Introduction Elliptic and Dressed Elliptic String Solutions Features of the Dressed Elliptic Strings Future Extensions Pohlmeyer Reduction, Dressing Method and Classical String Solutions on R × S 2 Georgios Pastras - NSCR Demokritos, Institute of Nuclear and Particle Physics based on arXiv:1805.09301 [hep-th], arXiv:1806.07730 [hep-th], arXiv:1903.01408 [hep-th] and arXiv:1903.01412 [hep-th] in collaboration with D. Katsinis and I. Mitsoulas HEP 2019 - Recent Developments in High Energy Physics and Cosmology - Athens, 19 April 2019 Georgios Pastras Pohlmeyer Reduction, Dressing and Classical Strings
Introduction Elliptic and Dressed Elliptic String Solutions Features of the Dressed Elliptic Strings Future Extensions Section 1 Introduction Georgios Pastras Pohlmeyer Reduction, Dressing and Classical Strings
Introduction Elliptic and Dressed Elliptic String Solutions Features of the Dressed Elliptic Strings Future Extensions Classical string solutions have shed light to several aspects of the holographic duality. The dispersion relations of the classical strings are related to the anomalous dimensions of operators in the dual CFT. 1 2 3 They also serve to develop some intuition on the dynamics of the classical system whose quantum version is the only known mathematically consistent theory of quantum gravity 1 S. Frolov and A. A. Tseytlin, Nucl. Phys. B 668, 77 (2003) [hep-th/0304255] 2 N. Beisert, J. A. Minahan, M. Staudacher and K. Zarembo, JHEP 0309, 010 (2003) [hep-th/0306139] 3 S. Frolov and A. A. Tseytlin, Phys. Lett. B 570, 96 (2003) [hep-th/0306143] Georgios Pastras Pohlmeyer Reduction, Dressing and Classical Strings
Introduction Elliptic and Dressed Elliptic String Solutions Features of the Dressed Elliptic Strings Future Extensions In this work We focus on strings propagating on R × S 2 , which are Pohlmeyer reducible to the sine-Gordon equation. We invert Pohlmeyer reduction and construct systematically the solutions with elliptic SG counterparts Then we perform a B¨ acklund transformation on the side of the SG equation and find new “dressed” string solutions The new solutions have several interesting features The dressed solution have interacting spikes. There are interesting interrelations between properties of the strings and their SG counterparts. The dressed solutions reveal the stability properties of their seeds. The energy and angular momenta of the dressed solutions have several qualitative features that could be detectable on the side of the boundary CFT. Georgios Pastras Pohlmeyer Reduction, Dressing and Classical Strings
Introduction Elliptic Solutions Elliptic and Dressed Elliptic String Solutions Properties of the Elliptic Solutions Features of the Dressed Elliptic Strings The Dressed Elliptic Solutions Future Extensions The Sine-Gordon Counterparts Section 2 Elliptic and Dressed Elliptic String Solutions Georgios Pastras Pohlmeyer Reduction, Dressing and Classical Strings
Introduction Elliptic Solutions Elliptic and Dressed Elliptic String Solutions Properties of the Elliptic Solutions Features of the Dressed Elliptic Strings The Dressed Elliptic Solutions Future Extensions The Sine-Gordon Counterparts The String action The action for strings propagating on R × S 2 , written as a Polyakov action is ∫︂ d 𝜊 + d 𝜊 − (︂ (︂ X − R 2 )︂)︂ ⃗ X · ⃗ S = T ( 𝜖 + X ) · ( 𝜖 − X ) + 𝜇 . The equations of motion are non linear and difficult to treat. 𝜖 + 𝜖 − X 0 = 0 ⇒ X 0 = f + 𝜊 + )︁ 𝜊 − )︁ (︁ + f − (︁ , X = − 1 (︂(︂ )︂ (︂ )︂)︂ 𝜖 + 𝜖 − ⃗ 𝜖 + ⃗ 𝜖 − ⃗ ⃗ X · X X . R 2 Additionally, the solution must satisfy the Virasoro constraints f ±′ )︁ 2 . (︂ )︂ (︂ )︂ 𝜖 ± ⃗ 𝜖 ± ⃗ (︁ X · X = Georgios Pastras Pohlmeyer Reduction, Dressing and Classical Strings
Introduction Elliptic Solutions Elliptic and Dressed Elliptic String Solutions Properties of the Elliptic Solutions Features of the Dressed Elliptic Strings The Dressed Elliptic Solutions Future Extensions The Sine-Gordon Counterparts The Pohlmeyer Reduction The system is integrable. A signature of the system’s integrability is the fact that can be reduced to an SSSG (symmetric space sine-Gordon), in our case the sine-Gordon equation itself 4 . We first take advantage of the diffeomorphism invariance and we select a linear gauge 𝜊 ± )︁ := m ± 𝜊 ± f ± (︁ and then we define as reduced field the angle between the vectors 𝜖 + ⃗ X and 𝜖 − ⃗ X (︂ )︂ (︂ )︂ := f + ′ f −′ cos 𝜚. 𝜖 + ⃗ 𝜖 − ⃗ X · X It is easy to show that the Pohlmeyer field obeys the sine-Gordon equation 𝜖 + 𝜖 − 𝜚 = 𝜈 2 sin 𝜚, where 𝜈 2 := − m + m − / R 2 . 4 K. Pohlmeyer, Commun. Math. Phys. 46, 207 (1976) Georgios Pastras Pohlmeyer Reduction, Dressing and Classical Strings
Introduction Elliptic Solutions Elliptic and Dressed Elliptic String Solutions Properties of the Elliptic Solutions Features of the Dressed Elliptic Strings The Dressed Elliptic Solutions Future Extensions The Sine-Gordon Counterparts Inversion of Pohlmeyer Reduction for Elliptic Solutions The Pohlmeyer reduction is a non-local and many-to-one mapping, making its inversion a non-trivial task. However, There is an advantage in finding a string solution given a solution of the reduced system; the equations of motion assume the form of linear differential equations. X = 𝜈 2 cos 𝜚⃗ 0 ⃗ 1 ⃗ − 𝜖 2 X + 𝜖 2 X , Using a solution of the reduced system that depends on only one world-sheet coordinate provides an extra advantage; these linear differential equations are solvable using separation of variables 5 , X i ( 𝜊 0 , 𝜊 1 ) := Σ i ( 𝜊 1 ) T i ( 𝜊 0 ) . − Σ i ′′ + (︂ (︂ 𝜊 1 + 𝜕 2 )︂ )︂ Σ i = 𝜆 i Σ i , 2 ℘ + x 1 T i = 𝜆 i T i . − ¨ 5 I. Bakas and G. Pastras, JHEP 1607 (2016) 070 [arXiv:1605.03920 [hep-th]] Georgios Pastras Pohlmeyer Reduction, Dressing and Classical Strings
Introduction Elliptic Solutions Elliptic and Dressed Elliptic String Solutions Properties of the Elliptic Solutions Features of the Dressed Elliptic Strings The Dressed Elliptic Solutions Future Extensions The Sine-Gordon Counterparts Elliptic Solutions of the SG Equation The solutions of the sine-Gordon equation that depend solely on one of the two variables are its elliptic solutions. They can be understood as the solutions of the equation of motion of the simple pendulum = − 1 (︃ + E )︃ 𝜊 0 − 𝜐 0 + 𝜕 2 ; g 2 ( E ) , g 3 ( E ) (︂ )︂ (︂ )︂ 𝜊 0 ; E cos 𝜚 0 2 ℘ . 𝜈 2 3 𝜚 0 E = − 9 𝜈 2 / 10 E = 0 4 𝜌 E = 9 𝜈 2 / 10 3 𝜌 E = 99 𝜈 2 / 100 2 𝜌 E = 𝜈 2 𝜌 E = 101 𝜈 2 / 100 𝜊 0 E = 5 𝜈 2 / 4 − 𝜌 E = 3 𝜈 2 / 2 Georgios Pastras Pohlmeyer Reduction, Dressing and Classical Strings
Introduction Elliptic Solutions Elliptic and Dressed Elliptic String Solutions Properties of the Elliptic Solutions Features of the Dressed Elliptic Strings The Dressed Elliptic Solutions Future Extensions The Sine-Gordon Counterparts The Elliptic String Solutions - Periodicity Without posting the details of the derivation, the Pohlmeyer reduction can be inverted in this case. In polar coordinates, the elliptic string solutions assume the form x 2 − ℘ ( a ) 𝜊 0 + R √︁ √︁ x 3 − ℘ ( a ) 𝜊 1 , t 0 / 1 = R √︄ 𝜊 0 / 1 + 𝜕 2 (︁ )︁ x 1 − ℘ cos 𝜄 0 / 1 = , x 1 − ℘ ( a ) x 1 − ℘ ( a ) 𝜊 1 / 0 − Φ (︂ )︂ 𝜊 0 / 1 ; a √︁ 𝜚 0 / 1 = − sgn ( Im a ) , where the quasi-periodic function Φ is defined as Φ ( 𝜊 ; a ) = − i 2 ln 𝜏 ( 𝜊 + 𝜕 2 + a ) 𝜏 ( 𝜕 2 − a ) 𝜏 ( 𝜊 + 𝜕 2 − a ) 𝜏 ( 𝜕 2 + a ) + i 𝜂 ( a ) 𝜊. Georgios Pastras Pohlmeyer Reduction, Dressing and Classical Strings
Introduction Elliptic Solutions Elliptic and Dressed Elliptic String Solutions Properties of the Elliptic Solutions Features of the Dressed Elliptic Strings The Dressed Elliptic Solutions Future Extensions The Sine-Gordon Counterparts The Elliptic String Solutions - Rigid Rotation - Spikes The elliptic string solutions can be written in the form f ( 𝜄, 𝜚 − 𝜕 t ) = 0 . Thus, the elliptic strings do not change shape with time. They just rotate as a rigid body. Writing down the Virasoro constraints in terms of the Pohlmeyer field, yields 2 = R 2 𝜈 2 cos 2 𝜚 2 = R 2 𝜈 2 sin 2 𝜚 ⃒ ⃒ ⃒ ⃒ ⃒ 𝜖 0 ⃗ ⃒ 𝜖 1 ⃗ X 2 , X 2 . ⃒ ⃒ ⃒ ⃒ ⃒ ⃒ Thus, whenever the Pohlmeyer field equals an integer multiple of 2 𝜌 , the derivative 𝜖 1 ⃗ X gets inverted and spikes emerge. The elliptic strings can be classified to four classes 6 , depending on which worldsheet coordinate the SG counterpart depends whether the SG counterpart is an oscillatory or librating pendulum solution 6 K. Okamura and R. Suzuki, Phys. Rev. D 75 (2007) 046001 [hep-th/0609026] Georgios Pastras Pohlmeyer Reduction, Dressing and Classical Strings
Introduction Elliptic Solutions Elliptic and Dressed Elliptic String Solutions Properties of the Elliptic Solutions Features of the Dressed Elliptic Strings The Dressed Elliptic Solutions Future Extensions The Sine-Gordon Counterparts The Elliptic String Solutions - Classification static oscillating static rotating counterpart counterpart translationally invariant oscillating translationally invariant rotating counterpart counterpart Georgios Pastras Pohlmeyer Reduction, Dressing and Classical Strings
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