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.
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Minimal nondegenerate extensions. With David Reutter. Journal of the American Mathematical Society, 2023. (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.
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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 Vf♮. 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 Vf♮. This surprise hints that there are as-yet-undiscovered features of
Conway Moonshine.
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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.
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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.
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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.
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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.
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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).
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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.
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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.
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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.
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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.
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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.
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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)
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)
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.
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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 VectR, 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 π1étSpec(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, π2étSpec(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)
(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 Ed-algebras in a symmetric monoidal (∞, n)-category C to an (∞, n+d)-category using the higher morphisms in C.
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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 R0|4//Spin(3), R0|7//G2, and R0|8//Spin(7), respectively.
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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.
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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 es, where f and s are complex polynomials of n variables and γ ⊆ Cn is an n-real-dimensional contour along which es 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.
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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)
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 Rd to the formality of the Ed operad, and conjecture a properadic description of the space of Ed formality quasiisomorphisms. (hide abstract)
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.
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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 ℝd 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.
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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 Vγ(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(Vγ) 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.
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