Categorical wall-crossing in Landau–Ginzburg models

We describe how categorical BPS data including chain complexes of solitons, CPT pairings, and interior amplitudes jump across a wall of marginal stability in two-dimensional $\mathcal{N}=(2,2)$ models. We show that our jump formulas hold if and only if the $A_{\infty}$-categories of half-BPS branes constructed on either side of the wall are homotopy equivalent. These results can be viewed as categorical enhancements of the Cecotti-Vafa wall-crossing formula.

[1]  A. Neitzke.  Lectures on BPS states and spectral networks , 2021, Quantum Field Theory and Manifold Invariants.

[2]  R. Bergamin,et al.  FQHE and tt* geometry , 2019, Journal of High Energy Physics.

[3]  F. Gesztesy,et al.  Trace Formulas for a Class of non-Fredholm Operators: A Review , 2016, 1610.04954.

[4]  E. Witten,et al.  An Introduction To The Web-Based Formalism , 2015, 1506.04086.

[5]  E. Witten,et al.  Algebra of the Infrared: String Field Theoretic Structures in Massive ${\cal N}=(2,2)$ Field Theory In Two Dimensions , 2015, 1506.04087.

[6]  M. Kapranov,et al.  Algebra of the infrared and secondary polytopes , 2014, 1408.2673.

[7]  Yuechen Zhu.  Donaldson–Thomas Invariants and Wall-Crossing Formulas , 2014, 1408.2671.

[8]  D. Auroux.  A Beginner’s Introduction to Fukaya Categories , 2013, 1301.7056.

[9]  Daniel Lowengrub,et al.  Deformation in Theory , 2014 .

[10]  G. Moore,et al.  Wall-crossing in coupled 2d-4d systems , 2011, 1103.2598.

[11]  B. Pioline.  Four ways across the wall , 2011, 1103.0261.

[12]  G. Moore,et al.  Wall-crossing from supersymmetric galaxies , 2010, 1008.0030.

[13]  G. Moore,et al.  Split states, entropy enigmas, holes and halos , 2007, hep-th/0702146.

[14]  G. Moore,et al.  Framed BPS states , 2010, 1006.0146.

[15]  M. Kontsevich,et al.  Motivic Donaldson-Thomas invariants: Summary of results , 2009, 0910.4315.

[16]  Yan Soibelman,et al.  Stability structures, motivic Donaldson-Thomas invariants and cluster transformations , 2008, 0811.2435.

[17]  G. Moore,et al.  Four-Dimensional Wall-Crossing via Three-Dimensional Field Theory , 2008, 0807.4723.

[18]  Paul Seidel,et al.  Fukaya Categories and Picard-Lefschetz Theory , 2008 .

[19]  A. Nelson.  Dynamical Breaking of Supersymmetry , 2008 .

[20]  J. Stasheff,et al.  Homotopy Algebras Inspired by Classical Open-Closed String Field Theory , 2004, math/0410291.

[21]  C. Weibel,et al.  An Introduction to Homological Algebra: References , 1960 .

[22]  N. Sakai,et al.  An Exact Solution of BPS Junctions and Its Properties , 2000, hep-th/0012182.

[23]  C. Vafa,et al.  D-Branes And Mirror Symmetry , 2000, hep-th/0005247.

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

[25]  S. Carroll,et al.  Domain wall junctions are 1/4 - BPS states , 1999, hep-th/9905217.

[26]  Bernhard Keller,et al.  Introduction to $A$-infinity algebras and modules , 1999, math/9910179.

[27]  G. Gibbons,et al.  Bogomol'nyi Equation for Intersecting Domain Walls , 1999, hep-th/9905196.

[28]  N. Dorey.  The BPS spectra of two-dimensional supersymmetric gauge theories with twisted mass terms , 1998, hep-th/9806056.

[29]  Kimyeong M. Lee,et al.  Dyons in N=4 supersymmetric theories and three-pronged strings , 1998, hep-th/9804174.

[30]  N. Seiberg.  The Power of Holomorphy -- Exact Results in 4D SUSY Field Theories , 1994, hep-th/9408013.

[31]  E. Witten,et al.  Electric - magnetic duality, monopole condensation, and confinement in N=2 supersymmetric Yang-Mills theory , 1994, hep-th/9407087.

[32]  C. Vafa,et al.  On classification ofN=2 supersymmetric theories , 1993 .

[33]  C. Vafa,et al.  Ising model andN=2 supersymmetric theories , 1993 .

[34]  C. Vafa,et al.  On Classification of N=2 Supersymmetric Theories, (e-mail uncorrupted version) , 1992, hep-th/9211097.

[35]  S.Cecotti,et al.  Ising Model and $N=2$ Supersymmetric Theories , 1992, hep-th/9209085.

[36]  C. Vafa,et al.  A New supersymmetric index , 1992, hep-th/9204102.

[37]  E. Witten.  Supersymmetry and Morse theory , 1982 .