The Coulomb Gas of Random Supergravities. David Marsh Cornell University
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1 The Coulomb Gas of Random Supergravities David Marsh Cornell University Based on: D.M., L. McAllister, T. Wrase, JHEP 3 (2012), 102, T. Bachlechner, D.M., L. McAllister, T. Wrase, to appear.
2 1. Random Supergravity 2. Coulomb gases and Random Matrix Theory 3. Supersymmetric vacua 4. de Sitter vacua
3 Statistical studies of the landscape Denef and Douglas, [hep-th/ ] JHEP 0405 (2004) 072, [hep-th/ ] JHEP 0503 (2005) 061.
4 Statistical studies of the landscape Denef and Douglas, [hep-th/ ] JHEP 0405 (2004) 072, [hep-th/ ] JHEP 0503 (2005) 061.
5 Random Supergravity At each critical point, evaluate: {W q, D a W q, D a D b W q,...} and {K q, a K q, a bk q,...}. Denef and Douglas, [hep-th/ ] JHEP 0405 (2004) 072, [hep-th/ ] JHEP 0503 (2005) 061.
6 Random Supergravity Fix diffeomorphism and Kähler invariance, K =0, K q a b = δ q a b, and construct the distributions: {W,W,...}, q=1 q=2 {D a W,D q=1 a W,...}, q=2 {D a D b W, D q=1 a D b W,...}, q=2... Denef and Douglas, [hep-th/ ] JHEP 0405 (2004) 072, [hep-th/ ] JHEP 0503 (2005) 061.
7 Random Supergravity Fix diffeomorphism and Kähler invariance, K =0, K q a b = δ q a b, and construct the distributions: {W q } Ω(µ, σ), {D a W q } Ω(µ, σ), {K a bc q } Ω(µ, σ), {K a bc d } Ω(µ, σ), q {D a D b W q } Ω(µ, σ), Denef and Douglas, [hep-th/ ] JHEP 0405 (2004) 072, [hep-th/ ] JHEP 0503 (2005) 061.
8 Random Supergravity These distributions can be computed in certain simple corners of the landscape (see e.g. Denef & Douglas), but for certain interesting questions, the details do not matter. prob Re[D a D b W ] Denef and Douglas, [hep-th/ ] JHEP 0405 (2004) 072, [hep-th/ ] JHEP 0503 (2005) 061.
9 Random Supergravity These distributions can be computed in certain simple corners of the landscape (see e.g. Denef & Douglas), but for certain interesting questions, the details do not matter. prob Re[D a D b W ] Denef and Douglas, [hep-th/ ] JHEP 0405 (2004) 072, [hep-th/ ] JHEP 0503 (2005) 061.
10 Random Supergravity Universality of random matrix theory ensures that questions about eigenvalues (and eigenvectors) only rely on the first few moments of the distribution. prob Re[D a D b W ]
11 Random Matrix Theory A random matrix model of a Hermitian matrix M can be defined through the partition function, Z = dm ab dm ab f 0 (M ab ), a,b=1 where the matrix elements are assumed to be independent and identically distributed (iid). Upon diagonalization and after integrating out the eigenvectors, the partition function can be written as, Z = dλ a f(λ 1,...,λ ), a=1 where f denotes the joint probability distribution of the eigenvalues.
12 Random Matrix Theory Expl: Wigner ensemble: M = M, f(λ 1,...,λ )=C exp β 2 1 2σ 2 i=1 λ 2 i 2 i<j ln λ i λ j.
13 Random Matrix Theory Expl: Wigner ensemble: M = M, f(λ 1,...,λ )=C exp β 2 1 2σ 2 i=1 λ 2 i 2 i<j ln λ i λ j.
14 Random Matrix Theory Expl: Wigner ensemble: M = M, f(λ 1,...,λ )=C exp β 2 1 2σ 2 i=1 λ 2 i 2 i<j ln λ i λ j.
15 Random Matrix Theory Expl: Wigner ensemble: M = M, f(λ 1,...,λ )=C exp β 2 1 2σ 2 i=1 λ 2 i 2 i<j ln λ i λ j.
16 Random Matrix Theory Expl: Wigner ensemble: M = M, f(λ 1,...,λ )=C exp β 2 1 2σ 2 i=1 λ 2 i 2 i<j ln λ i λ j. Expl: Wishart ensemble: M = XX, with X being L, f(µ 1,...,µ )=C exp β 2 1 σ 2 µ a 2 a=1 ln µ a µ b ξ a<b a=1 ln µ a, with ξ =(L +1 2 β ).
17 Random Matrix Theory Expl: Wigner ensemble: M = M, f(λ 1,...,λ )=C exp β 2 1 2σ 2 i=1 λ 2 i 2 i<j ln λ i λ j. Expl: Wishart ensemble: M = XX, with X being L, f(µ 1,...,µ )=C exp β 2 1 σ 2 µ a 2 a=1 ln µ a µ b ξ a<b a=1 ln µ a, with ξ =(L +1 2 β ).
18 Random Matrix Theory Expl: Wigner ensemble: M = M, f(λ 1,...,λ )=C exp β 2 1 2σ 2 i=1 λ 2 i 2 i<j ln λ i λ j. Expl: Wishart ensemble: M = XX, with X being L, f(µ 1,...,µ )=C exp β 2 1 σ 2 µ a 2 a=1 ln µ a µ b ξ a<b a=1 ln µ a, with ξ =(L +1 2 β ).
19 Random Matrix Theory Expl: Wigner ensemble: M = M, f(λ 1,...,λ )=C exp β 2 1 2σ 2 i=1 λ 2 i 2 i<j ln λ i λ j. Expl: Wishart ensemble: M = XX, with X being L, f(µ 1,...,µ )=C exp β 2 1 σ 2 µ a 2 a=1 ln µ a µ b ξ a<b a=1 ln µ a, with ξ =(L +1 2 β ).
20 Random Matrix Theory Expl: Wigner ensemble: M = M, f(λ 1,...,λ )=C exp β 2 1 2σ 2 i=1 λ 2 i 2 i<j ln λ i λ j. Expl: Wishart ensemble: M = XX, with X being L, f(µ 1,...,µ )=C exp β 2 1 σ 2 µ a 2 a=1 ln µ a µ b ξ a<b a=1 ln µ a, with ξ =(L +1 2 β ).
21 Random Matrix Theory Expl: Wigner ensemble: M = M, f(λ 1,...,λ )=C exp β 2 1 2σ 2 i=1 λ 2 i 2 i<j ln λ i λ j. Expl: Wishart ensemble: M = XX, with X being L, f(µ 1,...,µ )=C exp β 2 1 σ 2 µ a 2 a=1 ln µ a µ b ξ a<b a=1 ln µ a, with ξ =(L +1 2 β ). Expl: Altland-Zirnbauer CI: M = f(λ 1,...,λ )=C exp 1 2σ 2 0 Z Z 0 λ 2 i + i=1 with ln λ 2 i λ 2 j + i<j Z ab = Z ba, i=1 ln λ i.
22 Random Matrix Theory Expl: Wigner ensemble: M = M, f(λ 1,...,λ )=C exp β 2 1 2σ 2 i=1 λ 2 i 2 i<j ln λ i λ j. Expl: Wishart ensemble: M = XX, with X being L, f(µ 1,...,µ )=C exp β 2 1 σ 2 µ a 2 a=1 ln µ a µ b ξ a<b a=1 ln µ a, with ξ =(L +1 2 β ). Expl: Altland-Zirnbauer CI: M = f(λ 1,...,λ )=C exp 1 2σ 2 0 Z Z 0 λ 2 i + i=1 with ln λ 2 i λ 2 j + i<j Z ab = Z ba, i=1 ln λ i.
23 Random Matrix Theory Expl: Wigner ensemble: M = M, f(λ 1,...,λ )=C exp β 2 1 2σ 2 i=1 λ 2 i 2 i<j ln λ i λ j. Expl: Wishart ensemble: M = XX, with X being L, f(µ 1,...,µ )=C exp β 2 1 σ 2 µ a 2 a=1 ln µ a µ b ξ a<b a=1 ln µ a, with ξ =(L +1 2 β ). Expl: Altland-Zirnbauer CI: M = f(λ 1,...,λ )=C exp 1 2σ 2 0 Z Z 0 λ 2 i + i=1 with ln λ 2 i λ 2 j + i<j Z ab = Z ba, i=1 ln λ i.
24 Random Matrix Theory Expl: Wigner ensemble: M = M, f(λ 1,...,λ )=C exp β 2 1 2σ 2 i=1 λ 2 i 2 i<j ln λ i λ j. Expl: Wishart ensemble: M = XX, with X being L, f(µ 1,...,µ )=C exp β 2 1 σ 2 µ a 2 a=1 ln µ a µ b ξ a<b a=1 ln µ a, with ξ =(L +1 2 β ). Expl: Altland-Zirnbauer CI: M = f(λ 1,...,λ )=C exp 1 2σ 2 0 Z Z 0 λ 2 i + i=1 with ln λ 2 i λ 2 j + i<j Z ab = Z ba, i=1 ln λ i.
25 Random Matrix Theory Expl: Wigner ensemble: M = M, f(λ 1,...,λ )=C exp β 2 1 2σ 2 i=1 λ 2 i 2 i<j ln λ i λ j. Expl: Wishart ensemble: M = XX, with X being L, f(µ 1,...,µ )=C exp β 2 1 σ 2 µ a 2 a=1 ln µ a µ b ξ a<b a=1 ln µ a, with ξ =(L +1 2 β ). Expl: Altland-Zirnbauer CI: M = f(λ 1,...,λ )=C exp 1 2σ 2 0 Z Z 0 λ 2 i + i=1 with ln λ 2 i λ 2 j + i<j Z ab = Z ba, i=1 ln λ i.
26 Random Matrix Theory Expl: Wigner ensemble: Spectrum at large :
27 Random Matrix Theory Expl: Real Wishart ensemble: Spectrum at large = L-1 = 100:
28 Random Matrix Theory Expl: Altland-Zirnbauer CI: Spectrum at large :
29 Coulomb gases and Random Matrix Theory The joint pdf s all have logarithmic potentials, which corresponds to electrostatic repulsion in d=2. Z = dλ a f(λ 1,...,λ ), a=1 The partition function is that of a d=2 gas of charged classical particles confined to the real line and attracted to the origin by a quadratic or linear potential. We will soon report on how the Hessian matrix of =1 random supergravity (the Wigner+Wishart+Wishart model) can be understood as a particular Coulomb gas. F. Dyson, J. Math. Phys. 3, 140, (1962).
30 Supersymmetric vacua The Hessian of supersymmetric vacua is given by, H = 2 a b V 2 ā b V 2 ab V 2 āb V = Z Z WZ W Z ZZ 2 W 2, where, Z ab = Z ba = D a D b W.
31 Supersymmetric vacua The eigenvalues of the Hessian can be written exactly in terms of the eigenvalues of a complementary real Wishart matrix with L=+1, thus any question about the supersymmetric spectrum can be phrased as a question about a particular Wishart matrix. The spectrum depends on the relative size of the gravitino mass and the supersymmetric fermion masses: Z ab = D a D b W m susy Ẑ ab, where Ẑ ab = Ẑba Ω(0, 1/ ). T. Bachlechner, D.M., L. McAllister, T. Wrase, to appear (I).
32 The spectrum of supersymmetric vacua The spectrum consists of two branches. For m susy = 100 W : =5 m 2 BF = 9 4 W T. Bachlechner, D.M., L. McAllister, T. Wrase, to appear (I).
33 The spectrum of supersymmetric vacua The spectrum consists of two branches. For m susy = 10 W : =5 m 2 BF = 9 4 W T. Bachlechner, D.M., L. McAllister, T. Wrase, to appear (I).
34 The spectrum of supersymmetric vacua The spectrum consists of two branches. For m susy =2 W : =5 m 2 BF = 9 4 W T. Bachlechner, D.M., L. McAllister, T. Wrase, to appear (I).
35 The spectrum of supersymmetric vacua The spectrum consists of two branches. For m susy = W : =5 m 2 BF = 9 4 W T. Bachlechner, D.M., L. McAllister, T. Wrase, to appear (I).
36 The spectrum of supersymmetric vacua The spectrum consists of two branches. For m susy = 1 3 W : =5 m 2 BF = 9 4 W T. Bachlechner, D.M., L. McAllister, T. Wrase, to appear (I).
37 The spectrum of supersymmetric vacua The spectrum consists of two branches. For m susy = 1 5 W : =5 m 2 BF = 9 4 W T. Bachlechner, D.M., L. McAllister, T. Wrase, to appear (I).
38 The spectrum of supersymmetric vacua The spectrum consists of two branches. For m susy = 1 20 W : =5 m 2 BF = 2025 m 2 susy T. Bachlechner, D.M., L. McAllister, T. Wrase, to appear (I).
39 de Sitter vacua Finding metastable de Sitter vacua in string theory is hard [see e.g. talks by Shiu, McAllister, Grana, Rummel]. 1) Uplift from supersymmetric AdS. 2) Spontaneous supersymmetry breaking. T. Bachlechner, D.M., L. McAllister, T. Wrase, to appear (I).
40 de Sitter vacua Finding metastable de Sitter vacua in string theory is hard [see e.g. talks by Shiu, McAllister, Grana, Rummel]. 1) Uplift from supersymmetric AdS. 2) Spontaneous supersymmetry breaking [recall Liam s talk]. Aazami, Easther JCAP 0603 (2006) 013, Chen, Shiu, Sumitomo, Tye JHEP 1204 (2012) 026
41 de Sitter vacua Finding metastable de Sitter vacua in string theory is hard [see e.g. talks by Shiu, McAllister, Grana, Rummel]. 1) Uplift from supersymmetric AdS. 2) Spontaneous supersymmetry breaking [recall Liam s talk]. Aazami, Easther JCAP 0603 (2006) 013, Chen, Shiu, Sumitomo, Tye JHEP 1204 (2012) 026
42 de Sitter vacua Finding metastable de Sitter vacua in string theory is hard [see e.g. talks by Shiu, McAllister, Grana, Rummel]. 1) Uplift from supersymmetric AdS. 2) Spontaneous supersymmetry breaking [recall Liam s talk]. ot much is know in general about uplift potentials. In the past they have been modeled as real Wigner matrices, for most of the supersymmetric vacua however, we need only to assume that the uplift does not automatically cure all tachyons. Then we should study supersymmetric vacua with no or few tachyons. Aazami, Easther JCAP 0603 (2006) 013, Chen, Shiu, Sumitomo, Tye JHEP 1204 (2012) 026
43 Tachyon free supersymmetric vacua AdS vacua with m susy W typically have many BF-allowed tachyons, and these vacua can be hard to uplift to a metastable de Sitter vacua. =5 m susy = W : T. Bachlechner, D.M., L. McAllister, T. Wrase, to appear (I).
44 Tachyon free supersymmetric vacua AdS vacua with m susy W typically have many BF-allowed tachyons, and these vacua can be hard to uplift to a metastable de Sitter vacua. =5 m susy = W : Untypical supersymmetric vacua in this regime may still be tachyon free, but have a peculiar spectrum which can be computed by Coulomb gas techniques. =5 m susy = W : T. Bachlechner, D.M., L. McAllister, T. Wrase, to appear (I)
45 Tachyon free supersymmetric vacua The probability of such a fluctuation is given exactly by P (m 2 0) = exp ( 2 2 W 2 /m 2 susy). A. Edelman, SIAM J. Matrix Anal. Appl. 9 (Dec., 1988) For W m susy /, supersymmetric vacua without BFallowed tachyons are abundant. T. Bachlechner, D.M., L. McAllister, T. Wrase, to appear (I).
46 Tachyon free supersymmetric vacua Most tachyon free AdS vacua instead satisfy, m susy W. =5 m susy = 100 W : The expectation value of the smallest mass is of the order of m susy /, and these AdS vacua are not unlikely to be metastable after uplift to Minkowski space T. Bachlechner, D.M., L. McAllister, T. Wrase, to appear (I).
47 de Sitter vacua Summary: Spontaneous F-term supersymmetry breaking (Liam s talk): Generic regime: F m susy : Relative frequency of meta-stable critical points: p(m 2 min > 0) = e c 2
48 de Sitter vacua Summary: Spontaneous F-term supersymmetry breaking (Liam s talk): Approximately supersymmetric regime: F m susy : Relative frequency of meta-stable critical points: p(m 2 min > 0) = e c ln D.M., L. McAllister, T. Wrase, JHEP 3 (2012), 102, J. Bausch, to appear.
49 de Sitter vacua Summary: Supersymmetry breaking by uplifting : Regime I: m susy W : Relative frequency of tachyon free supersymmetric vacua: p(m 2 min > 0) = e 2 2 W 2 /m 2 susy AdS
50 de Sitter vacua Summary: Supersymmetry breaking by uplifting : Regime II: m susy W : Relative frequency of tachyon free supersymmetric vacua: p(m 2 min > 0) = O(1/2) AdS
51 Conclusions Random matrix theory offers powerful (and fun!) techniques to study a range of physically interesting questions.
52 Conclusions Random matrix theory offers powerful (and fun!) techniques to study a range of physically interesting questions. In vast regions of the string theory landscape, de Sitter vacua are exceedingly rare. Statistical studies can help in identifying more fertile regions of the landscape e.g. through decoupling.
53 Conclusions Random matrix theory offers powerful (and fun!) techniques to study a range of physically interesting questions. In vast regions of the string theory landscape, de Sitter vacua are exceedingly rare. Statistical studies can help in identifying more fertile regions of the landscape e.g. through decoupling. Thanks!
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