The Higher Riemann-Hilbert Correspondence and Multiholomorphic Mappings Aaron M. Smith A Dissertation in Mathematics Presented to the Faculties of the University of Pennsylvania in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy 2011 Jonathan Block Supervisor of Dissertation Jonathan Block Graduate Group Chairperson Dissertation Committee Jonathan Block Tony Pantev Florian Pop Acknowledgments This accomplishment –no matter how grand or how modest– could not have been reached without the support of many individuals who formed an important part of my life –both mathematical and otherwise. The most immediate thanks go out to Jonathan Block, my advisor, who gently pushed me towards new mathematical ventures and whose generosity and patience gave me some needed working-space. He shared his insights and knowledge freely, and gave me a clear picture of an ideal career as a mathematician. I must also thank him for his warm hospitality, and introducing me to his family. IamalsograciousforthepresenceofTonyPantev,whowasakindofadvisor-from-afar during my tenure, and whose students were often my secondary mathematical mentors. Collectively that are responsible for a significant amount of learning on my part. I also thank Denis Auroux, Calder Daenzer, Kevin Lin, Ricardo Mendes, Harshaaa Reddy, Jim Stasheff, Cliff Taubes, and Erik Van Erp on the same account. I cannot leave out the fine work of the office staff at Penn: Janet, Monica, Robin, Paula. They keep the machine running. The task of producing significant mathematical research was daunting for me, and brought me to the edge of burn-out on a number of occasions. I am infinitely grateful to have had classmates who were also solid friends. In this regard I am most in debt to Umut Isik, Ricardo Mendes, and Sohrab Shahshahani. But singling out a few doesn’t do ii justice to the warmth and cameraderie of the graduate department at Penn. I have saved the most important for last: my family. Without the friendship and suport of my parents and brothers it is unlikely that I could have succeeded in completing these graduate years. I am incredibly grateful to have been yoked to Liam and Evan, two admirable brothers that I can rely on, and to my parents who have always urged me to pursue my interests and passions where they lead. This thesis is dedicated to my parents, who let me play with fire. iii ABSTRACT The Higher Riemann-Hilbert Correspondence and Multiholomorphic Mappings Aaron M. Smith Jonathan Block, Advisor This thesis consists in two chapters. In the first part we describe an A -quasi- ∞ equivalenceofdg-categoriesbetweenBlock’sP ,correspondingtothedeRhamdgaAofa A compact manifold M and the dg-category of infinity-local systems on M. We understand this as a generalization of the Riemann-Hilbert correspondence to Z-graded connections (superconnections in some circles). In one formulation an infinity-local system is an (∞,1)-functor between the (∞,1)-categories π M and a repackaging of the dg-category ∞ of cochain complexes by virtue of the simplicial nerve and Dold-Kan. This theory makes crucial use of Igusa’s notion of higher holonomy transport for Z-graded connections which is a derivative of Chen’s main idea of generalized holonomy. In the appendix we describe some alternate perspectives on these ideas and some technical observations. The second chapter is concerned with the development of the theory of multiholomor- phic maps. This is a generalization in a particular direction of the theory of pseudoholo- morphic curves. We first present the geometric framework of compatible n-triads, from which follows naturally the definition of a multiholomorphic mapping. We develop some of the essential analytic and differential-geometric facts about these maps in general, and then focus on a special case of the theory which pertains to the calibrated geometry of G -manifolds. This work builds toward the realization of invariants generated by the 2 topology of the moduli spaces of multiholomorphic maps. Because this study is relatively fundamental, there will be many instances where questions/conjectures are put forward, or directions of further research are described. iv Contents 1 The Higher Riemann-Hilbert Correspondence 1 1.1 Introduction and Summary . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Infinity-Local Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2.1 ∞-categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2.2 Infinity Local Systems . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.3 Iterated Integrals and Holonomy of Z-graded Connections . . . . . . . . . 10 1.3.1 Sign Conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.3.2 Path Space Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.3.3 Iterated Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.3.4 Z-graded Connection Holonomy. . . . . . . . . . . . . . . . . . . . 15 1.3.5 Holonomy With Respect to the Pre-triangulated Structure . . . . 20 1.3.6 Cubes to Simplices . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 1.4 An A -quasi-equivalence . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 ∞ 1.4.1 The functor RH : P → LocC (π M) . . . . . . . . . . . . . . . . 25 A ∞ ∞ 1.4.2 RH is A -essentially surjective . . . . . . . . . . . . . . . . . . . . 29 ∞ 1.5 Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 1.5.1 Infinity-local systems taking values in codifferential-coalgebras and A -local systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ∞ 1.5.2 Coefficients in some P . . . . . . . . . . . . . . . . . . . . . . . . 38 B 1.6 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 1.6.1 The Simplicial Nerve . . . . . . . . . . . . . . . . . . . . . . . . . . 39 1.6.2 k-linear ∞-categories . . . . . . . . . . . . . . . . . . . . . . . . . . 41 1.6.3 Relevant Model Structures . . . . . . . . . . . . . . . . . . . . . . 45 1.6.4 The Linear Simplicial Nerve and C . . . . . . . . . . . . . . . . . 52 ∞ 1.6.5 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 2 The Theory of Multiholomorphic Maps 58 2.1 n-triads and the Multi-Cauchy-Riemann Equations . . . . . . . . . . . . . 58 2.1.1 n-triads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 2.1.2 The Multi-Cauchy-Riemann Equation . . . . . . . . . . . . . . . . 64 2.1.3 Endomorphisms and Quasi-regular mappings . . . . . . . . . . . . 67 2.1.4 The Energy Identity . . . . . . . . . . . . . . . . . . . . . . . . . . 74 2.1.5 Analytic Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 2.1.6 Variational Aspects . . . . . . . . . . . . . . . . . . . . . . . . . . 80 2.2 A Mapping Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 2.2.1 Main Goal. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 2.2.2 Foundational Matters . . . . . . . . . . . . . . . . . . . . . . . . . 88 2.2.3 Case: G -manifolds with the associative triad . . . . . . . . . . . . 89 2 2.2.4 Integrability and Existence . . . . . . . . . . . . . . . . . . . . . . 96 v 2.2.5 Coassociative Boundary conditions for the G -MCR equation . . . 100 2 2.2.6 Fukaya-like boundary conditions . . . . . . . . . . . . . . . . . . . 108 vi Chapter 1 The Higher Riemann-Hilbert Correspondence 1.1 Introduction and Summary Given a compact manifold M, the classical Riemann-Hilbert correspondence gives an equivalence of categories between Rep(π (M)) and the category Flat(M) of vector bun- 1 dles with flat connection on M. While beautiful, this correspondence has the primary drawback that it concerns the truncated object π , which in most cases contains only a 1 small part of the data which comprises the homotopy type of M. From the perspective of (smooth) homotopy theory the manifold M can be replaced by its infinity-groupoid Sing∞M := π M of smooth simplices. Considering the correct notion of a representa- • ∞ tion of this object will allow us to produce an untruncated Riemann-Hilbert theory. More specifically, we define an infinity-local system to be a map of simplicial sets which to each simplex of π M assigns a homotopy coherence in the category of chain complexes over ∞ R,C := Ch(R). Our main theorem is an A -quasi-equivalence ∞ RH : P → LocC(π M) (1.1.1) A ∞ 1 Where, P is the dg-category of graded bundles on M with flat Z-graded connection and A LocC(π M) is the dg-category of infinity-local systems on M. ∞ In the classical Riemann-Hilbert equivalence the map Flat(M) → Rep(π (M)) (1.1.2) 1 is developed by calculating the holonomy of a flat connection. The holonomy descends to a representation of π (M) as a result of the flatness. The other direction 1 Rep(π (M)) → Flat(M) (1.1.3) 1 is achieved by the associated bundle construction. In the first case our correspondence proceeds analogously by a calculation of the holonomy of a flat Z-graded connection. The technology of iterated integrals suggests a precise and rather natural notion of such holonomy. Given a vector bundle V over M with connection, the usual parallel transport can be understood as a form of degree 0 on the path space PM taking values in the bundle Hom(ev∗V,ev∗V). The higher holonomy 1 0 is then a string of forms of total degree 0 on the path space of M taking values in the same bundle. Such a form can be integrated over cycles in PM , and the flatness of the connection implies that such a pairing induces a representation of π as desired. This is ∞ the functor RH : P → LocC(π M). (1.1.4) A ∞ It would be an interesting problem in its own right to define an inverse functor which makes use of a kind of associated bundle construction. However we chose instead to prove quasi-essential surjectivity of the above functor. Given an infinity-local system (F,f) one can form a complex of sheaves over X by considering LocC(π U)(R,F). This complex ∞ is quasi-isomorphic to the sheaf obtained by extending by the sheaf of C∞ functions and then tensoring with the de Rham sheaf. Making use of a theorem of Illusie we construct 2 from this data a perfect complex of A0-modules quasi-isomorphic to the zero-component of the connection in RH(F). Finally we follow an argument of [Bl1] to complete this to an element of P which is quasi isomorphic to RH(F). A We reserve the appendix to work out some of the more conceptual aspects of the theory as it intersects with our understanding of homotopical/derived algebraic geometry (in the parlance of Lurie, Simpson, Toen-Vezzosi, et. al.). One straightforward extension of this theory is to take representations in any linear ∞-category. In fact, considering representations of π M in the category of A -algebra leads to a fruitful generalization ∞ ∞ of recent work of Emma Smith-Zbarsky [S-Z] who has considered the action of a group G on families of A algebras over a K(G,1). ∞ The authors extend thanks to Tobias Dyckerhoff, Pranav Pandit, Tony Pantev, and Jim Stasheff for helpful comments during the development this work, and especially Kiyoshi Igusa, who gratefully shared his ongoing work on integration of superconnec- tions. And lastly the authors are grateful to Camilo Aras Abad and Florian Schaetz for pointing out a significant oversight in our first draft. 1.2 Infinity-Local Systems 1.2.1 ∞-categories We first give a short commentary on some of the terminology of higher category theory that appears in this paper. All of the language and notation can be accessed in more completeformin[Lu2]. Initially, by∞-categoryonemeansanabstracthighercategorical structurewhichpossessesasetofobjectsandasetofk-morphismsforeachnaturalnumber k. The morphisms in the k-th level can be understood as morphisms between (k − 1)- morphisms and so on. Then by (∞,n)-category one denotes such a structure for which 3 all morphisms of level greater than n are in fact invertible up to higher level morphisms. One typically needs to deal with concrete mathematical objects which instantiate these kinds of structures. Hence there are several extant models for higher categories in the literature. We pick one for good so that there is no confusion as to what exactly we are dealing with when we talk about ∞-categories. By ∞-category we will henceforth mean a weak Kan complex (also called a quasicat- egory). This is a particularly concrete model for (∞,1)-categories. All higher categorical structures we will see here have the invertibility condition above level 1. Hopefully this will not cause confusion as this naming convention is relatively standard in the literature. Suppose that ∆k is the standard combinatorial k-simplex. By the qth k-horn Λk we mean q the simplicial set which consists of ∆k with its k-face and the codimension-1 face ∂ ∆k q removed. If q (cid:54)= 0,k we call Λk an inner horn. q Definition 1.2.1. A weak Kan complex is a simplicial set K which satisfies the property that any sSet map T: Λk → S from an inner horn to S, can be extended to a map from q the entire simplex Tˆ : ∆k → S. This weak Kan extension property elegantly marries homotopy theory to higher cat- egory theory. In the case of a two-simplex, the weak Kan extension criterion guarantees the existence of a kind of weak composition between the two faces that make up the inner horn. See [Lu2] for elaboration on this philosophy. In the same vein, Definition 1.2.2. A Kan complex is a simplicial set K which satisfies the Kan extension property of the previous definition for all horn inclusions, both inner and outer. A Kan complex models the theory of (∞,0)-categories because the outer horn exten- sion conditions imply invertibility for the 1-morphisms. Hence, these are also referred to 4