Investigation of Gauge/Gravity Correspondence Investigation of Gauge/Gravity Correspondence Including Higher Derivative Corrections Including Higher Derivative Corrections Yoshifumi Hyakutake (Osaka Univ. KEK) Based on arXiv:0811.3102 with Masanori Hanada, Jun Nishimura and Shingo Takeuchi and arXiv:0805.2005 ;JHEP 0807:066 with Kyosuke Hotta, Takahiro Kubota and Hiroaki Tanida,
1. Introduction 1. Introduction String theory offers an interesting framework to investigate gauge theory and gravity theory. Especially D-branes play important roles to connect these two theories. Typical example is realized by taking decoupling limit of D3-branes. Maldacena Supergravity approximation is valid when In order to investigate finite and region, we have to include higher derivative corrections to the supergravity. Gubser, Klebanov, Tseytlin
The purpose of my talk is to investigate the gauge/gravity correspondence including higher derivative corrections. Decoupling limit of D0-branes in type IIA : M5-branes wrapping on 4-cycles in Calabi-Yau 3-fold : After reducing to 3 dimensions
Plan Plan 1.Introduction 2.Higher derivative corrections in string theory 3. corrections to black hole thermodynamics from supersymmetric matrix quantum mechanics Masanori Hanada, YH, Jun Nishimura and Shingo Takeuchi 4.Brown-Henneaux’s canonical approach to topologically massive gravity Kyosuke Hotta, YH, Takahiro Kubota and Hiroaki Tanida 5.Summary
2. Higher Derivative Corrections in String Theory 2. Higher Derivative Corrections in String Theory Higher derivative corrections in string theories are considerably investigated in various ways Gross, Witten; Gross, Sloan • String scattering amplitude Grisaru, Zanon • Non linear sigma model • Superfield method • Duality • Noether’s method … and so on By combining all these results, we find that corrections start from order, and a part of bosonic terms in type IIA is written as SUGRA tree 1-loop
The complete structure of higher derivative terms will be determined by local supersymmetry. In fact, known terms can be derived completely. Ogushi, Hyakutake Local supersymmetry transformation (neglect flux dependence): Cancellation (neglect flux dependence): Solution is given by
Short Summary Short Summary • Higher derivative corrections start from order. • corrections contain topological term. Uplift to 11D 3-form Integrating out this part gives Gravitational Chern-Simons term in 3 dimensions.
3. corrections to black hole thermodynamics from 3. corrections to black hole thermodynamics from supersymmetric matrix quantum mechanics supersymmetric matrix quantum mechanics Let us consider the system of D0-branes in type IIA superstring theory, which provides a particularly simple example of gauge-gravity duality. D0-branes with open strings excited Decoupling limit Near horizon geometry of super matrix quantum non-extremal black 0-brane mechanics at finite temperature Strongly coupled Valid when In order to test the gauge-gravity duality, we need to know the gauge Hanada, Nishimura, Takeuchi theory at strongly coupled region. Anagnostopoulos, Hanada, Nishimura, Takeuchi Monte Carlo simulation by using a non-lattice regularization
Gravity theory Type IIA supergravity action Near horizon geometry of non-extremal black 0-brane Itzhaki, Maldacena, Sonnenschein Yankielowicz Hawking temperature, entropy and internal energy are calculated as
Supergravity approximation is valid when Let us take account of the correction to the supergravity. Then we should modify following things. action, solution, location of the horizon, temperature, entropy As a result, the internal energy is modified as Note that this result is understood by the dimensional analysis
Modifications Action : Solution : Horizon : temperature : entropy :
Gauge theory The worldvolume theory of D0-branes is given by the supersymmetric MQM defined by the action In the Monte Carlo simulation, we fix the gauge by the static diagonal gauge and introduce a UV cutoff Integration over the fermionic matrices yields a complex Pfaffian Without loss of generality, we set .
Monte Carlo results Slope 4.6 predicted by gravity 4 3 2 ln (7.41T 2.8 -E/N 2 ) 1 0 -1 -2 N=14, Λ =4 N=17, Λ =6 N=17, Λ =8 -3 -4 -1.0 -0.5 0.0 0.5 ln T The deviation of the internal energy from is plotted against the temperature in the log-log scale.
3.0 N=17, Λ =6 N=17, Λ =8 2.5 7.41T 2.8 7.41T 2.8 -5.58T 4.6 2.0 E/N 2 1.5 1.0 0.5 0.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 T Fitting the data within (with largest at each ) to , we obtain and . If we make a one-parameter fit with fixed, we obtain .
Short Summary Short Summary • Thermodynamic properties of the near horizon limit of non-extremal black 0-brane are studied including correction. • The power 4.6 is precisely reproduced by Monte Carlo data in gauge theory. is predicted. Gauge/Gravity correspondence is confirmed beyond supergravity approximation.
4. Brown 4. Brown- -Henneaux Henneaux’ ’s canonical approach to TMG s canonical approach to TMG Now we want to investigate the gauge/gravity correspondence including corrections. As mentioned before, there is a topological term which becomes gravitational Chern-Simons (GCS) term after the dimensional reduction. 3D Gravity + GCS = Topologically Massive Gravity (TMG) Deser, Jackiw The goal of this section is to generalize Brown-Henneaux’s canonical approach to TMG.
Equation of motion for TMG is expressed as Geometries which satisfy become solutions Global AdS 3 and BTZ black hole exist in TMG. global AdS 3 : BTZ black hole : Banados, Teitelboim, Zanelli
Thermodynamic entropy of BTZ BTZ black hole has inner and outer horizons : The entropy of the BTZ black hole is evaluated as Solodukhin; Tachikawa This is a thermodynamic entropy. Then it is natural to ask whether we can derive the above quantity from the statistical viewpoint.
Asymptotically symmetry group Brown, Henneaux Let us consider asymptotic behaviors of global AdS 3 and BTZ black hole at the boundary ( ). It is easy to see that those satisfy the following boundary condition. This b.c. is preserved under the coordinate transformations of Then Killing vector fields satisfy commutation relations of Thus the asymptotically AdS 3 spacetime is endowed with the 2D conformal symmetry on the boundary.
Hamiltonian formalism and central extension We want to evaluate the central extension of the Virasoro algebras in TMG. In order to do it, we execute the following procedure. A) Hamiltonian formalism. B) Calculate the variation of the Hamiltonian, and add surface term to obtain correct equations of motion. C) From this surface term, we obtain global charge. Possible to evaluate central charges.
A) Hamiltonian formalism. ADM decomposition of 3D metric. Then the Lagrangian of TMG is written as Canonical variables conjugate to and are given as : auxiliary fields Then Hamiltonian is constructed as
B) Add surface term to obtain correct equations of motion. Variations of the Hamiltonian is evaluated as Correct equations of motion can be obtained iff the total derivative part is cancelled. Thus we define a new generator for each Killing vector as Regge, Teitelboim where is defined so as to cancel the total derivative part.
The explicit expression is written as
C) Global charges are obtained by the surface term. defines a conserved quantity for each Killing vector . In particular, for and , in the background of BTZ black hole, mass and angular momentum can be obtained Algebraic structure of symmetric transformation group is given by the Poisson bracket of generators. The last term gives the central extension of the algebra
Now it is possible to evaluate mass and angular momentum of BTZ black hole, and central charges at the boundary. Mass Angular mom.
Central charges Thus we obtain left-right asymmetric central charges.
Statistical entropy of BTZ Note that correspond to the isometries . Therefore we obtain From Cardy's formula for counting the states in CFT, we obtain the statistical entropy for BTZ black holes. This agrees with the previous thermodynamic entropy. We have thus proven the agreement between the macroscopic entropy and the statistical entropy including higher derivative correction.
5. Summary 5. Summary Higher derivative corrections in string theory or M-theory are derived by imposing local 32 supersymmetry. Found only two candidates. Tree level : 1-loop : Gauge/gravity correspondence including corrections was tested by using Monte Calro simulation. AdS/CFT correspondence was confirmed in TMG. There is a warped AdS 3 vacuum in TMG. It is an interesting direction to test warped AdS 3 /CFT 2 correspondence. Li, Song, Strominger
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