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Copyright by Rustam Darius Riedel 2019 the Dissertation Committee for Rustam Darius Riedel Certifies That This Is the Approved Version of the Following Dissertation Copyright by Rustam Darius Riedel 2019 The Dissertation Committee for Rustam Darius Riedel certifies that this is the approved version of the following dissertation: Free loop spaces, Koszul duality, and shifted Poisson geometry Committee: David Ben-Zvi, Supervisor Andrew Neitzke Andrew Blumberg David Nadler Free loop spaces, Koszul duality, and shifted Poisson geometry by Rustam Darius Riedel DISSERTATION Presented to the Faculty of the Graduate School of The University of Texas at Austin in Partial Fulfillment of the Requirements for the Degree of DOCTOR OF PHILOSOPHY THE UNIVERSITY OF TEXAS AT AUSTIN December 2019 To my grandfather Dr. Pirozshaw Dara Antia, for whom I was able to finish this. \Monks, the All is burning. Which All is burning? The eye is burning. [...] The ear [...] nose [...] tongue [...] body is burning. [...] The mind is burning. Ideas are burning. Mind-consciousness is burning. Mind-contact is burning. And whatever there is that arises in dependence on mind- contact { experienced as pleasure, pain or neither-pleasure-nor-pain { that too is burning. Burning with what? Burning with the fire of passion, the fire of aver- sion, the fire of delusion. Burning, I say, with birth, aging & death, with sorrows, lamentations, pains, distresses, & despairs. Seeing thus, the instructed disciple of the noble ones [...] grows disenchanted with the mind, disenchanted with ideas, disenchanted with mind-consciousness, dis- enchanted with mind-contact. And whatever there is that arises in dependence on mind-contact, experienced as pleasure, pain or neither-pleasure-nor-pain: With that, too, he grows disenchanted. [...]" Aditta-pariy¯aya¯ Sutta (Samyutta Nik¯aya 35:28) Acknowledgments I was fortunate in David Ben-Zvi accepting me as a student after my long odyssey of trying to find an advisor. Without his patience, kindness and continued support through difficult times, I would have never been able to finish my PhD. Mathematically, besides the huge debt I owe my advisor, I am particularly grateful to Sam Raskin, Rok Gregoriˇc,and Pavel Safronov for their help. Andy Neitzke's kindness and the sympathetic ear he lent me on various occasions meant a lot to me. I am also immensely grateful to my undergraduate advisor Annette Werner for her continued caring support and mentorship. On a more purely personal level I would like to in addition thank my par- ents, my sister, my extended family, and all of the following people. Whether they are more casual or close friends or are or were part of my life in other ways, their presence (or just my interactions with them) has (have) been meaningful and valu- able to me in different ways, and in some cases very important even though this might not have been evident to them: The Venerable Bhante Ananda,¯ Nihal Arju, Behnam Arzaghi, Tobias Baumann, Santiago Benavides, Gaurav Chaudhary, Adrian Clough, Abhranil Das, Arun Debray, the Derryberrys: Richard, Dakota and Za- havi, Stefan Eccles, Diney Ether, Iordan Ganev, Tom Gannon, Rok Gregoriˇcand Neˇza Zagerˇ Korenjak, Anne Lise Haenni, Elham Heidari, Andy Hutchison, Annika Jansson, Dan Kaplan, Pat Koch, Max Kurandt, Jonathan Lai, Manasvi Lingam, vi Tim Magee, Tom Mainiero, George Miloshevich, Vaibhav Murali, the Otto/Otto- Sch¨odels: Matthias Siddhartha, Anja, and Jonna, Sumati Panicker and Wolfgang Kaim, Rebecca P. (please find happiness!), Travis Schedler and Marie-Amelie Lawn, Ellie Schmidt, Isabelle Scott, Akarsh Simha, the Stratis/Serwe clan: Yorgos, Sarah, and Lola, Sivaramakrishnanan Swaminathan, the Venerable Ajaan Geoff (T. h¯anissaro Bhikkhu) for his writings and dhamma talks, Megan Deepti Thomas, Haoran Wang, Michael Wong, Yan Zhou, and Yaoguang Zhu, and anyone I may have inadvertently omitted writing this, as is my wont, at the last minute and while sleep deprived. vii Free loop spaces, Koszul duality, and shifted Poisson geometry by Rustam Darius Riedel, Ph.D. The University of Texas at Austin, 2019 Supervisor: David Ben-Zvi Let X be a smooth affine scheme, and let LX be its (derived) free loop space. We have a Koszul duality equivalence between the 1-category of ind-coherent sheaves on LX, and the 1-category of sheaves on (the total space of) the shifted cotangent bundle T ∗X[2] given by modules for the commutative (differential graded) • algebra π (O ∗ ) ∼ Sym (T [−2]) (π is the projection of the 2-shifted cotangent ∗ T X[2] = OX X space T ∗X[2] to the base X). The action by loop rotation of S1 considered as a homotopy type on the category of ind-coherent sheaves on LX induces an action on the Koszul dual 1-category. This thesis describes this action on the Koszul dual 1-category in terms of the shifted Poisson structure on T ∗X[2]: The main result is that the action is the exponential of the Poisson bracket. viii Table of Contents Acknowledgments vi Abstract viii Chapter 1. Introduction 1 1.1 Introduction . .1 1.2 Notation and conventions. 14 Chapter 2. Preliminaries 15 2.1 A few words on our use of ∞−Categories . 15 2.2 Derived Algebraic Geometry . 20 2.3 Quasi-coherent and Ind-coherent Sheaves . 24 2.4 Shifted Symplectic and Poisson Geometry . 29 2.5 (Algebraic) Families of ∞−Categories . 34 2.6 Group actions on ∞−Categories . 39 2.7 Hochschild Homology and Cohomology . 48 2.8 Formal moduli problems . 54 2.9 Koszul duality . 59 2.10 Free derived loop spaces . 63 Chapter 3. The main theorem 67 Bibliography 69 Vita 85 ix Chapter 1 Introduction 1.1 Introduction Let X be a smooth scheme, and let LX be its (derived) free loop space (see below for an explanation of what this is). Using formality of cochains on the circle S1, we can identify LX with (the total space of) the (−1)-shifted tangent bundle TX[−1]. We have a Koszul duality equivalence between certain homotopical categories of sheaves on TX[−1] and on the Koszul dual space given by (the total space of) the shifted cotangent bundle T ∗X[2]. The S1-action by loop rotation on the category of sheaves on LX induces an action on the Koszul dual category of sheaves on T ∗X[2]. This thesis describes this action on the Koszul dual category in terms of the shifted Poisson structure on T ∗X[2]. 1.1.1 Free loop spaces The free loop space LX of a topological space X is the space of continuous maps from a circle S1 into X. Free loop spaces are of fundamental interest in topology and geometry. Let us mention some examples: In topology its rich structure is studied in string topology (see [35, 37, 38, 26]). In geometry one considers for X a smooth manifold only suitably regular maps from S1. The topology of LX is then of fundamental importance when studying the existence of closed geodesics on 1 Riemannian manifolds ([95]). Free loop spaces LG of compact Lie groups X = G are known as loop groups ([99]), and are the simplest example of infinite dimensional Lie groups. Loop groups (and their central extensions) have been studied both in geometry and representation theory, for example in the algebraic guise as mapping spaces out of formal disks through their connections to the affine Grassmannian and moduli of G-bundles on a curve (see e.g. [138] for an expository account), or through the theory of affine Kac-Moody algebras/groups ([64]). In symplectic geometry Viterbo identifies symplectic cohomology (a version of Hamiltonian Floer theory) of the cotangent space T ∗X with the homology of the free loop space ([130, 131, 2, 109, 3]). The circle acts on LX by loop rotation in such a way that the fixed point locus is X. In [132, 133] (see also [8] for an expository account) Witten showed that for a Spin manifold X equivariant localization to the fixed point locus in equivariant cohomology and K-theory yields the index of the Dirac operator and the elliptic genus (which is related to elliptic cohomology) of X, thereby connecting the study of LX to physics, geometry, and homotopy theory. A version of the free loop space in algebraic geometry was constructed in [124, 126] as a recipient of a Chern character for categorical sheaves [125]. This was further studied in [20] and applied to geometric representation theory in [19]. It is this algebro-geometrical version of the loop space, called the derived free loop space and its Koszul dual that we will consider in this thesis. 2 1.1.2 Derived free loop space Let us try to make sense of the free loop space LX of a smooth scheme X. We will consider S1 as a homotopy type, and think of it as two points connected by two arcs (i.e. we are given two essentially different identifications of the two points). Mapping the two points into X we have are free to choose any pair of two points x; y in X, that is the mapping space from the two points is X × X. The arcs tell us that for any pair (x; y) we want to identify x with y twice (once for each arc). Thus, the free loop space is given by imposing the equation x = y in X × X twice, i.e. it is the ∼ self intersection of the diagonal LX = X ×X×X X. This intersection is highly degenerate, and in the usual classical algebraic geometry setting just gives us back X. To make better sense of it one needs to embed the world of algebraic geometry into the more homotopical world of derived algebraic geometry. One of the major motivations for the development of derived algebra is to make sense of such highly degenerate intersections. The elementary building blocks of classical algebraic geometry are algebras, i.e.
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