## The Higher Riemann-Hilbert Correspondence and Multiholomorphic Mappings (2011)

### BibTeX

@MISC{Smith11thehigher,

author = {Aaron M. Smith and Aaron M. Smith and Jonathan Block and Tony Pantev},

title = {The Higher Riemann-Hilbert Correspondence and Multiholomorphic Mappings},

year = {2011}

}

### OpenURL

### Abstract

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. I am also gracious for the presence of Tony Pantev, who was a kind of advisor-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

### Citations

568 | Spin geometry
- Lawson, Michelsohn
- 1989
(Show Context)
Citation Context ...omposition with (−1) n J, (−1) n K. This example will become crucial because such a triadic manifold will be the domain in a mapping theory. Example 2.1.7. (Associatively calibrated G2-manifolds)(See =-=[HL]-=-) Consider ImO, and fix an identification with R 7 . a + bI + cJ + dK + e1 ′ + fI ′ + gJ ′ + hK ′ = (a, b, c, d, e, f, g, h) (2.1.14) This is equipped with the usual inner product g(, ), the octionic ... |

272 | Loop quantum gravity - Rovelli - 2003 |

261 |
Simplicial objects in algebraic topology
- May
- 1992
(Show Context)
Citation Context ... the following assumption: Categories enriched in simplicial vectorspaces model linear ∞-categories. (Note that these are a proper subset of sCat because all simplicial vectorspaces are Kan complexes =-=[Ma]-=-). One of the main goals of [Sm] is to check Speculation 1.6.9. The adjoint equivalence (C[], N ) restricts to an adjoint equivalence between sCat(k) and k − sSet. Proof. And furthermore, with respect... |

233 | Nomizu: Foundations of Differential Geometry, Vol. 1, Interscience Tracts in Pure and Applied Mathematics, Interscience - Kobayashi, K - 1963 |

158 | Homological mirror symmetry and torus fibration, in: Symplectic Geometry and Mirror Symmetry, (Seoul, 2000), eds by
- Kontsevich, Soibelman
- 2001
(Show Context)
Citation Context ...LocA∞(K) • (Fi, Fj) equipped 36with the direct sums of the above multiplications becomes an A∞-algebra, then LocA∞(K) is an A∞-category. In particular the one-point A∞-category is just an A∞-algebra.=-=[K-S]-=- Proposition 1.5.2. Loc C A∞ (K) is an A∞-category. In a recent pre-print [S-Z] Emma Smith-Zbarsky develops a closely related theory. She considers a Z-graded bundle V over a smooth manifold M and the... |

144 | Deformations of calibrated submanifolds
- McLean
- 1996
(Show Context)
Citation Context ...the branes for each of the distinct families of n-triad structures. There is preexisting work along these lines in [LL],[L1],[L2], (and other work of Leung), which in turn rests on the work of McLean =-=[McL]-=- with regard to the deformation theory of calibrated submanifolds and the associated branes. The main novelty of this paper’s framework is a PDE intertwining compatible structures on domain and target... |

137 |
Metrics with exceptional holonomy
- Bryant
- 1987
(Show Context)
Citation Context ...x 1 (dy 23 + dy 10 ) − dx 2 (dy 31 + dy 20 ) − dx 3 (dy 1 2 + dy 30 ). (2.1.21) 62This furnishes a well-known description of the exceptional Lie group G2: G2 := {σ ∈ GL(ImO)|σ ∗ ω0 = ω0} (2.1.22) In =-=[Br1]-=-, Bryant proved that G2 can also be described as the group which preserves the metric and vector cross product. For this reason, a Riemannian 7-manifold with G2holonomy is equipped with a parallel 3-f... |

137 |
D.: J-holomorphic Curves and Symplectic Topology
- McDuff, Salamon
- 2004
(Show Context)
Citation Context ... → Ω 1 (X, u ∗ T M) (2.1.74) Defining a vertical differential in general requires a connection on the bundle E. There is a natural way in which a connection on M induces a connection on E. (Following =-=[McDS]-=- sec. 3.1,) Suppose ∇M is the Levi-Civita connection on T M. A section αλ of E along a curve γ : R → B : λ ↦→ uλ is parallel with respect to the induced connection on E if the vector field λ ↦→ αλ(z; ... |

115 |
Iterated path integrals
- Chen
- 1997
(Show Context)
Citation Context ...with the given flat Z-connection. In [I] Kiyoshi Igusa presents from scratch a notion of higher parallel transport for a Z-connection. This is a tweaked example of Chen’s higher transport outlined in =-=[Ch]-=- which makes crucial use of his theory of iterated integrals. We slightly reformulate and extend this idea to produce a functor from PA to Loc C ∞(K) which is an A∞-quasi-equivalence. To start we pres... |

98 | The unregularized gradient flow of the symplectic action - Floer - 1988 |

90 | Gauge theory in higher dimensions, in The geometric universe - Donaldson, Thomas - 1998 |

69 |
Geometric Function Theory and Non-linear Analysis
- Iwaniec, Martin
- 2001
(Show Context)
Citation Context ...ngular. In fact the solutions in this case are a particular geometric generalization of the well-studied field of 1-quasiregular mappings. This issue is discussed in a proceeding section. We found in =-=[IM]-=- an excellent presentation of the of modern issues in this direction and a lot of inspiration for this more global differentialgeometric approach. In what remains in this section we describe the MCR e... |

64 |
Simplicial localization of categories
- Dwyer, Kan
- 1980
(Show Context)
Citation Context ...al fibrations. We can regard sVect as a model category with the model structure coming from the Kan structure on sSet. Then, Proposition 1.6.12. Vectk is excellent. Proof. Follows from the results of =-=[DK]-=-. 47Lurie [Lu1], and later Dugger-Spivak [DSp],show that the functors N and C constitute a Quillen equivalence N : sCatB ⇆ sSetJ : C (1.6.31) And we expect, [Sm], a Quillen equivalence N : sCat(k)BT ... |

60 | The homotopy theory of dg-categories and derived Morita theory - Toën |

50 | Representation Theory. A First - Fulton, Harris - 1996 |

48 | A model category structure on the category of simplicial categories
- Bergner
(Show Context)
Citation Context ... form the (∞, 1)-category of (∞, 1)categories. There are explicit model structures on each of these categories which present these structures. These are the model structure on sCat defined by Bergner =-=[Be]-=-, and the model structure of Joyal on sSet. These model categories are denoted sCatB and sSetJ respectively. Likewise there is a model structures on sCat(k) and k − sSet which are related to these two... |

46 | Twisted connected sums and special Riemannian holonomy
- Kovalev
(Show Context)
Citation Context ...p1 ∼ = SU2. Constructions of G2-manifolds We mention that there are well-known methods of constructing G2-manifolds with full G2-holonomy, first famously due to Joyce [J3], and another due to Kovalev =-=[Ko]-=-. We leave this exploration to the reader. Multiholomorphic images are associative Coming back to the main thread, suppose a closed, oriented, Riemannian 3-manifold X maps by u into a closed G2-manifo... |

31 |
Segal topoi and stacks over Segal categories. Available for download: math.AG/0212330
- Toën, Vezzosi
(Show Context)
Citation Context ...ce to its (linear) ∞-category 10(alternately its dg-category) of “homotopy-locally constant” sheaves of vector spaces (in our setup M ↦→ (PA)∞). In fact Toen and Vezzosi present something similar in =-=[TV1]-=- and [To1]. Their primary concern is a Segal Tannakian theory, and they show that for a CW complex X, π∞X can be recovered from the category of simplicial local systems. Our Riemann-Hilbert correspond... |

30 |
Gauge theory in higher dimensions
- Donaldson, Thomas
- 1996
(Show Context)
Citation Context ...on an oriented 7-manifold M is a principal, G2subbundle of the oriented frame bundle of M. One can regard the frame bundle over some point x ∈ M as consisting of isomorphisms φ : TxM → R7 . Following =-=[DS]-=- and [J1], we describe an equivalent notion of a G2-structure more explicitly. Consider V a 7-dimensional real vectorspace with orientation O. There is an open GL+(V )-orbit P3 ⊂ Λ3V ∗ each element of... |

29 | Duality and equivalence of module categories in noncommutative geometry I,” arXiv:math/0509284
- Block
(Show Context)
Citation Context .... Making use of a theorem of Illusie we construct 2from 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 PA which is quasi isomorphic to RH(F ). We reserve the appendix to work out some of the more conceptual aspects of the theory as it intersects with our understanding... |

21 | Some remarks on G2-structures
- Bryant
- 2006
(Show Context)
Citation Context ...x 1 (dy 23 + dy 10 ) − dx 2 (dy 31 + dy 20 ) − dx 3 (dy 1 2 + dy 30 ). (2.1.21) 62This furnishes a well-known description of the exceptional Lie group G2: G2 := {σ ∈ GL(ImO)|σ ∗ ω0 = ω0} (2.1.22) In =-=[Br1]-=-, Bryant proved that G2 can also be described as the group which preserves the metric and vector cross product. For this reason, a Riemannian 7-manifold with G2holonomy is equipped with a parallel 3-f... |

20 |
Michelsohn: Spin Geometry
- Lawson, M-L
- 1989
(Show Context)
Citation Context ...may be divergent from theirs. Deformation Index We collect here some facts with are of central importance. Mostly these are from [McL] and [HL], although the former derives heavily from the essential =-=[LM]-=-. To start, it is useful to recall that Sp1 ∼ = Su2 ∼ = Spin3, which may be used interchangeably based on the context. Suppose X is an associative submanifold embedded in M, a G2-manifold. 101There i... |

17 | Homotopical and higher categorical structures in algebraic geometry, Hablitation Thesis available at math.AG/0312262 - Toën |

16 |
Riemannian Holonomy Groups and Calibrated Geometry, Oxford Graduate Texts
- Joyce
- 2007
(Show Context)
Citation Context ... in the theory of pseudoholomorphic curves between Kähler and almost-Kähler manifolds. First we describe more fully the subject of G2-manifolds and collect some relevant facts. An excellent source is =-=[J1]-=-. 89G2-manifolds Definition 2.2.2. A G2-structure on an oriented 7-manifold M is a principal, G2subbundle of the oriented frame bundle of M. One can regard the frame bundle over some point x ∈ M as c... |

16 | De Rham model for string topology - Merkulov |

15 | Generalized Classical Mechanics and Field Theory - Léon, Rodrigues - 1985 |

15 | A Darboux Theorem for Multi-Symplectic Manifolds - Martin - 1988 |

13 |
la notion de diagramme homotopiquement cohérent. Cahiers Topologie Géom. Différentielle 23
- Cordier
- 1982
(Show Context)
Citation Context ...of sheaves with coherent cohomology on Σ, [Bl1]. 1.6 Appendix 1.6.1 The Simplicial Nerve We describe here a construction called the simplicial nerve functor which was originally introduced by Cordier =-=[Co]-=- and appears in Lurie’s book on higher Topoi [Lu1]. The simplicial nerve is a functor N : sCat → sSet (1.6.1) which is defined by the adjunction property: sSet(∆ n , N(C)) = sCat(C[∆ n ], C) (1.6.2) w... |

11 | Iterated integrals and algebraic cycles: examples and prospects, Contemporary trends in algebraic geometry and algebraic topology
- Hain
(Show Context)
Citation Context ... reasonable definition of vector bundles over a differentiable space as well as differential forms. One can likewise define an exterior differential, and subsequently a so-called Chen de Rham complex =-=[Ha]-=-. We will try to make transparent use of these constructions, but we defer the reader to the existing discussions of these matters in [Ch],[Ha],[I],[BH]. The primary reason that path-space calculus is... |

11 | Floer homology and Novikov - Hofer, Salamon - 1995 |

10 | Categorified symplectic geometry and the classical string
- Baez, Hoffnung, et al.
(Show Context)
Citation Context ...ing such a form an n-plectic triad. The definition of n-plectic or multi-(sym)plectic appears in work of Gotay, Isenberg, Marsden, Montgomery, [GIMM], and some recent papers by Baez, Hoffnung, Rogers =-=[BHR]-=-. (motivated by the canonical n-plectic form on an n-form bundle). Symplectic manifolds are 1-plectic. And because a triadic manifold is Riemannian it makes sense to consider the covariant derivatives... |

9 | Enriched model categories and an application to additive endomorphism spectra
- Dugger, Shipley
(Show Context)
Citation Context ...es with another precedent in the literature of describing connective dg-categories (and their A∞-cousins) as models for linear ∞- categories. The model equivalence of Tabuada [Ta], and Dugger-Shipley =-=[DSh]-=- (described below) lends weight to this precedent. In this paper we will not speculate about what should really model linear ∞-categories. Instead we will start with the following assumption: Categori... |

9 | Model Categories and Their Localizations, Mathematical Surveys and Monographs - Hirschhorn |

8 |
A new proof of local C 1,α regularity for solutions of certain degenerate elliptic PDE
- Evans
- 1982
(Show Context)
Citation Context ...d it becomes an interesting question of what regularity we can cull from this fact. Regularity for the Homogeneous p-Laplace equation In R n+1 it has been shown in the work of Uhlenbeck [U] and Evans =-=[E]-=-, and related work in [T], that a priori a solution to the (n + 1)-Laplace equation on real functions (with constant coefficients) on R n+1 must have C 1,α -regularity. In principle this is the best o... |

8 |
Théorie des intersections et théorème de
- Berthelot, Grothendieck, et al.
- 1971
(Show Context)
Citation Context ...erived category of perfect complexes of sheaves D perf (Mod-S X) is equivalent the derived category of perfect complexes of modules D perf (Mod-S X(X)). Proof. See Proposition 2.3.2, Exposé II, SGA6, =-=[SGA6]-=-. Theorem 1.4.5. The functor RH : PA → Loc C ∞(π∞M) is A∞-essentially surjective. Proof. By the Proposition , there is a (strictly) perfect complex (E • , E 0 ) of A-modules and quasi-isomorphism e 0 ... |

8 | Duality and intersection theory in complex manifolds. II. The holomorphic Lefschetz formula - Toledo, Tong - 1978 |

8 | Differential Analysis on
- Wells
- 1980
(Show Context)
Citation Context ...ecomes a ringed space. For an open subset U ⊂ M, let (CF (U), D) = (Loc C ∞(π∞U)(R|U, F |U), D). Let (CF , D) denote the associated complex of sheaves. Then CF is soft; see the proof of Theorem 3.15, =-=[W]-=-. By corollary 1.2.11, C F is a perfect complex of sheaves over R. Let A M denote the sheaf of C ∞ functions and (A • , d) denote the dg sheaf of C ∞ forms on M. Set C ∞ F = C F ⊗R A M. By the flatnes... |

7 |
Grothendieck-Riemann-Roch for complex manifolds
- O’Brian, Toledo, et al.
- 1981
(Show Context)
Citation Context ...plex (E • , E 0 ) of A-modules and quasi-isomorphism e 0 : (E • , E 0 ) → (X • , X 0 ) := (Γ(M, C ∞ F ), D). Following the argument of Theorem 3.2.7 of [Bl1], which in turn is based on arguments from =-=[OTT]-=-, we construct the higher components Ei of a Z-graded connection along with the higher components of a morphism ei at the same time. We have a Z-graded connection on X• by X := D ⊗ 1 + 1 ⊗ d : X • → X... |

6 | A parametrix for ∂ and Riemann-Roch in Čech theory. Topology 15 - Toledo, Tong - 1976 |

5 | Deformations of associative submanifolds with boundary, Adv
- Gayet, Witt
(Show Context)
Citation Context ... conditions is divided into components by energy levels. In the next section we describe normal deformations of such configurations of submanifolds. This issue has been addressed by Gayet and Witt in =-=[GW]-=-. Our study of this proceeded without this knowledge at first, but was broken off after finding their prior work. We include a somewhat loose discussion of these results noting that our definition of ... |

5 | theory, the Conley index and Floer - Salamon, Morse - 1990 |

4 |
Differential graded versus Simplicial categories. Available at arXiv:0711.3845
- Tabuada
(Show Context)
Citation Context ...icategorical picture agrees with another precedent in the literature of describing connective dg-categories (and their A∞-cousins) as models for linear ∞- categories. The model equivalence of Tabuada =-=[Ta]-=-, and Dugger-Shipley [DSh] (described below) lends weight to this precedent. In this paper we will not speculate about what should really model linear ∞-categories. Instead we will start with the foll... |

2 | Akbulut and Sema Salur, Calibrated manifolds and gauge theory - Selman - 2008 |

2 | Enhanced Triangulated Categories, Math USSR Sbornik, Vol 70 - Bondal, Kapranov - 1991 |

2 |
Twisting Cochains and Higher Torsion, arXiv: 0303047v1
- Igusa
(Show Context)
Citation Context ...ds. Later on we will refer to the ∞-groupoid of a space M. 1.2.2 Infinity Local Systems Now we develop an higher version of a local system. These objects will be almost the same as the A∞-functors of =-=[Ig02]-=-, but tailored to suit our equivalence result. We want to emphasize the analogy with classical local systems. It will also be clear that there is yet another perspective in which these could be regard... |

2 |
Stable Infinity Categories, arXiv: 0608228v5
- Lurie
(Show Context)
Citation Context ...-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 complete form in =-=[Lu2]-=-. Initially, by ∞-category one means an abstract higher categorical structure which possesses a set of objects and a set of k-morphisms for each natural number k. The morphisms in the k-th level can b... |

2 |
Liouville's theorem on conformal mappings under minimal regularity assumptions
- Reshetnyak
- 1967
(Show Context)
Citation Context ...ard G2-structure (coming from the spin structure on R8 , and incorporating the round metric [Lo]). These kinds of examples are properly situated in the G2-geometry sections. 71The work of Reshetnyak =-=[Re]-=- established the following strong rigidity result for solutions of the Cauchy-Riemann system in R n , Theorem 2.1.21. (Reshetnyak) Given u ∈ W 1,n (Ω, R n ) satisfying the Euclidean CRsystem, then u i... |

1 | Akbulut and Sema Salur, Deformations in G2 manifolds - Selman |

1 |
Akbulut and Sema Salur, Associative manifolds of a G2-manifold. arxiv: 0412032v3
- Selman
(Show Context)
Citation Context ...3-dimensional manifold inside a G2-manifold. Given such a Spin4-structure one can define S and E as above, realizing that in the case where the submanifold is associative, S becomes its spinor bundle =-=[AS3]-=-, [Br1]. On the boundary ν has additional interesting structures. Proposition 2.2.11. A boundary component Σ has an outward-facing normal ˆn which determines a complex structure on each boundary compo... |

1 | Griffiths,and Daniel Grossman, Exterior Differential Systems and Euler-Lagrange Partial Differential Systems - Bryant, Phillip - 2003 |