old supersymmetry as new mathematics

Transcription

1 old supersymmetry as new mathematics PILJIN YI Korea Institute for Advanced Study with help from Sungjay Lee

2 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)

3 quantum and geometry glued by superstring theory when can we perform path integrals exactly? counting geometry with supersymmetric path integrals

4 quantum and geometry glued by superstring theory

5 Einstein

6 this theory famously resisted quantization, however

7 on the other hand, five superstring theories, with a consistent quantum gravity inside, live in 10 dimensional spacetime

8 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,

9 that is, more precisely,

10 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

11 a particle located somewhere in our visible space

12 a particle located somewhere in our visible space a wrapped brane in the hidden Calabi-Yau at that point

13 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

14 particle geometry..

15 particle geometry

16 particle geometry

17 quantum bound state of particle geometry

18 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

19 more generally, d-dimensional brane wrapping c-dimensional cycles (d-c) dimensional object in the noncompact spacetime

20 which also allows realization of an instanton in our visible space as wrapped string/brane in the hidden Calabi-Yau

21 d-dimensional brane wrapping d-dimensional supersymmetric cycles supersymmetric instanton in the noncompact spacetime

22 for example, string world-sheet wrapping 2-cycles

23 string world-sheet wrapping 2-cycles correct low energy effective theory in the remaining 4d spacetime

24 but life is never that simple: this picture translates to reliable physical facts only upon appropriate restrictions on both sides

25 Calabi-Yau manifold & calibrated 3-cycles

26 a geometer asks: for a Calabi-Yau, which topological 3-cycles can be calibrated?

27 a string theorist answers: quantum supersymmetric states calibrated 3-cycles

28 a string theorist answers: no quantum supersymmetric states no calibrated 3-cycles

29 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

30 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

31 when can we perform the relevant path integrals exactly?

32 a prototype: supersymmetric harmonic oscillators 32

33 supersymmetric harmonic oscillators 33

34 witten index = twisted partition function of supersymmetric harmonic oscillators 34

35 witten index = twisted partition function of supersymmetric harmonic oscillators bosonic fermionic 35

36 witten index = twisted partition function of supersymmetric harmonic oscillators fermionic bosonic 36

37 how do we get the same thing from path integral? 37

38 how do we get the same thing from path integral? 38

39 boson 39

40 boson 40

41 fermion 41

42 fermion 42

43 combining the two 43

44 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

45 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?

46 trivially because free of interactions because of supersymmetry because of the simplicity of the quantity being computed

47 or, more interestingly, because free of interactions because of supersymmetry because of the simplicity of the quantity being computed

48 so why was the simple computation possible, really? hopeless 48

49 so why was the simple computation possible, really? perhaps 49

50 so why was the simple computation possible, really? perhaps 50

51 so why was the simple computation possible, really? bosonic fermionic 51

52 so why was the simple computation possible, really? formally independent of this parameter 52

53 so why was the simple computation possible, really? formally independent of this parameter whereby the path integral becomes one-loop exact 53

54 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

55 example : nonlinear sigma model for Euler index 55

56 example : nonlinear sigma model for Euler index 56

57 example : nonlinear sigma model for Euler index 57

58 example : nonlinear sigma model for Euler index 58

59 Euler index 59

60 counting geometry with supersymmetric path integrals

61 counting rational Gromov-Witten invariant 2 with S partition function of d=2 GLSM

62 rational Gromov-Witten invariants are encoded in the low energy effective metric of CY3 moduli

63 which can be sometimes counted via mirror symmetry mirror symmetry between a pair of CY3

64 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

65 localization in a nutshell

66 localization in a nutshell supersymmetry + localization claims = original path integral which is fully quantum deformed path integral which is Gaussian

67 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

68 2 localization for S partition function of 2d (2,2) theories Benini & Cremonesi; Doroud,Gomis,Le Floch, & Sungay Lee

69 2 localization for S partition function of 2d (2,2) theories A twisted flat cylinder anti-a twisted Jaume Gomis & Sungay Lee

70 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

71 which can also do all examples with known mirror pair =

72 counting special Lagrangian submanifolds of CY3 1 with S partition function of d=1 GLSM

73 1d N=4 Gauged Linear Sigma Models gauge fields FI constants for U(1) s chiral matter LG /NLSM NLSM/LG

74 wall-crossing of special Lagrange 3-cycles in CY 3-fold

75 depends on the precise shape of the CY 3-fold

76 Kachru + McGreevy 1999 Denef 2002 quiver quantum mechanics

77 equivariant Witten index of d=1 N=4 GLSM K.Hori + H.Kim + P. Y. 2014

78 N=4, compact and geometric

79

80

81 then, a localization procedure produces in the end

82 K.Hori + H.Kim + P. Y the Jeffrey-Kirwan residue

83 K.Hori + H.Kim + P. Y the Jeffrey-Kirwan residue

84 the Jeffrey-Kirwan residue tagged by FI constant K.Hori + H.Kim + P. Y cf) Hwang + Kim + Kim + Park, Cordova + Shao 2014

85 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

86 N = 4 (3,1,1) triangle quiver

87 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)

88 what else?

89 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)

90 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)