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Beyond Algebraic Superstring Compactification

Tristan Hübsch

TL;DR

This work argues for extending Calabi–Yau compactifications beyond purely algebraic spaces by embracing non-algebraic deformations that fit within mirror symmetry. It develops an explicit, GLSM-motivated double deformation framework for Calabi–Yau hypersurfaces in Hirzebruch scrolls, interprets deformations torically via a stripe-based transpolar operation, and connects these to generalized complete intersections and Tyurin degenerations. It then expands mirror symmetry to non-algebraic ambient spaces using transposition mirrors, simplicial reductions, and flip-folded multifans, introducing VEX multitopes and a local transpolar involution. Finally, it discusses Laurent deformations, an intrinsic-limit completion, and an algebraic Cox-type alternative that regularizes Laurent systems, outlining open questions about topology, cohomology, and moduli in this broadened landscape.

Abstract

Superstring compactifications have been vigorously studied for over four decades, and have flourished involving an active iterative feedback between physics and (complex) algebraic geometry. This led to an unprecedented wealth of constructions, virtually all of which are "purely" algebraic. Recent developments however indicate many more possibilities to be afforded by including certain generalizations that, at first glance at least, are not algebraic -- yet fit remarkably well within an overall mirror-symmetric framework and are surprisingly amenable to standard computational analysis upon certain mild but systematic modifications.

Beyond Algebraic Superstring Compactification

TL;DR

This work argues for extending Calabi–Yau compactifications beyond purely algebraic spaces by embracing non-algebraic deformations that fit within mirror symmetry. It develops an explicit, GLSM-motivated double deformation framework for Calabi–Yau hypersurfaces in Hirzebruch scrolls, interprets deformations torically via a stripe-based transpolar operation, and connects these to generalized complete intersections and Tyurin degenerations. It then expands mirror symmetry to non-algebraic ambient spaces using transposition mirrors, simplicial reductions, and flip-folded multifans, introducing VEX multitopes and a local transpolar involution. Finally, it discusses Laurent deformations, an intrinsic-limit completion, and an algebraic Cox-type alternative that regularizes Laurent systems, outlining open questions about topology, cohomology, and moduli in this broadened landscape.

Abstract

Superstring compactifications have been vigorously studied for over four decades, and have flourished involving an active iterative feedback between physics and (complex) algebraic geometry. This led to an unprecedented wealth of constructions, virtually all of which are "purely" algebraic. Recent developments however indicate many more possibilities to be afforded by including certain generalizations that, at first glance at least, are not algebraic -- yet fit remarkably well within an overall mirror-symmetric framework and are surprisingly amenable to standard computational analysis upon certain mild but systematic modifications.

Paper Structure

This paper contains 19 sections, 4 theorems, 38 equations, 4 figures.

Table of Contents

  1. Introduction, Rationale and Summary
  2. A Showcasing Deformation Family
  3. From GLSM to Toric
  4. The Superpotential
  5. The Transpolar Operation
  6. From Toric to Generalized Intersections
  7. The Deformation Family Picture
  8. $\vec{\epsilon}\mkern2mu{=}\mkern2mu{(0,0,0)}$:
  9. $\vec{\epsilon}\mkern2mu{=}\mkern2mu(1,0,0)$:
  10. $\vec{\epsilon}\mkern2mu{=}\mkern2mu(1,1,0)$:
  11. $\vec{\epsilon}\mkern2mu{=}\mkern2mu(1,1,1)$:
  12. Black Sheep in the Deformation Family
  13. Transposition Mirrors
  14. Transposition as Multitope Swapping
  15. Simplicial Reductions
  16. ...and 4 more sections

Key Result

Corollary 1

For a list of (chiral superfield) variables and their $U(1;\mathbb{C})^n$-charges as in e:X1-6, the most general superpotential, $W$ in e:U, is an $X_0$-multiple of a deformation the fundamental monomial, $\Pi{x}$. The lattice of all candidate monomials, $\Delta(X)$, has hyperplanes at 1-derivative

Figures (4)

  • Figure 1: Some of the $c_1(F^{(2)}_{m})$-degree monomials plotted to indicate the "strips" discussed in the text; the fundamental monomial, $x_1x_2x_5x_6$ is boxed
  • Figure 2: Combinatorial data for $\mathbb{P}^2$ as a toric variety, and for ${}^\triangledown\!\mathbb{P}^2$: the Newton polygon $\Delta$ of (regular) anticanonical monomials (a), in Cox coordinates specified by the spanning polygon $\Delta\!^{\star}$ and fan $\Sigma(\mathbb{P}^2)$, (b). Similarly: Newton polygon $\Delta({}^\triangledown\!\mathbb{P}^2)$ in (c) and spanning polygon $\Delta\!^{\star\!}({}^\triangledown\!\mathbb{P}^2)$ and fan $\Sigma({}^\triangledown\!\mathbb{P}^2)$ in (d).
  • Figure 3: The distinct types of transverse cubics rAGZV-Sing1, classified as distinct simplicial 0-enclosing reductions of the complete Newton polytope
  • Figure 4: Combinatorial data for $F^{(2)}_{3}$ and ${}^\triangledown\!F^{(2)}_{3}$: $\Delta(F^{(2)}_{3})\mkern2mu{=}\mkern2mu\Delta\!^{\star\!}({}^\triangledown\!F^{(2)}_{3})$ at left and $\Delta\!^{\star\!}(F^{(2)}_{3})\mkern2mu{=}\mkern2mu\Delta({}^\triangledown\!F^{(2)}_{3})$ at right; the transpose pair of "cornerstone" defining polynomials (top right)

Theorems & Definitions (19)

  • Corollary 1
  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Remark 5
  • Remark 6
  • Definition 1: Ref. rBH-gB
  • Remark 7
  • Conjecture 1
  • ...and 9 more