Research papers

Published and accepted for publication

1. (3+1)D topological orders with only a Z₂-charged particle. Commun. Contemp. Math., 2024. (abstract, arXiv:2011.11165.)

   Abstract: There is exactly one bosonic (3+1)-dimensional topological order whose only nontrivial particle is an emergent boson: pure Z₂ gauge theory. There are exactly two (3+1)-dimensional topological orders whose only nontrivial particle is an emergent fermion: pure "spin-Z₂" gauge theory, in which the dynamical field is a spin structure; and an anomalous version thereof. I give three proofs of this classification, varying from hands-on to abstract. Along the way, I provide a detailed study of the braided fusion 2-category Z₍₁₎(ΣSVec) of string and particle operators in pure spin-Z₂ gauge theory. (hide abstract)

2. Mock modularity and a secondary elliptic genus. With Davide Gaiotto. Journal of High Energy Physics, 2023, 94. (abstract, arXiv:1904.05788. DOI:10.1007/JHEP08(2023)094)

   Abstract: The theory of Topological Modular Forms suggests the existence of deformation invariants for two-dimensional supersymmetric field theories that are more refined than the standard elliptic genus. In this note we give a physical definition of some of these invariants. The theory of mock modular forms makes a surprise appearance, shedding light on the integrality properties of some well-known examples. (hide abstract)

3. Minimal nondegenerate extensions. With David Reutter. Journal of the American Mathematical Society, Volume 37, Number 1, January 2024, Pages 81–150. (abstract, arXiv:2105.15167, DOI:10.1090/jams/1023.)

   Abstract: We prove that every slightly degenerate braided fusion category admits a minimal nondegenerate extension. As a corollary, every pseudounitary super modular tensor category admits a minimal modular extension. This completes the program of characterizing minimal nondegenerate extensions of braided fusion categories.
   
   Our proof relies on the new subject of fusion 2-categories. We study in detail the Drinfel'd centre Z(Mod-B) of the fusion 2-category Mod-B of module categories of a braided fusion 1-category B. We show that minimal nondegenerate extensions of B correspond to certain trivializations of Z(Mod-B). In the slightly degenerate case, such trivializations are obstructed by a class in H⁵(K(Z₂, 2); k^×) and we use a numerical invariant — defined by evaluating a certain two-dimensional topological field theory on a Klein bottle — to prove that this obstruction always vanishes.
   
   Along the way, we develop techniques to explicitly compute in braided fusion 2-categories which we expect will be of independent interest. In addition to the known model of Z(Mod-B) in terms of braided B-module categories, we introduce a new computationally useful model in terms of certain algebra objects in B. We construct an S-matrix pairing for any braided fusion 2-category, and show that it is nondegenerate for Z(Mod-B). As a corollary, we identify components of Z(Mod-B) with blocks in the annular category of B and with the homomorphisms from the Grothendieck ring of the Müger centre of B to the ground field. (hide abstract)

4. Ground-state degeneracy of twisted sectors of Conway Moonshine SCFT. With Alissa Furet. Commun. Contemp. Math., 2024. (abstract, arXiv:2305.05081.)

   Abstract: We calculate the ground state degeneracies of all twisted sectors in the “Conway Moonshine” holomorphic SCFT V^f♮. We find that almost all sectors have ground states of only a single parity: specifically, 66 twisted sectors have nontrivial ground states of a single parity, 39 twisted sectors have spontaneous supersymmetry breaking, and only 6 twisted sectors have ground states of both parities. Although “nontrivial ground states, all of the same parity” is the expected behavior for a generic SQM model without symmetry protection, it is surprising in the presence of a large symmetry group, as is the case in V^f♮. This surprise hints that there are as-yet-undiscovered features of Conway Moonshine. (hide abstract)

5. Topological Orders in (4+1)-Dimensions. With Matthew Yu. SciPost Physics, 13, 068 (2022). (abstract, arXiv:2104.04534, DOI:10.21468/SciPostPhys.13.3.068.)

   Abstract: We investigate the Morita equivalences of (4+1)-dimensional topological orders. We show that any (4+1)-dimensional super (fermionic) topological order admits a gapped boundary condition -- in other words, all (4+1)-dimensional super topological orders are Morita trivial. As a result, there are no inherently gapless super (3+1)-dimensional theories. On the other hand, we show that there are infinitely many algebraically Morita-inequivalent bosonic (4+1)-dimensional topological orders. (hide abstract)

6. On the classification of topological orders. Communications in Mathematical Physics, 393, pp 989–1033 (2022). (abstract, arXiv:2003.06663, DOI:10.1007/s00220-022-04380-3, full-text published version.)

   Abstract: We axiomatize the extended operators in topological orders (possibly gravitationally anomalous, possibly with degenerate ground states) in terms of monoidal Karoubi-complete n-categories which are mildly dualizable and have trivial centre. Dualizability encodes the word "topological," and we take it as the definition of "multifusion n-category"; triviality of the centre implements the physical principle of "remote detectability." We show that such n-categorical algebras are Morita-invertible (in the appropriate higher Morita category), thereby identifying topological orders with anomalous fully-extended TQFTs. We identify centreless fusion n-categories (i.e. multifusion n-categories with indecomposable unit) with centreless braided fusion (n-1)-categories. We then discuss the classification in low spacetime dimension, proving in particular that all (1+1)- and (3+1)-dimensional topological orders, with arbitrary symmetry enhancement, are suitably-generalized topological sigma models. These mathematical results confirm and extend a series of conjectures and proposals by X.G. Wen et al. (hide abstract)

7. Fusion 2-categories with no line operators are grouplike. With Matthew Yu. Bulletin of the Australian Mathematical Society, vol 104, issue 3, pp 434–442, December 2021. (abstract, arXiv:2010.07950, DOI:10.1017/S0004972721000095.)

   Abstract: We show that if C is a fusion 2-category in which the endomorphism category of the unit object is Vec or SVec, then the indecomposable objects of C form a finite group. (hide abstract)

8. Holomorphic SCFTs with small index. With Davide Gaiotto. Canadian Journal of Mathematics, 2021:1-29. (abstract, arXiv:1811.00589. DOI:10.4153/S0008414X2100002X.)

   Abstract: We observe that every self-dual ternary code determines a holomorphic N=1 superconformal field theory. This provides ternary constructions of some well-known holomorphic N=1 SCFTs, including Duncan's "supermoonshine" model and the fermionic "beauty and the beast" model of Dixon, Ginsparg, and Harvey. Along the way, we clarify some issues related to orbifolds of fermionic holomorphic CFTs. We give a simple coding-theoretic description of the supersymmetric index and conjecture that for every self-dual ternary code this index is divisible by 24; we are able to prove this conjecture except in the case when the code has length 12 mod 24. Lastly, we discuss a conjecture of Stolz and Teichner relating N=1 SCFTs with Topological Modular Forms. This conjecture implies constraints on the supersymmetric indexes of arbitrary holomorphic SCFTs, and suggests (but does not require) that there should be, for each k, a holomorphic N=1 SCFT of central charge 12k and index 24/gcd(k,24). We give ternary code constructions of SCFTs realizing this suggestion for k \leq 5. (hide abstract)

9. Supersymmetry and the Suzuki chain. Tunisian Journal of Mathematics, Vol 3, No 2, pp 309–359, 2021. (abstract, arXiv:1908.11012, DOI:10.2140/tunis.2021.3.309.)

   Abstract: We classify N=1 SVOAs with no free fermions and with bosonic subalgebra a simply connected WZW algebra which is not of type E. The latter restriction makes the classification tractable; the former restriction implies that the N=1 automorphism groups of the resulting SVOAs are finite. We discover two infinite families and nine exceptional examples. The exceptions are all related to the Leech lattice: their automorphism groups are the larger groups in the Suzuki chain (Co₁, Suz:2, G₂(4):2, J₂:2, U₃(3):2) and certain large centralizers therein (2¹⁰:M₁₂:2, M₁₂:2, U₄(3):D₈, M₂₁:2²). (hide abstract)

10. A note on some minimally supersymmetric models in two dimensions. With Davide Gaiotto and Edward Witten. In Integrability, Quantization, and Geometry II. Quantum Theories and Algebraic Geometry: Dedicated to the Memory of Boris Dubrovin 1950–2019, volume 103.2 of Proceedings of Symposia in Pure Mathematics, pp 203–222, Amer. Math. Soc., Providence, RI, 2021. (abstract, arXiv:1902.10249, AMS Bookstore.)

    Abstract: We explore the dynamics of a simple class of two-dimensional models with (0,1) supersymmetry, namely sigma-models with target S³ and the minimal possible set of fields. For any nonzero value of the Wess–Zumino coupling k, we describe a superconformal fixed point to which we conjecture that the model flows in the infrared. For k=0, we conjecture that the model spontaneously breaks supersymmetry. We further explore the question of whether this model can be continuously connected to one that spontaneously breaks supersymmetry by "flowing up and down the renormalization group trajectories," in a sense that we describe. We show that this is possible if k is a multiple of 24, or equivalently if the target space with its B-field is the boundary of a "string manifold." The mathematical theory of "topological modular forms" suggests that this condition is necessary as well as sufficient. (hide abstract)

11. Galois action on VOA gauge anomalies. Progress in Mathematics volume in honour of Kolya Reshetikhin, 2020. (abstract, arXiv:1811.06495.)

    Abstract: Assuming regularity of the fixed subalgebra, any action of a finite group G on a holomorphic VOA V determines a gauge anomaly α∈H³(G;μ), where μ⊂C^× is the group of roots of unity. We show that under Galois conjugation V↦^γV, the gauge anomaly transforms as α↦γ²(α). This provides an a priori upper bound of 24 on the order of anomalies of actions preserving a Q-structure, for example the Monster group M acting on its Moonshine VOA V^♮. We speculate that each field K should have a "vertex Brauer group" isomorphic to H³(Gal(‾K);μ^⊗2). In order to motivate our constructions and speculations, we warm up with a discussion of the ordinary Brauer group, emphasizing the analogy between VOA gauging and quantum Hamiltonian reduction. (hide abstract)

12. Heisenberg-picture quantum field theory. Progress in Mathematics volume in honour of Kolya Reshetikhin, 2020. (abstract, arXiv:1508.05908.)

    Abstract: This paper discusses what we should mean by "Heisenberg-picture quantum field theory." Atiyah–Segal-type axioms do a good job of capturing the "Schrödinger picture": these axioms define a "d-dimensional quantum field theory" to be a symmetric monoidal functor from an (∞,d)-category of "spacetimes" to an (∞,d)-category which at the second-from-top level consists of vector spaces, so at the top level consists of numbers. This paper argues that the appropriate parallel notion "Heisenberg picture" should also be defined in terms of symmetric monoidal functors from the category of spacetimes, but the target should be an (∞,d)-category that in top dimension consists of pointed vector spaces instead of numbers; the second-from-top level can be taken to consist of associative algebras or of pointed categories. The paper ends by outlining two sources of such Heisenberg-picture field theories: factorization algebras and skein theory. (hide abstract)

13. Third homology of some sporadic finite groups. With David Treumann. Symmetry, Integrability and Geometry: Methods and Applications 15 (2019), 059. (abstract, arXiv:1810.00463, DOI: 10.3842/SIGMA.2019.059.)

    Abstract: We compute the integral third homology of most of the sporadic finite simple groups and of their central extensions. (hide abstract)

14. Symmetry protected topological phases and generalized cohomology. With Davide Gaiotto. Journal of High Energy Physics. May 2019. (abstract, arXiv:1712.07950, DOI: 10.1007/JHEP05(2019)007.)

    Abstract: We discuss the classification of SPT phases in condensed matter systems. We review Kitaev's argument that SPT phases are classified by a generalized cohomology theory, valued in the spectrum of gapped physical systems. We propose a concrete description of that spectrum and of the corresponding cohomology theory. We compare our proposal to pre-existing constructions in the literature. (hide abstract)

15. The Moonshine Anomaly. Communications in Mathematical Physics. February 2019, Volume 365, Issue 3, pp 943–970. (abstract. DOI: 10.1007/s00220-019-03300-2. Published PDF available at https://rdcu.be/bjHMt. arXiv:1707.08388.)

    Abstract: By definition, the anomaly for the Monster group M acting on its natural (aka moonshine) representation V^♮ is a particular cohomology class ω^♮ \in H^3(M,U(1)) that arises as a conformal field theoretic generalization of the second Chern class of a representation. We show in this paper that ω^♮ has order exactly 24 and is not a Chern class. In order to perform this computation, we introduce a finite-group version of T-duality, which we use to relate ω^♮ to the anomaly for the Leech lattice CFT. (hide abstract)

16. H⁴(Co₀;Z)=Z/24. With David Treumann. International Mathematics Research Notices, 2020, no. 21, 7873–7907. (abstract. DOI: 10.1093/imrn/rny219. arXiv:1707.07587.)

    Abstract: We show that the fourth integral cohomology of Conway's group Co₀ is a cyclic group of order 24, generated by the first fractional Pontryagin class of the 24-dimensional representation. (hide abstract)

17. How to derive Feynman diagrams for finite-dimensional integrals directly from the BV formalism. With Owen Gwilliam. Topology and quantum theory in interaction, 175–185, Contemp. Math., 718, Amer. Math. Soc., Providence, RI, 2018. (abstract. AMS bookstore. arXiv:1202.1554.)

    Abstract: The Batalin-Vilkovisky formalism in quantum field theory was originally invented to avoid the difficult problem of finding diagrammatic descriptions of oscillating integrals with degenerate critical points. But since then, BV algebras have become interesting objects of study in their own right, and mathematicians sometimes have good understanding of the homological aspects of the story without any access to the diagrammatics. In this note we reverse the usual direction of argument: we begin by asking for an explicit calculation of the homology of a BV algebra, and from it derive WickÕs Theorem and the other Feynman rules for finite-dimensional integrals. (hide abstract)

18. Spin, statistics, orientations, unitarity. Algebraic & Geometric Topology 17 (2017) 917–956. (abstract, arXiv:1507.06297, DOI: 10.2140/agt.2017.17.917.)

    Abstract: A topological quantum field theory is Hermitian if it is both oriented and complex-valued, and orientation-reversal agrees with complex-conjugation. A field theory satisfies spin-statistics if it is both spin and super, and 360 degree rotation of the spin structure agrees with the operation of flipping the signs of all fermions. We set up a framework in which these two notions are precisely analogous. In this framework, field theories are defined over Vect_R, but rather than being defined in terms of a single tangential structure, they are defined in terms of a bundle of tangential structures over Spec(R). Bundles of tangential structures may be étale-locally equivalent without being equivalent, and Hermitian field theories are nothing but the field theories controlled by the unique nontrivial bundle of tangential structures that is étale-locally equivalent to Orientations. This bundle owes its existence to the fact that π₁^étSpec(R) = π₁BO(∞). We interpret Deligne's "existence of super fiber functors" theorem as implying that in a categorification of algebraic geometry in which symmetric monoidal categories replace commutative rings, π₂^étSpec(R) = π₂BO(∞). One finds that there are eight bundles of tangential structures étale-locally equivalent to Spins, one of which is distinguished; upon unpacking the meaning of a field theory with that distinguished tangential structure, one arrives at a field theory that is both Hermitian and satisfies spin-statistics. Finally, we formulate in our framework a notion of "reflection-positivity" and prove that if an "étale-locally-oriented" field theory is reflection-positive then it is necessarily Hermitian, and if an "étale-locally-spin" field theory is reflection-positive then it necessarily both satisfies spin-statistics and is Hermitian. The latter result is a topological version of the famous Spin-Statistics Theorem. (hide abstract)

19. (Op)lax natural transformations, twisted field theories, and the "even higher" Morita categories. With Claudia Scheimbauer. Advances in Mathematics, 307 (2017) 147–223. (abstract, arXiv:1502.06526, DOI: 10.1016/j.aim.2016.11.014.)

    Abstract: Motivated by the challenge of defining twisted quantum field theories in the context of higher categories, we develop a general framework for lax and oplax transformations and their higher analogs between strong (∞, n)-functors. We construct a double (∞, n)-category built out of the target (∞, n)-category governing the desired diagrammatics. We define (op)lax transformations as functors into parts thereof, and an (op)lax twisted field theory to be a symmetric monoidal (op)lax natural transformation between field theories. We verify that lax trivially-twisted relative field theories are the same as absolute field theories. As a second application, we extend the higher Morita category of E_d-algebras in a symmetric monoidal (∞, n)-category C to an (∞, n+d)-category using the higher morphisms in C. (hide abstract)

20. The quaternions and Bott periodicity are quantum Hamiltonian reductions. Symmetry, Integrability and Geometry: Methods and Applications, 12 (2016), 116, 6 pages. (abstract, arXiv:1603.06603, DOI: 10.3842/SIGMA.2016.116.)

    Abstract: We show that the Morita equivalences Cliff(4) ≈ H, Cliff(7) ≈ Cliff(−1), and Cliff(8) ≈ R arise from quantizing the Hamiltonian reductions R^0|4//Spin(3), R^0|7//G₂, and R^0|8//Spin(7), respectively. (hide abstract)

21. Tree- versus graph-level quasilocal Poincaré duality on S¹. Journal of homotopy and related structures, June 2016, Volume 11, Issue 2, pp 333–374. (abstract, arXiv:1412.4664, DOI: 10.1007/s40062-015-0110-2.)

    Abstract: Among its many corollaries, Poincaré duality implies that the de Rham cohomology of a compact oriented manifold is a commutative Frobenius algebra. Focusing on the case of S¹, this paper studies the question of whether this commutative Frobenius algebra structure lifts to a "homotopy" commutative Frobenius algebra structure at the cochain level, under a mild locality-type condition called "quasilocality". The answer turns out to depend on the choice of context in which to do homotopy algebra — there are two reasonable worlds in which to study structures (like Frobenius algebras) that involve many-to-many operations. If one works at "tree level", we prove that there is a homotopically-unique quasilocal cochain-level homotopy Frobenius algebra structure lifting the Frobenius algebra structure on cohomology. However, if one works instead at "graph level", we prove that a quasilocal lift does not exist. (hide abstract)

22. Homological perturbation theory for nonperturbative integrals. Letters in Mathematical Physics, November 2015, Volume 105, Issue 11, pp 1605–1632. (abstract, arXiv:1206.5319, DOI: 10.1007/s11005-015-0791-9.)

    Abstract: In this paper we study integrals of the form ∫_γ f e^s, where f and s are complex polynomials of n variables and γ ⊆ Cⁿ is an n-real-dimensional contour along which e^s enjoys exponential decay. Suppose s is generic of degree d. Using homological algebra, we automate the method of ``integration by parts,'' and show how to express any such integral as a linear combination of integrals of monomials which are of degree < d-1 in each variable. We conjecture that for generic contour γ the values of these (d-1)ⁿ integrals are inaccessible to pure algebra.
    
    More generally, we explain how homological algebra allows to ``integrate out the high-energy modes'' to turn any such integral problem into an integral over the scheme-theoretic critical locus {d s = 0}. Thus concentration onto the critical locus is not only a perturbative phenomenon. Our primary tool in this paper is the Homological Perturbation Lemma, which when applied to perturbative integrals recovers the method of Feynman diagrams --- ``perturbation theory'' is another not-only-perturbative phenomenon. Our motivation for this paper is to better understand the ``path'' integrals that appear in quantum field theory, and we make a few brief comments about these at the end. (hide abstract)

23. Reflexivity and dualizability in categorified linear algebra. With Martin Brandenburg and Alexandru Chirvasitu. Theory and Applications of Categories, Vol. 30, No. 23, 2015, pp. 808–835. (abstract, arXiv:1409.5934, published version (open access).)

    Abstract: The "linear dual" of a cocomplete linear category C is the category of all cocontinuous linear functors C→VECT. We study the questions of when a cocomplete linear category is reflexive (equivalent to its double dual) or dualizable (the pairing with its dual comes with a corresponding copairing). Our main results are that the category of comodules for a countable-dimensional coassociative coalgebra is always reflexive, but (without any dimension hypothesis) dualizable if and only if it has enough projectives, which rarely happens. Along the way, we prove that the category Qcho(X)$ of quasi-coherent sheaves on a stack X is not dualizable if X is the classifying stack of a semisimple algebraic group in positive characteristic or if X is a scheme containing a closed projective subscheme of positive dimension, but is dualizable if X is the quotient of an affine scheme by a virtually linearly reductive group. Finally we prove tensoriality (a type of Tannakian duality) for affine ind-schemes with countable indexing poset. (hide abstract)

24. Poisson AKSZ theories and their quantizations. In Proceedings of the conference String-Math 2013, volume 88 of Proceedings of Symposia in Pure Mathematics, pages 291–306, Providence, RI, 2014. Amer. Math. Soc. (abstract, PDF (published version), arXiv:1307.5812, DOI: 10.1090/pspum/088.)

    Abstract: We generalize the AKSZ construction of topological field theories to allow the target manifolds to have possibly-degenerate (homotopy) Poisson structures. Classical AKSZ theories, which exist for all oriented spacetimes, are described in terms of dioperads. The quantization problem is posed in terms of extending from dioperads to properads. We conclude by relating the quantization problem for AKSZ theories on R^d to the formality of the E_d operad, and conjecture a properadic description of the space of E_d formality quasiisomorphisms. (hide abstract)

25. The fundamental pro-groupoid of an affine 2-scheme. With Alex Chirvasitu. Applied Categorical Structures, Vol 21, Issue 5 (2013), pp. 469–522. (abstract, arXiv:1105.3104, DOI: 10.1007/s10485-011-9275-y).

    Abstract: A natural question in the theory of Tannakian categories is: What if you don't remember Forget? Working over an arbitrary commutative ring R, we prove that an answer to this question is given by the functor represented by the étale fundamental groupoid π₁(spec(R))$, i.e. the separable absolute Galois group of R when it is a field. This gives a new definition for étale π₁(spec(R))$ in terms of the category of R-modules rather than the category of étale covers. More generally, we introduce a new notion of "commutative 2-ring" that includes both Grothendieck topoi and symmetric monoidal categories of modules, and define a notion of π₁ for the corresponding "affine 2-schemes." These results help to simplify and clarify some of the peculiarities of the étale fundamental group. For example, étale fundamental groups are not "true" groups but only profinite groups, and one cannot hope to recover more: the "Tannakian" functor represented by the étale fundamental group of a scheme preserves finite products but not all products. (hide abstract)

26. The formal path integral and quantum mechanics. Journal of Mathematical Physics, 51, 122103 (2010). (abstract, published PDF, DOI:10.1063/1.3503472, arXiv:1004.4305, equation and theorem numbering differs between preprint and published versions).

    Abstract: Given an arbitrary Lagrangian function on ℝ^d and a choice of classical path, one can try to define Feynman's path integral supported near the classical path as a formal power series parameterized by "Feynman diagrams," although these diagrams may diverge. We compute this expansion and show that it is (formally, if there are ultraviolet divergences) invariant under volume-preserving changes of coordinates. We prove that if the ultraviolet divergences cancel at each order, then our formal path integral satisfies a "Fubini theorem" expressing the standard composition law for the time evolution operator in quantum mechanics. Moreover, we show that when the Lagrangian is inhomogeneous-quadratic in velocity such that its homogeneous-quadratic part is given by a matrix with constant determinant, then the divergences cancel at each order. Thus, by "cutting and pasting" and choosing volume-compatible local coordinates, our construction defines a Feynman-diagrammatic "formal path integral" for the nonrelativistic quantum mechanics of a charged particle moving in a Riemannian manifold with an external electromagnetic field. (hide abstract)

27. Feynman-diagrammatic description of the asymptotics of the time evolution operator in quantum mechanics. Letters in Mathematical Physics, November 2010, Volume 94, Issue 2, pp 123–149. (abstract, arXiv:1003.1156, available Open Access from Springer Link at DOI: 10.1007/s11005-010-0424-2).

    Abstract: We describe the "Feynman diagram" approach to nonrelativistic quantum mechanics on ℝⁿ, with magnetic and potential terms. In particular, for each classical path γ connecting points q₀ and q₁ in time t, we define a formal power series V_γ(t, q₀, q₁) in Planck's constant h, given combinatorially by a sum of diagrams that each represent finite-dimensional convergent integrals. We prove that exp(V_γ) satisfies Schrödinger's equation, and explain in what sense the t → 0 limit approaches the δ distribution. As such, our construction gives explicitly the full h → 0 asymptotics of the fundamental solution to Schrödinger's equation in terms of solutions to the corresponding classical system. These results justify the heuristic expansion of Feynman's path integral in diagrams. (hide abstract)

Submitted for Publication

• Topological Mathieu Moonshine. 2020. (abstract, arXiv:2006.02922.)

  Abstract: We explore the Atiyah–Hirzebruch spectral sequence for the tmf^•[1/2]-cohomology of the classifying space BM₂₄ of the largest Mathieu group M₂₄, twisted by a class ω ∈ H⁴(BM₂₄; Z[1/2]) = Z₃. Our exploration includes detailed computations of the F₃-cohomology of M₂₄ and of the first few differentials in the AHSS. We are specifically interested in the value of tmf^•_ω(BM₂₄)[1/2] in cohomological degree -27. Our main computational result is that tmf^-27_ω(BM₂₄)[1/2] = 0 when ω ≠ 0. For comparison, the restriction map tmf^-3_ω(BM₂₄)[1/2] → tmf^-3(pt)[1/2] = Z₃ is nonzero for one of the two nonzero values of ω. Our motivation comes from Mathieu Moonshine. Assuming a well-studied conjectural relationship between TMF and supersymmetric quantum field theory, there is a canonically-defined Co₁-twisted-equivariant lifting [V^f♮] of the class {24Δ} ∈ TMF^-24(pt), for a specific value ω of the twisting, where Co₁ denotes Conway's largest sporadic group. We conjecture that the product [V^f♮]ν, where ν ∈ TMF^-3(pt) is the image of the generator of tmf^-3(pt) = Z₂₄, does not vanish Co₁-equivariantly, but that its restriction to M₂₄-twisted-equivariant TMF does vanish. We explain why this conjecture answers some of the questions in Mathieu Moonshine: it implies the existence of a minimally supersymmetric quantum field theory with M₂₄ symmetry, whose twisted-and-twined partition functions have the same mock modularity as in Mathieu Moonshine. Our AHSS calculation establishes this conjecture "perturbatively" at odd primes. An appendix included mostly for entertainment purposes discusses "l-complexes," in which the differential D satisfies D^l=0 rather than D²=0, and their relation to SU(2) Verlinde rings. The case l=3 is used in our AHSS calculations. (hide abstract)

• Condensations in higher categories. With Davide Gaiotto. 2019. (abstract, arXiv:1905.09566.)

  Abstract: We present a higher-categorical generalization of the "Karoubi envelope" construction from ordinary category theory, and prove that, like the ordinary Karoubi envelope, our higher Karoubi envelope is the closure for absolute limits. Our construction replaces the idempotents in the ordinary version with a notion that we call "condensations." The name is justified by the direct physical interpretation of the notion of condensation: it encodes a general class of constructions which produce a new topological phase of matter by turning on a commuting projector Hamiltonian on a lattice of defects within a different topological phase, which may be the trivial phase. We also identify our higher Karoubi envelopes with categories of fully-dualizable objects. Together with the Cobordism Hypothesis, we argue that this realizes an equivalence between a very broad class of gapped topological phases of matter and fully extended topological field theories, in any number of dimensions. (hide abstract)

Other Preprints

• Exact triangles, Koszul duality, and coisotopic boundary conditions. 2016. (abstract, arXiv:1608.08598.)

  Abstract: We develop a theory of "arrowed" (operads and) dioperads, which are to exact triangles as dioperads are to vector spaces. A central example to this paper is the arrowed operad controlling "derived ideals" for any operad. The Koszul duality theory of arrowed dioperads interacts well with rotation of exact triangles, and in particular with "exact Stars of David," which are pairs of exact triangles drawn on top of each other in an interesting way. Using this framework, we give a cochain-level lift of the "relative Poincar\'e duality" enjoyed by oriented manifolds with boundary; moreover, our cochain-level lift satisfies a natural locality-type condition, and is uniquely determined by this property. We discuss the meaning of the words "relative orientation" and "coisotropic." We extend the AKSZ construction to bulk-boundary settings with Poisson bulk fields and coisotropic boundary conditions. (hide abstract)

• Chains(R) does not admit a geometrically meaningful properadic homotopy Frobenius algebra structure. 2013. (abstract, arXiv:1308.3423.) Published as Tree- versus graph-level quasilocal Poincaré duality on S¹

  Abstract: The embedding of Chains(R) into Cochains(R) as the compactly supported cochains might lead one to expect Chains(R) to carry a nonunital commutative Frobenius algebra structure, up to a degree shift and some homotopic weakening of the axioms. We prove that under reasonable "locality" conditions, a cofibrant resolution of the dioperad controlling nonunital shifted-Frobenius algebras does act on Chains(R), and in a homotopically-unique way. But we prove that this action does not extend to a homotopy Frobenius action at the level of properads or props. This gives an example of a geometrically meaningful algebraic structure on homology that does not lift in a geometrically meaningful way to the chain level.
  
  Note: An expanded retelling of the story in this paper, with different conventions and clearer results, is available in "Tree- versus graph-level quasilocal Poincaré duality on S¹." (hide abstract)

• Peturbative techniques in path integration. Ph.D. Thesis. 2013. (abstract, PDF.) Published as: Feynman-diagrammatic description of the asymptotics of the time evolution operator in quantum mechanics, The formal path integral and quantum mechanics

  Abstract: This dissertation addresses a number of related questions concerning perturbative "path" integrals. Perturbative methods are one of the few successful ways physicists have worked with (or even defined) these infinite-dimensional integrals, and it is important as mathematicians to check that they are correct. Chapter 0 provides a detailed introduction.
  
  We take a classical approach to path integrals in Chapter 1. Following standard arguments, we posit a Feynman-diagrammatic description of the asymptotics of the time-evolution operator for the quantum mechanics of a charged particle moving nonrelativistically through a curved manifold under the influence of an external electromagnetic field. We check that our sum of Feynman diagrams has all desired properties: it is coordinate-independent and well-defined without ultraviolet divergences, it satisfies the correct composition law, and it satisfies Schrödinger's equation thought of as a boundary-value problem in PDE.
  
  Path integrals in quantum mechanics and elsewhere in quantum field theory are almost always of the shape ∫ f e^s for some functions f (the "observable") and s (the "action"). In Chapter 2 we step back to analyze integrals of this type more generally. Integration by parts provides algebraic relations between the values of ∫ (-) e^s for different inputs, which can be packaged into a Batalin–Vilkovisky-type chain complex. Using some simple homological perturbation theory, we study the version of this complex that arises when f and s are taken to be polynomial functions, and power series are banished. We find that in such cases, the entire scheme-theoretic critical locus (complex points included) of s plays an important role, and that one can uniformly (but noncanonically) integrate out in a purely algebraic way the contributions to the integral from all "higher modes," reducing ∫ f e^s to an integral over the critical locus. This may help explain the presence of analytic continuation in questions like the Volume Conjecture.
  
  We end with Chapter 3, in which the role of integration is somewhat obscured, but perturbation theory is prominent. The Batalin–Vilkovisky homological approach to integration illustrates that there are generalizations of the notion of "integral" analogous to the generalization from cotangent bundles to Poisson manifolds. The AKSZ construction of topological quantum field theories fits into this approach; in what is usually called "AKSZ theory," everything is still required to be symplectic. Using factorization algebras as a framework for (topological) quantum field theory, we construct a one-dimensional Poisson AKSZ field theory for any formal Poisson manifold M. Quantizations of our field theory correspond to formal star-products on M. By using a ``universal'' formal Poisson manifold and abandoning configuration-space integrals in favor of other homological-perturbation techniques, we construct a universal formal star-product all of whose coefficients are manifestly rational numbers. (hide abstract)

• On the coordinate (in)dependence of the formal path integral. 2010. (abstract, PDF, arXiv:1003.5730).

  Abstract: When path integrals are discussed in quantum field theory, it is almost always assumed that the fields take values in a vector bundle. When the fields are instead valued in a possibly-curved fiber bundle, the independence of the formal path integral on the coordinates becomes much less obvious. In this short note, aimed primarily at mathematicians, we first briefly recall the notions of Lagrangian classical and quantum field theory and the standard coordinate-full definition of the "formal" or "Feynman-diagrammatic" path integral construction. We then outline a proof of the following claim: the formal path integral does not depend on the choice of coordinates, but only on a choice of fiberwise volume form. Our outline is an honest proof when the formal path integral is defined without ultraviolet divergences. (hide abstract)