## Final Report Summary - STONE (String-gauge correspondence as a tool for Nature)

Work performed

During the period of the project, I worked on the following directions, which are sub-projects of the main project.

Integrability of gauge and string theories and holographic three-point functions of semiclassical states

Integrability has been the driving force behind the recent progress in studying the spectral problem in the context of the AdS/CFT correspondence, which concerns the important question of obtaining the exact spectrum of anomalous dimensions of gauge-invariant operators of the theory under examination. The best studied example involves the correspondence between N=4 Super Yang Mills (SYM) theory in four dimensions and type IIB superstring theory on AdS5XS5. Recently a novel explicit example of a gauge/string duality of the type AdS4/CFT3 has emerged and a great deal of work has been done in also examining the question of integrability in this case. The study of the spectral problem allowed us to compute the two-point correlation functions of gauge invariant operators from their anomalous dimension. To completely solve the theory, however, one should also know the set of all three-point correlation functions. Much progress in this direction has been done recently, starting with the work of Zarembo, which proposes a prescription for the computation of the holographic three-point functions in N=4 SYM theory in the case when two of the operators correspond to semi-classical states and one to a supergravity state. A very interesting question that I investigated is how to compare the strong and weak coupling results for the three-point correlation functions using the AdS/CFT correspondence for two semi-classical operators and one light chiral primary operator. This study has been recently performed between the tree-level three-point function with the two semi-classical operators described by coherent states, while on the string side the three-point function is found in the Frolov-Tseytlin limit. In this context, I computed the one-loop correction to the three-point function on the gauge theory side and compared this to the corresponding correction on the string theory side, finding that the corrections do not match.

In other publications, I found a new Penrose limit of AdS5×S5 giving the maximally supersymmetric pp-wave background with two explicit space-like isometries. This is an important missing piece of information in studying the AdS/CFT correspondence in certain subsectors. Our analysis completes the study of all possible pp-wave backgrounds that can be obtained as Penrose limits of the AdS5×S5 geometry. It also represents a further step in the investigation of matching the strongly coupled gauge theory and string theory in certain sectors, which are relevant for describing non-perturbative physics of type IIB string theory on AdS5×S5. In particular, where the Penrose limit giving one space-like isometry is useful for analysing the SU(2) sector of N=4 SYM, this new Penrose limit is instead useful for studying the SU(2|3) and SU(1, 2|3) sectors.

The new Penrose limit of AdS5×S5 that we found is also relevant for studying the finite temperature behaviour of AdS/CFT. It is conjectured that the confinement/deconfinement transition temperature of planar N=4 SYM on RxS3 is dual to the Hagedorn temperature of type IIB string theory on AdS5×S5. Using the Penrose limit, this was shown quantitatively to be true (see work by Orselli and Harmark) by matching the confinement/deconfinement temperature of planar N=4 SYM on RxS3 in a limit with R-charge chemical potentials to the Hagedorn temperature of type IIB string on the pp-wave background. We furthermore expect that these results could help in understanding more generally the behaviour of string theory above the Hagedorn temperature and to study the connection between gauge theory and black holes in AdS5×S5, since the decoupling limit to the SU(1; 2,3) subsector is relevant for studying (nearly) supersymmetric black holes in AdS5×S5. This is part of a work in progress.

In addition to the new Penrose limit of AdS5×S5, we also find a new Penrose limit of AdS4 × CP3 in the context of the newly proposed example of AdS/CFT correspondence relating type IIA string theory on AdS4 × CP3 and N=6 Superconformal field theory (ABJM theory). Here two different classes of Penrose limits have been found: one in which there are no explicit space-like isometries and another in which there are two explicit space-like isometries that makes it suitable for studying the SU(2) X SU(2) sector of ABJM theory. We found a new Penrose limit of AdS4 × CP3 background that gives a pp-wave background with one explicit space-like isometry.

In the context of the type IIA/ABJM correspondence recently proposed, we found the full interacting Lagrangian and Hamiltonian for quantum strings in a near plane wave limit of AdS4 × CP3. The leading curvature corrections give rise to cubic and quartic terms in the Lagrangian and Hamiltonian that we have computed in full. The Lagrangian is found as the type IIA Green-Schwarz superstring in the light-cone gauge employing a superspace construction with 32 grassmann-odd coordinates. The light-cone gauge for the fermions is nontrivial since it should commute with the supersymmetry condition. We provide a prescription to properly fix the kappa-symmetry gauge condition to make it consistent with a light-cone gauge.

Using an effective vertex method we explicitly derived the two-loop dilatation generator of ABJM theory in its SU(2)×SU(2) sector, including all non-planar corrections. Subsequently, we applied this generator to a series of finite length operators as well as to two different types of BMN operators. As in N=4 SYM, the finite length operators at the planar level are found to exhibit a degeneracy between certain pairs of operators with opposite parity - a degeneracy which can be attributed to the existence of an extra conserved charge and thus to the integrability of the planar theory. When non-planar corrections are taken into account, the degeneracies between parity pairs disappear, hinting at the absence of higher conserved charges. The analysis of the BMN operators resembles that of N=4 SYM. Additional non-planar terms appear for BMN operators of finite length but once the strict BMN limit is taken these terms disappear.

D-branes in thermal backgrounds

Another project that I worked on and that has started a new line of research involves a very recent proposal, formulated by me in collaboration with scientists from Denmark, Sweden and Italy, for a novel approach to black hole physics at finite temperature. This might lead to important and revolutionary results in the investigation and understanding of the quark-gluon plasma physics and related topics.

The main feature of the new method we propose is that it describes a brane probe consisting of a large number of coincident non-extremal D-branes, so that the probe is in thermal equilibrium with the background. We employed this new method to study the BIon solution of Callan and Maldacena in the background of hot flat space. Doing this, we found that the finite temperature BIon behaves qualitatively different to its zero-temperature counterpart.

A key observation is that the requirement of thermal equilibrium between the probe and the background changes the probe, not only globally but also locally, in that it changes the energy-momentum tensor in the equations of motion of the probe. This is due to the fact that the degrees of freedom living on the brane get heated up. While this is taken into account in our new method it is, however, not taken into account in the previously used approach for describing D-brane probes in thermal background, which employed the classical Dirac-Born-Infeld (DBI) action. Thus, the previously used approach does not provide an accurate description of D-brane probes in thermal backgrounds. Our method could thus potentially open up new insights and developments in the study of D-branes as probes of thermal backgrounds, particularly with applications to the AdS/CFT correspondence.

In the case of D3-branes, the extremal BIon solution either takes the form of an infinite spike with an F-string charge coming out of the D3-brane, describing a number of coincident F-strings ending on the D3-brane, or the form of a D3-brane and a parallel anti-D3-brane connected by a 'wormhole' with F-string charge, which we dubbed the brane-antibrane-wormhole configuration. The finite temperature BIon solution that we found is a generalisation of the brane-antibrane-wormhole configuration, which is found employing a non-extremal D3-F1 probe brane system curved in hot flat space. Note that, while the extremal BIon solution of the DBI action is found for a system with one D3-brane (and anti-D3-brane) in the regime of weak string coupling, the finite temperature BIon solution is instead found in a regime with N D3-branes (and anti-D3-branes) with N >>1 and at strong coupling.

By examining the structure of the phases of the brane-antibrane-wormhole configuration at finite temperature, we found that there are up to three available phases for a given temperature and separation between the D3-branes and anti-D3-branes, in contrast with the two available phases for the extremal BIon. We also investigated the possibility of constructing a finite temperature generalisation of the infinite spike configuration of the extremal BIon.

During the period of the project, I worked on the following directions, which are sub-projects of the main project.

Integrability of gauge and string theories and holographic three-point functions of semiclassical states

Integrability has been the driving force behind the recent progress in studying the spectral problem in the context of the AdS/CFT correspondence, which concerns the important question of obtaining the exact spectrum of anomalous dimensions of gauge-invariant operators of the theory under examination. The best studied example involves the correspondence between N=4 Super Yang Mills (SYM) theory in four dimensions and type IIB superstring theory on AdS5XS5. Recently a novel explicit example of a gauge/string duality of the type AdS4/CFT3 has emerged and a great deal of work has been done in also examining the question of integrability in this case. The study of the spectral problem allowed us to compute the two-point correlation functions of gauge invariant operators from their anomalous dimension. To completely solve the theory, however, one should also know the set of all three-point correlation functions. Much progress in this direction has been done recently, starting with the work of Zarembo, which proposes a prescription for the computation of the holographic three-point functions in N=4 SYM theory in the case when two of the operators correspond to semi-classical states and one to a supergravity state. A very interesting question that I investigated is how to compare the strong and weak coupling results for the three-point correlation functions using the AdS/CFT correspondence for two semi-classical operators and one light chiral primary operator. This study has been recently performed between the tree-level three-point function with the two semi-classical operators described by coherent states, while on the string side the three-point function is found in the Frolov-Tseytlin limit. In this context, I computed the one-loop correction to the three-point function on the gauge theory side and compared this to the corresponding correction on the string theory side, finding that the corrections do not match.

In other publications, I found a new Penrose limit of AdS5×S5 giving the maximally supersymmetric pp-wave background with two explicit space-like isometries. This is an important missing piece of information in studying the AdS/CFT correspondence in certain subsectors. Our analysis completes the study of all possible pp-wave backgrounds that can be obtained as Penrose limits of the AdS5×S5 geometry. It also represents a further step in the investigation of matching the strongly coupled gauge theory and string theory in certain sectors, which are relevant for describing non-perturbative physics of type IIB string theory on AdS5×S5. In particular, where the Penrose limit giving one space-like isometry is useful for analysing the SU(2) sector of N=4 SYM, this new Penrose limit is instead useful for studying the SU(2|3) and SU(1, 2|3) sectors.

The new Penrose limit of AdS5×S5 that we found is also relevant for studying the finite temperature behaviour of AdS/CFT. It is conjectured that the confinement/deconfinement transition temperature of planar N=4 SYM on RxS3 is dual to the Hagedorn temperature of type IIB string theory on AdS5×S5. Using the Penrose limit, this was shown quantitatively to be true (see work by Orselli and Harmark) by matching the confinement/deconfinement temperature of planar N=4 SYM on RxS3 in a limit with R-charge chemical potentials to the Hagedorn temperature of type IIB string on the pp-wave background. We furthermore expect that these results could help in understanding more generally the behaviour of string theory above the Hagedorn temperature and to study the connection between gauge theory and black holes in AdS5×S5, since the decoupling limit to the SU(1; 2,3) subsector is relevant for studying (nearly) supersymmetric black holes in AdS5×S5. This is part of a work in progress.

In addition to the new Penrose limit of AdS5×S5, we also find a new Penrose limit of AdS4 × CP3 in the context of the newly proposed example of AdS/CFT correspondence relating type IIA string theory on AdS4 × CP3 and N=6 Superconformal field theory (ABJM theory). Here two different classes of Penrose limits have been found: one in which there are no explicit space-like isometries and another in which there are two explicit space-like isometries that makes it suitable for studying the SU(2) X SU(2) sector of ABJM theory. We found a new Penrose limit of AdS4 × CP3 background that gives a pp-wave background with one explicit space-like isometry.

In the context of the type IIA/ABJM correspondence recently proposed, we found the full interacting Lagrangian and Hamiltonian for quantum strings in a near plane wave limit of AdS4 × CP3. The leading curvature corrections give rise to cubic and quartic terms in the Lagrangian and Hamiltonian that we have computed in full. The Lagrangian is found as the type IIA Green-Schwarz superstring in the light-cone gauge employing a superspace construction with 32 grassmann-odd coordinates. The light-cone gauge for the fermions is nontrivial since it should commute with the supersymmetry condition. We provide a prescription to properly fix the kappa-symmetry gauge condition to make it consistent with a light-cone gauge.

Using an effective vertex method we explicitly derived the two-loop dilatation generator of ABJM theory in its SU(2)×SU(2) sector, including all non-planar corrections. Subsequently, we applied this generator to a series of finite length operators as well as to two different types of BMN operators. As in N=4 SYM, the finite length operators at the planar level are found to exhibit a degeneracy between certain pairs of operators with opposite parity - a degeneracy which can be attributed to the existence of an extra conserved charge and thus to the integrability of the planar theory. When non-planar corrections are taken into account, the degeneracies between parity pairs disappear, hinting at the absence of higher conserved charges. The analysis of the BMN operators resembles that of N=4 SYM. Additional non-planar terms appear for BMN operators of finite length but once the strict BMN limit is taken these terms disappear.

D-branes in thermal backgrounds

Another project that I worked on and that has started a new line of research involves a very recent proposal, formulated by me in collaboration with scientists from Denmark, Sweden and Italy, for a novel approach to black hole physics at finite temperature. This might lead to important and revolutionary results in the investigation and understanding of the quark-gluon plasma physics and related topics.

The main feature of the new method we propose is that it describes a brane probe consisting of a large number of coincident non-extremal D-branes, so that the probe is in thermal equilibrium with the background. We employed this new method to study the BIon solution of Callan and Maldacena in the background of hot flat space. Doing this, we found that the finite temperature BIon behaves qualitatively different to its zero-temperature counterpart.

A key observation is that the requirement of thermal equilibrium between the probe and the background changes the probe, not only globally but also locally, in that it changes the energy-momentum tensor in the equations of motion of the probe. This is due to the fact that the degrees of freedom living on the brane get heated up. While this is taken into account in our new method it is, however, not taken into account in the previously used approach for describing D-brane probes in thermal background, which employed the classical Dirac-Born-Infeld (DBI) action. Thus, the previously used approach does not provide an accurate description of D-brane probes in thermal backgrounds. Our method could thus potentially open up new insights and developments in the study of D-branes as probes of thermal backgrounds, particularly with applications to the AdS/CFT correspondence.

In the case of D3-branes, the extremal BIon solution either takes the form of an infinite spike with an F-string charge coming out of the D3-brane, describing a number of coincident F-strings ending on the D3-brane, or the form of a D3-brane and a parallel anti-D3-brane connected by a 'wormhole' with F-string charge, which we dubbed the brane-antibrane-wormhole configuration. The finite temperature BIon solution that we found is a generalisation of the brane-antibrane-wormhole configuration, which is found employing a non-extremal D3-F1 probe brane system curved in hot flat space. Note that, while the extremal BIon solution of the DBI action is found for a system with one D3-brane (and anti-D3-brane) in the regime of weak string coupling, the finite temperature BIon solution is instead found in a regime with N D3-branes (and anti-D3-branes) with N >>1 and at strong coupling.

By examining the structure of the phases of the brane-antibrane-wormhole configuration at finite temperature, we found that there are up to three available phases for a given temperature and separation between the D3-branes and anti-D3-branes, in contrast with the two available phases for the extremal BIon. We also investigated the possibility of constructing a finite temperature generalisation of the infinite spike configuration of the extremal BIon.