arXiv subjects

Robert de Mello Koch

Publications and source records attributed to Robert de Mello Koch.

At least 91 records · Page 5Linked to original sources

arXiv2014

Finite N Quiver Gauge Theory

At finite N the number of restricted Schur polynomials is greater than or equal to the number of generalized restricted Schur polynomials. In this note we study this discrepancy and explain its origin. We conclude that, for quiver gauge theories, in general, the generalized restricted Schur polynomials correctly account for the complete set of finite N constraints and they provide a basis, while the restricted Schur polynomials only account for a subset of the finite N constraints and are thus overcomplete. We identify several situations in which the restricted Schur polynomials do in fact account for the complete set of finite N constraints. In these situations the restricted Schur polynomials and the generalized restricted Schur polynomials both provide good bases for the quiver gauge theory. Finally, we demonstrate situations in which the generalized restricted Schur polynomials reduce to the restricted Schur polynomials.

hep-th↗

arXiv2014

Subleading corrections to the Double Coset Ansatz preserve integrability

In this article we compute the anomalous dimensions for a class of operators, belonging to the SU(3) sector of the theory, that have a bare dimension of order N. For these operators the large N limit and the planar limit are distinct and summing only the planar diagrams will not capture the large N dynamics. Although the spectrum of anomalous dimensions has been computed for this class of operators, previous studies have neglected certain terms which were argued to be small. After dropping these terms diagonalizing the dilatation operator reduces to diagonalizing a set of decoupled oscillators. In this article we explicitly compute the terms which were neglected previously and show that diagonalizing the dilatation operator still reduces to diagonalizing a set of decoupled oscillators.

hep-th↗

arXiv2013

Higher Loop Nonplanar Anomalous Dimensions from Symmetry

In this article we study the action of the one loop dilatation operator on operators with a classical dimension of order N. These operators belong to the su(2) sector and are constructed using two complex fields Y and Z. For these operators non-planar diagrams contribute already at the leading order in N and the planar and large N limits are distinct. The action of the one loop and the two loop dilatation operator reduces to a set of decoupled oscillators and factorizes into an action on the Z fields and an action on the Y fields. Direct computation has shown that the action on the Y fields is the same at one and two loops. In this article, using the su(2) symmetry algebra as well as structural features of field theory, we give compelling evidence that the factor in the dilatation operator that acts on the Ys is given by the one loop expression, at any loop order.

hep-th↗

arXiv2013

Operators, Correlators and Free Fermions for SO(N) and Sp(N)

Using the recently constructed basis for local operators in free SO(N) gauge theory we derive an exact formula for the correlation functions of multi trace operators. This formula is used to obtain a simpler form and a simple product rule for the operators in the SO(N) basis. The coefficients of the product rule are the Littlewood-Richardson numbers which determine the corresponding product rule in free U(N) gauge theory. SO(N) gauge theory is dual to a non-oriented string theory on the AdS_5xRP^5 geometry. To explore the physics of this string theory we consider the limit of the gauge theory that, for the U(N) gauge theory, is dual to the pp-wave limit of AdS_5xS^5. Non-planar unoriented ribbon diagrams do not survive this limit. We give arguments that the number of operators in our basis matches counting using the exact free field partition function of free SO(N) gauge theory. We connect the basis we have constructed to free fermions, which has a natural interpretation in terms of a class of 1/2-BPS bubbling geometries, which arise as orientifolds of type IIB string theory. Finally, we obtain a complete generalization of these results to Sp(N) gauge theory by proving that the finite N physics of SO(N) and Sp(N) gauge theory are related by exchanging symmetrizations and antisymmetrizations and replacing N by -N.

hep-th↗

arXiv2013

On the refined counting of graphs on surfaces

Ribbon graphs embedded on a Riemann surface provide a useful way to describe the double line Feynman diagrams of large N computations and a variety of other QFT correlator and scattering amplitude calculations, e.g in MHV rules for scattering amplitudes, as well as in ordinary QED. Their counting is a special case of the counting of bi-partite embedded graphs. We review and extend relevant mathematical literature and present results on the counting of some infinite classes of bi-partite graphs. Permutation groups and representations as well as double cosets and quotients of graphs are useful mathematical tools. The counting results are refined according to data of physical relevance, such as the structure of the vertices, faces and genus of the embedded graph. These counting problems can be expressed in terms of observables in three-dimensional topological field theory with S_d gauge group which gives them a topological membrane interpretation.

hep-th↗

arXiv2013

A basis for large operators in N=4 SYM with orthogonal gauge group

We develop techniques to study the correlation functions of "large operators" whose bare dimension grows parametrically with N, in SO(N) gauge theory. We build the operators from a single complex matrix. For these operators, the large N limit of correlation functions is not captured by summing only the planar diagrams. By employing group representation theory we are able to define local operators which generalize the Schur polynomials of the theory with gauge group U(N). We compute the two point function of our operators exactly in the free field limit showing that they diagonalize the two point function. We explain how these results can be used to obtain the exact free field answers for correlators of operators in the trace basis.

hep-th↗

arXiv2012

Large N anomalous dimensions for large operators in Leigh-Strassler deformed SYM

We study the large N anomalous dimensions of operators in a Leigh-Strassler deformation of N=4 super Yang-Mills theory. The operators that we study have a bare dimension of order N (so that the large N limit is not captured by planar diagrams) and are AdS/CFT dual to giant gravitons. The diagonalization of the dilatation operator factorizes into two problems. One of these problems is solved using a double coset ansatz. The second problem is equivalent to a set of decoupled harmonic oscillators.

hep-th↗

arXiv2012

Restricted Schur Polynomials for Fermions and integrability in the su(2|3) sector

We define restricted Schur polynomials built using both fermionic and bosonic fields which transform in the adjoint of the gauge group U(N). We show that these operators diagonalize the free field two point function to all orders in 1/N. As an application of our new operators, we study the action of the one loop dilatation operator in the su(2|3) sector in a large N but non-planar limit. The restricted Schur polynomials we study are dual to giant gravitons. We find that the one loop dilatation operator can be diagonalized using a double coset ansatz. The resulting spectrum of anomalous dimensions matches the spectrum of a set of decoupled oscillators. Finally, in an Appendix we study the action of the one loop dilatation operator in an sl(2) sector. This action is again diagonalized by a double coset ansatz.

hep-th↗

arXiv2012

Nonplanar integrability at two loops

In this article we compute the action of the two loop dilatation operator on restricted Schur polynomials that belong to the su(2) sector, in the displaced corners approximation. In this non-planar large N limit, operators that diagonalize the one loop dilatation operator are not corrected at two loops. The resulting spectrum of anomalous dimensions is related to a set of decoupled harmonic oscillators, indicating integrability in this sector of the theory at two loops. The anomalous dimensions are a non-trivial function of the 't Hooft coupling, with a spectrum that is continuous and starting at zero at large N, but discrete at finite N.

hep-th↗

arXiv2012

S=1 in O(N)/HS duality

Following the work of Maldacena and Zhiboedov, we study the implementation of the Coleman-Mandula theorem in the free O(N)/Higher Spin correspondence. In the bi-local framework we first define an S-matrix for scattering of collective dipoles. Its evaluation in the case of free UV fixed point theory leads to the result S=1 stated in the title. We also present an appropriate field transformation that is seen to transform away all the non-linear 1/N interactions of this theory. A change of boundary conditions and/or external potentials results in a nontrivial S-matrix.

hep-th↗

arXiv2012

Beyond the Planar Limit in ABJM

In this article we consider gauge theories with a U(N)X U(N) gauge group. We provide, for the first time, a complete set of operators built from scalar fields that are in the bi fundamental of the two groups. Our operators diagonalize the two point function of the free field theory at all orders in 1/N. We then use this basis to investigate non-planar anomalous dimensions in the ABJM theory. We show that the dilatation operator reduces to a set of decoupled harmonic oscillators, signaling integrability in a nonplanar large N limit.

hep-th↗

arXiv2012

Extremal vs. Non-Extremal Correlators with Giant Gravitons

We consider extremal and non-extremal three-point functions of two giant gravitons and one point-like graviton using Schur polynomials in N=4 super Yang-Mills theory and holographically, using a semiclassical Born-Infeld analysis as well as bubbling geometries. For non-extremal three-point functions our computations using all three approaches are in perfect agreement. For extremal correlators we find that our results from the bubbling geometry analysis agree with existing results from the gauge theory. The semiclassical Born-Infeld computation for the extremal case is known to give a different answer, which we interpret as a manifestation of the known subtlety of holography for extremal correlators.

hep-th↗

arXiv2012

A double coset ansatz for integrability in AdS/CFT

We give a proof that the expected counting of strings attached to giant graviton branes in AdS_5 x S^5, as constrained by the Gauss Law, matches the dimension spanned by the expected dual operators in the gauge theory. The counting of string-brane configurations is formulated as a graph counting problem, which can be expressed as the number of points on a double coset involving permutation groups. Fourier transformation on the double coset suggests an ansatz for the diagonalization of the one-loop dilatation operator in this sector of strings attached to giant graviton branes. The ansatz agrees with and extends recent results which have found the dynamics of open string excitations of giants to be given by harmonic oscillators. We prove that it provides the conjectured diagonalization leading to harmonic oscillators.

hep-th↗

arXiv2012

Strings from Feynman Graph counting : without large N

A well-known connection between n strings winding around a circle and permutations of n objects plays a fundamental role in the string theory of large N two dimensional Yang Mills theory and elsewhere in topological and physical string theories. Basic questions in the enumeration of Feynman graphs can be expressed elegantly in terms of permutation groups. We show that these permutation techniques for Feynman graph enumeration, along with the Burnside counting lemma, lead to equalities between counting problems of Feynman graphs in scalar field theories and Quantum Electrodynamics with the counting of amplitudes in a string theory with torus or cylinder target space. This string theory arises in the large N expansion of two dimensional Yang Mills and is closely related to lattice gauge theory with S_n gauge group. We collect and extend results on generating functions for Feynman graph counting, which connect directly with the string picture. We propose that the connection between string combinatorics and permutations has implications for QFT-string dualities, beyond the framework of large N gauge theory.

hep-th↗

arXiv2011

On subgroup adapted bases for representations of the symmetric group

The split basis of an irreducible representation of the symmetric group, $S_{n+m}$, is the basis which is adapted to direct product subgroups of the form $S_{n} \times S_{m}$. In this article we have calculated symmetric group subduction coefficients relating the standard Young-Yamanouchi basis for the symmetric group to the split basis by means of a novel version of the Schur-Weyl duality. We have also directly obtained matrix representations in the split basis using these techniques. The advantages of this approach are that we obtain analytical expressions for the subduction coefficients and matrix representations and directly resolve issues of multiplicity in the subduction of $S_{n} \times S_{m}$ irreducible representations from $S_{n+m}$ irreducible representations. Our method is applicable to $S_{n+m}$ irreducible representations labelled by Young diagrams with large row length differences. We provide evidence for a systematic expansion in the inverse row length difference and tools to compute the leading contribution, which is exact when row length differences are infinite.

math-ph↗

arXiv2011

Non-planar Anomalous Dimensions in the sl(2) Sector

In this note we compute the non-planar one loop anomalous dimension of restricted Schur polynomials that belong to the sl(2) sector of N=4 super Yang-Mills theory and have a bare dimension of order N. Although the details are rather different, ultimately the problem of diagonalizing the dilatation operator in the sl(2) sector can be reduced to the su(2) sector problem. In this way we establish the expected dynamical emergence of the Gauss Law for giant gravitons and further show that the dilatation operator reduces to a set of decoupled harmonic oscillators.

hep-th↗

arXiv2011

From Large N Nonplanar Anomalous Dimensions to Open Spring Theory

In this note we compute the non-planar one loop anomalous dimension of restricted Schur polynomials that have a bare dimension of O(N). This is achieved by mapping the restricted Schur polynomials into states of a specific U(N) irreducible representation. In this way the dilatation operator is mapped into a u(n) valued operator and, as a result, can easily be diagonalized. The resulting spectrum is reproduced by a classical model of springs between masses.

hep-th↗

arXiv2011

Giant Graviton Oscillators

We study the action of the dilatation operator on restricted Schur polynomials labeled by Young diagrams with p long columns or p long rows. A new version of Schur-Weyl duality provides a powerful approach to the computation and manipulation of the symmetric group operators appearing in the restricted Schur polynomials. Using this new technology, we are able to evaluate the action of the one loop dilatation operator. The result has a direct and natural connection to the Gauss Law constraint for branes with a compact world volume. We find considerable evidence that the dilatation operator reduces to a decoupled set of harmonic oscillators. This strongly suggests that integrability in N=4 super Yang-Mills theory is not just a feature of the planar limit, but extends to other large N but non-planar limits.

hep-th↗