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IntroductionMain correspondence
DualitiesConclusions
Compactified strings and Quantum Mechanicson the Jacobian torus
Roberto Volpato
Dipartimento di Fisica G.Galileo and INFNUniversità di Padova
Miniworkshop “Supersymmetry, Supergravity, Superstrings”
Pisa, March 20 2007
Roberto Volpato Compactified strings and QM on the Jacobian
IntroductionMain correspondence
DualitiesConclusions
Outline
1 Introduction
2 Main correspondenceFree field theory on a Riemann surfaceQuantum mechanics on the Jacobian torus
3 DualitiesT-duality vs high/low temperature dualityTarget space vs worldsheetDimensionality vs topology
4 Conclusions
Roberto Volpato Compactified strings and QM on the Jacobian
IntroductionMain correspondence
DualitiesConclusions
Outline
1 Introduction
2 Main correspondenceFree field theory on a Riemann surfaceQuantum mechanics on the Jacobian torus
3 DualitiesT-duality vs high/low temperature dualityTarget space vs worldsheetDimensionality vs topology
4 Conclusions
Roberto Volpato Compactified strings and QM on the Jacobian
IntroductionMain correspondence
DualitiesConclusions
Main results
Bosonic free field theory
World-sheet: RS ofgenus g
Target space: S1 (Td )
Classical action of aninstantonic solution
QM of a free particle
Space: 2g dimensionaltorus
Finite temperature
Eigenvalue of theLaplacian
Smn = λmnyZcl(β) = Zstat(β)
Roberto Volpato Compactified strings and QM on the Jacobian
IntroductionMain correspondence
DualitiesConclusions
Main results
Bosonic free field theory
World-sheet: RS ofgenus g
Target space: S1 (Td )
Classical action of aninstantonic solution
QM of a free particle
Space: 2g dimensionaltorus
Finite temperature
Eigenvalue of theLaplacian
Smn = λmn
yZcl(β) = Zstat(β)
Roberto Volpato Compactified strings and QM on the Jacobian
IntroductionMain correspondence
DualitiesConclusions
Main results
Bosonic free field theory
World-sheet: RS ofgenus g
Target space: S1 (Td )
Classical action of aninstantonic solution
QM of a free particle
Space: 2g dimensionaltorus
Finite temperature
Eigenvalue of theLaplacian
Smn = λmnyZcl(β) = Zstat(β)
Roberto Volpato Compactified strings and QM on the Jacobian
IntroductionMain correspondence
DualitiesConclusions
Motivations
Immediate consequences
T-duality and High/Low Temperature duality
Worldsheet/Target Space duality [Giveon, Porrati, Rabinovici ’94]
Related topics
Dimensionality vs topology and negative curvaturetarget space [Silverstein ’05]
Spectral problem on Riemann surfaces
KMS states and non-commutative geometry[Connes, Marcolli, Ramachandran ’05]
Roberto Volpato Compactified strings and QM on the Jacobian
IntroductionMain correspondence
DualitiesConclusions
Free field theory on a Riemann surfaceQuantum mechanics on the Jacobian torus
Outline
1 Introduction
2 Main correspondenceFree field theory on a Riemann surfaceQuantum mechanics on the Jacobian torus
3 DualitiesT-duality vs high/low temperature dualityTarget space vs worldsheetDimensionality vs topology
4 Conclusions
Roberto Volpato Compactified strings and QM on the Jacobian
IntroductionMain correspondence
DualitiesConclusions
Free field theory on a Riemann surfaceQuantum mechanics on the Jacobian torus
Definitions
Notation
Riemann surface Σ of genus g
Symplectic basis of 1-cycles α1, . . . , αg , β1, . . . , βg with
#(αi , αj) = 0 = #(βi , βj) , #(αi , βj) = δij
Normalized basis of holomorphic 1-differentials∫αi
ωj = δij , Ωij =
∫βi
ωj
Roberto Volpato Compactified strings and QM on the Jacobian
IntroductionMain correspondence
DualitiesConclusions
Free field theory on a Riemann surfaceQuantum mechanics on the Jacobian torus
Definitions
Free field X on Σ with
Target space S1 = R/2πRZ
Action S[X ] =1
4πR2
∫Σ∂X ∂X
Partition function Z (β) =
∫(Σ,S1)
dX e−βS[X ]
Bosonic strings β = R2/α′
Roberto Volpato Compactified strings and QM on the Jacobian
IntroductionMain correspondence
DualitiesConclusions
Free field theory on a Riemann surfaceQuantum mechanics on the Jacobian torus
Classical solutions
Splitting
X = X cl + X q
Equation of motion
∂∂X cl = 0
Multivaluedness
X clmn(z + p · α+ q · β) = X cl
mn(z) + 2πR(m · q − n · p)ySolution
X clmn(z) =
πRi
[(m + nΩ)(Im Ω)−1
∫ z
ω − c.c.], m,n ∈ Zg
Roberto Volpato Compactified strings and QM on the Jacobian
IntroductionMain correspondence
DualitiesConclusions
Free field theory on a Riemann surfaceQuantum mechanics on the Jacobian torus
Partition function
S[X ] = S[X cl ] + S[Xq]yZ (β) = Zcl(β)Zq(β)
Smn ≡ S[X clmn] = π(m + nΩ)(Im Ω)−1(m + Ωn)
Zcl(β) =∑
m,n∈Zg
e−βSmn
Roberto Volpato Compactified strings and QM on the Jacobian
IntroductionMain correspondence
DualitiesConclusions
Free field theory on a Riemann surfaceQuantum mechanics on the Jacobian torus
The complex torus JΩ
JΩ = Cg/(Zg + ΩZg)
Hodge metric on JΩ ds2 =∑
(2 Im Ω)−1ij dzidzj
Laplacian on JΩ ∆ =∑
(2 Im Ω)ij∂
∂zi
∂
∂zj
Hamiltonian H =∆
2π
Partition function
Zstat(β) = Tr e−βH , β =1
kT
Roberto Volpato Compactified strings and QM on the Jacobian
IntroductionMain correspondence
DualitiesConclusions
Free field theory on a Riemann surfaceQuantum mechanics on the Jacobian torus
Statistical partition function
Eigenfunctions ψm,n = expπ[(m + nΩ)(Im Ω−1)z − c.c.]
Eigenvalues λm,n = π(m + nΩ)(Im Ω−1)(m + Ωn)yλmn = Smny
Zstat(β) =∑
m,n∈Zg
e−βλmn =∑
m,n∈Zg
e−βSmn = Zcl(β)
Roberto Volpato Compactified strings and QM on the Jacobian
IntroductionMain correspondence
DualitiesConclusions
T-duality vs high/low temperature dualityTarget space vs worldsheetDimensionality vs topology
Outline
1 Introduction
2 Main correspondenceFree field theory on a Riemann surfaceQuantum mechanics on the Jacobian torus
3 DualitiesT-duality vs high/low temperature dualityTarget space vs worldsheetDimensionality vs topology
4 Conclusions
Roberto Volpato Compactified strings and QM on the Jacobian
IntroductionMain correspondence
DualitiesConclusions
T-duality vs high/low temperature dualityTarget space vs worldsheetDimensionality vs topology
T-duality vs high/low temperature duality
Duality relation
Z (β) = β−gZ (1/β)
Free field theory on Σ ←→ QM on Jacobian torusy yR → Rsd/R T → Tsd/Ty y
T-duality ←→ High-Low T duality
Roberto Volpato Compactified strings and QM on the Jacobian
IntroductionMain correspondence
DualitiesConclusions
T-duality vs high/low temperature dualityTarget space vs worldsheetDimensionality vs topology
T-duality vs high/low temperature duality
Duality relation
Z (β) = β−gZ (1/β)
Free field theory on Σ
←→
QM on Jacobian torus
y yR → Rsd/R T → Tsd/Ty y
T-duality ←→ High-Low T duality
Roberto Volpato Compactified strings and QM on the Jacobian
IntroductionMain correspondence
DualitiesConclusions
T-duality vs high/low temperature dualityTarget space vs worldsheetDimensionality vs topology
T-duality vs high/low temperature duality
Duality relation
Z (β) = β−gZ (1/β)
Free field theory on Σ
←→
QM on Jacobian torusy
y
R → Rsd/R
T → Tsd/Ty yT-duality ←→ High-Low T duality
Roberto Volpato Compactified strings and QM on the Jacobian
IntroductionMain correspondence
DualitiesConclusions
T-duality vs high/low temperature dualityTarget space vs worldsheetDimensionality vs topology
T-duality vs high/low temperature duality
Duality relation
Z (β) = β−gZ (1/β)
Free field theory on Σ
←→
QM on Jacobian torusy
y
R → Rsd/R
T → Tsd/T
y
y
T-duality
←→ High-Low T duality
Roberto Volpato Compactified strings and QM on the Jacobian
IntroductionMain correspondence
DualitiesConclusions
T-duality vs high/low temperature dualityTarget space vs worldsheetDimensionality vs topology
T-duality vs high/low temperature duality
Duality relation
Z (β) = β−gZ (1/β)
Free field theory on Σ
←→
QM on Jacobian torusy yR → Rsd/R T → Tsd/Ty
y
T-duality
←→ High-Low T duality
Roberto Volpato Compactified strings and QM on the Jacobian
IntroductionMain correspondence
DualitiesConclusions
T-duality vs high/low temperature dualityTarget space vs worldsheetDimensionality vs topology
T-duality vs high/low temperature duality
Duality relation
Z (β) = β−gZ (1/β)
Free field theory on Σ
←→
QM on Jacobian torusy yR → Rsd/R T → Tsd/Ty y
T-duality
←→
High-Low T duality
Roberto Volpato Compactified strings and QM on the Jacobian
IntroductionMain correspondence
DualitiesConclusions
T-duality vs high/low temperature dualityTarget space vs worldsheetDimensionality vs topology
T-duality vs high/low temperature duality
Duality relation
Z (β) = β−gZ (1/β)
Free field theory on Σ ←→ QM on Jacobian torusy yR → Rsd/R T → Tsd/Ty y
T-duality ←→ High-Low T duality
Roberto Volpato Compactified strings and QM on the Jacobian
IntroductionMain correspondence
DualitiesConclusions
T-duality vs high/low temperature dualityTarget space vs worldsheetDimensionality vs topology
Toroidal compactification
Free FT on ΣΩ → T ≡ Rd/Λ, with metric Gab
Smn = πGab(ma + naΩ)(Im Ω)−1(mb + Ωnb)
= λmn
where m,n ∈ Λ⊗ Zg
QM on JΩ ⊗ T∗ = Cgd/ΛΩ ⊗ Λ∗
Laplacian ∆ = G−1ab (2 Im Ω)−1
ij∂
∂zai∂
∂zbj
Eigenfunctions ψmn = expπ[Gab(ma + naΩ)(Im Ω)−1zb − c.c.]
Roberto Volpato Compactified strings and QM on the Jacobian
IntroductionMain correspondence
DualitiesConclusions
T-duality vs high/low temperature dualityTarget space vs worldsheetDimensionality vs topology
Toroidal compactification
Free FT on ΣΩ → T ≡ Rd/Λ, with metric Gab
Smn = πGab(ma + naΩ)(Im Ω)−1(mb + Ωnb)
= λmn
where m,n ∈ Λ⊗ Zg
QM on JΩ ⊗ T∗ = Cgd/ΛΩ ⊗ Λ∗
Laplacian ∆ = G−1ab (2 Im Ω)−1
ij∂
∂zai∂
∂zbj
Eigenfunctions ψmn = expπ[Gab(ma + naΩ)(Im Ω)−1zb − c.c.]
Roberto Volpato Compactified strings and QM on the Jacobian
IntroductionMain correspondence
DualitiesConclusions
T-duality vs high/low temperature dualityTarget space vs worldsheetDimensionality vs topology
Toroidal compactification
Free FT on ΣΩ → T ≡ Rd/Λ, with metric Gab
Smn = πGab(ma + naΩ)(Im Ω)−1(mb + Ωnb) = λmn
where m,n ∈ Λ⊗ Zg
QM on JΩ ⊗ T∗ = Cgd/ΛΩ ⊗ Λ∗
Laplacian ∆ = G−1ab (2 Im Ω)−1
ij∂
∂zai∂
∂zbj
Eigenfunctions ψmn = expπ[Gab(ma + naΩ)(Im Ω)−1zb − c.c.]
Roberto Volpato Compactified strings and QM on the Jacobian
IntroductionMain correspondence
DualitiesConclusions
T-duality vs high/low temperature dualityTarget space vs worldsheetDimensionality vs topology
Target space vs worldsheet
FT on ΣΩ (genus g) → Jτ (2h-dim) with metric (Im τ)−1xyQM on R4gh/ΛΩ ⊗ Λτ with metric 1
2(Im Ω)−1 ⊗ (Im τ)−1 ⊗ 14
xyFT on Στ (genus h) → JΩ (2g-dim) with metric (Im Ω)−1
Target-space↔Worldsheet [Giveon, Porrati, Rabinovici ’94]
Roberto Volpato Compactified strings and QM on the Jacobian
IntroductionMain correspondence
DualitiesConclusions
T-duality vs high/low temperature dualityTarget space vs worldsheetDimensionality vs topology
Target space vs worldsheet
FT on ΣΩ (genus g) → Jτ (2h-dim) with metric (Im τ)−1xyQM on R4gh/ΛΩ ⊗ Λτ with metric 1
2(Im Ω)−1 ⊗ (Im τ)−1 ⊗ 14
xyFT on Στ (genus h) → JΩ (2g-dim) with metric (Im Ω)−1
Target-space↔Worldsheet [Giveon, Porrati, Rabinovici ’94]
Roberto Volpato Compactified strings and QM on the Jacobian
IntroductionMain correspondence
DualitiesConclusions
T-duality vs high/low temperature dualityTarget space vs worldsheetDimensionality vs topology
Target space vs worldsheet
FT on ΣΩ (genus g) → Jτ (2h-dim) with metric (Im τ)−1xyQM on R4gh/ΛΩ ⊗ Λτ with metric 1
2(Im Ω)−1 ⊗ (Im τ)−1 ⊗ 14
xyFT on Στ (genus h) → JΩ (2g-dim) with metric (Im Ω)−1
Target-space↔Worldsheet [Giveon, Porrati, Rabinovici ’94]
Roberto Volpato Compactified strings and QM on the Jacobian
IntroductionMain correspondence
DualitiesConclusions
T-duality vs high/low temperature dualityTarget space vs worldsheetDimensionality vs topology
Compactification on Riemann surfaces
Target space RD
Density of states
Target space Σ
Density of (winding) states
ρ(m) ∼ eA√
α′m√
D−2 ρ(m) ∼ e√
α′m
r(2g−2)l2s
VΣ
In the limit VΣ ∼ l2s
D = 2gyDimensionality ↔ Topology
Roberto Volpato Compactified strings and QM on the Jacobian
IntroductionMain correspondence
DualitiesConclusions
T-duality vs high/low temperature dualityTarget space vs worldsheetDimensionality vs topology
Compactification on Riemann surfaces
Target space RD
Density of states
Target space Σ
Density of (winding) states
ρ(m) ∼ eA√
α′m√
D−2 ρ(m) ∼ e√
α′m
r(2g−2)l2s
VΣ
In the limit VΣ ∼ l2s
D = 2gyDimensionality ↔ Topology
Roberto Volpato Compactified strings and QM on the Jacobian
IntroductionMain correspondence
DualitiesConclusions
T-duality vs high/low temperature dualityTarget space vs worldsheetDimensionality vs topology
Compactification on Riemann surfaces
In the limit VΣ ∼ l2s
D = 2gyDimensionality ↔ Topology
yConjecture
Target space is the Jacobian torus [McGreevy, Silverstein, Starr ’06]
Roberto Volpato Compactified strings and QM on the Jacobian
IntroductionMain correspondence
DualitiesConclusions
Outline
1 Introduction
2 Main correspondenceFree field theory on a Riemann surfaceQuantum mechanics on the Jacobian torus
3 DualitiesT-duality vs high/low temperature dualityTarget space vs worldsheetDimensionality vs topology
4 Conclusions
Roberto Volpato Compactified strings and QM on the Jacobian
IntroductionMain correspondence
DualitiesConclusions
Conclusions
2D bosonic free FT ↔ QM on higher dim. space
Instantonic action ↔ Energy eigenvalue
Compactification radius ↔ Temperature
T-duality ↔ High/Low Temperature duality
Target Space vs World sheet duality
Topology ↔ Dimensionality
Roberto Volpato Compactified strings and QM on the Jacobian
References
References
M. Matone, P. Pasti, S. Shadchin and R. Volpato,
Phys. Rev. Lett. 97 (2006) 261601
[arXiv:hep-th/0607133]
M. Matone, P. Pasti and R. Volpato,
work in progress.
Roberto Volpato Compactified strings and QM on the Jacobian