QUANTUM ALGEBRAIC CURVE FOR N=4 SYM

Nikolay Gromov

Based on works with V.Kazakov, S.Leurent, D.Volin 1305.1939 F. Levkovich-Maslyuk, G. Sizov 1305.1944

IGST 2013Utrecht, Netherlands

Historical overview

According to Beisert, Kazakov, Sakai and Zarembo, we can map a classical string motion to an 8-sheet Riemann surface

Bohr-Sommerfeld quantization condition:

Classical spectral curve

ABA – Large Length spectrum

[Minahan, Zarembo;Beisert, Staudacher;Beisert, Hernandez, Lopez;Beisert, Eden, Staudacher]

[N.G., Kazakov, VieiraBombardelli, Fioravanti, TateoN.G., Kazakov, Kozak, Vieira

Arutynov, Frolov]

Thermodynamic Bethe Ansatz

Quantum spectral curve

The system reduced to 4+5 functions:

Analytical continuation to the next sheet:

Quadratic branch cuts:

Definition of the quantum curve

• Any T-function can be found• Any Y-function can be found• Y-functions automatically satisfy TBA equations!

In particular:

Relation to TBA

encodes anomalous dimension

Useful: [Cavaglia, Fioravanti, Tateo]

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

The correct combination has an asymptotic

Good for positive integer S, but is obviously symmetric S -> -1-S. So cannot givea singularity at S=-1 [Janik (1,2 loops); NG, Kazakov (1 loop)]

Asymptotic

There are two independent solutions of sl2 Baxter equation (see Janik’s talk):

[Derkachov,Korchemsky,Manashov 2003]

Near BPS limit – can solve analytically:

1.

2.

3.

Main simplification are small

Derivation

Remember the algebraic constraint:

[NG. Sizov, Valatka, Levkovich-Maslyuk in prog.]

All are trivial

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]

Solution

Example2: Slope-to-Slope

Small P’s imply small discontinuity of mu:

Next order

Small P’s imply small discontinuity of mu:

Result: [NG. Sizov, Valatka, Levkovich-Maslyuk in prog.]

Dressing phase!

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

Our result contains an essential part of the dressing phase:

Dressing phase, asymptotic limit

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

[NG., Kazakov, Leurent, Volin to appear]

is an analog of Baxter equation from which ABA follows as an analyticityCondition.

Example3: Wilson line with cusp

21

[Drukker, Forrini 2011]

Which is in fact log derivative of expectation value of a circular WL [Ericson, Semenoff, Zarembo 2000; Drukker, Gross 2000; Pestun 2010]

For L=0 the result is known from localization: [Corea,Maldacena,Sever 2012]

Near BPS limit – can solve analytically:

1.

2.

3.

Main simplification are small All are trivial

[NG, Sizov, Levkovich-Maslyuk 2013]

The R-charges of the state are encoded in the asymptotics

Angle and the energy are in the coefficients of the expansion

Is a polynomial of degree L

Hilbert transform of the r.h.s.

For L=0, is a constant and

Quantum Classical

[Valatka, Sizov 2013]

Wave functionIn 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 wavefunction is complitely factorized

A natural conjecture that for AdS/CFT the wave function can be build in terms of Ps and fermonic Qs

This construction is known explicitly in some cases

[Sklyanin 1985; Smirnov 1998; Lukyanov 2000]

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 Pin 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