Final Report Summary - D2DFT (Defects in two dimensional field theories)
One of the most outstanding problems of modern theoretical physics is to find a fully general and consistent theory of quantum gravity. Unfortunately, in strong contrast to the situation which was prevalent during the development of the 'Standard Model' of particle physics, we do not expect any quantum gravity regime to be in reach of current experimental techniques and therefore we cannot rely on experiments to guide us in the construction of the theory.
This means that we have to build it relying only on its self-consistency. That such a task should be possible at all is far from obvious. After all, the 'Standard Model' is just one member, singled out by experiments, of a potentially infinite number of quantum field theories. The main surprise which came out of string theory is that consistency arguments single out a unique and still largely unknown structure, M-theory, of which the various string theories and eleven dimensional supergravity are weakly coupled descriptions. In this context, it is crucial to identify and test systematically the consistency conditions that we expect quantum theories of gravity to satisfy.
In this project, we investigated a type of consistency conditions which has not yet been explored thoroughly: the cancellation of global gravitational anomalies in the effective field theory to which the quantum gravity theory reduces at low energies. Global gravitational anomalies are a subtle breaking of the invariance of the quantum field theory partition function under the action of the group of diffeomorphisms. Such a breaking impedes the gauging of the metric and in consequence cannot occur if the quantum field theory is obtained as a low energy limit of a consistent theory of quantum gravity. The cancellation of global gravitational anomalies is familiar to many theoretical physicists in the case of two-dimensional conformal field theories, where it goes under the name of modular invariance.
In the latter case, one often can find closed expressions for the partition function of the quantum field theory, what allows to study the anomaly with modular forms. This is not the case in general and we have to use more elaborate mathematical techniques. We see the partition function as the section of a line bundle over the space of metrics modulo diffeomorphisms, the anomaly bundle. Although the partition function is often too complicated to study directly, one can often define a natural connection on the anomaly bundle. By studying this connection, one can determine if the partition function can be defined as a true function over the space of metrics modulo diffeomorphisms. More precisely, all of the holonomies of the connection must be trivial. This is the condition of vanishing of the global gravitational anomaly.
In the supergravity, string theory and M-theory context, there are essentially two types of chiral quantum field theories which can give rise to gravitational anomalies. The first are chiral fermionic field theories, for which an anomaly formula has been known since the mid-80's. The other are self-dual abelian gauge field theories, for which a complete anomaly formula was not known.
Our contribution has been to determine a formula for the global gravitational anomaly of the self-dual field theories.
Mathematically, our derivation features a beautiful mix of different mathematical subjects: index theory, modular geometry and algebraic topology. There is no doubt that there is a rich mathematical structure underlying it and remaining to be uncovered. Some of these mathematical aspects have already been explored in a publication and yielded a new topological invariant of spin manifolds.
Physically, we have only begun to explore the many situations in which we can now check the cancellation of global gravitational anomalies and test our current understanding of supergravity, string theory and M-theory. We already managed to check the cancellation of global gravitational anomalies in a certain version of type IIB supergravity. Forthcoming work will explore the cancellation in the version of type IIB supergravity describing the low energy limit of the type IIB superstring, in six dimensional supergravities and for the worldvolume theories of the five-branes in M-theory, type IIA string theory and heterotic strings. Given all we already learned about quantum field theory, supergravity and string theory from anomalies, we can hope that the cancellation of global gravitational anomalies will provide us with further decisive insights into these theories and the nature of quantum gravity.
This means that we have to build it relying only on its self-consistency. That such a task should be possible at all is far from obvious. After all, the 'Standard Model' is just one member, singled out by experiments, of a potentially infinite number of quantum field theories. The main surprise which came out of string theory is that consistency arguments single out a unique and still largely unknown structure, M-theory, of which the various string theories and eleven dimensional supergravity are weakly coupled descriptions. In this context, it is crucial to identify and test systematically the consistency conditions that we expect quantum theories of gravity to satisfy.
In this project, we investigated a type of consistency conditions which has not yet been explored thoroughly: the cancellation of global gravitational anomalies in the effective field theory to which the quantum gravity theory reduces at low energies. Global gravitational anomalies are a subtle breaking of the invariance of the quantum field theory partition function under the action of the group of diffeomorphisms. Such a breaking impedes the gauging of the metric and in consequence cannot occur if the quantum field theory is obtained as a low energy limit of a consistent theory of quantum gravity. The cancellation of global gravitational anomalies is familiar to many theoretical physicists in the case of two-dimensional conformal field theories, where it goes under the name of modular invariance.
In the latter case, one often can find closed expressions for the partition function of the quantum field theory, what allows to study the anomaly with modular forms. This is not the case in general and we have to use more elaborate mathematical techniques. We see the partition function as the section of a line bundle over the space of metrics modulo diffeomorphisms, the anomaly bundle. Although the partition function is often too complicated to study directly, one can often define a natural connection on the anomaly bundle. By studying this connection, one can determine if the partition function can be defined as a true function over the space of metrics modulo diffeomorphisms. More precisely, all of the holonomies of the connection must be trivial. This is the condition of vanishing of the global gravitational anomaly.
In the supergravity, string theory and M-theory context, there are essentially two types of chiral quantum field theories which can give rise to gravitational anomalies. The first are chiral fermionic field theories, for which an anomaly formula has been known since the mid-80's. The other are self-dual abelian gauge field theories, for which a complete anomaly formula was not known.
Our contribution has been to determine a formula for the global gravitational anomaly of the self-dual field theories.
Mathematically, our derivation features a beautiful mix of different mathematical subjects: index theory, modular geometry and algebraic topology. There is no doubt that there is a rich mathematical structure underlying it and remaining to be uncovered. Some of these mathematical aspects have already been explored in a publication and yielded a new topological invariant of spin manifolds.
Physically, we have only begun to explore the many situations in which we can now check the cancellation of global gravitational anomalies and test our current understanding of supergravity, string theory and M-theory. We already managed to check the cancellation of global gravitational anomalies in a certain version of type IIB supergravity. Forthcoming work will explore the cancellation in the version of type IIB supergravity describing the low energy limit of the type IIB superstring, in six dimensional supergravities and for the worldvolume theories of the five-branes in M-theory, type IIA string theory and heterotic strings. Given all we already learned about quantum field theory, supergravity and string theory from anomalies, we can hope that the cancellation of global gravitational anomalies will provide us with further decisive insights into these theories and the nature of quantum gravity.