algebra, geometry, and schroedinger...
TRANSCRIPT
![Page 1: algebra, geometry, and Schroedinger atomshome.kias.re.kr/psec/QUC/PY-2014_QUC_Inauguration_Piljin... · 2014-02-28 · quantum physics count states, gravity makes geometry, and superstring](https://reader034.vdocument.in/reader034/viewer/2022042400/5f0f4f6a7e708231d4438707/html5/thumbnails/1.jpg)
algebra, geometry, and Schroedinger atoms
PILJIN YIKorea Institute for Advanced Study
QUC Inauguration ConferenceFebruary 2014
![Page 2: algebra, geometry, and Schroedinger atomshome.kias.re.kr/psec/QUC/PY-2014_QUC_Inauguration_Piljin... · 2014-02-28 · quantum physics count states, gravity makes geometry, and superstring](https://reader034.vdocument.in/reader034/viewer/2022042400/5f0f4f6a7e708231d4438707/html5/thumbnails/2.jpg)
Atiyah-Singer Index Theorem ~ 1963Calabi-Yau ~ 1978
Calibrated Geometry ~ 1982(Harvey & Lawson)
.
.
.Homological Mirror Symmetry ~ 1994
(Kontsevich)....
Stability & Derived Category ~ 2000.....
Wall-Crossing Conjecture~ 2008(Conjecture by Kontsevich & Soibelman)
.
.
.
.
1975 ~ Bogomolnyi-Prasad-Sommerfeld (BPS)1977 ~ Supersymmetry1983 ~ Superstring Theory.1985 ~ Calabi-Yau Compactification1988 ~ Mirror Symmetry.1994 ~ Wall-Crossing Discovered
(Seiberg & Witten)1995 ~ Dirichlet Branes1998 ~ Wall-Crossing is Bound State
Dissociation (Lee & P.Y.).2001 ~ Wall-Crossing for Black Holes
(Denef)....2008 ~ Konstevich-Soibelman Explained
(Gaiotto & Moore & Neitzke).2011 ~ KS Wall-Crossing proved
via Quatum Mechanics Manschot , Pioline & Sen /Kim , Park, Wang & P.Y. / Sen
![Page 3: algebra, geometry, and Schroedinger atomshome.kias.re.kr/psec/QUC/PY-2014_QUC_Inauguration_Piljin... · 2014-02-28 · quantum physics count states, gravity makes geometry, and superstring](https://reader034.vdocument.in/reader034/viewer/2022042400/5f0f4f6a7e708231d4438707/html5/thumbnails/3.jpg)
quantum is algebraic
spacetime is geometric
a little bit of superstring theory
how quantum mechanics solved a modern geometry problem
![Page 4: algebra, geometry, and Schroedinger atomshome.kias.re.kr/psec/QUC/PY-2014_QUC_Inauguration_Piljin... · 2014-02-28 · quantum physics count states, gravity makes geometry, and superstring](https://reader034.vdocument.in/reader034/viewer/2022042400/5f0f4f6a7e708231d4438707/html5/thumbnails/4.jpg)
Quantum Mechanics
Algebra
![Page 5: algebra, geometry, and Schroedinger atomshome.kias.re.kr/psec/QUC/PY-2014_QUC_Inauguration_Piljin... · 2014-02-28 · quantum physics count states, gravity makes geometry, and superstring](https://reader034.vdocument.in/reader034/viewer/2022042400/5f0f4f6a7e708231d4438707/html5/thumbnails/5.jpg)
quantum is algebraic
![Page 6: algebra, geometry, and Schroedinger atomshome.kias.re.kr/psec/QUC/PY-2014_QUC_Inauguration_Piljin... · 2014-02-28 · quantum physics count states, gravity makes geometry, and superstring](https://reader034.vdocument.in/reader034/viewer/2022042400/5f0f4f6a7e708231d4438707/html5/thumbnails/6.jpg)
Balmer / Rydberg ~ 1880’s
![Page 7: algebra, geometry, and Schroedinger atomshome.kias.re.kr/psec/QUC/PY-2014_QUC_Inauguration_Piljin... · 2014-02-28 · quantum physics count states, gravity makes geometry, and superstring](https://reader034.vdocument.in/reader034/viewer/2022042400/5f0f4f6a7e708231d4438707/html5/thumbnails/7.jpg)
Balmer / Rydberg
![Page 8: algebra, geometry, and Schroedinger atomshome.kias.re.kr/psec/QUC/PY-2014_QUC_Inauguration_Piljin... · 2014-02-28 · quantum physics count states, gravity makes geometry, and superstring](https://reader034.vdocument.in/reader034/viewer/2022042400/5f0f4f6a7e708231d4438707/html5/thumbnails/8.jpg)
Schroedinger Atoms
![Page 9: algebra, geometry, and Schroedinger atomshome.kias.re.kr/psec/QUC/PY-2014_QUC_Inauguration_Piljin... · 2014-02-28 · quantum physics count states, gravity makes geometry, and superstring](https://reader034.vdocument.in/reader034/viewer/2022042400/5f0f4f6a7e708231d4438707/html5/thumbnails/9.jpg)
Schroedinger Atoms
![Page 10: algebra, geometry, and Schroedinger atomshome.kias.re.kr/psec/QUC/PY-2014_QUC_Inauguration_Piljin... · 2014-02-28 · quantum physics count states, gravity makes geometry, and superstring](https://reader034.vdocument.in/reader034/viewer/2022042400/5f0f4f6a7e708231d4438707/html5/thumbnails/10.jpg)
Schroedinger Atoms
![Page 11: algebra, geometry, and Schroedinger atomshome.kias.re.kr/psec/QUC/PY-2014_QUC_Inauguration_Piljin... · 2014-02-28 · quantum physics count states, gravity makes geometry, and superstring](https://reader034.vdocument.in/reader034/viewer/2022042400/5f0f4f6a7e708231d4438707/html5/thumbnails/11.jpg)
General Relativity
Geometry
![Page 12: algebra, geometry, and Schroedinger atomshome.kias.re.kr/psec/QUC/PY-2014_QUC_Inauguration_Piljin... · 2014-02-28 · quantum physics count states, gravity makes geometry, and superstring](https://reader034.vdocument.in/reader034/viewer/2022042400/5f0f4f6a7e708231d4438707/html5/thumbnails/12.jpg)
spacetime is geometric
![Page 13: algebra, geometry, and Schroedinger atomshome.kias.re.kr/psec/QUC/PY-2014_QUC_Inauguration_Piljin... · 2014-02-28 · quantum physics count states, gravity makes geometry, and superstring](https://reader034.vdocument.in/reader034/viewer/2022042400/5f0f4f6a7e708231d4438707/html5/thumbnails/13.jpg)
Einstein ~ 1915
![Page 14: algebra, geometry, and Schroedinger atomshome.kias.re.kr/psec/QUC/PY-2014_QUC_Inauguration_Piljin... · 2014-02-28 · quantum physics count states, gravity makes geometry, and superstring](https://reader034.vdocument.in/reader034/viewer/2022042400/5f0f4f6a7e708231d4438707/html5/thumbnails/14.jpg)
Kaluza & Klein (~1921) asked :
can the world has more than 3 spatial dimensions or 4 spacetime dimensions
with the extra directions curled up so small to be practically invisibleand, if so, what are the physical consequences ?
after all, space (& time) is supposed to curved
![Page 15: algebra, geometry, and Schroedinger atomshome.kias.re.kr/psec/QUC/PY-2014_QUC_Inauguration_Piljin... · 2014-02-28 · quantum physics count states, gravity makes geometry, and superstring](https://reader034.vdocument.in/reader034/viewer/2022042400/5f0f4f6a7e708231d4438707/html5/thumbnails/15.jpg)
five superstring theories live in 10 dimensional spacetime
![Page 16: algebra, geometry, and Schroedinger atomshome.kias.re.kr/psec/QUC/PY-2014_QUC_Inauguration_Piljin... · 2014-02-28 · quantum physics count states, gravity makes geometry, and superstring](https://reader034.vdocument.in/reader034/viewer/2022042400/5f0f4f6a7e708231d4438707/html5/thumbnails/16.jpg)
superstring theory says :
spacetime is composed of 4+6 dimensions withvery small & tightly-curved (say, Calabi-Yau) 6D manifold
sitting at each and every point of usual 3D space,
![Page 17: algebra, geometry, and Schroedinger atomshome.kias.re.kr/psec/QUC/PY-2014_QUC_Inauguration_Piljin... · 2014-02-28 · quantum physics count states, gravity makes geometry, and superstring](https://reader034.vdocument.in/reader034/viewer/2022042400/5f0f4f6a7e708231d4438707/html5/thumbnails/17.jpg)
6차원
4차원
roughly,
![Page 18: algebra, geometry, and Schroedinger atomshome.kias.re.kr/psec/QUC/PY-2014_QUC_Inauguration_Piljin... · 2014-02-28 · quantum physics count states, gravity makes geometry, and superstring](https://reader034.vdocument.in/reader034/viewer/2022042400/5f0f4f6a7e708231d4438707/html5/thumbnails/18.jpg)
or, more precisely,
![Page 19: algebra, geometry, and Schroedinger atomshome.kias.re.kr/psec/QUC/PY-2014_QUC_Inauguration_Piljin... · 2014-02-28 · quantum physics count states, gravity makes geometry, and superstring](https://reader034.vdocument.in/reader034/viewer/2022042400/5f0f4f6a7e708231d4438707/html5/thumbnails/19.jpg)
a little bit of superstring theoryparticles from geometry / geometry from particles
![Page 20: algebra, geometry, and Schroedinger atomshome.kias.re.kr/psec/QUC/PY-2014_QUC_Inauguration_Piljin... · 2014-02-28 · quantum physics count states, gravity makes geometry, and superstring](https://reader034.vdocument.in/reader034/viewer/2022042400/5f0f4f6a7e708231d4438707/html5/thumbnails/20.jpg)
Quantum Mechanics
General Relativity
Geometry
Superstring
Algebra
![Page 21: algebra, geometry, and Schroedinger atomshome.kias.re.kr/psec/QUC/PY-2014_QUC_Inauguration_Piljin... · 2014-02-28 · quantum physics count states, gravity makes geometry, and superstring](https://reader034.vdocument.in/reader034/viewer/2022042400/5f0f4f6a7e708231d4438707/html5/thumbnails/21.jpg)
basic building blocks in superstring theory
fundamentalstrings
light particles some of which mediate “forces”such as gravity & electromagnetism
![Page 22: algebra, geometry, and Schroedinger atomshome.kias.re.kr/psec/QUC/PY-2014_QUC_Inauguration_Piljin... · 2014-02-28 · quantum physics count states, gravity makes geometry, and superstring](https://reader034.vdocument.in/reader034/viewer/2022042400/5f0f4f6a7e708231d4438707/html5/thumbnails/22.jpg)
basic building blocks in superstring theory
fundamentalstrings
Dirichletbranes
(as in membrane)
![Page 23: algebra, geometry, and Schroedinger atomshome.kias.re.kr/psec/QUC/PY-2014_QUC_Inauguration_Piljin... · 2014-02-28 · quantum physics count states, gravity makes geometry, and superstring](https://reader034.vdocument.in/reader034/viewer/2022042400/5f0f4f6a7e708231d4438707/html5/thumbnails/23.jpg)
basic building blocks in superstring theory
heavy particles, typically with electric/magnetic charges
fundamentalstrings
Dirichletbranes
![Page 24: algebra, geometry, and Schroedinger atomshome.kias.re.kr/psec/QUC/PY-2014_QUC_Inauguration_Piljin... · 2014-02-28 · quantum physics count states, gravity makes geometry, and superstring](https://reader034.vdocument.in/reader034/viewer/2022042400/5f0f4f6a7e708231d4438707/html5/thumbnails/24.jpg)
a particle located somewhere in our visible space
![Page 25: algebra, geometry, and Schroedinger atomshome.kias.re.kr/psec/QUC/PY-2014_QUC_Inauguration_Piljin... · 2014-02-28 · quantum physics count states, gravity makes geometry, and superstring](https://reader034.vdocument.in/reader034/viewer/2022042400/5f0f4f6a7e708231d4438707/html5/thumbnails/25.jpg)
a particle located somewhere in our visible space= a wrapped brane in the hidden Calabi-Yau at that point
![Page 26: algebra, geometry, and Schroedinger atomshome.kias.re.kr/psec/QUC/PY-2014_QUC_Inauguration_Piljin... · 2014-02-28 · quantum physics count states, gravity makes geometry, and superstring](https://reader034.vdocument.in/reader034/viewer/2022042400/5f0f4f6a7e708231d4438707/html5/thumbnails/26.jpg)
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
![Page 27: algebra, geometry, and Schroedinger atomshome.kias.re.kr/psec/QUC/PY-2014_QUC_Inauguration_Piljin... · 2014-02-28 · quantum physics count states, gravity makes geometry, and superstring](https://reader034.vdocument.in/reader034/viewer/2022042400/5f0f4f6a7e708231d4438707/html5/thumbnails/27.jpg)
particle vs. geometry
..
![Page 28: algebra, geometry, and Schroedinger atomshome.kias.re.kr/psec/QUC/PY-2014_QUC_Inauguration_Piljin... · 2014-02-28 · quantum physics count states, gravity makes geometry, and superstring](https://reader034.vdocument.in/reader034/viewer/2022042400/5f0f4f6a7e708231d4438707/html5/thumbnails/28.jpg)
particle vs. geometry
![Page 29: algebra, geometry, and Schroedinger atomshome.kias.re.kr/psec/QUC/PY-2014_QUC_Inauguration_Piljin... · 2014-02-28 · quantum physics count states, gravity makes geometry, and superstring](https://reader034.vdocument.in/reader034/viewer/2022042400/5f0f4f6a7e708231d4438707/html5/thumbnails/29.jpg)
particle vs. geometry
![Page 30: algebra, geometry, and Schroedinger atomshome.kias.re.kr/psec/QUC/PY-2014_QUC_Inauguration_Piljin... · 2014-02-28 · quantum physics count states, gravity makes geometry, and superstring](https://reader034.vdocument.in/reader034/viewer/2022042400/5f0f4f6a7e708231d4438707/html5/thumbnails/30.jpg)
particle vs. geometry
![Page 31: algebra, geometry, and Schroedinger atomshome.kias.re.kr/psec/QUC/PY-2014_QUC_Inauguration_Piljin... · 2014-02-28 · quantum physics count states, gravity makes geometry, and superstring](https://reader034.vdocument.in/reader034/viewer/2022042400/5f0f4f6a7e708231d4438707/html5/thumbnails/31.jpg)
existence of “quantum BPS (bound) states”= existence of “calibrated 3-cycles”
![Page 32: algebra, geometry, and Schroedinger atomshome.kias.re.kr/psec/QUC/PY-2014_QUC_Inauguration_Piljin... · 2014-02-28 · quantum physics count states, gravity makes geometry, and superstring](https://reader034.vdocument.in/reader034/viewer/2022042400/5f0f4f6a7e708231d4438707/html5/thumbnails/32.jpg)
non-existence of “quantum BPS (bound) states”= non-existence of “calibrated 3-cycles”
![Page 33: algebra, geometry, and Schroedinger atomshome.kias.re.kr/psec/QUC/PY-2014_QUC_Inauguration_Piljin... · 2014-02-28 · quantum physics count states, gravity makes geometry, and superstring](https://reader034.vdocument.in/reader034/viewer/2022042400/5f0f4f6a7e708231d4438707/html5/thumbnails/33.jpg)
how quantum mechanics solved a modern geometry problem
![Page 34: algebra, geometry, and Schroedinger atomshome.kias.re.kr/psec/QUC/PY-2014_QUC_Inauguration_Piljin... · 2014-02-28 · quantum physics count states, gravity makes geometry, and superstring](https://reader034.vdocument.in/reader034/viewer/2022042400/5f0f4f6a7e708231d4438707/html5/thumbnails/34.jpg)
generic BPS particle is a multi-center bound state
![Page 35: algebra, geometry, and Schroedinger atomshome.kias.re.kr/psec/QUC/PY-2014_QUC_Inauguration_Piljin... · 2014-02-28 · quantum physics count states, gravity makes geometry, and superstring](https://reader034.vdocument.in/reader034/viewer/2022042400/5f0f4f6a7e708231d4438707/html5/thumbnails/35.jpg)
thus, wall-crossing has a very simple and interpretationin the particle / quantum mechanics viewpoint
as bound states becoming unbound
![Page 36: algebra, geometry, and Schroedinger atomshome.kias.re.kr/psec/QUC/PY-2014_QUC_Inauguration_Piljin... · 2014-02-28 · quantum physics count states, gravity makes geometry, and superstring](https://reader034.vdocument.in/reader034/viewer/2022042400/5f0f4f6a7e708231d4438707/html5/thumbnails/36.jpg)
quantum mechanics for many such charged particlesKim+Park+P.Y.+Wang 2011
![Page 37: algebra, geometry, and Schroedinger atomshome.kias.re.kr/psec/QUC/PY-2014_QUC_Inauguration_Piljin... · 2014-02-28 · quantum physics count states, gravity makes geometry, and superstring](https://reader034.vdocument.in/reader034/viewer/2022042400/5f0f4f6a7e708231d4438707/html5/thumbnails/37.jpg)
repulsive or attractive depending on a sign
![Page 38: algebra, geometry, and Schroedinger atomshome.kias.re.kr/psec/QUC/PY-2014_QUC_Inauguration_Piljin... · 2014-02-28 · quantum physics count states, gravity makes geometry, and superstring](https://reader034.vdocument.in/reader034/viewer/2022042400/5f0f4f6a7e708231d4438707/html5/thumbnails/38.jpg)
3n-dim dynamics 3 + 2(n-1) dim nonlinear sigma modelvia deformation & localization that preserve the index
![Page 39: algebra, geometry, and Schroedinger atomshome.kias.re.kr/psec/QUC/PY-2014_QUC_Inauguration_Piljin... · 2014-02-28 · quantum physics count states, gravity makes geometry, and superstring](https://reader034.vdocument.in/reader034/viewer/2022042400/5f0f4f6a7e708231d4438707/html5/thumbnails/39.jpg)
index theorem
![Page 40: algebra, geometry, and Schroedinger atomshome.kias.re.kr/psec/QUC/PY-2014_QUC_Inauguration_Piljin... · 2014-02-28 · quantum physics count states, gravity makes geometry, and superstring](https://reader034.vdocument.in/reader034/viewer/2022042400/5f0f4f6a7e708231d4438707/html5/thumbnails/40.jpg)
Bose/Fermi statistics is essential
![Page 41: algebra, geometry, and Schroedinger atomshome.kias.re.kr/psec/QUC/PY-2014_QUC_Inauguration_Piljin... · 2014-02-28 · quantum physics count states, gravity makes geometry, and superstring](https://reader034.vdocument.in/reader034/viewer/2022042400/5f0f4f6a7e708231d4438707/html5/thumbnails/41.jpg)
universal wall-crossing formulae from quantum mechanics of BPS particles
Kim+Park+P.Y.+Wang 2011
![Page 42: algebra, geometry, and Schroedinger atomshome.kias.re.kr/psec/QUC/PY-2014_QUC_Inauguration_Piljin... · 2014-02-28 · quantum physics count states, gravity makes geometry, and superstring](https://reader034.vdocument.in/reader034/viewer/2022042400/5f0f4f6a7e708231d4438707/html5/thumbnails/42.jpg)
2000 Stern + P.Y.wall-crossing formula for simple magnetic charges; weak coupling regime
2002 Denefquiver dynamics representation of N=2 supergravity BH’s
2008 Kontsevich + Soibelman
2010/2011 Manschot + Pioline + Sengeneral n-particle conjecture for Quantum Mechanics Counting
2011 Lee+P.Y. / Kim+Park+Wang+P.Y.general n-particle solution to Quantum Mechanics Counting
2011 SenQuantum Mechanics Counting = Kontsevich-Soibelman Conjecture
![Page 43: algebra, geometry, and Schroedinger atomshome.kias.re.kr/psec/QUC/PY-2014_QUC_Inauguration_Piljin... · 2014-02-28 · quantum physics count states, gravity makes geometry, and superstring](https://reader034.vdocument.in/reader034/viewer/2022042400/5f0f4f6a7e708231d4438707/html5/thumbnails/43.jpg)
Quantum Mechanics
General Relativity
Geometry
Superstring
Algebra
![Page 44: algebra, geometry, and Schroedinger atomshome.kias.re.kr/psec/QUC/PY-2014_QUC_Inauguration_Piljin... · 2014-02-28 · quantum physics count states, gravity makes geometry, and superstring](https://reader034.vdocument.in/reader034/viewer/2022042400/5f0f4f6a7e708231d4438707/html5/thumbnails/44.jpg)
existence of “quantum BPS (bound) states”= existence of “calibrated 3-cycles”
![Page 45: algebra, geometry, and Schroedinger atomshome.kias.re.kr/psec/QUC/PY-2014_QUC_Inauguration_Piljin... · 2014-02-28 · quantum physics count states, gravity makes geometry, and superstring](https://reader034.vdocument.in/reader034/viewer/2022042400/5f0f4f6a7e708231d4438707/html5/thumbnails/45.jpg)
Calabi-Yau manifold & calibrated 3-cycles
![Page 46: algebra, geometry, and Schroedinger atomshome.kias.re.kr/psec/QUC/PY-2014_QUC_Inauguration_Piljin... · 2014-02-28 · quantum physics count states, gravity makes geometry, and superstring](https://reader034.vdocument.in/reader034/viewer/2022042400/5f0f4f6a7e708231d4438707/html5/thumbnails/46.jpg)
given a family of Calabi-Yau manifold with a fixed topology,which topological 3-cycles can be calibrated ?
![Page 47: algebra, geometry, and Schroedinger atomshome.kias.re.kr/psec/QUC/PY-2014_QUC_Inauguration_Piljin... · 2014-02-28 · quantum physics count states, gravity makes geometry, and superstring](https://reader034.vdocument.in/reader034/viewer/2022042400/5f0f4f6a7e708231d4438707/html5/thumbnails/47.jpg)
minimal volume condition+
an extra “minimization” condition
given a family of Calabi-Yau manifold with a fixed topology,which topological 3-cycles can be calibrated ?
![Page 48: algebra, geometry, and Schroedinger atomshome.kias.re.kr/psec/QUC/PY-2014_QUC_Inauguration_Piljin... · 2014-02-28 · quantum physics count states, gravity makes geometry, and superstring](https://reader034.vdocument.in/reader034/viewer/2022042400/5f0f4f6a7e708231d4438707/html5/thumbnails/48.jpg)
which is the reason why topology alonecannot guarantee existence of such cycles
![Page 49: algebra, geometry, and Schroedinger atomshome.kias.re.kr/psec/QUC/PY-2014_QUC_Inauguration_Piljin... · 2014-02-28 · quantum physics count states, gravity makes geometry, and superstring](https://reader034.vdocument.in/reader034/viewer/2022042400/5f0f4f6a7e708231d4438707/html5/thumbnails/49.jpg)
Kontsevich-Soibelman conjecture :a universal algebra for wall-crossing ?
![Page 50: algebra, geometry, and Schroedinger atomshome.kias.re.kr/psec/QUC/PY-2014_QUC_Inauguration_Piljin... · 2014-02-28 · quantum physics count states, gravity makes geometry, and superstring](https://reader034.vdocument.in/reader034/viewer/2022042400/5f0f4f6a7e708231d4438707/html5/thumbnails/50.jpg)
Kontsevich-Soibelman conjecture :a universal algebra for wall-crossing ?
![Page 51: algebra, geometry, and Schroedinger atomshome.kias.re.kr/psec/QUC/PY-2014_QUC_Inauguration_Piljin... · 2014-02-28 · quantum physics count states, gravity makes geometry, and superstring](https://reader034.vdocument.in/reader034/viewer/2022042400/5f0f4f6a7e708231d4438707/html5/thumbnails/51.jpg)
Kontsevich-Soibelman conjecture :a universal algebra for wall-crossing ?
![Page 52: algebra, geometry, and Schroedinger atomshome.kias.re.kr/psec/QUC/PY-2014_QUC_Inauguration_Piljin... · 2014-02-28 · quantum physics count states, gravity makes geometry, and superstring](https://reader034.vdocument.in/reader034/viewer/2022042400/5f0f4f6a7e708231d4438707/html5/thumbnails/52.jpg)
Kontsevich-Soibelman conjecture :a universal algebra for wall-crossing ?
![Page 53: algebra, geometry, and Schroedinger atomshome.kias.re.kr/psec/QUC/PY-2014_QUC_Inauguration_Piljin... · 2014-02-28 · quantum physics count states, gravity makes geometry, and superstring](https://reader034.vdocument.in/reader034/viewer/2022042400/5f0f4f6a7e708231d4438707/html5/thumbnails/53.jpg)
the “quantum degeneracy” of a given species of cycle / particle;non-zero if and only if such a cycle exists in the geometric sense
Kontsevich-Soibelman conjecture :a universal algebra for wall-crossing ?
![Page 54: algebra, geometry, and Schroedinger atomshome.kias.re.kr/psec/QUC/PY-2014_QUC_Inauguration_Piljin... · 2014-02-28 · quantum physics count states, gravity makes geometry, and superstring](https://reader034.vdocument.in/reader034/viewer/2022042400/5f0f4f6a7e708231d4438707/html5/thumbnails/54.jpg)
the “quantum degeneracy” of a given species of cycle / particle;non-zero if and only if such a cycle exists in the geometric sense
Kontsevich-Soibelman conjecture :a universal algebra for wall-crossing ?
![Page 55: algebra, geometry, and Schroedinger atomshome.kias.re.kr/psec/QUC/PY-2014_QUC_Inauguration_Piljin... · 2014-02-28 · quantum physics count states, gravity makes geometry, and superstring](https://reader034.vdocument.in/reader034/viewer/2022042400/5f0f4f6a7e708231d4438707/html5/thumbnails/55.jpg)
intersection number between the two cycles
Euler number of the moduli space of the cycle
mathematicians say, from 6d viewpoint
![Page 56: algebra, geometry, and Schroedinger atomshome.kias.re.kr/psec/QUC/PY-2014_QUC_Inauguration_Piljin... · 2014-02-28 · quantum physics count states, gravity makes geometry, and superstring](https://reader034.vdocument.in/reader034/viewer/2022042400/5f0f4f6a7e708231d4438707/html5/thumbnails/56.jpg)
Schwinger product
on [ a spin ½ + two spin 0 ] x [ angular momentum multiplet ]l
2nd helicity trace = the “number” of species of such particles
physicists say, from 4d viewpoint
![Page 57: algebra, geometry, and Schroedinger atomshome.kias.re.kr/psec/QUC/PY-2014_QUC_Inauguration_Piljin... · 2014-02-28 · quantum physics count states, gravity makes geometry, and superstring](https://reader034.vdocument.in/reader034/viewer/2022042400/5f0f4f6a7e708231d4438707/html5/thumbnails/57.jpg)
an infinite dimensional representation of Kontsevich-Soibelman wall-crossing algebra
![Page 58: algebra, geometry, and Schroedinger atomshome.kias.re.kr/psec/QUC/PY-2014_QUC_Inauguration_Piljin... · 2014-02-28 · quantum physics count states, gravity makes geometry, and superstring](https://reader034.vdocument.in/reader034/viewer/2022042400/5f0f4f6a7e708231d4438707/html5/thumbnails/58.jpg)
-+ sideside =
the conjecture : given the left-hand-side, the right-hand-side is entirely determined via the algebraic identity as follows
![Page 59: algebra, geometry, and Schroedinger atomshome.kias.re.kr/psec/QUC/PY-2014_QUC_Inauguration_Piljin... · 2014-02-28 · quantum physics count states, gravity makes geometry, and superstring](https://reader034.vdocument.in/reader034/viewer/2022042400/5f0f4f6a7e708231d4438707/html5/thumbnails/59.jpg)
=
for example,
![Page 60: algebra, geometry, and Schroedinger atomshome.kias.re.kr/psec/QUC/PY-2014_QUC_Inauguration_Piljin... · 2014-02-28 · quantum physics count states, gravity makes geometry, and superstring](https://reader034.vdocument.in/reader034/viewer/2022042400/5f0f4f6a7e708231d4438707/html5/thumbnails/60.jpg)
=
for example,
![Page 61: algebra, geometry, and Schroedinger atomshome.kias.re.kr/psec/QUC/PY-2014_QUC_Inauguration_Piljin... · 2014-02-28 · quantum physics count states, gravity makes geometry, and superstring](https://reader034.vdocument.in/reader034/viewer/2022042400/5f0f4f6a7e708231d4438707/html5/thumbnails/61.jpg)
2000 Stern + P.Y.wall-crossing formula for simple magnetic charges; weak coupling regime
2002 Denefquiver dynamics representation of N=2 supergravity BH’s
2008 Kontsevich + Soibelman
2010/2011 Manschot + Pioline + Sengeneral n-particle conjecture for Quantum Mechanics Counting
2011 Lee+P.Y. / Kim+Park+Wang+P.Y.general n-particle solution to Quantum Mechanics Counting
2011 SenQuantum Mechanics Counting = Kontsevich-Soibelman Conjecture
![Page 62: algebra, geometry, and Schroedinger atomshome.kias.re.kr/psec/QUC/PY-2014_QUC_Inauguration_Piljin... · 2014-02-28 · quantum physics count states, gravity makes geometry, and superstring](https://reader034.vdocument.in/reader034/viewer/2022042400/5f0f4f6a7e708231d4438707/html5/thumbnails/62.jpg)
Atiyah-Singer Index Theorem ~ 1963Calabi-Yau ~ 1978
Calibrated Geometry ~ 1982(Harvey & Lawson)
.
.
.Homological Mirror Symmetry ~ 1994
(Kontsevich)....
Stability & Derived Category ~ 2000.....
Wall-Crossing Conjecture~ 2008(Conjecture by Kontsevich & Soibelman)
.
.
.
.
1975 ~ Bogomolnyi-Prasad-Sommerfeld (BPS)1977 ~ Supersymmetry1983 ~ Superstring Theory.1985 ~ Calabi-Yau Compactification1988 ~ Mirror Symmetry.1994 ~ Wall-Crossing Discovered
(Seiberg & Witten)1995 ~ Dirichlet Branes1998 ~ Wall-Crossing is Bound State
Dissociation (Lee & P.Y.).2001 ~ Wall-Crossing for Black Holes
(Denef)....2008 ~ Konstevich-Soibelman Explained
(Gaiotto & Moore & Neitzke).2011 ~ KS Wall-Crossing proved
via Quatum Mechanics Manschot , Pioline & Sen /Kim , Park, Wang & P.Y. / Sen
![Page 63: algebra, geometry, and Schroedinger atomshome.kias.re.kr/psec/QUC/PY-2014_QUC_Inauguration_Piljin... · 2014-02-28 · quantum physics count states, gravity makes geometry, and superstring](https://reader034.vdocument.in/reader034/viewer/2022042400/5f0f4f6a7e708231d4438707/html5/thumbnails/63.jpg)
quantum physics count states, gravity makes geometry, and superstring theory combines quantum & gravity
quantum mechanics “count” geometry via superstring theory
quantum mechanical proof of the Kontsevich-Soibelman conjecturewhich solves, partially, a 30-year-old geometry problem
![Page 64: algebra, geometry, and Schroedinger atomshome.kias.re.kr/psec/QUC/PY-2014_QUC_Inauguration_Piljin... · 2014-02-28 · quantum physics count states, gravity makes geometry, and superstring](https://reader034.vdocument.in/reader034/viewer/2022042400/5f0f4f6a7e708231d4438707/html5/thumbnails/64.jpg)
geometry as mathematical tools for modern physics
string theory as a physical tool for modern geometry