Laurent Exponential Gauss--Manin Systems: Exact Quantum Curves, Completion, and Hard-Edge Degeneration
Year: 2026
Type: preprint
Abstract: For polynomial--Laurent exponential phases with independentinverse-power coefficients, we construct split twisted normal formsand flat meromorphic connections, uniformly in the formalsemiclassical parameter. The localized quotient has rank $N=d+k$;the constant amplitude is cyclic and satisfies an exact monic quantumcurve of order $N$, with its operator-ordering correction determinedby twisted integration by parts. Exact normal-form trace identitiesgive the full period determinant$C_\Gamma\hbar^{N/2}b_k^{(k+1)/2}e^{-\Sigma/\hbar}$,where the total critical action $\Sigma$ separates into polynomialand inverse-power coefficient contributions. This identity includesthe determinant monodromy and a Gaussian thimble normalization.For a single pole coefficient the connection is Fuchsian, withmoment-frame residue spectrum $0$ (multiplicity $d$) and$1/k,\ldots,k/k$. A normalized completion--Tor--monodromy theorem,transferred with its full proof from the GHEA research, identifiesthe completed comparison kernel with derived Tor layers and thelocal monodromy image. Pole removal gives exact Weyl factorizations,rank loss, and a flat but non-finitely-generated unlocalized family.A Newton-gap theorem determines the number of collapsing criticalpoints and their singular action and Hessian scales when a highestpole disappears above a lower nonzero pole. A rank-five positive-rayexample has exponential irregular growth although its determinantconnection is regular singular. We also prove endpointPuiseux--logarithmic expansions and the large-parameter splittinginto polynomial and hard-edge saddles. Full signed Stokes matricesand a simultaneous multipole Tor--monodromy theorem remain open;historical novelty of the individual formulas is not asserted.
Author: Forjan, Jesse
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225632445
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