Classical and Quantum Aspects of the String Double Sigma Model Franco Pezzella Classical and Quantum Aspects of Introduction and Motivation the String Double Sigma Model Hodge-Dual Symmetric Free Scalar Fields in 2D Double Sigma Franco Pezzella Model (Closed Strings) Duality INFN - Naples Division Symmetric Free Closed Strings STRINGY GEOMETRY - Mainz - September, 15th 2015 Quantization of the Double String Model Conclusion and Perspectives 1 / 44
Based on: Classical and Quantum Aspects of the String Double Sigma Model Franco F. P., arXiv:1503.01709. Pezzella L. De Angelis, G. Gionti, R. Marotta and F. P., JHEP 1404 Introduction and Motivation (2014) 171, arXiv:1312.7367. Hodge-Dual F. Rennecke, JHEP 1410 (2014) 69 . Symmetric Free Scalar Fields in 2D M. Duff, Nucl. Phys. B335 (1990) 610. Double Sigma A. A. Tseytlin, Phys.Lett. B242 (1990) 163-174 ; Nucl.Phys. Model (Closed Strings) B350 (1991) 395-440. Duality Symmetric R. Floreanini and R. Jackiw, Phys. Rev. Lett. 59 (1987) 1873. Free Closed Strings C. Hull, JHEP 0510 (2005) 065, hep-th/0406102 . Quantization of the Double String Model Conclusion and Perspectives 2 / 44
Plan of the talk Classical and Quantum Aspects of the String Double 1 Introduction and Motivation Sigma Model Franco Pezzella Introduction and Motivation Hodge-Dual Symmetric Free Scalar Fields in 2D Double Sigma Model (Closed Strings) Duality Symmetric Free Closed Strings Quantization of the Double String Model Conclusion and Perspectives 3 / 44
Plan of the talk Classical and Quantum Aspects of the String Double 1 Introduction and Motivation Sigma Model Franco Pezzella 2 Hodge-Dual Symmetric Free Scalar Fields Introduction in 2D and Motivation Hodge-Dual Symmetric Free Scalar Fields in 2D Double Sigma Model (Closed Strings) Duality Symmetric Free Closed Strings Quantization of the Double String Model Conclusion and Perspectives 3 / 44
Plan of the talk Classical and Quantum Aspects of the String Double 1 Introduction and Motivation Sigma Model Franco Pezzella 2 Hodge-Dual Symmetric Free Scalar Fields Introduction in 2D and Motivation Hodge-Dual Symmetric Free Scalar 3 Double Sigma Model (Closed Strings) Fields in 2D Double Sigma Model (Closed Strings) Duality Symmetric Free Closed Strings Quantization of the Double String Model Conclusion and Perspectives 3 / 44
Plan of the talk Classical and Quantum Aspects of the String Double 1 Introduction and Motivation Sigma Model Franco Pezzella 2 Hodge-Dual Symmetric Free Scalar Fields Introduction in 2D and Motivation Hodge-Dual Symmetric Free Scalar 3 Double Sigma Model (Closed Strings) Fields in 2D Double Sigma Model (Closed 4 Duality Symmetric Free Closed Strings Strings) Duality Symmetric Free Closed Strings Quantization of the Double String Model Conclusion and Perspectives 3 / 44
Plan of the talk Classical and Quantum Aspects of the String Double 1 Introduction and Motivation Sigma Model Franco Pezzella 2 Hodge-Dual Symmetric Free Scalar Fields Introduction in 2D and Motivation Hodge-Dual Symmetric Free Scalar 3 Double Sigma Model (Closed Strings) Fields in 2D Double Sigma Model (Closed 4 Duality Symmetric Free Closed Strings Strings) Duality Symmetric Free Closed 5 Quantization of the Double String Model Strings Quantization of the Double String Model Conclusion and Perspectives 3 / 44
Plan of the talk Classical and Quantum Aspects of the String Double 1 Introduction and Motivation Sigma Model Franco Pezzella 2 Hodge-Dual Symmetric Free Scalar Fields Introduction in 2D and Motivation Hodge-Dual Symmetric Free Scalar 3 Double Sigma Model (Closed Strings) Fields in 2D Double Sigma Model (Closed 4 Duality Symmetric Free Closed Strings Strings) Duality Symmetric Free Closed 5 Quantization of the Double String Model Strings Quantization of the Double 6 Conclusion and Perspectives String Model Conclusion and Perspectives 3 / 44
Reminding T-Duality Classical and Quantum T-duality is an old subject in string theory. It implies that in Aspects of the String Double many cases two different geometries for the extra-dimensions are Sigma Model physically equivalent. Franco Pezzella Introduction and Motivation Hodge-Dual Symmetric Free Scalar Fields in 2D Double Sigma Model (Closed Strings) Duality Symmetric Free Closed Strings Quantization of the Double String Model Conclusion and Perspectives 4 / 44
Reminding T-Duality Classical and Quantum T-duality is an old subject in string theory. It implies that in Aspects of the String Double many cases two different geometries for the extra-dimensions are Sigma Model physically equivalent. Franco Pezzella T-duality is a discrete symmetry. It implies that string physics at Introduction a very small scale cannot be distinguished from the one at a and Motivation large scale. It is also a clear indication that ordinary geometric Hodge-Dual Symmetric concepts can break down in string theory at the string scale. Free Scalar Fields in 2D Double Sigma Model (Closed Strings) Duality Symmetric Free Closed Strings Quantization of the Double String Model Conclusion and Perspectives 4 / 44
Reminding T-Duality Classical and Quantum T-duality is an old subject in string theory. It implies that in Aspects of the String Double many cases two different geometries for the extra-dimensions are Sigma Model physically equivalent. Franco Pezzella T-duality is a discrete symmetry. It implies that string physics at Introduction a very small scale cannot be distinguished from the one at a and Motivation large scale. It is also a clear indication that ordinary geometric Hodge-Dual Symmetric concepts can break down in string theory at the string scale. Free Scalar Fields in 2D Double Sigma In the simplest case of circular compactifications, T-duality is Model (Closed Strings) encoded, for bosonic closed strings, in the simultaneous transformations R ↔ α ′ / R and p a ↔ w a /α ′ under which Duality Symmetric X a = X a R , with w a playing the role of R ↔ ˜ Free Closed L + X a X a ≡ X a L − X a Strings momentum mode for ˜ X a . These transformations leave the mass Quantization of the Double spectrum invariant. String Model Conclusion and Perspectives 4 / 44
Reminding T-Duality Classical and Quantum T-duality is an old subject in string theory. It implies that in Aspects of the String Double many cases two different geometries for the extra-dimensions are Sigma Model physically equivalent. Franco Pezzella T-duality is a discrete symmetry. It implies that string physics at Introduction a very small scale cannot be distinguished from the one at a and Motivation large scale. It is also a clear indication that ordinary geometric Hodge-Dual Symmetric concepts can break down in string theory at the string scale. Free Scalar Fields in 2D Double Sigma In the simplest case of circular compactifications, T-duality is Model (Closed Strings) encoded, for bosonic closed strings, in the simultaneous transformations R ↔ α ′ / R and p a ↔ w a /α ′ under which Duality Symmetric X a = X a R , with w a playing the role of R ↔ ˜ Free Closed L + X a X a ≡ X a L − X a Strings momentum mode for ˜ X a . These transformations leave the mass Quantization of the Double spectrum invariant. String Model Conclusion and In toroidal compactifications (with constant backgrounds G µν Perspectives and B µν ) T-duality is described by O ( D , D ; Z ) transformations. 4 / 44
O(D,D) Duality in String Theory Classical and Quantum Already on the classical level the indefinite orthogonal group Aspects of the String Double O ( D , D ; R ) appears naturally in the Hamiltonian description of Sigma Model the usual bosonic string model. Franco Pezzella With ∗ the Hodge operator with respect to h = diag( − 1 , 1), the Introduction action is: and Motivation Hodge-Dual Symmetric S [ X ; G , B ] = T � � G ab ( X ) dX a ∧ ∗ dX b + B ab ( X ) dX a ∧ dX b � Free Scalar Fields in 2D 2 Double Sigma Model (Closed Strings) Duality Symmetric Free Closed Strings Quantization of the Double String Model Conclusion and Perspectives 5 / 44
O(D,D) Duality in String Theory Classical and Quantum Already on the classical level the indefinite orthogonal group Aspects of the String Double O ( D , D ; R ) appears naturally in the Hamiltonian description of Sigma Model the usual bosonic string model. Franco Pezzella With ∗ the Hodge operator with respect to h = diag( − 1 , 1), the Introduction action is: and Motivation Hodge-Dual Symmetric S [ X ; G , B ] = T � � G ab ( X ) dX a ∧ ∗ dX b + B ab ( X ) dX a ∧ dX b � Free Scalar Fields in 2D 2 Double Sigma Model (Closed Strings) Varying S with respect to X a yields the equation of motion: Duality Symmetric Free Closed Strings bc dX b ∧ ∗ dX c = 1 d ∗ dX a + Γ a 2 G am H mbc dX b ∧ dX c (1) Quantization of the Double String Model bc = 1 Conclusion and with H = dB and Γ a 2 G am ( ∂ b G mc + ∂ c G mb − ∂ m G bc ) the Perspectives coefficients of the Levi Civita connection. 5 / 44
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