned} \Vert \phi \Vert _{W^{1,p}(U)}+\Vert \psi \Vert _{W^{1,p/2}(U)}\le C\left( \Vert \phi \Vert _{W^{1,2}(B_1)}+\Vert \psi \Vert _{L^4(B_1)}\right) \end{aligned}$$
 (2)If \(4<p\le p_0\), then there exist some \(t_*=t_*(p)\in (4,\infty )\) and \(q_*=q_*(p)\in (2,\frac{2p}{p2})\) such that \(\psi \in W^{1,\frac{2t_*}{2+t_*}o}_{loc}(B_1)\hookrightarrow L^{t_*o}_{loc}(B_1)\) and \(\phi \in W^{1,q_*o}(B_1)\). Furthermore, there exists an \(\varepsilon =\varepsilon (p)>0\) such that whenever \(\Vert \phi \Vert _{W^{1,2}(B_1)}+\Vert \psi \Vert _{L^4(B_1)}\le \varepsilon \), then for any \(U\Subset B_1\), and for any \(t<t_*\) and \(q<q_*\),for some constant \(C=C(p,q,t,U,\Vert Q\chi \Vert _{L^p(B_1)})>0\).$$\begin{aligned} \Vert \phi \Vert _{W^{1,q}(U)}+\Vert \psi \Vert _{W^{1,\frac{2t}{2+t}}(U)}\le C\left( \Vert \phi \Vert _{W^{1,2}(B_1)}+\Vert \psi \Vert _{L^4(B_1)}\right) \end{aligned}$$
The methods used here are quite typical in the analysis of geometric partial differential equations. As we are dealing with a critical case for the Sobolev framework, we need a little Morrey space theory. Then Rivière’s regularity theory [21] and its extensions in e.g. [22, 23, 24, 25] enable us to utilize the antisymmetric structure of the equations for \(\phi \) to improve the regularity. Using similar methods, regularity results for weak solutions of the simpler models, namely Diracharmonic maps and Diracharmonic maps with curvature terms, are achieved in [3, 9, 27, 28]. Here in this more general model, the structure of the system is even more complicated because of the divergence terms and the appearance of the gravitinos. In the present work, we obtain regularity results for weak solutions for the case of coarse gravitinos.
With this result in hand, we turn to the system (2) and (3). Now we may make use of the concrete expressions of the coefficients \(\Omega ^{ab}\)’s and \(A^{ab}\)’s. That is, by Theorem 1.1, \(\phi '\) and \(\psi '\) now have better integrability properties, hence so do the corresponding \(\Omega ^{ab}\)’s and \(A^{ab}\)’s. A more precise analysis of these coefficients will then lead to our main result.
Theorem 1.2
Let \((\phi ,\psi )\in \mathcal {X}^{1,2}_{1,4/3}(M,N)\) be a critical point of the action functional \(\mathbb {A}\). Suppose the gravitino \(\chi \in \Gamma ^p(S\otimes TM)\) for some \(p\in (4,\infty ]\). Then \(\phi \in W^{1,p}(M,N)\) and \(\psi \in \Gamma ^{1,p/2}(S\otimes \phi ^*TN)\). In particular, they are Hölder continuous.
The article is organized as follows. We first prepare some lemmata to handle the equations for \(\psi \) and \(\phi \) separately. Then we can use an iteration procedure to improve the regularity of the solutions to the system (4) and (5) step by step. One can directly start from the section of iterations, skipping the two sections in which the lemmata are prepared, and refer to it back when necessary. In the final section we analyze the original system (2) and (3) and prove Theorem 1.2. Unlike many other problems where the coupling of variables causes additional problems, here the coupling behavior helps to achieve our goals.
Before start we would like to express our thanks to Marius Yamakou for producing the nice graphs with MATLAB.
2 Preparation lemma for spinor components
Lemma 2.1
We remark that \(\mathbb {R}^L\otimes \mathbb {R}^K\) represents the typical fiber of a twisted spinor bundle over the mdimensional unit ball \(B_1\), which is trivial. By this lemma we see that, as long as B in (7) has better regularity than \(M^{\frac{4}{3},2}\), the integrability of \(\varphi \) can be improved. Arguments of this type have been used to show the regularities for Dirac type equations in various contexts, see e.g. [26] in dimension \(m\ge 2\) and see e.g. [3, 25] in dimension \(m=2\). The above result improves that in Lemma 6.1 in [19], where the case of \(s=2\) was done and we include the sketch of the proof here only for the convenience of readers.
Proof
Since the case \(s=2\) has been shown in [19, Lemma 6.1], here we consider \(s\in (\frac{4}{3},2)\).
3 Preparation lemma for map components
Now the Eq. (4) for \(\phi \) are almost away from being critical, and we will show that the map has better regularity than \(W^{1,2}(B_1,\mathbb {R}^K)\). Note that \(\Omega ^{ab}\nabla \phi ^b\in L^1(B_1)\) and \({{\mathrm{div}}}V^a\in W^{1,2}(B_1)\), and both of them may cause trouble. The following lemma, which is a combination of Campanato regularity theory and Rivière’s regularity theory, will be useful for handling these problems.
Lemma 3.1
Remark
Proof
4 Improvement of regularity by an iteration procedure
In this section we prove Theorem 1.1, and in the end we give two examples of different values of p and different terminating values \(q_*\).
Proof of Theorem 1.1
Before dealing with the general solutions, let’s consider some particular cases.
Second, when \(p=\infty \), the situation is almost trivial. Actually, now \(B^a=e_\alpha \cdot \nabla \phi ^a\cdot \chi ^\alpha \in L^2(B_1)\) for each a. From Lemma 6.1 in [19] it follows that \(\psi \in L^{\infty o}_{loc}(B_1)\). Then applying Lemma 3.1 we get \(\phi \in W^{1,po}_{loc}(B_1)\). This returns to the situation above, and also finishes the proof for the case \(p=\infty \).
 \(\textcircled {1}\)

Suppose it has been shown that \(\psi \in L^t_{loc}(B_1)\) and \(\nabla \phi \in L^q_{loc}(B_1)\) for some \(t>4\) and \(q>2\).
 \(\textcircled {2}\)

Then \(B\in L^s_{loc}(B_1)\) with \(s=s(q)=\frac{pq}{p+q}>\frac{4}{3}\). If \(s\ge 2\), then as before we immediately get \(\psi \in L^{\infty o}_{loc}(B_1)\) and \(\nabla \phi \in L^{po}_{loc}(B_1)\). The desired result follows. Thus we may take \(q<\frac{2p}{p2}\equiv Q_0(p)\) in \(\textcircled {1}\) so that \(s<2\).
 \(\textcircled {3}\)
 By Lemma 2.1, \(\psi \in L^{T(q)o}_{loc}(B_1)\) with$$\begin{aligned} T\equiv T(q)=\frac{8}{63s(q)}=\frac{8(p+q)}{6p+6q3pq}\in (4,\infty ). \end{aligned}$$
 \(\textcircled {4}\)
 To determine the value of \(\sigma \), we need to compareand$$\begin{aligned} \frac{T}{4}=\frac{2(p+q)}{6p+6q3pq} \end{aligned}$$A simple calculation shows that$$\begin{aligned} \frac{2pT}{2(p+T)+pT}=\frac{8p(p+q)}{(3p^2+10p+8)q+(10p^2+8p)}. \end{aligned}$$Since \(q>2\) while \(\frac{14p^28p}{9p^214p+8}<2\) (since \(p>4\) by assumption), the value of \(\sigma \) is determined by$$\begin{aligned} \frac{T}{4}\ge \frac{2pT}{2(p+T)+pT}\Leftrightarrow q\ge \frac{14p^28p}{9p^214p+8}. \end{aligned}$$For \(q\in (2,\frac{2p}{p2})\), \(\sigma \) lies in the interval$$\begin{aligned} \sigma =\frac{2pT}{2(p+T)+pT}\wedge \frac{T}{4}=\frac{2pT}{2(p+T)+pT} =\frac{8p(p+q)}{(3p^2+10p+8)q+(10p^2+8p)}. \end{aligned}$$which is a proper subinterval of (1, 2). In particular, \(\sigma <2\) and$$\begin{aligned} \left( \frac{2p(p+2)}{p^2+7p+4}, \frac{2p}{p+2}\right) , \end{aligned}$$$$\begin{aligned} \frac{2\sigma }{2\sigma }=\frac{pT}{p+T}=\frac{8p(p+q)}{(3p^2+6p+8)q+(6p^2+8p)}=:Q(q)\equiv Q. \end{aligned}$$
 \(\textcircled {5}\)

Lemma 3.1 then shows that \(\nabla \phi \in L^{Qo}_{loc}(B_1)\).
 \(\textcircled {6}\)

Compare the value of q and Q(q). Case 1: \(q<Q(q)<Q_0=\frac{2p}{p2}\). Then go to \(\textcircled {1}\) with \(\psi \in L^{T(q)o}_{loc}(B_1)\) and \(\nabla \phi \in L^{Q(q)o}_{loc}(B_1)\), and then go through the procedure again. Case 2: \(Q(q)\ge Q_0\). Then \(B\in L^2_{loc}(B_1)\). The desired result is obtained as before. Case 3: \(Q(q)\le q\). Then this procedure also terminates, with \(t_*=T(q)\) and \(q_*=Q(q)\) in the statement of Theorem 1.1.
Next we analyze the limiting behavior of such an iteration. It turns out that this is determined by p.
Thus the improvement will not work at \(q_*=q_(p)\) for \(p\le p_0\). The corresponding \(t_*\) is given by \(T(q_*)\). On the other hand if \(p>p_0\), then one can easily get the regularity improved to the expected level (Fig. 2).
The desired estimates follows from an iterated combination of (9) and (11). The proof of Theorem 1.1 is completed. \(\square \)
Here the horizontal lines stand for the barrier \(Q_0(p)=\frac{2p}{p2}\).
5 Regularity of the critical points of the action functional
We can now turn to the regularity of the critical points of the action functional (1), or equivalently the solutions of the Euler–Lagrange equations (2) and (3). In contrast to Theorem 1.1, the solutions of the Euler–Lagrange equations have the expected regularities, due to the structure of the equations.
Proof of Theorem 1.2
Footnotes
Notes
Acknowledgements
Open access funding provided by Max Planck Society. Ruijun Wu thanks the International Max Planck Research School Mathematics in the Sciences for financial support. Miaomiao Zhu was supported in part by the National Natural Science Foundation of China (No. 11601325).
References
 1.Ammann, B.: A Variational Problem in Conformal Spin Geometry. Habilitation. Hamburg University, Hamburg (2003)Google Scholar
 2.Adams, D.R.: A note on Riesz potentials. Duke Math. J. 42(4), 765–778 (1975)CrossRefzbMATHMathSciNetGoogle Scholar
 3.Branding, V.: Some aspects of Diracharmonic maps with curvature term. Differ. Geom. Appl. 40, 1–13 (2015)CrossRefzbMATHMathSciNetGoogle Scholar
 4.Branding, V.: Energy estimates for the supersymmetric nonlinear sigma model and applications. Potential Anal. 45(2016), 737–754 (2016)CrossRefzbMATHMathSciNetGoogle Scholar
 5.Brink, L., Di Vecchia, P., Howe, P.: A locally supersymmetric and reparametrization invariant action for the spinning string. Phys. Lett. B 65(5), 471–474 (1976)CrossRefGoogle Scholar
 6.Chen, Q., Jost, J., Li, J., Wang, J.: Regularity theorems and energy identities for Diracharmonic maps. Math. Z. 251(1), 61–84 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
 7.Chen, Q., Jost, J., Li, J., Wang, G.: Diracharmonic maps. Math. Z. 254(2), 409–432 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
 8.Chen, Q., Jost, J., Wang, G.: Liouville theorems for Diracharmonic maps. J. Math. Phys. 48(11), 113517 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
 9.Chen, Q., Jost, J., Wang, G., Zhu, M.: The boundary value problem for Diracharmonic maps. J. Eur. Math. Soc. 15(3), 997–1031 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
 10.Chen, Y.Z, Wu, L.C.: Second order elliptic equations and elliptic systems. Translations of mathematical monographs 174. American Mathematical Society, Providence (1998)Google Scholar
 11.Deligne, P., et al.: Quantum Fields and Strings: A Course for Mathematicians. American Mathematical Society, Providence (1999)Google Scholar
 12.Deser, S., Zumino, B.: A complete action for the spinning string. Phys. Lett. B 65(4), 369–373 (1976)CrossRefMathSciNetGoogle Scholar
 13.Giaquinta, M.: Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems. Princeton University Press, Princeton (1983)zbMATHGoogle Scholar
 14.Hélein, F.: Régularité des applications faiblement harmoniques entre une surface et une varieté riemannienne. CR Acad. Sci. Paris 312, 591–596 (1991)zbMATHGoogle Scholar
 15.Jost, J.: Riemannian Geometry and Geometric Analysis, 6th edn. Springer, Berlin (2011)CrossRefzbMATHGoogle Scholar
 16.Jost, J.: Geometry and Physics. Springer, Berlin (2009)CrossRefzbMATHGoogle Scholar
 17.Jost, J., Keßler, E., Tolksdorf J.: Super riemann surfaces, metrics, and gravitinos. 2014, to appear in advances in theoretical and mathematical physics. arXiv:1412.5146 [mathph] (2014)
 18.Jost, J., Liu, L., Zhu, M.: Geometric analysis of the action functional of the nonlinear supersymmetric sigma model. 2015, MPI MIS Preprint: 77/2015Google Scholar
 19.Jost, J., Keßler, B., Tolksdorf, J., Wu, R., Zhu, M.: Regularity of solutions of the nonlinear sigma model with gravitino. 2016, to appear in Communication Mathematical Physics. arXiv:1610.02289 [math.DG] (2016)
 20.Lawson, H.B., Michelsohn, M.L.: Spin Geometry. Princeton University Press, New Jersey (1989)zbMATHGoogle Scholar
 21.Rivière, T.: Conservation laws for conformally invariant variational problems. Invent. Math. 168(1), 1–22 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
 22.Rivière, T.: Conformally Invariant 2Dimensional Variational Problems. Cours joint de l’Institut Henri Poincarè, Paris (2010)Google Scholar
 23.Rivière, T., Struwe, M.: Partial regularity for harmonic maps, and related problems. Commun. Pure Appl. Math. 61(4), 0451–0463 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
 24.Sharp, B., Topping, P.: Decay estimates for Riviere’s equation, with applications to regularity and compactness. Trans. Am. Math. Soc. 365(5), 2317–2339 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
 25.Sharp, B., Zhu, M.: Regularity at the free boundary for Diracharmonic maps from surfaces. Calc. Var. Partial Differ. Equ. 55(2), 27–55 (2016)CrossRefzbMATHMathSciNetGoogle Scholar
 26.Wang, C.: A remark on nonlinear Dirac equations. Proceed. Am. Math. Soc. 138(10), 3753–3758 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
 27.Wang, C., Deliang, X.: Regularity of Diracharmonic maps. Int. Math. Res. Not. 20, 3759–3792 (2009)zbMATHMathSciNetGoogle Scholar
 28.Zhu, M.: Regularity for weakly Diracharmonic maps to hypersurfaces. Ann. Glob. Anal. Geom. 35(4), 405–412 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
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