Strings in curved spaces, integrability and duality

Lead Research Organisation: Imperial College London
Department Name: Physics

Abstract

One of the important directions of current research in theoretical physics of fundamental interactions is the study of AdS/CFT duality involving strings in curved backgrounds and gauge theories. Integrability plays prominent role in making possible to solve such models. The project is aimed at finding new examples of integrable models describing string motion in curved spaces. It will be explored if strings on certain conifolds may be integrable. Possible connections to AdS/CFT and string dualities will be investigated. Another direction will be a study of tensionless limits of string models in various curved backgrounds. Tensionless limit of strings in AdS should be related to free adjoint conformal models at the boundary, with U(N) Maxwell model being an example.

Publications

10 25 50

Studentship Projects

Project Reference Relationship Related To Start End Student Name
EP/N509486/1 01/10/2016 31/03/2022
1994333 Studentship EP/N509486/1 01/10/2017 30/09/2021 Nathaniel LEVINE
 
Description The spectrum of string theory on curved spaces in generally very hard to compute. A notable exception is integrable examples such as Ads5xS5 where integrability tools have allowed great progress in the solution of the theory. This motivates the general study of integrable 2d sigma models (a particular type of quantum field theory). A general question is which target space geometries correspond to integrable sigma models.

An initial question was whether strings on certain conifolds are integrable, which was quickly answered in the negative.

The first part of the project focussed on a possible way to "diagnose" integrability, by computing an S-matrix for scattering around the trivial vacuum. In general, the S-matrix carries clear signatures of integrability (factorization and no particle production). However the particles around this trivial vacuum are all massless, and since massless 2d scalars suffer from IR problems, we found that even the tree-level (classical) S-matrix suffers from IR singularities, which give ambiguous results. We studied various ways of resolving these ambiguities. We concluded that, possibly except for specially chosen scattering amplitudes that are ambiguity-free, the massless 2d S-matrix does not have the usual integrability properties, due to the IR ambiguities.

The second part of the project focussed on renormalizability as a possible diagnostic for integrability. It is a curious fact that classically integrable sigma models seem to be stable under the 1-loop RG flow, or "renormalizable", with only finitely many couplings running. No general proof is known, but we are not aware of any exceptions to this rule. We asked the question of whether this relationship extended further beyond the 1-loop order. We found indeed that it does extend to higher loops, provided the integrable target space geometries are supplemented with specific quantum corrections. In the case of the integrable lambda-model, we expressed the theory on a "tripled" configuration space, where no corrections are needed, and renormalizability follows automatically (at all orders) due to extra symmetries becoming manifest.

A third part uncovered a new and different link between integrability and the RG flow in the context of sigma-models with 'local couplings' depending on the 2d co-ordinates.
Starting from an integrable sigma-model (admitting a flat Lax connection), promoting its couplings to functions of 2d time and demanding that it still admits a Lax connection, we found in various examples that the time dependence is forced to follow the 1-loop RG flow of the original model.
This link between classical integrability and the 1-loop quantum RG flow is remarkable and, as yet, unexplained.
Such sigma-models depending on time according to the RG flow arise naturally in a certain class of string models through a gauge-fixing procedure. This suggests a new way to embed integrable models into string theory, and that these may be a large, new class of solvable string theories.
Exploitation Route Massless 2d S-matrix: Our findings clarified various confusions in the literature, so that in future people will be very careful when making conclusions -- particularly about integrability -- from massless 2d S-matrix computations, even at tree-level. The project may be taken forward by formalising the idea that certain specially chosen amplitudes are IR ambiguity-free (at tree-level), and so may be used as a diagnostic for integrability. It would be interesting to try to understand in general which amplitudes are not ambiguous, so that this could become a useful tool for classifying integrable sigma models.

Integrability and RG flow: Our findings may be relevant for anyone studying integrable sigma models beyond the 1-loop order. We suggested a particular choice of bare action with finite counterterms chosen so that the theory remains renormalizable beyond 1-loop. In the spirit of the conjectured relationship between integrability and renormalizability, this may imply that this bare action is the natural way to define the quantum integrable theory. I would like to explore this further in the future -- e.g. computing Ward identities for hidden integrable symmetries and arguing that our finite counterterms are serving the role of restoring integrability at the quantum level. Another open question is to understand the structure of the counterterms in generality, or even in a general class of integrable models such as the lambda-model.
We also clarified that the RG flow commutes with non-abelian duality (adding suitable quantum corrections to the non-abelian dual model), clearing up a long-standing confusion in the literature, and this should be useful for others in the future.
This work also has implications for string theory (in particular bosonic string theory), in the special case of a conformal sigma model -- which is then also renormalizable. The quantum corrections then become corrections required to preserve conformality at the quantum level beyond 1-loop.

Time-dependent sigma-models: It would be interesting to investigate the new and intriguing relation between integrability and RG flow in more examples -- e.g. exotic integrable sigma-models derived from affine Gaudin models, or even massive integrable theories like Toda theories. Other open questions include: embedding these time-dependent integrable models into the 4d Chern-Simons construction; lifting up the integrable structure to the corresponding conformal sigma-model with 2 extra scalars; explaining or deriving this new relationship.

Integrable GxG and GxG/H sigma-models: In ongoing work (soon to be published) with Arkady Tseytlin and Nat Levine, we are studying the relationship between integrability and RG flow in the context of sigma-models with GxG and GxG/H target spaces and left-invariant metrics. Integrable examples of such models have been derived from affine Gaudin models. We find that the special integrable loci are stable at the leading 1-loop order (for GxG this was already known, and for GxG/H it is a new result). At the 2-loop order, we find that integrable GxG models are automatically stable in a particular subtraction scheme, while the GxG/H models are not, indicating that they will require some additional finite counterterms to restore 2-loop RG stability.
Sectors Education,Other
URL https://www.imperial.ac.uk/people/n.levine17/research.html