Brane quantization · Brane quantization • Complexify (M, ) to (Y, ) • Define A-model on (Y,...

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Brane quantization Work in progress with E.Witten

Transcript of Brane quantization · Brane quantization • Complexify (M, ) to (Y, ) • Define A-model on (Y,...

Page 1: Brane quantization · Brane quantization • Complexify (M, ) to (Y, ) • Define A-model on (Y, ): symplectic form Im , B-field Re • Define A-brane Bcc: support Y, trivial CP

Brane quantizationWork in progress with E.Witten

Page 2: Brane quantization · Brane quantization • Complexify (M, ) to (Y, ) • Define A-model on (Y, ): symplectic form Im , B-field Re • Define A-brane Bcc: support Y, trivial CP

Quantization• Naive:

• Real phase space M -> Hilbert space H

• Real functions on M -> self-adjoint operators on H

• Slightly better:

• Real phase space M + pre-quantum line bundle L (curvature )

• Define path integral

[f , g] = �i~\{f, g}

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Page 3: Brane quantization · Brane quantization • Complexify (M, ) to (Y, ) • Define A-model on (Y, ): symplectic form Im , B-field Re • Define A-brane Bcc: support Y, trivial CP

The art of quantization

• Anomalies: operator algebra is deformed, symmetries broken, not unique

• Perturbative approximation: deformation quantization

• M = T*N (shifted) : half-densities on N, differential operators (twisted)

• M complex, (1,1) symplectic form: holomorphic sections of L

• General: quantize patches and hope they glue well…

Page 4: Brane quantization · Brane quantization • Complexify (M, ) to (Y, ) • Define A-model on (Y, ): symplectic form Im , B-field Re • Define A-brane Bcc: support Y, trivial CP

Brane quantization• Complexify (M, ) to (Y, )

• Define A-model on (Y, ): symplectic form Im , B-field Re

• Define A-brane Bcc: support Y, trivial CP bundle

• Define A brane B: support M, CP bundle wt curvature -Re

• H = Hom(B,Bcc)

• Holomorphic functions on Y quantized by Hom(Bcc,Bcc)

!

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Page 5: Brane quantization · Brane quantization • Complexify (M, ) to (Y, ) • Define A-model on (Y, ): symplectic form Im , B-field Re • Define A-brane Bcc: support Y, trivial CP

BBcc

f

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Page 6: Brane quantization · Brane quantization • Complexify (M, ) to (Y, ) • Define A-model on (Y, ): symplectic form Im , B-field Re • Define A-brane Bcc: support Y, trivial CP

Features

• Depends on Y

• Hyperkähler guarantees physical (4,4) 2d sigma model

• Bcc, B are ABA branes

• Sometimes (GL, cluster, category O) higher-dimensional UV completion

• Agrees with deformation quantization

Page 7: Brane quantization · Brane quantization • Complexify (M, ) to (Y, ) • Define A-model on (Y, ): symplectic form Im , B-field Re • Define A-brane Bcc: support Y, trivial CP

Quantizing Lagrangians

• Very naive: equations which define L annihilate state

• Slightly better: define path integral b.c.

• Either case: family of states from functions (sections) on L

• Deformation quantization story already rich!

|Li

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|L; fi

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Page 8: Brane quantization · Brane quantization • Complexify (M, ) to (Y, ) • Define A-model on (Y, ): symplectic form Im , B-field Re • Define A-brane Bcc: support Y, trivial CP

Lagrangian brane quantization

• Complexify L to complex Lagrangian X in Y

• Define A-brane S (BAA) supported on X (choose CP bundle)

• Hom(S,Bcc) x Hom(B,S) -> Hom(B,Bcc) produces states

• Left input: Hom(S,Bcc) ~ holomorphic sections on X.

• Right input: Hom(B,S) more mysterious.

|S; fL; fRi

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Page 9: Brane quantization · Brane quantization • Complexify (M, ) to (Y, ) • Define A-model on (Y, ): symplectic form Im , B-field Re • Define A-brane Bcc: support Y, trivial CP

BBcc

f

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Page 10: Brane quantization · Brane quantization • Complexify (M, ) to (Y, ) • Define A-model on (Y, ): symplectic form Im , B-field Re • Define A-brane Bcc: support Y, trivial CP

Features• Correspondence -> BAA interface -> operator

• Composition of correspondences -> composition of interfaces

• Interface compose classically (CP bundle?)

• Corners composition?

•

S

S

S’

Page 11: Brane quantization · Brane quantization • Complexify (M, ) to (Y, ) • Define A-model on (Y, ): symplectic form Im , B-field Re • Define A-brane Bcc: support Y, trivial CP

Complex quantization

• Can apply to

• Doubling trick: Hilbert space

• Action of holomorphic and anti-holomorphic algebra

• Quantization of complex Lagrangians in Y

• No mysterious corners

Y ⇢ Y ⇥ Y

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Hom(Bcc, Bcc)

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S

Bcc

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Bcc

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Page 12: Brane quantization · Brane quantization • Complexify (M, ) to (Y, ) • Define A-model on (Y, ): symplectic form Im , B-field Re • Define A-brane Bcc: support Y, trivial CP

Example: Higgs bundles• 4d origin

• Analytic GL

• H = half-densities on Bun[G,C]

• Commuting Hitchin Hamiltonians

• Integral Hecke operators

• Canonical corner

Hecke

Bcc

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Bcc

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Hamiltonian

Page 13: Brane quantization · Brane quantization • Complexify (M, ) to (Y, ) • Define A-model on (Y, ): symplectic form Im , B-field Re • Define A-brane Bcc: support Y, trivial CP

S-duality

• 4d dictionary

• B-model side

• H = real opers

• Wilson line operators spectrum:

• Other real branes?

(s, s)

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Wilson

Hamiltonian

Oper Anti-Oper

Page 14: Brane quantization · Brane quantization • Complexify (M, ) to (Y, ) • Define A-model on (Y, ): symplectic form Im , B-field Re • Define A-brane Bcc: support Y, trivial CP

Example: Flat connections, quantum Teichmuller

Bcc

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Hitchin Section

Aq

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Aq

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Theodore N. Tomaras- Brane-world evolution with brane-bulk energy exchange

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Geometric Quantization 1 Introduction - UGCSugcs.caltech.edu/~mainiero/articles/geom_quant_mainiero_hassan.pdf · Geometric Quantization 1 Introduction The aim of the geometric quantization

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Contents - fmt.if.usp.brgtlandi/courses/second-quantization-4.pdf · quantization. Second quantization is extremely powerful. We will learn in these notes to think about second quantization

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Brane Black Holes

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3) Vector Quantization (VQ) and Learning Vector Quantization (LVQ)

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Brane-World Gravity

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Metaplectic quantization

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String/Brane Cosmology

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6D Brane Cosmological Solutions

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Geometric Quantization 1 Introductionsites.ugcs.caltech.edu/~mainiero/articles/geom_quant_mainiero... · Geometric Quantization 1 Introduction The aim of the geometric quantization

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Dilaton gravity at the brane with general matter-dilaton ...€¦ · Dilaton gravity at the brane with general matter-dilaton coupling. introductionat the brane: effective Einstein-like

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