Final Report Summary - STRING (String theory and noncommutative geometry)
This project has studied various algebraic-geometric aspects of string theory and its implications for gravity, cosmology and gauge field theories, in particular investigating how different kinds of algebraic structures (e.g. Lie groups, Hopf algebras, supersymmetry) act as symmetries of different physical models. The aim is to determine the physical properties of these systems by analyzing the representation theory of their symmetry algebras/groups and viceversa to get insight on the differential-geometric structure of the algebras/groups through the spaces they act on.
A primary focus has been the mapping of the geometry of exceptional Lie groups. One of the first main results of this project is a method to parametrize the group manifold [arXiv:0906.0121] which for the compact forms generalizes the Euler angles for SU(2), while for the non compact forms it is an application of the Iwasawa decomposition. It is a technique based on the fibration structure and as such it gives information on the quotient space as well. I have used this method to study the calorons related to the deconfinement phase transition in gauge theories with exceptional Lie groups. Our techniques have dramatically reduced the computing time necessary to run Montecarlo simulations of lattice gauge theories with exceptional Lie groups as a symmetry.
Non compact forms of exceptional Lie groups describe the U-duality of supergravity theories (SUGRA) as well as the stabilizers of the vector multiplets' scalar manifold. With S. Ferrara and B. Zumino we have applied this knowledge of their geometry to classify in terms of the invariant of the group different types of black holes orbits with respect to the attractor mechanism in 4 [arXiv:0902.3973] as well as in 5 dimensions [arXiv:1006.3101].
In the past few years there has been a strong development in supergravity, due to the unexpected discovery that N=8 supergravity in 4 dimensions is finite in the ultraviolet up to four loops [arXiv:1103.1848] while the N=4 theory is finite up to 3 loops [arXiv:1209.2472] which is not a priori guaranteed by supersymmetry, where a valid counterterm could in principle exist.
Since perturbative finiteness of N = 8 d=4 SUGRA is possible only because the E7 symmetry is anomaly-free, a verification of the claims that it is strong enough to make the theory perturbatively finite to every order [arXiv:1009.1135] requires calculations analogous to instanton computations, using the global properties of the group manifold. Since the coset representative directly enters in the Lagrangian of the corresponding supergravity theory, the Iwasawa parametrization provides a way to choose the fields so that they enter in a polynomial way in the Lagrangian. In collaboration with S. Ferrara, S. Cacciatori, A. Marrani we have analyzed these polynomials through the principal SL(2) subgroup [Kostant, Am.J.Math. 81 (1959) p. 973] which determines the structure of the nilpotent subalgebras and have been able to determine their maximal degrees for any faithful representation.
Moreover, we have computed the Iwasawa parametrization for the symmetric spaces E7(7)/(SU(8)/Z2) relevant for 4-dim. N=8 SUGRA [arXiv:1005.2231 arXiv:1202.3055] and E7(-25)/(E6xU(1)/Z3) relevant for N=2 exceptional SUGRA [arXiv:1201.6314 arXiv:1201.6667]. For the latter case we have also found a manifestly U-duality covariant frame, which is an analogue of the Calabi-Vesentini basis.
One of the ways to define exceptional Lie groups is as symmetries of rank 3 Jordan algebras by means of the Tits or Vinberg formula. These can then be arranged into remarkable arrays, known as Freudenthal-Tits magic squares, which highlight the various embeddings/branchings among the groups. Another important finding of this project is that we have constructed and classified [arXiv:1208.6153] all possible Magic Squares (MS's) related to Euclidean or Lorentzian rank-3 simple Jordan algebras, both on normed division algebras and split composition algebras and described their role for the truncation and reductions of the corresponding supergravity theories. Considering Lorentzian Jordan algebras automatically produces the real forms F4(-20) and E6(-14) as well, which cannot be constructed in this way otherwise. Our result has been applied in [arXiv:1301.4176] to explain the structure of the SUGRA multiplets as a double copy of the Super-Yang-Mills (SYM) ones, which is a first step towards understanding the remarkable ultraviolet behaviour of N=8 and N=4 SUGRA, which seems to be inherited from the integrability of N=4 SYM.
A major result of this project is the Mathematica program I have developed to explicitly compute the generators of the Lie algebras entering the magic square. It is one of the main tools which has allowed me to perform this analysis of the exceptional Lie groups. It can be used to construct the fundamental as well as the adjoint representations of all the real forms of the exceptional Lie groups, starting from G2 all the way up to E8. It is freely available on my webpage: http://www.mat.unimi.it/users/cerchiai/MathematicaProgram/(opens in new window)
The same mathematical structures relevant for SUGRA play a major role in quantum information theory [quant-ph/0609227], relating entanglement measures for qubits to black hole entropy, which in a certain case involves the quartic invariant of the Lie group E7. A further important outcome of this project is that with B. Van Geemen we have found [arXiv:1003.4255] a relatively straightforward manner in which qutrits lead to the Weyl group of E7. This result has been used in quantum information theory as a basis for finding error-free channels of transmission [arXiv:1009.1195] solving for those channels a problem which had been long open. Groups at the Institute for Quantum Computing in Waterloo and at Imperial (UK) are actively working on extending this to more channels. As a consequence our article has been selected by the American Physical Society and the American Institute of Physics for publication in the "Virtual Journal of Quantum Information", Jan 2011.
Another main line of research of this project is the study of applications of noncommutative geometry to physical systems, in particular to various kinds of gauge field theories. The discovery of a noncommutative nonassociative structure for closed strings [arXiv: 1106.0316] is a recent interesting result. A first step to understand it is the study of the existence of a map similar to the Seiberg-Witten map relating the Weyl-Moyal star product for open strings to the dual theory on the ordinary space-time [hep-th/9908142]. This could be most easily done through the cohomological approach I have developed in [hep-th/0105192] with B. Zumino, which corresponds to a BRST quantization and is based on the rigidity of the corresponding algebroid structure. Ambiguities of the Seiberg-Witten map may then explain the field redefinitions discussed e.g. in [arXiv:1211.6437] in the framework of non-geometric backgrounds. Combined with the noncommutative Cartan frame formalism I have developed in [math.QA/0002007] it could lead to a better algebraic understanding of the differential geometric structure of the theory.
A primary focus has been the mapping of the geometry of exceptional Lie groups. One of the first main results of this project is a method to parametrize the group manifold [arXiv:0906.0121] which for the compact forms generalizes the Euler angles for SU(2), while for the non compact forms it is an application of the Iwasawa decomposition. It is a technique based on the fibration structure and as such it gives information on the quotient space as well. I have used this method to study the calorons related to the deconfinement phase transition in gauge theories with exceptional Lie groups. Our techniques have dramatically reduced the computing time necessary to run Montecarlo simulations of lattice gauge theories with exceptional Lie groups as a symmetry.
Non compact forms of exceptional Lie groups describe the U-duality of supergravity theories (SUGRA) as well as the stabilizers of the vector multiplets' scalar manifold. With S. Ferrara and B. Zumino we have applied this knowledge of their geometry to classify in terms of the invariant of the group different types of black holes orbits with respect to the attractor mechanism in 4 [arXiv:0902.3973] as well as in 5 dimensions [arXiv:1006.3101].
In the past few years there has been a strong development in supergravity, due to the unexpected discovery that N=8 supergravity in 4 dimensions is finite in the ultraviolet up to four loops [arXiv:1103.1848] while the N=4 theory is finite up to 3 loops [arXiv:1209.2472] which is not a priori guaranteed by supersymmetry, where a valid counterterm could in principle exist.
Since perturbative finiteness of N = 8 d=4 SUGRA is possible only because the E7 symmetry is anomaly-free, a verification of the claims that it is strong enough to make the theory perturbatively finite to every order [arXiv:1009.1135] requires calculations analogous to instanton computations, using the global properties of the group manifold. Since the coset representative directly enters in the Lagrangian of the corresponding supergravity theory, the Iwasawa parametrization provides a way to choose the fields so that they enter in a polynomial way in the Lagrangian. In collaboration with S. Ferrara, S. Cacciatori, A. Marrani we have analyzed these polynomials through the principal SL(2) subgroup [Kostant, Am.J.Math. 81 (1959) p. 973] which determines the structure of the nilpotent subalgebras and have been able to determine their maximal degrees for any faithful representation.
Moreover, we have computed the Iwasawa parametrization for the symmetric spaces E7(7)/(SU(8)/Z2) relevant for 4-dim. N=8 SUGRA [arXiv:1005.2231 arXiv:1202.3055] and E7(-25)/(E6xU(1)/Z3) relevant for N=2 exceptional SUGRA [arXiv:1201.6314 arXiv:1201.6667]. For the latter case we have also found a manifestly U-duality covariant frame, which is an analogue of the Calabi-Vesentini basis.
One of the ways to define exceptional Lie groups is as symmetries of rank 3 Jordan algebras by means of the Tits or Vinberg formula. These can then be arranged into remarkable arrays, known as Freudenthal-Tits magic squares, which highlight the various embeddings/branchings among the groups. Another important finding of this project is that we have constructed and classified [arXiv:1208.6153] all possible Magic Squares (MS's) related to Euclidean or Lorentzian rank-3 simple Jordan algebras, both on normed division algebras and split composition algebras and described their role for the truncation and reductions of the corresponding supergravity theories. Considering Lorentzian Jordan algebras automatically produces the real forms F4(-20) and E6(-14) as well, which cannot be constructed in this way otherwise. Our result has been applied in [arXiv:1301.4176] to explain the structure of the SUGRA multiplets as a double copy of the Super-Yang-Mills (SYM) ones, which is a first step towards understanding the remarkable ultraviolet behaviour of N=8 and N=4 SUGRA, which seems to be inherited from the integrability of N=4 SYM.
A major result of this project is the Mathematica program I have developed to explicitly compute the generators of the Lie algebras entering the magic square. It is one of the main tools which has allowed me to perform this analysis of the exceptional Lie groups. It can be used to construct the fundamental as well as the adjoint representations of all the real forms of the exceptional Lie groups, starting from G2 all the way up to E8. It is freely available on my webpage: http://www.mat.unimi.it/users/cerchiai/MathematicaProgram/(opens in new window)
The same mathematical structures relevant for SUGRA play a major role in quantum information theory [quant-ph/0609227], relating entanglement measures for qubits to black hole entropy, which in a certain case involves the quartic invariant of the Lie group E7. A further important outcome of this project is that with B. Van Geemen we have found [arXiv:1003.4255] a relatively straightforward manner in which qutrits lead to the Weyl group of E7. This result has been used in quantum information theory as a basis for finding error-free channels of transmission [arXiv:1009.1195] solving for those channels a problem which had been long open. Groups at the Institute for Quantum Computing in Waterloo and at Imperial (UK) are actively working on extending this to more channels. As a consequence our article has been selected by the American Physical Society and the American Institute of Physics for publication in the "Virtual Journal of Quantum Information", Jan 2011.
Another main line of research of this project is the study of applications of noncommutative geometry to physical systems, in particular to various kinds of gauge field theories. The discovery of a noncommutative nonassociative structure for closed strings [arXiv: 1106.0316] is a recent interesting result. A first step to understand it is the study of the existence of a map similar to the Seiberg-Witten map relating the Weyl-Moyal star product for open strings to the dual theory on the ordinary space-time [hep-th/9908142]. This could be most easily done through the cohomological approach I have developed in [hep-th/0105192] with B. Zumino, which corresponds to a BRST quantization and is based on the rigidity of the corresponding algebroid structure. Ambiguities of the Seiberg-Witten map may then explain the field redefinitions discussed e.g. in [arXiv:1211.6437] in the framework of non-geometric backgrounds. Combined with the noncommutative Cartan frame formalism I have developed in [math.QA/0002007] it could lead to a better algebraic understanding of the differential geometric structure of the theory.