Model structures and quantum cohomology of higher orbifolds
aa r X i v : . [ m a t h . AG ] J u l Model Structures and Quantum Cohomology ofHigher Orbifolds
Jiajun DaiJuly 24, 2020
This is an informal essay, though propped with detailed explanations insome parts, recording some thoughts of mine mainly about model structureon higher orbifolds and its application in quantum cohomology of higherorbifolds in recent months, developing Chen-Ruan’s orbifold quantum coho-mology. The thoughts originated from two ideas:One is owing to the Taylor series expansion of the exponential map inRiemmanian geometry. Now that this formula which is so important incalculus can be transplanted to geometry, can it be generalized to othercontexts? What will happen if sum became direct sum, and power becamemultiple (direct, tensor or other) product?
Graded algebra like universal en-veloping algebra, tensor algebra and exterior algebra provides a suitable an-swer, and graded category fits in some sense. Extending it a bit, how aboutgraded geometry, graded topology, and so forth? Besides, can we define bi-graded, multi-graded, ∞ -graded, graded of graded? Moreover, as the gradedalgebra universal enveloping algebra is universal, does “graded” have anyconnection with “universal” or with somewhat related “completed”? If does,how? There is a rather simplified example: whether universal space EG ob-tained from principal G -bundle is graded or not? If so, what actually is thegraded stuff and how? HomotopyHypothesis which asserts that n-groupoidsare equivalent to homotopy n - types for all extended natural numbers n ∈ N casts an illuminating model. Furthermore, what if the modifier graded isweakened to be stratified, sliced or “locally structured” in other forms? ...The other one arose from a reflection on a mistake I made to take a point(0 , g , g , ..., g n ) where g , g , ..., g n are elements in topological group G inthe base space of a principal G -bundle as intrinsically possessing a structureisomorphic to G . The fact that the structure is encoded in the correspond-ing fiber reveals an approach to “resolving” singularities and “condensed1bjects” by endowing them with an extra structure attached to “fiber-like things”. Setting aside techniques for resolution in singularity theory, Y oneda lemma along with representation theory partly annotates it, while sheaf theory and (topological, algebraic, equivariant, ´etale, etc.) K - theory are known to be good supplements to fiber bundles. Therefore, it is naturalto expect an appropriate method to unify these theories together. What’smore, once feasible, will the unifying be possible to collide with the formeridea of the “graded”? For example, can “loop sheaf” connected with fun-damental group be generalized to “higher loop sheaf” associated to higherhomotopy group? Cohomology theory should be an easy found case thatcounts and deserves to be further studied under these assumptions. Afterall, homotopy theory and algebraic topology can be enlarged with a numberof new elements and be excavated for profounder connotation.Above thoughts, albeit superficial and naive without neither strict logicnor axiomatized semantics, propelled me to search for clues and hints insymposia, books, seminars, minicourses, and other accessible ways. As newingredients from a variety of theories add to them, some questions weresolved or partly solved, and new questions emerged, mixing and reactingwith those partly unsolved old ones. The process circles over and over tomake inspirations explode in my mind, driving me overwhelmed and lost inthese endless puzzles. Grothendieck’s theories which construct a marvelousempire with solid foundation of mathematical logic reorganized and enrichedmy unfettered thoughts well but don’t clear my confusions on some key is-sues. Thanks to the series of mini courses on Derived Algebraic Geometry(abbreviated as DAG ) which was given by Bertrand To¨en et cetera in Octo-ber, 2019, presenting to me concrete illustration to a number of the riddles.Consequently, I proceeded to learn
DAG , starting with preliminaries likehigher category theory, higher topos theory, and higher algebra, which con-stitute the logical basis of the renewed systematic theoretical frameworks.Before diving deeper into
DAG , I am going to set out to explore some topicson higher orbifolds in next section. Since it is not finished yet despite thata plenty of work has been done, only certain parts of axiomatized seman-tics will be interpreted. Meanwhile, definitions, lemmas, propositions andtheorems given or proved in bibliographies cited in this article would notbe repeated unless necessary. Notwithstanding the fact that
DAG theoriesinvolved in this essay are based on both Jacob Lurie’s, Bertrand To¨en andGabriele Vezzosi’s, for convenience, the terminology and notations involvedwill be borrowed from Jacob Lurie’s if there are ambiguities or discrepanciesbetween the two. 2
Model Structure on Higher Orbifolds
Orbifolds were known as singular spaces that are locally modelled on quo-tients of open subsets of R n by finite group actions. The orbifold structureencodes not only the structure of the underlying quotient space, but alsothat of the isotropy subgroups. Atlases are used to describe the orbifoldstructure. However, it is complicated and inconvenient to define the mor-phisms, not to mention the composition of morphisms, hindering categoricaloperations on orbifolds and further steps.I. Moerdijk and D. A. Pronk has shown in [MP97] that orbifolds areessentially the same as certain “proper” groupoids. E. Lerman mentionedin [Ler10] that a proper ´etale Lie groupoid is locally isomorphic to an finiteaction of groupoid. From then on, geometers are used to define orbifolds asproper ´etale Lie groupoids [Moe02]. What’s more, as groupoids can also besupposed to be atlases on orbifolds, there is a way of thinking of a groupoidas coordinates on a corresponding stack. In another word, orbifolds can alsobe seen as stacks.The concept of stack (i.e. 2- sheaf ) is the categorical analogue of sheaf,which is a generalization of principal bundle. Recall that a principal G -bundle can be defined as a collection of transition functions g ij : U ij → G on the double intersections U ij of some open covering { U i } i ∈ I of base man-ifold M , satisfying the cocycle condition g ij g jk = g ik compatible with thesingular cohomology group H ( M ; G ). These transition functions constructmorphisms of groupoids from the C`ech groupoid ` U ij ⇒ ` U i associated tothe open covering { U i } i ∈ I to the Lie groupoid G ⇒ ∗ . Meanwhile, a princi-pal G -bundle P over a manifold M canonically determines a homotopy classof maps from M to the classifying space BG of the group G . In effect, theset of isomorphism classes of G -principal bundles over M is in bijection withthe set of homotopy classes of maps M → BG ≡ K ( G, K ( G, G -torsor and H -torsor which is gen-erally called 2-morphisms, we can define bibundles [Bre10] ,2-groupoids and2-stacks satisfying certain descent condition. In particular, G -gerbes areequivalent to AU T ( G )- principal bundles , for AU T ( G ) the automorphism2-group of G [GS15]. Take G -bibundle as a prototype, we can construct asimilar classifying stack BBG ≡ K ( G,
2) associated with 2-C`ech coveringsatisfying 2-cocycle condition, which is compatible with H ( M ; G ). As for H n ( M ; G ), where n ≥ K ( G, n ) is a higher n -stack [Sim96] classifying“higher” pincipal G -bundles [Vez11], whose n -C`ech covering (essentially thesame with hyper-covering) nerve is n -hypergoupoid [Dus79]. Equivalently,3he singular cohomology groups H n ( M ; G ) of a nice topological space M with coefficients in an abelian group G (sheaf cohomology H sheaf ( M ; G )of M with coecients in the constant sheaf G associated to G for a gen-eral space M ) is actually a representable functor of M . That is, thereexists an Eilenberg-MacLane space K ( G, n ) and a universal cohomologyclass η ∈ H n ( K ( G, n ); G ) such that, for any nice topological space X , pull-back of η determines a bijection [ X ; K ( G, n )] → H n ( X ; G ) [Lur09a]. Here[ X ; K ( G, n )] denotes the set of homotopy classes of maps from X to K ( G, n ).Extending n to be ∞ , ∞ -groupoid can be defined to be an ∞ -category[Lur09a] (quasi-category by Joyal and weak Kan complex by Boardman andVogt) in higher category [Lur20] to be an abstract homotopical model fortopological spaces. Therefore, stacks over ∞ -groupoid generalize those overgroupoid, requiring a new topology corresponding to higher category, whichis formally named higher topos [Lur09a], represented by model topos.Back to orbifolds which have been previously mentioned to be proper´etale Lie groupoids. As Lie groupoids are essentially differentiable stacksup to Morita equivalence [BX06], it is reasonable to assume higher orbifoldsto be ´etale differentiable higher stacks. Start from higher ( n -)stack whichcan also be regarded as ( n -)truncated ∞ -stack (also denoted by ( ∞ ,n)-stack), or n -geometric D -stack in [BT08]. Since ∞ -stack (also known as( ∞ , Kan complexes (whichare fibrant-cofibrant objects), there exists a n -truncated geometry. We mayliterally transplant Jacob Lurie’s frameworks in [Lur09a] to obtain local andglobal model structures on ( ∞ , n )-stacks, i.e. n -orbifolds, although detailedmachinery is rather complicated. David Carchedi gave another descriptionin [Car15b] [Car15a] [Car13]. W. Chen and Y. Ruan have asserted in [CR00] [CR04] that an importantfeature of orbifold cohomology groups is degree shifting, i.e. shifting up thedegree of cohomology classes of X ( g ) by 2 ι ( g ) , defining the orbifold cohomol-ogy group of degree d to be the direct sum H dorb ( X ; Q ) = ⊕ ( g ) ∈ T H d − ι ( g ) (cid:0) X ( g ) ; Q (cid:1) , for any rational number d ∈ [0 , n ], where X is a closed almost complexorbifold with dim C X = n , X ( g ) = n(cid:0) p, ( g ) G p (cid:1) ∈ ˜ X | ( g ) G p ∈ ( g ) o called atwisted sector for ( g ) = (1), and ι ( g ) called degree shifting numbers. More-over, Y. Ruan explored twisted orbifold cohomology and its relation to dis-4rete torsion in [Rua00].Now that the geometry and topology behavior of higher orbifolds can bedetected in section 2, it should be possible to compute their quantum coho-mology but will obviously be a significant amount of work. However, whenit comes to derived orbifolds, things should be radically different. And whatwill derived twisted orbifolds look like? How would the twisted sector anddegree shifting numbers change? Will them get simpler compared with thegeneral case? Techniques are supposed to be totally different with existingliteratures’.I apologize to suspend it here because of time. References [Bre10] Lawrence Breen.
Notes on 1- and 2-Gerbes , pages 193–235.Springer New York, New York, NY, 2010.[BT08] G.V. Bertrand Ton.
Homotopical Algebraic Geometry II: Geomet-ric Stacks and Applications . American Mathematical Soc., 2008.[BX06] Kai Behrend and Ping Xu. Differentiable stacks and gerbes, 2006.[Car13] David Carchedi. Higher orbifolds and deligne-mumford stacks asstructured infinity topoi, 2013.[Car15a] David Carchedi. On the homotopy type of higher orbifolds andhaefliger classifying spaces, 2015.[Car15b] David Carchedi. On the tale homotopy type of higher stacks, 2015.[CR00] Weimin Chen and Yongbin Ruan. Orbifold quantum cohomology,2000.[CR04] Weimin Chen and Yongbin Ruan. A new cohomology theory oforbifold.
Communications in Mathematical Physics , 248(1):131,May 2004.[Dus79] John Williford Duskin. Higher dimensional torsors and the coho-mology of topoi : The abelian theory. 1979.[GS15] Gr´egory Ginot and Mathieu Sti´enon. G-gerbes, principal 2-groupbundles and characteristic classes.
Journal of Symplectic Geome-try , 13(4):1001–1048, January 2015.[Ler10] Eugene Lerman. Orbifolds as stacks?
Enseign. Math. (2) , 56(3-4):315–363, 2010. 5Lur09a] J. Lurie.
Higher Topos Theory (AM-170) . Annals of MathematicsStudies. Princeton University Press, 2009.[Lur09b] Jacob Lurie. Derived algebraic geometry v: Structured spaces,2009.[Lur20] Jacob Lurie. Kerodon. https://kerodon.net , 2020.[Moe02] Ieke Moerdijk. Orbifolds as groupoids: an introduction, 2002.[MP97] I. Moerdijk and Dorette Pronk. Orbifolds, sheaves and groupoids.