Complex Path Integrals and String Glasses

8/4/2019 Complex Path Integrals and String Glasses

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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