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8/4/2019 Complex Path Integrals and String Glasses
http://slidepdf.com/reader/full/complex-path-integrals-and-string-glasses 1/1
Complex Path Integrals &String Glasses
Daniel Ferrante, Gerald Guralnik
High Energy Theory, Brown University
Z [ J] =
C
e S[φ] e J φDφ < ∞ : Integration Cycles
δφS[φ → − δ J]
Z [ J] = 0 : Boundary Conditions
Complexification of the Path Integral.
Riemann Sphere: Modular symmetry ∼ dualities.
φ, J: R-, C-, Matrix-, L ie Algebra-valued.
Matrix models: integrability (Lax pairs), M-theory (BFSS).
L ie Algebras: Group Field Theory (LQG, CDT, Spin Foams).
Quantum Phases: structure cnts
cycles
←−−→ Matrix ↔ Lie algebra.
Integro-Differential Problem
Wall-Crossing: Stokes Phenomena for SD Op.
SD Op: meromorphic connection on C-plane.
SD Eq: isomonodromic transfs of non-linear d
eq in C⇒ Stokes phenomena.
Hodge Theory: phase factors of meromorph
connections.
FPI ⇔ integral discriminant (Lagrangian ∼ Quadrics).
SUSY: ‘det’ as ‘exp’ of Grassmann variables.
Wall-Crossing
Linear Canonical Transformation (Modular Group SL(2,R)):
F bc d
[ J] = − e π J2 d/ b
∞
−∞
F[φ] e π ϕ2 / b e−2 π φ J / b
F 0
c d
[ J] =
d e π J2 c d F[ J d]
Special Cases:
• Fourier Transform (FPI):
bc d
=
0 1−1 0
;
• Fractional FT:
bc d
=
cos ω sin ω− sin ω cos ω
;
• Fresnel Transform:
bc d
=
1 Λ0 1
.
Range of
bc d
∈ SL(2,R) selects quantum phase.
Quantum Phase (integration cycle): Stokes phase factor.
Modular Transform: preserves Riemann Sphere.
Integral Transform
Schwinger-Dyson operator: SD = δφS[φ → − δ J].
Fredholm Kernel: K [ J, J] =
Ω[ J] Ω[ J] ∼ δ[ J− J].
Fredholm Integral: G[ J] =
K [ J, J] F[ J]D J ; SD[G] = F.
“Quantum Phase Coherent State quantization”.
Fredholm Theory Stratification of (super-)Quadrics ∼ Lagrangia
parameter dependent quadratic forms (2-tensor).
Strata: multiplicities of the eigenvalues.
Curvature of SD-connection (Jacobi fields, Euler charact
jumps at Wall-Crossings, Stokes phenomena.
Quantum Phases: filtration of the space of S
operators by multiplicity of vacuum state.
Quantum Phases
D0-brane with S[φ] = φ3 / 3.
Z [ J] =
C
e φ3 / 3 e J φDφ < ∞⇒ C:Arg(φ)=0,±2 π/ 3
∂2
J− J
Z [ J] = 0⇒ Z [ J] = Ai[ J] + b Bi[ J]
Quantum Phases: Ai and Bi.
C:
bc d
= 0 −2 π g
12π g
0
; F[φ] = e φ3 / 3
Example: Cubic Action
D0-brane with S[φ] = φ2 / 2 + g φ4 / 4, g = λ/ 2.
Z [ J] =
C
e (φ2 / 2+g φ4 / 4) e J φDφ < ∞⇒ C:Arg(φ)=0,±π/ 2
g ∂3
J+ ∂ J − J
Z [ J] = 0⇒ Z [ J]= U[g, J]+b V [g, J]+c W [g, J]
Quantum Phases (ground states wrt g): U, V and W .
C:
bc d
= −g −2 π
1−g2π
−1
; F[φ] = e φ4 / 4
Example: Quartic Action