Modeling General Asymptotic Calabi-Yau Periods

In the quests to uncovering the fundamental structures that underlie some of the asymptotic Swampland conjectures we initiate the general study of asymptotic period vectors of CalabiYau manifolds. Our strategy is to exploit the constraints imposed by completeness, symmetry, and positivity, which are formalized in asymptotic Hodge theory. We use these general principles to study the periods near any boundary in complex structure moduli space and explain that near most boundaries leading exponentially suppressed corrections must be present for consistency. The only exception are period vectors near the well-studied large complex structure point. Together with the classification of possible boundaries, our procedure makes it possible to construct general models for these asymptotic periods. The starting point for this construction is the sl(2)-data classifying the boundary, which we use to construct the asymptotic Hodge decomposition known as the nilpotent orbit. We then use the latter to determine the asymptotic period vector. We explicitly carry out this program for all possible oneand two-moduli boundaries in Calabi-Yau threefolds and write down general models for their asymptotic periods. b.bastian@uu.nl t.w.grimm@uu.nl d.t.e.vandeheisteeg@uu.nl 1 ar X iv :2 10 5. 02 23 2v 2 [ he pth ] 3 S ep 2 02 1

[1]  M. Graña Flux compactifications in string theory: A Comprehensive review , 2005, hep-th/0509003.

[2]  E. Cattani,et al.  Infinitesimal variations of Hodge structure at infinity , 2008, 0810.0151.

[3]  Singularities of variations of mixed Hodge structure , 2000, math/0007040.

[4]  E. Cattani,et al.  DEGENERATING VARIATIONS OF HODGE STRUCTURE , 1989 .

[5]  E. Palti,et al.  Infinite distance networks in field space and charge orbits , 2018, Journal of High Energy Physics.

[6]  Sheldon Katz,et al.  Mirror symmetry and algebraic geometry , 1999 .

[7]  Asymptotic Hodge theory and quantum products , 2000, math/0011137.

[8]  I. Valenzuela,et al.  Asymptotic flux compactifications and the swampland , 2019, 1910.09549.

[9]  Thomas W. Grimm,et al.  The Swampland Distance Conjecture for Kähler moduli , 2018, Journal of High Energy Physics.

[10]  Kazuya Kato,et al.  (2)-orbit theorem for degeneration of mixed Hodge structure , 2008 .

[11]  On the algebraicity of the zero locus of an admissible normal function , 2009, Compositio Mathematica.

[12]  R. Blumenhagen,et al.  The refined Swampland Distance Conjecture in Calabi-Yau moduli spaces , 2018, Journal of High Energy Physics.

[13]  S. Cecotti Moduli spaces of Calabi-Yau d-folds as gravitational-chiral instantons , 2020, Journal of High Energy Physics.

[14]  I. Valenzuela,et al.  Merging the weak gravity and distance conjectures using BPS extremal black holes , 2020, Journal of High Energy Physics.

[15]  E. Palti,et al.  The Swampland: Introduction and Review , 2019, Fortschritte der Physik.

[16]  A. Landman On the Picard-Lefschetz transformation for algebraic manifolds acquiring general singularities , 1973 .

[17]  S. Ferrara,et al.  N = 2 supergravity and N = 2 super Yang-Mills theory on general scalar manifolds: Symplectic covariance gaugings and the momentum map , 1996, hep-th/9605032.

[18]  E. Plauschinn,et al.  Swampland conjectures for type IIB orientifolds with closed-string U(1)s , 2020, Journal of High Energy Physics.

[19]  T. Weigand,et al.  Modular fluxes, elliptic genera, and weak gravity conjectures in four dimensions , 2019, Journal of High Energy Physics.

[20]  E. Palti,et al.  Infinite distances in field space and massless towers of states , 2018, Journal of High Energy Physics.

[21]  W. Schmid,et al.  DEGENERATION OF HODGE-STRUCTURES , 1986 .

[22]  Thomas W. Grimm,et al.  Special points of inflation in flux compactifications , 2014, 1412.5537.

[23]  Steven Zucker,et al.  Variation of mixed Hodge structure. I , 1985 .

[24]  S. Cecotti Swampland geometry and the gauge couplings , 2021, Journal of High Energy Physics.

[25]  N. Cribiori,et al.  The web of swampland conjectures and the TCC bound , 2020, Journal of High Energy Physics.

[26]  Fabian Ruehle,et al.  Classifying Calabi–Yau Threefolds Using Infinite Distance Limits , 2019, Communications in Mathematical Physics.

[27]  S. Kachru,et al.  Nonperturbative results on the point particle limit of N=2 heterotic string compactifications , 1995, hep-th/9508155.

[28]  F. Carta,et al.  The String Landscape, the Swampland, and the Missing Corner , 2017, 1711.00864.

[29]  G. G. Stokes "J." , 1890, The New Yale Book of Quotations.

[30]  Cumrun Vafa,et al.  Mirror Symmetry , 2000, hep-th/0002222.

[31]  Thomas W. Grimm Moduli space holography and the finiteness of flux vacua , 2020, Journal of High Energy Physics.

[32]  Mirror Symmetry and Integral Variations of Hodge Structure Underlying One Parameter Families of Calabi-Yau Threefolds , 2005, math/0505272.

[33]  T. Weigand,et al.  Quantum corrections in 4d N = 1 infinite distance limits and the weak gravity conjecture , 2020, 2011.00024.

[34]  The e ective action of N = 1 Calabi - Yau orientifolds , 2004, hep-th/0403067.

[35]  W. Schmid Variation of hodge structure: The singularities of the period mapping , 1973 .

[36]  T. Weigand,et al.  Tensionless strings and the weak gravity conjecture , 2018, Journal of High Energy Physics.

[37]  What is special Kähler geometry , 1997, hep-th/9703082.

[38]  Green,et al.  Finite distance between distinct Calabi-Yau manifolds. , 1989, Physical review letters.

[39]  S. Yau,et al.  Mirror symmetry, mirror map and applications to Calabi-Yau hypersurfaces , 1993, hep-th/9308122.

[40]  Flux Compactification , 2006, hep-th/0610102.

[41]  L. McAllister,et al.  Conifold Vacua with Small Flux Superpotential , 2020, Fortschritte der Physik.

[42]  F. Marchesano,et al.  Instantons and infinite distances , 2019, Journal of High Energy Physics.

[43]  A. Font,et al.  The Swampland Distance Conjecture and towers of tensionless branes , 2019, Journal of High Energy Physics.

[44]  Bernd S Siebert,et al.  Period integrals from wall structures via tropical cycles, canonical coordinates in mirror symmetry and analyticity of toric degenerations , 2019, Publications mathématiques de l'IHÉS.

[45]  E. Witten,et al.  CFT's from Calabi–Yau four-folds , 1999, hep-th/9906070.

[46]  Xenia de la Ossa,et al.  A Pair of Calabi-Yau manifolds as an exactly soluble superconformal theory , 1991 .

[47]  A. Tyurin Fano versus Calabi - Yau , 2003, math/0302101.

[48]  Tsuyoshi Murata,et al.  {m , 1934, ACML.

[49]  L. McAllister,et al.  Vacua with Small Flux Superpotential. , 2019, Physical review letters.

[50]  Xenia de la Ossa,et al.  Mirror Symmetry for Two Parameter Models -- I , 1993, hep-th/9308083.

[51]  M. Graña,et al.  Algorithmically Solving the Tadpole Problem , 2021, Advances in Applied Clifford Algebras.

[52]  T. Weigand,et al.  A stringy test of the Scalar Weak Gravity Conjecture , 2018, Nuclear Physics B.

[53]  Infinite distance and zero gauge coupling in 5D supergravity , 2020, Physical Review D.

[54]  Abhinav Joshi,et al.  Swampland distance conjecture for one-parameter Calabi-Yau threefolds , 2019, Journal of High Energy Physics.

[55]  P. Griffiths,et al.  NÉRON MODELS AND BOUNDARY COMPONENTS FOR DEGENERATIONS OF HODGE STRUCTURE OF MIRROR QUINTIC TYPE , 2008 .

[56]  R. Blumenhagen,et al.  Small Flux Superpotentials for Type IIB Flux Vacua Close to a Conifold , 2020, Fortschritte der Physik.

[57]  T. Weigand,et al.  Emergent strings, duality and weak coupling limits for two-form fields , 2019, Journal of High Energy Physics.