“Momentum” research team
Zoltán Bajnok, János Balog, Tamás Gombor#, Árpád Hegedűs, László Holló#, Minkyoo Kim, József Konczer#, Gábor Pusztai, Gábor Zsolt Tóth
Richard Feynman: “If you want to learn about nature, to appreciate nature, it is necessary to understand the language that she speaks in.”
According to our present knowledge the language of Nature is gauge theories: The electromagnetic, weak, strong and gravitational forces are the four fundamental interactions of Nature. The electromagnetic interaction and the weak interaction are unified by the SU(2)
× U(1) electro-weak quantum gauge theory and have been tested with very high precision.
The strong interaction is formulated by an SU(3) quantum gauge theory but tested analytically at high energies only. The gravitational interaction can be formulated as a classical gauge theory, but does not allow a satisfactory quantum field-theoretical formulation. Thus, although the language of Nature seems to be gauge theories, we do not have a satisfactory understanding of strongly interacting quantum gauge theories.
The primary aim of our community is to solve a strongly coupled 4D quantum gauge theory exactly. The simplest interacting 4D SU(N) gauge theory has the maximal amount of (super) symmetry and is considered to be the hydrogen atom of all gauge theories. This theory is invariant not only under scale transformations but also under conformal transformations, i.e.
coordinate transformations that preserve angles. The large amount of conformal symmetry completely fixes the coordinate dependence of the two- and three-point functions in terms of the scaling dimensions of the fields. The multiplicative constant of the two-point function can be absorbed into the normalization of fields, but the three-point (3pt) coupling is a highly nontrivial dynamical quantity. Its importance lies in the fact that all higher-point correlation functions can be expressed in terms of the scaling dimensions and these three-point couplings, recursively. The unexpected opportunity to calculate the scaling dimensions and 3pt couplings of this theory came from a holographic duality.
Holographic dualities connect gauge theories with string theories including gravity. In a broad sense, holography is an equivalence between gravity in a d+1 dimensional open curved space and a strongly-coupled d-dimensional gauge theory living on the boundary of this space, in a way that is reminiscent of an optical hologram which stores a 3D image on a 2D holographic plate. Although such relations seem surprising, many holographic correspondences have been conjectured by now, the best established being the one proposed by Juan Maldacena in 1998. Maldacena's holographic correspondence relates the maximally supersymmetric 4D quantum gauge theory to superstring/gravity theory on the maximally symmetric negatively curved 5D anti de Sitter (AdS) space. Anti de Sitter space is the simplest solution of Einstein's equation with a negative cosmological constant, and is nothing but the Lorentzian analog of the Bolyai-Lobachevsky plane.
# Ph.D. student
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The holographic correspondence is a kind of duality, as it connects strongly-coupled gauge theories to semi-classical string theory, and it relates the deeply quantum string theory (gravity) to perturbative gauge theory. This makes holography hard to prove – it remains a conjecture.
Heroic efforts have been undertaken in order to test Maldacena's conjecture. From the many case studies, the following consistent holographic dictionary has been set up: The energies of string states are related to the scaling dimensions of local gauge-invariant operators. The t' Hooft coupling of the gauge theory is proportional to the inverse of the string tension, while the number of colors is proportional to the inverse of the string coupling. In the planar (large color) limit, strings do not interact and one has to evaluate the string action on a two-dimensional cylindrically-shaped worldsheet to calculate its spectrum. The leading 1/N correction relates the 3pt couplings of the gauge theory to the amplitude of splitting a string into two other strings (see Figure 1).
Due to the high number of symmetries, the effective two-dimensional field theory, namely the worldsheet string theory, turned out to be integrable.
Consequently, the holographic description allows us to use tools and methods that were developed for two-dimensional integrable quantum field theories to study the four-dimensional maximally symmetric gauge theory and the ten dimensional quantum string theory including quantum gravity.
It is instructive to recall how the two point functions were solved using the integrable S-matrix bootstrap method, since we follow a similar strategy to determine the 3pt functions. First, integrability was shown both at weak and at strong couplings and exploited in calculating systematic expansions. As these approximations didn't have any overlapping domain, bootstrap started to play the leading role. Assuming integrability for any coupling, powerful functional equations were formulated for the scattering matrix allowing the complete solution. This infinite volume S-matrix was then used to calculate the polynomially (Asymptotic Bethe Ansatz) and exponentially small (Luscher type) finite size corrections.
These corrections were finally summed up by the Thermodynamic Bethe Ansatz (TBA) equations, which were nicely reformulated in a compact form via the quantum spectral curve.
The 3pt functions of the maximally supersymmetric 4D gauge theory are important for several reasons. First, they provide the missing fundamental conformal data for its complete solution.
Second, they correspond, on the string theory side, to the amplitude of the process in which one big string splits into two small ones, or alternatively when two small strings join into a big one (see Figure 1). Thus it provides the exact string interactions of the theory, which is required to develop the quantum theory of strings, namely string field theory (SFT). The amplitude is called the SFT vertex, which hasn't been calculated explicitly except for the flat background. There have been already developments using integrability to calculate the 3pt functions both at weak and at strong couplings, together with their systematic expansions, but no overlapping domains have been found. Thus, just as it was the case for the spectral problem, bootstrap approach started to play a role. In our two very recent publications we advertised the adaptation of the well developed 2D integrable form factor bootstrap program
Figure 1. Spacetime process in which a big string #3 splits into strings #1 and #2.
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to calculate the 3pt functions in the following way: We first decompactify the process in which a string of size L3 splits into two smaller strings of sizes L1 and L2 in two different ways by sending either L3 and L2 or L3 and L1 to infinity, see Figure 2. In this limit the finite size string serves as a non-local operator insertion and we derived restrictive functional relations for its matrix elements between the asymptotic states of the decompactified strings. These equations differ from the form factor equations of local operators due to the “missing space-time of the finite size string”, but reduce to them in the limit, when the small string shrinks to zero. This shrinking is a very singular limit, in which some nontrivial operator insertion can remain and the analysis requires a special care, what we started last year. In the case when both L1 and L2 are large the way that we can decompactify the problem in two alternative ways gives severe restrictions on the allowed solutions of the functional equations.
Figure 2. The figure shows the splitting of string #3 into strings #1 and #2 and the program for obtaining the finite volume string field theory vertex via two alternative decompactifications and systematically including finite size corrections.
Once all the amplitudes of the operator insertions are calculated in the decompactified case one has to take into account the finite size corrections. The polynomial corrections in the inverse of the sizes are related to the normalization of states and can be calculated from the Asymptotic Bethe Ansatz equations relying on the S-matrix. The exponentially small finite size corrections are not even known in the usual 2D integrable setting and we plan to advance into this direction in the future.
Grants
OTKA K 81461: Two dimensional quantum field theories and their applications (Z. Bajnok 2010-2015)
OTKA K 83267: Relativistic particle systems (J. Balog 2011-2015)
OTKA K 109312: Holographic solution to measure theories (Á. Hegedűs 2013-2015)
HoloGrav ESF Network: Holographic methods for strongly coupled systems (Z. Bajnok 2012-2016)
“Momentum” Program of the HAS (Z. Bajnok 2012-2017)
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International cooperation
MTA Hungarian-Japanese bilateral: Integrability in gauge gravity duality and strong coupling dynamics of gauge theory; Kyoto, Tokyo and Tsukuba (Z. Bajnok, 2013-2014)
TÉT French-Hungarian bilateral: Application of spin chains and super strings to study fundamental interactions: the integrability side of the AdS/CFT correspondence; Paris Saclay and ENS (J. Balog, 2013-2014)
MTA Hungarian-Polish bilateral: Gauge string duality and its applications; Krakow (Z. Bajnok 2013-2015)
Hungarian-Korean-Polish collaboration: On the Gauge-string duality, Seoul, Krakow (Z. Bajnok 2013-2014)
Long term visitors
Chaiho Rim, Seoul (Z. Bajnok, 2014.12.21-2015.02.31)
Publications
Articles
1. Bajnok Z, el Deeb O, Pearce PA: Finite-volume spectra of the Lee-Yang model. J HIGH ENERGY PHYS 2015:(4) Paper 073. 43 p. (2015)
2. Bajnok Z, Janik RA: String field theory vertex from integrability. J HIGH ENERGY PHYS 2015:(4) Paper 042. 50 p. (2015)
3. Hegedűs Á: Extensive numerical study of a D-brane, anti-D-brane system in AdS5CFT4. J HIGH ENERGY PHYS 2015:(4) Paper 107. 47 p. (2015)
4. Holló L, Jiang YF, Petrovskii A: Diagonal form factors and heavy-heavy-light three-point functions at weak coupling. J HIGH ENERGY PHYS (9) Paper 125. 37 p. (2015)
See also: R-A.7
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