Abstract
We consider possible discretizations for a gauge-fixed Green-Schwarz action of Type IIB superstring. We use them for measuring the action, from which we extract the cusp anomalous dimension of planar $\mathcal{N}=4$ SYM as derived from AdS/CFT, as well as the mass of the two $AdS$ excitations transverse to the relevant null cusp classical string solution. We perform lattice simulations employing a Rational Hybrid Monte Carlo (RHMC) algorithm and two Wilson-like fermion discretizations, one of which preserves the global $SO(6)$ symmetry of the model. We compare our results with the expected behavior at various values of $g=\frac{\sqrt{\lambda}}{4\pi}$. For both the observables, we find a good agreement for large $g$, which is the perturbative regime of the sigma-model. For smaller values of $g$, the expectation value of the action exhibits a deviation compatible with the presence of quadratic divergences. After their non-perturbative subtraction the continuum limit can be taken, and suggests a qualitative agreement with the non-perturbative expectation from AdS/CFT. Furthermore, we detect a phase in the fermion determinant, whose origin we explain, that for very small $g$ leads to a sign problem not treatable via standard reweigthing. The continuum extrapolations of the observables in the two different discretizations agree within errors, which is strongly suggesting that they lead to the same continuum limit. Part of the results discussed here were presented earlier in arXiv:1601.04670.
Highlights
We use them for measuring the action, from which we extract the cusp anomalous dimension of planar N = 4 SYM as derived from AdS/CFT, as well as the mass of the two AdS excitations transverse to the relevant null cusp classical string solution
The continuum extrapolations of the observables in the two different discretizations agree within errors, which is strongly suggesting that they lead to the same continuum limit
In appendix B we present simulations obtained with another fermionic discretization — see (B.1)–(B.2) — consistent only with lattice perturbation theory performed around vacua coinciding with one of six cartesian coordinates uM, M = 1, · · ·, 6 it breaks explicitly the SO(6) invariance of the model (again, the U(1) symmetry is broken down as in the previous case)
Summary
The AdS5 × S5 superstring “cusp” action, which describes quantum fluctuations above the null cusp background can be written after Wick-rotation as [8]. In (2.1), local bosonic (diffeomorphism) and fermionic (κ-) symmetries originally present in the Type IIB superstring action on AdS5 × S5 [56] have been fixed in a “AdS light-cone gauge” [54, 55]. The first one is the SU(4) ∼ SO(6) symmetry originating from the isometries of S5, which is unaffected by the gauge fixing Under this symmetry the fields zM change in the 6 representation (vector representation), the fermions {ηi, θi} and {ηi, θi} transform in the 4 and (fundamental and anti-fundamental) respectively, whereas the fields x and x∗ are neutral. Symmetry in the two AdS5 directions orthogonal to AdS3 (i.e. transverse to the classical solution) and contrary to the previous case, the fields x and x∗ are charged (with charges 1 and −1 respectively) while the zM are neutral. Where the second equivalence obviously ignores potential phases or anomalies
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