The following are brief notes on a variety of subjects related to open/closed string duality that weren't substantial enough to make it into their own papers.
Relation of c=1 to A model on CP^1 via Hurwitz numbers
In Complex matrix model duality I fleshed out an equality noticed earlier in the literature between certain correlation functions of half-BPS operators in N=4 SYM, encapsulated by a complex matrix model, and amplitudes of tachyons in the non-critical c=1 string compactified at the self-dual radius. I also showed how to express these amplitudes as sums over Hurwitz numbers that count holomorphic maps from the worldsheet to the sphere CP^1 with three branch points. Each bipartite Feynman diagram of the complex matrix model is a dessin d'enfant for a Belyi map.
One natural question to ask is: Is the c=1 string then related to the A model on CP^1?
- From c=1 to Hurwitz to localization on CP1 string moduli space details some partial answers. The Casimirs used in the expansion of the complex matrix model are related to the completed cycles that appear in the studes of the CP^1 A model by Okounkov and Pandharipande. Furthermore, as detailed in Section 4, the bipartite graphs of the complex matrix model are the same as those graphs that label fixed points in Kontsevich's localization procedure for the A model on CP^1.
Another natural question: If the c=1 string is some kind of half-BPS reduction of (free) N=4, can we embed it in some bigger model with full PSU(2,2|4) symmetry that might correspond to the zero radius limit of AdS_5 x S^5? One could for example try to write the c=1, R=1 string with its tachyons in terms of some coset model with PSU(1|1) symmetry and then extend it.
Discrete gauge/string duality from localization
- Feynman diagrams as fixed points in moduli space - a conjecture about how the discrete nature of gauge theories arises in their dual string theories. The idea is that a correlation function in the string theory reduces to an integral over some moduli space (e.g. of a punctured Riemann surface into some embedding space). A torus action on the embedding space then acts on the moduli space such that we can use localization. The fixed points of the torus action are labelled by the different Feynman diagrams, which are summed to get the corrrelation function in the gauge theory. The precedent is the relation of the c=1 non-critical string to the CP^1 A model to the complex matrix model, see above.
Open-closed-open duality
In a talk in 2010 Open-Closed-Open String Duality at the Second Jo'burg Workshop on String Theory, Gopakumar suggested that there might exist dualities between field theories which correspond to graph dualities of their Feynman diagrams. He called this open-open duality, in contrast to open-closed string duality. He explored an example for Hermitian matrix models. In Complex matrix model duality I explored implications for a matrix model with a single complex matrix and its relations to the c=1, R=1 string.
The reason this is interesting is that it might be easier to see how open-closed duality works from the open-open dual of a field theory.
The following notes extend the complex matrix model duality to a multiple-matrix model and one with spacetime dependence.
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Multiple fields and loop corrections - complex matrix model duality with multiple matrices and the effective one-loop interaction from N=4 SYM.
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Incorporating space-time - incorporating spacetime dependence give a dual matrix model similar to the IIB IKKT matrix model.
Lorentzian open-closed duality
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Open-closed duality in Lorentzian signature (to arrive soon) - The basic idea here is that in a Lorentzian target space the issue of whether you can interpret a cylindrical worldsheet as a loop of open string or a closed string propagating depends on your reference frame. For example, imagine the space around you and all the virtual particle vacuum bubbles. If we lift this to an open string theory, then these bubbles are virtual cylinders. But given any cylinder oriented in spacetime we can always boost to a frame in which the surface of simultaneity intersects the cylinder as a circle, i.e. a closed string. What does this mean in terms of things we can measure?
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Worldline instanton in a gravitational field (to arrive soon) - Idea here would be to repeat the worldline instanton calculation of the Schwinger effect by Affleck, Alvarez and Manton, lifted to open string theory by Bachas and Porrati, but in a strong gravitational field instead of a strong electric field. Could one reproduce the gravitational analogue of the Schwinger effect, i.e. Unruh/Hawking radiation? Would this have any interpretation in terms of open/closed string duality?
Mathematics problems
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U(n) to S_n - some notes on how one might systematically reduce representations of
U(n)to representations of its subgroupS_n, with examples. -
Schur-Weyl for the singleton of SO(2,4) - includes onshell fields.