Gauge Theory Operators Research Articles

Gauge theory loop operators and Liouville theory

We propose a correspondence between loop operators in a family of four dimensional $$ \mathcal{N} $$ = 2 gauge theories on S 4 — including Wilson, ‘t Hooft and dyonic operators — and Liouville theory loop operators on a Riemann surface. This extends the beautiful relation between the partition function of these $$ \mathcal{N} $$ = 2 gauge theories and Liouville correlators found by Alday, Gaiotto and Tachikawa. We show that the computation of these Liouville correlators with the insertion of a Liouville loop operator reproduces Pestun’s formula capturing the expectation value of a Wilson loop operator in the corresponding gauge theory. We prove that our definition of Liouville loop operators is invariant under modular transformations, which given our correspondence, implies the conjectured action of S-duality on the gauge theory loop operators. Our computations in Liouville theory make an explicit prediction for the exact expectation value of ’t Hooft and dyonic loop operators in these $$ \mathcal{N} $$ = 2 gauge theories. The Liouville loop operators are also found to admit a simple geometric interpretation within quantum Teichmüller theory as the quantum operators representing the length of geodesics. We study the algebra of Liouville loop operators and show that it gives evidence for our proposal as well as providing definite predictions for the operator product expansion of loop operators in gauge theory.

Charges of monopole operators in Chern-Simons Yang-Mills theory

We calculate the non-abelian R-charges of BPS monopole operators in three-dimensional gauge theories with N=3 supersymmetry. This class of models includes ABJM theory, the proposed gauge theory dual of M-theory on AdS_4 x S^7/Z_k, as a special case. In the UV limit of the N=3 theories the Yang-Mills coupling becomes weak and the monopole operators are described by classical backgrounds. This allows us to find their SU(2)_R charges in a one-loop computation which by virtue of the non-renormalization of non-abelian R-charges yields the exact result for any value of the coupling. The spectrum of SU(2)_R charges is found by quantizing the SU(2)/U(1) collective coordinate of the BPS background, whose dynamics is that of a charged particle on a sphere with a Wess-Zumino term representing a magnetic monopole at its center. If the Wess-Zumino coefficient is h, then the smallest possible SU(2)_R representation for BPS monopole operators has spin |h|/2. We find, in agreement with earlier proposals, that h is proportional to the sum of the U(1)_R charges of all the fermion fields weighted by the effective monopole charges determined by their gauge representations. The field content of ABJM theory is such that h=0. This proves for any Chern-Simons level k the existence of monopole operators which are singlets under all global symmetries and have vanishing scaling dimensions. These operators are essential for matching the spectrum of the ABJM theory with supergravity and for the supersymmetry enhancement to N=8.

A NOTE ON STRINGS IN DEFORMED AdS4 × ℂℙ3: GIANT MAGNON AND SINGLE SPIKE SOLUTIONS

In this paper we study the solitonic string solutions of magnon and single spike type in the beta-deformed AdS4 × ℂℙ3 background. We find the dispersion relations which are supposed to give the anomalous dimension of the gauge theory operators.

Double-helix Wilson loops: case of two angular momenta

Recently, Wilson loops with the shape of a double helix have played an important role in studying large spin operators in gauge theories. They correspond to a quark and an anti-quark moving in circles on an S3 (and therefore each of them describes a helix in R × S3). In this paper we consider the case where the particles have two angular momenta on the S3. The string solution corresponding to such Wilson loop can be found using the relation to the Neumann-Rosochatius system allowing the computation of the energy and angular momenta of the configuration. The particular case of only one angular momentum is also considered. It can be thought as an analytic continuation of the rotating strings which are dual to operators in the SL(2) sector of = 4 SYM.

Quantum strings inAdS5×S5: strong-coupling corrections to dimension of Konishi operator

We consider leading strong coupling corrections to the energy of the lightest massive string modes in AdS_5 x S^5, which should be dual to members of the Konishi operator multiplet in N=4 SYM theory. This determines the general structure of the strong-coupling expansion of the anomalous dimension of the Konishi operator. We use 1-loop results for several semiclassical string states to extract information about the leading coefficients in this expansion. Our prediction is Delta= 2 lambda^{1/4} + b_0 + b_1 lambda^{-1/4} + b_3 lambda^{-3/4} +..., where b_0 and b_1 are rational while b_3 is transcendental containing zeta(3). Explicitly, we argue that b_0= Delta_0 - 4 (where Delta_0 is the canonical dimension of the corresponding gauge-theory operator in the Konishi multiplet) and b_1=1. Our conclusions are sensitive to few assumptions, implied by a correspondence with flat-space expressions, on how to translate semiclassical quantization results into predictions for the exact quantum string spectrum.

Surface operators in abelian gauge theory

We consider arbitrary embeddings of surface operators in a pure, non-supersymmetric abelian gauge theory on spin (non-spin) four-manifolds. For any surface operator with a priori simultaneously non-vanishing parameters, we explicitly show that the parameters transform naturally under an SL(2, Z) (or a congruence subgroup thereof) duality of the theory. However, for non-trivially-embedded surface operators, exact S-duality holds only if the quantum parameter effectively vanishes, while the overall SL(2, Z) (or a congruence subgroup thereof) duality holds up to a c-number at most, regardless. Via the formalism of duality walls, we furnish an alternative derivation of the transformation of parameters - found also to be consistent with a switch from Wilson to 't Hooft loop operators under S-duality. With any background embedding of surface operators, the partition function and the correlation functions of non-singular, gauge-invariant local operators on any curved four-manifold, are found to transform like modular forms under the respective duality groups.

Fine structure of anomalous dimensions in [formula omitted] super-Yang–Mills theory

Anomalous dimensions of high-twist Wilson operators in generic gauge theories occupy a band of width growing logarithmically with their conformal spin. We perform a systematic study of its fine structure in the autonomous SL(2) subsector of the dilatation operator of planar N=4 super-Yang–Mills theory which is believed to be integrable to all orders in 't Hooft coupling. We resort in our study on the framework of the Baxter equation to unravel the properties of the ground state trajectory and the excited trajectories in the spectrum. We use two complimentary approaches in our analysis based on the asymptotic solution of the Baxter equation and on the semiclassical expansion to work out the leading asymptotic expression for the trajectories in the upper and lower part of the band and to find how they are modified by the perturbative corrections.

Quantum corrections to energy of short spinning string inAdS5

Motivated by a desire to shed light on the strong-coupling behavior of dimensions of short gauge-theory operators, we consider the famous example of folded spinning string in AdS{sub 5} in the limit of small semiclassical spin parameter S=(S/{radical}({lambda})). In this limit the string becomes short and is moving in a near-flat central region of AdS{sub 5}. Its energy scales with spin as E={radical}(2S)[a{sub 0}+a{sub 1}S+a{sub 2}S{sup 2}+...]. We explicitly compute the leading 1-loop quantum AdS{sub 5}xS{sup 5} superstring correction to the short string energy and show that the coefficient a{sub 0} is not renormalized from its classical value while a{sub 1} receives a nontrivial contribution containing {zeta}(3)

A tunneling picture of dual giant Wilson loop

We further discuss a rotating dual giant Wilson loop (D3-brane) solution constructed in Lorentzian AdS by Drukker et al. The solution is shown to be composed of a dual giant Wilson loop and a dual giant graviton by minutely examining its shape. Hence it may be supposed that the dual gauge-theory operator should be a k-th symmetric Wilson loop with insertions of dual giant graviton operators. To give a convincing argument for this observation, the classical action of the solution should be evaluated in Euclidean signature. For this purpose we perform a Wick rotation to the Lorentzian solution by following the tunneling prescription and obtain Euclidean solutions attaching to a circle or a straight line on the AdS boundary. The classical action contains a logarithmic divergence which is proportional to the R-charge. It is consistent with the correlation function of the dual giant graviton operators. Because of ambiguities concerning regularization, finite terms of the action do not lead to conclusive evidence. However we may expect that the ambiguities would cancel by subtracting the action of the D3-brane with a straight-line boundary from that with a circular boundary. It is really shown that the difference is consistent with the expectation value of the k-th symmetric Wilson loop. Although our analysis does not specify the unique dual gauge-theory operator, we discuss possible candidates implied by the D3-brane computation.

Rigid Surface Operators

Surface operators in gauge theory are analogous to Wilson and ’t Hooft line operators except that they are supported on a two-dimensional surface rather than a one-dimensional curve. In a previous paper, we constructed a certain class of half-BPS surface operators in N = 4 super Yang–Mills theory, and determined how they transform under S-duality. Those surface operators depend on a relatively large number of freely adjustable parameters. In the present paper, we consider the opposite case of half-BPS surface operators that are “rigid” in the sense that they do not depend on any parameters at all. We present some simple constructions of rigid half-BPS surface operators and attempt to determine how they transform under duality. This attempt is only partially successful, suggesting that our constructions are not the whole story. The partial match suggests interesting connections with quantization. We discuss some possible refinements and some string theory constructions which might lead to a more complete picture.

Giant gravitons - with strings attached (III)

We study the one-loop anomalous dimensions of operators in the ${\cal N}=4$ super Yang-Mills theory that are dual to open strings ending on giant gravitons. We consider both AdS and sphere giants as well as boundstates of them. The open strings we consider carry angular momentum on an S^3 embedded in the S^5 of the AdS$_5\times$S^5 background. The main result of this article is that we derive a bosonic lattice Hamiltonian that describes the one loop mixing of the operators dual to the general excited giant graviton system. A semiclassical analysis of the Hamiltonian allows us to give a geometrical interpretation for the labeling used to describe the gauge theory operators. We also argue that AdS giant gravitons are unstable against the excitations considered.

On the anatomy of multi-spin magnon and single spike string solutions

We study rigid string solutions rotating in AdS 5 × S 5 background. For particular values of the parameters of the solutions we find multispin solutions corresponding to giant magnons and single spike strings. We present an analysis of the dispersion relations in the case of three spin solutions distributed only in S 5 and the case of one spin in AdS 5 and two spins in S 5 . The possible relation of these string solutions to gauge theory operators and spin chains are briefly discussed.

Counting chiral operators in quiver gauge theories

We discuss in detail the problem of counting BPS gauge invariant operators in the chiral ring of quiver gauge theories living on D-branes probing generic toric CY singularities. The computation of generating functions that include counting of baryonic operators is based on a relation between the baryonic charges in field theory and the Kahler moduli of the CY singularities. A study of the interplay between gauge theory and geometry shows that given geometrical sectors appear more than once in the field theory, leading to a notion of ``multiplicities. We explain in detail how to decompose the generating function for one D-brane into different sectors and how to compute their relevant multiplicities by introducing geometric and anomalous baryonic charges. The Plethystic Exponential remains a major tool for passing from one D-brane to arbitrary number N of D-branes. Explicit formulae are given for few examples, including 3/3, 0, and dP1.

Branes, anti-branes and Brauer algebras in gauge-gravity duality

We propose gauge theory operators built using a complex Matrix scalar which are dual to brane-anti-brane systems in $AdS_5 \times S^5 $, in the zero coupling limit of the dual Yang-Mills. The branes involved are half-BPS giant gravitons. The proposed operators dual to giant-anti-giant configurations satisfy the appropriate orthogonality properties. Projection operators in Brauer algebras are used to construct the relevant multi-trace Matrix operators. These are related to the ``coupled representations'' which appear in 2D Yang-Mills theory. We discuss the implications of these results for the quantum mechanics of a complex matrix model, the counting of non-supersymmetric operators and the physics of brane-anti-brane systems. The stringy exclusion principle known from the properties of half-BPS giant gravitons, has a new incarnation in this context. It involves a qualitative change in the map between brane-anti-brane states to gauge theory operators. In the case of a pair of sphere giant and anti-giant this change occurs when the sum of the magnitudes of their angular momenta reaches $N$.

Counting BPS operators in gauge theories: quivers, syzygies and plethystics

We develop a systematic and efficient method of counting single-trace and multi-trace BPS operators with two supercharges, for world-volume gauge theories of $N$ D-brane probes for both $N \to \infty$ and finite $N$. The techniques are applicable to generic singularities, orbifold, toric, non-toric, complete intersections, et cetera, even to geometries whose precise field theory duals are not yet known. The so-called ``Plethystic Exponential'' provides a simple bridge between (1) the defining equation of the Calabi-Yau, (2) the generating function of single-trace BPS operators and (3) the generating function of multi-trace operators. Mathematically, fascinating and intricate inter-relations between gauge theory, algebraic geometry, combinatorics and number theory exhibit themselves in the form of plethystics and syzygies.

Giant gravitons—with strings attached (II)

We study the one-loop anomalous dimensions of operators in the ${\cal N}=4$ super Yang-Mills theory that are dual to open strings ending on giant gravitons. We consider both AdS and sphere giants as well as boundstates of them. The open strings we consider carry angular momentum on an S^3 embedded in the S^5 of the AdS$_5\times$S^5 background. The main result of this article is that we derive a bosonic lattice Hamiltonian that describes the one loop mixing of the operators dual to the general excited giant graviton system. A semiclassical analysis of the Hamiltonian allows us to give a geometrical interpretation for the labeling used to describe the gauge theory operators. We also argue that AdS giant gravitons are unstable against the excitations considered.

Counting BPS baryonic operators in CFTs with Sasaki-Einstein duals

We study supersymmetric D3 brane configurations wrapping internal cycles of type II backgrounds AdS(5) x H for a generic Sasaki-Einstein manifold H. These configurations correspond to BPS baryonic operators in the dual quiver gauge theory. In each sector with given baryonic charge, we write explicit partition functions counting all the BPS operators according to their flavor and R-charge. We also show how to extract geometrical information about H from the partition functions; in particular, we give general formulae for computing volumes of three cycles in H.

Surface operators in gauge theory

AbstractI discuss certain surface operators in gauge theory and their relevance to the ramified case of the geometric Langlands program.

Fat magnon

We consider a D-brane type state which shares the characteristic of the recently found giant magnon of Hofman and Maldacena. More specifically we find a bound state of giant graviton (D3-brane) and giant magnon (F-string), which has exactly the same anomalous dimension as that of the giant magnon. It is described by the D3-brane with electric flux which is topologically a $S^3$ elongated by the electric flux. The angular momentum and energy are infinite, but split sensibly into two parts -- the infinite part precisely the same as that of the giant magnon and the finite part which can be identified as the contribution from the giant graviton. We discuss that the corresponding dual gauge theory operator is not a simple chain type but rather admixture of the (sub-)determinant and chain types.

HEAT KERNEL OF NONMINIMAL GAUGE FIELD KINETIC OPERATORS ON MOYAL PLANE

We generalize the Endo formula1 originally developed for the computation of the heat kernel asymptotic expansion for nonminimal operators in commutative gauge theories to the noncommutative case. In this way, the first three nonzero heat trace coefficients of the nonminimal U (N) gauge field kinetic operator on the Moyal plane taken in an arbitrary background are calculated. We show that the nonplanar part of the heat trace asymptotics is determined by U (1) sector of the gauge model. The nonplanar or mixed heat kernel coefficients are shown to be gauge-fixing dependent in any dimension of space–time. In the case of the degenerate deformation parameter the lowest mixed coefficients in the heat expansion produce nonlocal gauge-fixing dependent singularities of the one-loop effective action that destroy the renormalizability of the U (N) model at one-loop level. Such phenomenon was observed at first in Ref. 2 for spacelike noncommutative ϕ4 scalar and U (1) gauge theories. The twisted-gauge transformation approach is discussed.