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12 Application to Superstring Theory
Fig. 12.7 Data for training. The x axis is the quark mass [GeV] and the y axis is the chiral
condensate [GeV 3 ], where the blue dots indicate positive data and the orange dots indicate negative
data
To map the data to asymptotic information of the scalar field φ in the asymptotic
AdS spacetime, we again use the AdS/CFT dictionary [139]. Scalar field theory in
the bulk is the φ 4 theory, and the potential is
V [φ] =
λ
4
φ
4 .
(12.18)
This coupling constant λ is also subject to learning (learned values must be positive).
Since the dimension of the chiral condensate operator qq is 3, the formula
O ≡ (d/2) +
d 2 /4 + m 2 R 2
(12.19)
gives the mass of the scalar field as m 2 = −3/R 2 . Here R is the AdS radius, which
is also the subject of learning. At the asymptotic boundary of AdS, h(η) ≈ 4/R, so
the equation of motion of the bulk scalar field there is
∂
2
η φ +
4
R
∂ η φ −
3
R 2 φ − λφ
3
= 0 .
(12.20)
The solution is as follows:
R
3/2 φ ≈ αe
−η/R
+ βe
−3η/R
−
λα 3
2R 2 η e
−3η/R .
(12.21)