Transcript Integrability in string theory

Nikolay Gromov
Based on works with V.Kazakov, S.Leurent, D.Volin 1305.1939
F. Levkovich-Maslyuk, G. Sizov 1305.1944
IGST 2013
Utrecht, Netherlands
Historical overview
Classical spectral curve
According to Beisert, Kazakov, Sakai and Zarembo, we can map a classical string motion
to an 8-sheet Riemann surface
Bohr-Sommerfeld quantization condition:
ABA – Large Length spectrum
[Minahan, Zarembo;
Beisert, Staudacher;
Beisert, Hernandez, Lopez;
Beisert, Eden, Staudacher]
Thermodynamic Bethe Ansatz
[N.G., Kazakov, Vieira
Bombardelli, Fioravanti, Tateo
N.G., Kazakov, Kozak, Vieira
Arutynov, Frolov]
Quantum spectral curve
Definition of the quantum curve
The system reduced to 4+5 functions:
Analytical continuation to the next sheet:
Quadratic branch cuts:
Relation to TBA
• Any T-function can be found
• Any Y-function can be found
• Y-functions automatically satisfy TBA equations!
Useful: [Cavaglia, Fioravanti, Tateo]
In particular:
encodes anomalous dimension
Global charges
Asymptotic can be read off from the relation to the quasi-momenta
General equation see Dima’s talk. For sl2 sector:
Having fixed asymptotic for P what are the possible asymptotic for mu:
We identify by comparing with TBA:
Example: Basso’s Slope function
Asymptotic
There are two independent solutions of sl2 Baxter equation (see Janik’s talk):
[Derkachov,Korchemsky,Manashov 2003]
Good for positive integer S, but is obviously symmetric S -> -1-S. So cannot give
a singularity at S=-1
[Janik (1,2 loops); NG, Kazakov (1 loop)]
The correct combination has an asymptotic
Derivation
Near BPS limit – can solve analytically:
[NG. Sizov, Valatka, Levkovich-Maslyuk in prog.]
1.
2.
3.
Main simplification
are small
Remember the algebraic constraint:
All
are trivial
Solution
The R-charges of the state are encoded in the asymptotics. For twist L=2:
Angle and the energy are in the coefficients of the expansion
[Basso]
Example2: Slope-to-Slope
Next order
Small P’s imply small discontinuity of mu:
Small P’s imply small discontinuity of mu:
Result:
Dressing phase!
[NG. Sizov, Valatka, Levkovich-Maslyuk in prog.]
Tests
Weak coupling we get:
In agreement with:
[Kotikov, Lipatov, Onishenko, Velizhanin]
[Moch, Vermaseren, Vogt] [Staudacher]
[Kotikov, Lipatov, Rej, Staudacher, Velizhanin]
[Bajnok, Janik, Lukowski]
[Lukowski, Rej, Velizhanin]
Tests
Strong coupling we get (so far only numerically):
Basso
Basso Folded
string
0-loop
Folded string
1-loop
[NG. Valatka]
Example3: ABA
Dressing phase, asymptotic limit
Our result contains an essential part of the dressing phase:
These integrals appear in our result.
In general we derive the ABA of Beisert-Eden-Staudacher in full generality from
System in asymptotic limit when
is an analog of Baxter equation from which ABA follows as an analyticity
Condition.
[NG., Kazakov, Leurent, Volin to appear]
Example3: Wilson line with cusp
[Drukker, Forrini 2011]
21
For L=0 the result is known from localization:
[Corea,Maldacena,Sever 2012]
Which is in fact log derivative of expectation value of a circular WL
[Ericson, Semenoff, Zarembo 2000; Drukker, Gross 2000; Pestun 2010]
Near BPS limit – can solve analytically:
[NG, Sizov, Levkovich-Maslyuk 2013]
1.
2.
3.
Main simplification
are small
All
are trivial
Hilbert transform of the r.h.s.
Is a polynomial of degree L
The R-charges of the state are encoded in the asymptotics
Angle and the energy are in the coefficients of the expansion
For L=0,
is a constant and
Quantum
Classical
[Valatka, Sizov 2013]
Wave function
In separated variables
Simple relation to the quasi-momenta:
exactly like:
Quantum
Classical
In integrable models it is possible to make a canonical transformation so that the wave
function is complitely factorized
This construction is known explicitly in some cases
[Sklyanin 1985; Smirnov 1998; Lukyanov 2000]
A natural conjecture that for AdS/CFT the wave function can be build in terms of
Ps and fermonic Qs
Measure could be complicated
Conclusions/Open questions
• Too soon to draw any conclusions
• New exact and simple formulation for all operators
• There should be a relation to the wave function in the
separated Skylanin variables
• Could a similar set of equations be found for correlation
functions? Relation to Thermodynamic Bubble Ansatz?
• Solve P𝜇 in different regimes – strong coupling
systematic expansion, BFKL…
• More observables can be studied (available on the
market already see Zoltan Bajnok talk)
• Systematic (diagrammatic?) all loop expansion around
BPS