old supersymmetry as new mathematics PILJIN YI Korea Institute for Advanced Study with help from Sungjay Lee Atiyah-Singer Index Theorem ~ 1963 1975 ~ Bogomolnyi-Prasad-Sommerfeld (BPS) Calabi-Yau ~ 1978 1977 ~ Supersymmetry Calibrated Geometry ~ 1982 1982 ~ Index Thm by Path Integral (Alvarez-Gaume) (Harvey & Lawson) 1985 ~ Calabi-Yau Compactification 1988 ~ Mirror Symmetry 1992~ 2d Wall-Crossing / tt* (Cecotti & Vafa etc) Homological Mirror Symmetry ~ 1994 1994 ~ 4d Wall-Crossing (Seiberg & Witten) (Kontsevich) 1998 ~ Wall-Crossing is Bound State Dissociation (Lee & P.Y.) Stability & Derived Category ~ 2000 2000 ~ Path Integral Proof of Mirror Symmetry (Hori & Vafa) Wall-Crossing Conjecture ~ 2008 2008 ~ Konstevich-Soibelman Explained (conjecture by Kontsevich & Soibelman) (Gaiotto & Moore & Neitzke) 2011 ~ KS Wall-Crossing proved via Quatum Mechanics (Manschot , Pioline & Sen / Kim , Park, Wang & P.Y. / Sen) 2012 ~ S2 Partition Function as tt* (Jocker, Kumar, Lapan, Morrison & Romo /Gomis & Lee) quantum and geometry glued by superstring theory when can we perform path integrals exactly ? counting geometry with supersymmetric path integrals quantum and geometry glued by superstring theory Einstein this theory famously resisted quantization, however on the other hand, five superstring theories, with a consistent quantum gravity inside, live in 10 dimensional spacetime these superstring theories say, spacetime is composed of 4+6 dimensions with very small & tightly-curved (say, Calabi-Yau) 6D manifold sitting at each and every point of usual 3D space, that is, more precisely, these superstring theories say, spacetime is composed of 4+6 dimensions with very small & tightly-curved (say, Calabi-Yau) 6D manifold sitting at each and every point of usual 3D space, which implies ………… a particle located somewhere in our visible space a particle located somewhere in our visible space a wrapped brane in the hidden Calabi-Yau at that point this means that we can actually detect geometry (loops, holes, cavities, ……) of the hidden 6D space by detecting what kind particles exist in visible 3D world particle geometry .. particle geometry particle geometry quantum bound state of particle geometry meaning, spectrum of certain 4d quantum theory obtained via Calabi-Yau compatification of type II string theory submanifolds (and their topological properties) of the same Calabi-Yau 3-fold more generally, d-dimensional brane wrapping c-dimensional cycles (d-c) dimensional object in the noncompact spacetime which also allows realization of an instanton in our visible space as wrapped string/brane in the hidden Calabi-Yau d-dimensional brane wrapping d-dimensional supersymmetric cycles supersymmetric instanton in the noncompact spacetime for example, string world-sheet wrapping 2-cycles string world-sheet wrapping 2-cycles correct low energy effective theory in the remaining 4d spacetime but life is never that simple: this picture translates to reliable physical facts only upon appropriate restrictions on both sides Calabi-Yau manifold & calibrated 3-cycles a geometer asks: for a Calabi-Yau, which topological 3-cycles can be calibrated ? a string theorist answers: “quantum supersymmetric states” “calibrated 3-cycles” a string theorist answers: no quantum supersymmetric states no calibrated 3-cycles such strange maps between 4d quantum objects and 6d classical geometries are possible because superstring theories have extended objects in the form of fundamental strings, D-branes, and NS-branes in the end, existence and counting of such a special submanifold manifests in some quantum path integral such as Witten index, or more generally supersymmetric partition functions when can we perform the relevant path integrals exactly ? a prototype: supersymmetric harmonic oscillators 32 supersymmetric harmonic oscillators 33 witten index = twisted partition function of supersymmetric harmonic oscillators 34 witten index = twisted partition function of supersymmetric harmonic oscillators bosonic fermionic 35 witten index = twisted partition function of supersymmetric harmonic oscillators fermionic bosonic 36 how do we get the same thing from path integral ? 37 how do we get the same thing from path integral ? 38 boson 39 boson 40 fermion 41 fermion 42 combining the two 43 why did the example of harmonic oscillators work so well? because free of interactions because of supersymmetry because of the simplicity of the quantity being computed why did the example of harmonic oscillators work so well? because free of interactions because of supersymmetry because of the simplicity of the quantity being computed all of above……but any one of these can be relaxed? trivially because free of interactions because of supersymmetry because of the simplicity of the quantity being computed or, more interestingly, because free of interactions because of supersymmetry because of the simplicity of the quantity being computed so why was the simple computation possible, really ? hopeless 48 so why was the simple computation possible, really ? perhaps 49 so why was the simple computation possible, really ? perhaps 50 so why was the simple computation possible, really ? bosonic fermionic 51 so why was the simple computation possible, really ? formally independent of this parameter 52 so why was the simple computation possible, really ? formally independent of this parameter whereby the path integral becomes one-loop exact 53 so why was the simple computation possible, really ? formally independent of this parameter find V and a limit of t, whereby the path integral becomes one-loop exact 54 example : nonlinear sigma model for Euler index 55 example : nonlinear sigma model for Euler index 56 example : nonlinear sigma model for Euler index 57 example : nonlinear sigma model for Euler index 58 Euler index 59 counting geometry with supersymmetric path integrals counting rational Gromov-Witten invariant with S 2 partition function of d=2 GLSM rational Gromov-Witten invariants are encoded in the low energy effective metric of CY3 moduli which can be sometimes counted via mirror symmetry mirror symmetry between a pair of CY3 2d N=(2,2) GLSM for CY NLSM gauge fields FI constants for U(1)’s chiral matter Theta angles for U(1)’s LG Calabi-Yau NLSM localization in a nutshell localization in a nutshell supersymmetry + localization claims = original path integral deformed path integral which is fully quantum which is Gaussian localization in a nutshell typical results where obey (1) equation of motion of the deformed theory (2) supersymmetric condition Benini & Cremonesi; Doroud,Gomis,Le Floch, & Sungay Lee localization for S 2 partition function of 2d (2,2) theories Benini & Cremonesi; Doroud,Gomis,Le Floch, & Sungay Lee localization for S 2 partition function of 2d (2,2) theories A twisted flat cylinder anti-A twisted Jaume Gomis & Sungay Lee Gromov-Witten Invariants without mirror symmetry rational Gromov-Witten invariants are nothing but the number of S2 holomorphically embedded in the CY3 in question, which manfest in the 4d spacetime low energy effective action Gulliksen-Negard Determinantal CY Jockers,Kumar,Lapan,Morrison,Romo = which can also do all examples with known mirror pair mirror which can also do all examples with known counting special Lagrangian submanifolds of CY3 with S1 partition function of d=1 GLSM 1d N=4 Gauged Linear Sigma Models gauge fields FI constants for U(1)’s chiral matter LG’/NLSM’ NLSM/LG wall-crossing of special Lagrange 3-cycles in CY 3-fold depends on the precise shape of the CY 3-fold quiver quantum mechanics Kachru + McGreevy 1999 Denef 2002 equivariant Witten index of d=1 N=4 GLSM K.Hori + H.Kim + P. Y. 2014 N=4, compact and geometric then, a localization procedure produces in the end the Jeffrey-Kirwan residue K.Hori + H.Kim + P. Y. 2014 the Jeffrey-Kirwan residue K.Hori + H.Kim + P. Y. 2014 the Jeffrey-Kirwan residue tagged by FI constant K.Hori + H.Kim + P. Y. 2014 cf) Hwang + Kim + Kim + Park, Cordova + Shao 2014 which can also count, with some more effort, the entire Hodge diamond of arbitrary compact Kaehler manifold that emerges as classical moduli space of quiver GLSM this includes any manifold that can be obtained from symplectic reduction of C n , and holomorphically embedded submanifold thereof N = 4 (3,1,1) triangle quiver many more subtleties with non-compact theories or with theories at non-compact parameter values (where localization is always tricky; see Lee+P.Y. 2016) what else? exact path integral via localization supersymmetric partition functions for d = 1,2,3,4,5 supersymmetric field theories various M-theory/F-theory ramifications for d = 5,6 (local community leading the effort) (0,2) GLSM/heterotic theories needs to explored further black hole microstates for lower supersymmetry (goes beyond wall-crossing/Kontsevich-Soibelman) Atiyah-Singer Index Theorem ~ 1963 1975 ~ Bogomolnyi-Prasad-Sommerfeld (BPS) Calabi-Yau ~ 1978 1977 ~ Supersymmetry Calibrated Geometry ~ 1982 1982 ~ Index Thm by Path Integral (Alvarez-Gaume) (Harvey & Lawson) 1985 ~ Calabi-Yau Compactification 1988 ~ Mirror Symmetry 1992~ 2d Wall-Crossing / tt* (Cecotti & Vafa etc) Homological Mirror Symmetry ~ 1994 1994 ~ 4d Wall-Crossing (Seiberg & Witten) (Kontsevich) 1998 ~ Wall-Crossing is Bound State Dissociation (Lee & P.Y.) Stability & Derived Category ~ 2000 2000 ~ Path Integral Proof of Mirror Symmetry (Hori & Vafa) Wall-Crossing Conjecture ~ 2008 2008 ~ Konstevich-Soibelman Explained (conjecture by Kontsevich & Soibelman) (Gaiotto & Moore & Neitzke) 2011 ~ KS Wall-Crossing proved via Quatum Mechanics (Manschot , Pioline & Sen / Kim , Park, Wang & P.Y. / Sen) 2012 ~ S2 Partition Function as tt* (Jocker, Kumar, Lapan, Morrison & Romo /Gomis & Lee).