old supersymmetry as new mathematics PILJIN YI Korea Institute for Advanced Study with help from Sungjay Lee
Atiyah-Singer Index Theorem ~ 1963 Calabi-Yau ~ 1978 Calibrated Geometry ~ 1982 (Harvey & Lawson) 1975 ~ Bogomolnyi-Prasad-Sommerfeld (BPS) 1977 ~ Supersymmetry 1982 ~ Index Thm by Path Integral (Alvarez-Gaume) 1985 ~ Calabi-Yau Compactification 1988 ~ Mirror Symmetry 1992~ 2d Wall-Crossing / tt* (Cecotti & Vafa etc) Homological Mirror Symmetry ~ 1994 (Kontsevich) 1994 ~ 4d Wall-Crossing (Seiberg & Witten) 1998 ~ Wall-Crossing is Bound State Dissociation (Lee & P.Y.) Stability & Derived Category ~ 2000 Wall-Crossing Conjecture ~ 2008 (conjecture by Kontsevich & Soibelman) 2000 ~ Path Integral Proof of Mirror Symmetry (Hori & Vafa) 2008 ~ Konstevich-Soibelman Explained (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 2 with S 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 chiral matter FI constants for U(1) s Theta angles for U(1) s LG Calabi-Yau NLSM
localization in a nutshell
localization in a nutshell supersymmetry + localization claims = original path integral which is fully quantum deformed path integral 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
2 localization for S partition function of 2d (2,2) theories Benini & Cremonesi; Doroud,Gomis,Le Floch, & Sungay Lee
2 localization for S 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 =
counting special Lagrangian submanifolds of CY3 1 with S 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
Kachru + McGreevy 1999 Denef 2002 quiver quantum mechanics
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
K.Hori + H.Kim + P. Y. 2014 the Jeffrey-Kirwan residue
K.Hori + H.Kim + P. Y. 2014 the Jeffrey-Kirwan residue
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 n C, 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 Calabi-Yau ~ 1978 Calibrated Geometry ~ 1982 (Harvey & Lawson) 1975 ~ Bogomolnyi-Prasad-Sommerfeld (BPS) 1977 ~ Supersymmetry 1982 ~ Index Thm by Path Integral (Alvarez-Gaume) 1985 ~ Calabi-Yau Compactification 1988 ~ Mirror Symmetry 1992~ 2d Wall-Crossing / tt* (Cecotti & Vafa etc) Homological Mirror Symmetry ~ 1994 (Kontsevich) 1994 ~ 4d Wall-Crossing (Seiberg & Witten) 1998 ~ Wall-Crossing is Bound State Dissociation (Lee & P.Y.) Stability & Derived Category ~ 2000 Wall-Crossing Conjecture ~ 2008 (conjecture by Kontsevich & Soibelman) 2000 ~ Path Integral Proof of Mirror Symmetry (Hori & Vafa) 2008 ~ Konstevich-Soibelman Explained (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)