Search

Abstract The giant graviton expansion of the line defect Schur index in four dimensional N 4 $$ \mathcal{N}4 $$ U(N) SYM was recently proposed in arXiv:2403.11543 to be captured in the dual string theory by counting fluctuations states of two half-infinite fundamental strings in AdS 5 × S 5 ending on the line defect and D3 brane giant. However, agreement with the gauge theory data for the defect line index at finite N required the inclusion of ad hoc extra contributions with unclear origin. We discuss the large N leading order contribution of the giant graviton expansion of the defect line index by a direct analysis of semiclassical string partition function in a twisted background. We discuss supersymmetric boundary conditions in the presence of the D3 brane and evaluate the quadratic fluctuations effective action by introducing a suitable projection of fluctuation modes. We show that the extra contributions to the single giant graviton correction found in arXiv:2403.11543 correspond to a supersymmetric Casimir energy contribution.

Abstract We consider Schur line defect correlators in four dimensional N $$ \mathcal{N} $$ = 4 U(N) SYM and their giant graviton expansion encoding finite N corrections to the large N limit. We compute in closed form the single giant graviton contribution to correlators with general insertions of 1 2 $$ \frac{1}{2} $$ -BPS charged Wilson lines. For the 2-point function with fundamental and anti-fundamental Wilson lines, we match the result from fluctuations of two half-infinite strings ending on the giant graviton, recently proposed in arXiv:2403.11543. In particular, we prove exact factorization of the defect contribution with respect to wrapped D3 brane fluctuations representing the single giant graviton correction to the undecorated Schur index. This follows from a finite-difference representation of the Schur line defect index in terms of the index without defects, and similar factorization holds quite generally for more complicated defect configurations. In particular, the single giant graviton contribution to the 4-point function with two fundamental and two anti-fundamental lines is computed and discussed in this perspective.

Abstract The flavored superconformal Schur index of N $$ \mathcal{N} $$ = 4 U(N) SYM has finite N corrections encoded in its giant graviton expansion in terms of D3 branes wrapped in AdS 5 × S 5. The key element of this decomposition is the non-trivial index of the theory living on the wrapped brane system. A remarkable feature of the Schur limit is that the brane index is an analytic continuation of the flavored index of N $$ \mathcal{N} $$ = 4 U(n) SYM, where n is the total brane wrapping number. We exploit recent exact results about the Schur index of N $$ \mathcal{N} $$ = 4 U(N) SYM to evaluate the closed form of the brane indices appearing in the giant graviton expansion. Away from the unflavored limit, they are characterized by quasimodular forms providing exact information at all orders in the index universal fugacity. As an application of these results, we present novel exact expressions for the giant graviton expansion of the unflavored Schur index in a class of four dimensional N $$ \mathcal{N} $$ = 2 theories with equal central charges a = c, i.e. the non-Lagrangian theories Γ ̂ SU N $$ \hat{\Gamma}\left(\textrm{SU}(N)\right) $$ with Γ = E 6, E 7, E 8.

Abstract We consider the refined Schur superconformal index of 4d N $$ \mathcal{N} $$ = 4 U(N) SYM and the first term of its giant-graviton expansion, first predicted in arXiv:2001.11667 using indirect superconformal algebra considerations and analytic continuation of fugacities. This correction is the leading non-perturbative correction to the index at large N and we reproduce it from the semiclassical partition function of quantum D3 brane wrapped on S 1 × S 3 in a twisted modification of the AdS 5 × S 5 string background, depending on the index R-symmetry fugacity. Our calculation does not exploit directly supersymmetry. It is based on the determination of the partition function of the various bosonic and fermionic fluctuations on the wrapped brane whose action is conformal with specific constant holonomies along thermal cycle. We show how those partition functions may be obtained by adapting the operator counting method of Cardy to the twisted background.

Abstract We study four-dimensional $$\mathcal{N}$$ = 2 superconformal circular, cyclic symmetric quiver theories which are planar equivalent to $$\mathcal{N}$$ = 4 super Yang-Mills. We use localization to compute nonplanar corrections to the free energy and the circular half-BPS Wilson loop in these theories for an arbitrary number of nodes, and examine their behaviour in the limit of long quivers. Exploiting the relationship between the localization quiver matrix integrals and an integrable Bessel operator, we find a closed-form expression for the leading nonplanar correction to both observables in the limit when the number of nodes and ’t Hooft coupling become large. We demonstrate that it has different asymptotic behaviour depending on how the two parameters are compared, and interpret this behaviour in terms of properties of a lattice model defined on the quiver diagram.

We consider 4d N=4 U(N) SYM and the leading giant graviton correction at large N to the Schur defect 2-point functions of 12-BPS Wilson lines in rank-k symmetric and antisymmetric representations. We study in particular the large k limit for the symmetric case and the regime 1≪k≪N in the antisymmetric one. In both cases we present exact results for the correction. Wilson lines in symmetric/antisymmetric representations admit a description in terms of D3k and D5k brane probes representing a collection of k fundamental strings. In this picture, giant graviton corrections come from fluctuations of brane probes in the presence of a wrapped D3 brane giant graviton. In the antisymmetric case, our leading correction matches the half-index of the 4d N=4 Maxwell theory living on the disk which is a part of the giant graviton divided out by the D5k probe, as recently proposed in arXiv:2404.08302. For the symmetric case at large k, we derive an explicit exact residue formula for the leading large N correction.

It was shown in arXiv:2309.10786 that the leading non-perturbative contribution to the large N expansion of the superconformal index of the (2,0) 6d theory (which describes low-energy dynamics of N coincident M5 branes) is reproduced by the semiclassical partition function of quantum M2 brane wrapped on S1×S2 in a twisted version of AdS×7S4 background. Here we demonstrate an analogous relation for the leading non-perturbative contribution to the large N expansion of the superconformal index of the N=8 supersymmetric level-one U(N)×U(N) ABJM theory (which describes low-energy dynamics of N coincident M2 branes). The roles of M2 and M5 branes get effectively interchanged. Namely, the large N correction to the ABJM index is found to be given by the semiclassical partition function of quantum M5 brane wrapped on S1×S5 in a twisted version of AdS×4S7 background. This effectively confirms the suggestion for the “M5 brane index” made in arXiv:2007.05213 on the basis of indirect superconformal algebra considerations.

Abstract We present a quantum M2 brane computation of the instanton prefactor in the leading non-perturbative contribution to the ABJM 3-sphere free energy at large N and fixed level k. Using supersymmetric localization, such instanton contribution was found earlier to take the form F inst N k = − sin 2 2 π k − 1 exp − 2 π 2 N k + . … $$ {F}^{inst}\left(N,k\right)=-{\left({\sin}^2\frac{2\pi }{k}\right)}^{-1}\exp \left(-2\pi \sqrt{\frac{2N}{k}}\right)+.\dots $$ The exponent comes from the action of an M2 brane instanton wrapped on S 3/ℤ k , which represents the M-theory uplift of the ℂP1 instanton in type IIA string theory on AdS4 × ℂP3. The IIA string computation of the leading large k term in the instanton prefactor was recently performed in arXiv:2304.12340. Here we find that the exact value of the prefactor sin 2 2 π k − 1 $$ {\left({\sin}^2\frac{2\pi }{k}\right)}^{-1} $$ is reproduced by the 1-loop term in the M2 brane partition function expanded near the S 3/ℤ k instanton configuration. As in the Wilson loop example in arXiv:2303.15207, the quantum M2 brane computation is well defined and produces a finite result in exact agreement with localization.

Abstract We consider the Schur index of N $$ \mathcal{N} $$ = 4 U(N) SYM theory in 4d and its holographic giant graviton-type expansion at finite N. We compute the world-volume brane superconformal index by a recently proposed definition of the gauge holonomy integral as a multivariate residue. This is evaluated by a novel deformation algorithm that avoids Gröbner basis methods. Various terms of the brane expansion are computed and their sum is shown to be free of wall-crossing singularities to the order we explored. The relation between the brane expansion and previous giant graviton-type represenations of the Schur index is clarified.

The superconformal index Z of the 6d (2,0) theory on S5×S1 (which is related to the localization partition function of 5d SYM on S5) should be captured at large N by the quantum M2 brane theory in the dual M-theory background. Generalizing the type IIA string theory limit of this relation discussed in arXiv:2111.15493 and arXiv:2304.12340, we consider semiclassically quantized M2 branes in a half-supersymmetric 11d background which is a twisted product of thermal AdS7 and S4. We show that the leading non-perturbative term at large N is reproduced precisely by the 1-loop partition function of an “instanton” M2 brane wrapped on S1×S2 with S2⊂S4. Similarly, the (2,0) theory analog of the BPS Wilson loop expectation value is reproduced by the partition function of a “defect” M2 brane wrapped on thermal AdS⊂3 AdS7. We comment on a curious analogy of these results with similar computations in arXiv:2303.15207 and arXiv:2307.14112 of the partition function of quantum M2 branes in AdS×4S7/Zk which reproduced the corresponding localization expressions in the ABJM 3d gauge theory.

Abstract We study non-planar corrections in two special N $$ \mathcal{N} $$ = 2 superconformal SU(N) gauge theories that are planar-equivalent to N $$ \mathcal{N} $$ = 4 SYM theory: two-nodes quiver model with equal couplings and N $$ \mathcal{N} $$ = 2 vector multiplet coupled to two hypermultiplets in rank-2 symmetric and antisymmetric representations. We focus on two observables in these theories that admit representation in terms of localization matrix model: free energy on 4-sphere and the expectation value of half-BPS circular Wilson loop. We extend the methods developed in arXiv:2207.11475 to derive a systematical expansion of non-planar corrections to these observables at strong ’t Hooft coupling constant λ. We show that the leading non- planar corrections are given by a power series in λ 3/2 /N 2 with rational coefficients. Sending N and the coupling constant λ to infinity with λ 3/2 /N 2 kept fixed corresponds to the familiar double scaling limit in matrix models. We find that in this limit the observables in the two models are related in a remarkably simple way: the free energies differ by the factor of 2, whereas the Wilson loop expectation values coincide. Surprisingly, these relations hold only at strong coupling, they are not valid in the weak coupling regime. We also discuss a dual string theory interpretation of the leading corrections to the free energy in the double scaling limit suggesting their relation to curvature corrections in type IIB string effective action.

Abstract We consider four dimensional U(N) N $$ \mathcal{N} $$ = 4 SYM theory interacting with a 3d N $$ \mathcal{N} $$ = 4 theory living on a codimension-one interface and holographically dual to the D3-D5 system without flux. Localization captures several observables in this dCFT, including its free energy, related to the defect expectation value, and single trace 1 2 $$ \frac{1}{2} $$ -BPS composite scalars. These quantities may be computed in a hermitian one-matrix model with non-polynomial single-trace potential. We exploit the integrable Volterra hierarchy underlying the matrix model and systematically study its 1/N expansion at any value of the ’t Hooft coupling. In particular, the strong coupling regime is determined — up to non-perturbative exponentially suppressed corrections — by differential relations that constrain higher order terms in the 1/N expansion. The analysis is extended to the model with SU(N) gauge symmetry by resorting to the more general Toda lattice equations.

Abstract We apply the localization technique to compute the free energy on four-sphere and the circular BPS Wilson loop in the four-dimensional N $$ \mathcal{N} $$ = ∈ superconformal Sp(2N) gauge theory containing vector multiplet coupled to four hypermultiplets in fundamental representation and one hypermultiplet in rank-2 antisymmetric representation. This theory is unique among similar N $$ \mathcal{N} $$ = ∈ superconformal models that are planar-equivalent to N $$ \mathcal{N} $$ = ∆ SYM in that the corresponding localization matrix model has the interaction potential containing single-trace terms only. We exploit this property to show that, to any order in large N expansion and an arbitrary ’t Hooft coupling λ, the free energy and the Wilson loop satisfy Toda lattice equations. Solving these equations at strong coupling, we find remarkably simple expressions for these observables which include all corrections in 1/N and 1/ λ $$ \sqrt{\lambda } $$ . We also compute the leading exponentially suppressed O e − λ $$ \mathcal{O}\left({e}^{-\sqrt{\lambda }}\right) $$ corrections and consider a generalization to the case when the fundamental hypermultiplets have a non-zero mass. The string theory dual of this N $$ \mathcal{N} $$ = ∈ gauge theory should be a particular orientifold of AdS5 × S 5 type IIB string and we discuss the string theory interpretation of the obtained strong-coupling results.

We compare large N expansion of the localization result for the free energy F in the 3d N=6 superconformal U(N)k×U(N)−k Chern-Simons-matter theory to its AdS/CFT counterpart, i.e. to the perturbative expansion of M-theory partition function on AdS×4S7/Zk and to the weak string coupling expansion of type IIA effective action on AdS×4CP3. We show that the general form of the perturbative expansions of F on the two sides of the AdS/CFT duality is indeed the same. Moreover, the transcendentality properties of the coefficients in the large N, large k expansion of F match those in the corresponding M-theory or string theory expansions. To shed light on the structure of the 1-loop M-theory partition function on AdS×4S7/Zk we use the expression for the 1-loop 4-graviton scattering amplitude in the 11d supergravity. We also use the known information about the transcendental coefficients of the leading curvature invariants in the low-energy effective action of type II string theory. Matching of the remaining rational factors in the coefficients requires a precise information about currently unknown RR field strength terms in the corresponding superinvariants.

Abstract We consider strong ’t Hooft coupling expansion in special four-dimensional 𝒩 = 2 superconformal models that are planar-equivalent to 𝒩 = 4 super Yang-Mills theory. Various observables in these models that admit localization matrix model representation can be expressed at large N in terms of a Fredholm determinant of a Bessel operator. The latter previously appeared in the study of level spacing distributions in matrix models and, more recently, in four-point correlation functions of infinitely heavy half-BPS operators in planar 𝒩 = 4 SYM. We use this relation and a suitably generalized Szegő-Akhiezer-Kac formula to derive the strong ’t Hooft coupling expansion of the leading corrections to free energy, half-BPS circular Wilson loop, and certain correlators of chiral primaries operators in the 𝒩 = 2 models. This substantially generalizes partial results in the literature and represents a challenge for dual string theory calculations in AdS/CFT context. We also demonstrate that the resulting strong-coupling expansions suffer from Borel singularities and require adding non-perturbative, exponentially suppressed corrections. As a byproduct of our analysis, we determine the non-perturbative correction to the above mentioned four-point correlator in planar 𝒩 = 4 SYM.

Abstract Extending earlier work, we find the two-loop term in the beta-function for the scalar coupling ζ in a generalized Wilson loop operator of the N $$ \mathcal{N} $$ = 4 SYM theory, working in the planar weak-coupling expansion. The beta-function for ζ has fixed points at ζ = ±1 and ζ = 0, corresponding respectively to the supersymmetric Wilson-Maldacena loop and to the standard Wilson loop without scalar coupling. As a consequence of our result for the beta-function, we obtain a prediction for the two-loop term in the anomalous dimension of the scalar field inserted on the standard Wilson loop. We also find a subset of higher-loop contributions (with highest powers of ζ at each order in ‘t Hooft coupling λ) coming from the scalar ladder graphs determining the corresponding terms in the five-loop beta-function. We discuss the related structure of the circular Wilson loop expectation value commenting, in particular, on consistency with a 1d defect version of the F-theorem. We also compute (to two loops in the planar ladder model approximation) the two-point correlators of scalars inserted on the Wilson line.

Abstract As a continuation of the study (in arXiv:2102.07696 and arXiv:2104.12625 ) of strong-coupling expansion of non-planar corrections in N $$ \mathcal{N} $$ = 2 4d superconformal models we consider two special theories with gauge groups SU(N) and Sp(2N). They contain N-independent numbers of hypermultiplets in rank 2 antisymmetric and fundamental representations and are planar-equivalent to the corresponding N $$ \mathcal{N} $$ = 4 SYM theories. These N $$ \mathcal{N} $$ = 2 theories can be realised on a system of N D3-branes with a finite number of D7-branes and O7-plane; the dual string theories should be particular orientifolds of AdS5 × S 5 superstring. Starting with the localization matrix model representation for the N $$ \mathcal{N} $$ = 2 partition function on S 4 we find exact differential relations between the 1/N terms in the corresponding free energy F and the 1 2 $$ \frac{1}{2} $$ -BPS Wilson loop expectation value W $$ \left\langle \mathcal{W}\right\rangle $$ and also compute their large ’t Hooft coupling (λ » 1) expansions. The structure of these expansions is different from the previously studied models without fundamental hypermultiplets. In the more tractable Sp(2N) case we find an exact resummed expression for the leading strong coupling terms at each order in the 1/N expansion. We also determine the exponentially suppressed at large λ contributions to the non-planar corrections to F and W $$ \left\langle \mathcal{W}\right\rangle $$ and comment on their resurgence properties. We discuss dual string theory interpretation of these strong coupling expansions.

Abstract We consider the expectation value W $$ \left\langle \mathcal{W}\right\rangle $$ of the circular BPS Wilson loop in N $$ \mathcal{N} $$ = 2 superconformal SU(N) gauge theory containing a vector multiplet coupled to two hypermultiplets in rank-2 symmetric and antisymmetric representations. This theory admits a regular large N expansion, is planar-equivalent to N $$ \mathcal{N} $$ = 4 SYM theory and is expected to be dual to a certain orbifold/orientifold projection of AdS5 × S 5 superstring theory. On the string theory side W $$ \left\langle \mathcal{W}\right\rangle $$ is represented by the path integral expanded near the same AdS2 minimal surface as in the maximally supersymmetric case. Following the string theory argument in [5], we suggest that as in the N $$ \mathcal{N} $$ = 4 SYM case and in the N $$ \mathcal{N} $$ = 2 SU(N) × SU(N) superconformal quiver theory discussed in [19], the coefficient of the leading non-planar 1/N 2 correction in W $$ \left\langle \mathcal{W}\right\rangle $$ should have the universal λ 3/2 scaling at large ’t Hooft coupling. We confirm this prediction by starting with the localization matrix model representation for W $$ \left\langle \mathcal{W}\right\rangle $$ . We complement the analytic derivation of the λ 3/2 scaling by a numerical high-precision resummation and extrapolation of the weak-coupling expansion using conformal mapping improved Padé analysis.

Abstract We consider the N $$ \mathcal{N} $$ = 2 SYM theory with gauge group SU(N) and a matter content consisting of one multiplet in the symmetric and one in the anti-symmetric representation. This conformal theory admits a large-N ’t Hooft expansion and is dual to a particular orientifold of AdS5 × S 5. We analyze this gauge theory relying on the matrix model provided by localization à la Pestun. Even though this matrix model has very non-trivial interactions, by exploiting the full Lie algebra approach to the matrix integration, we show that a large class of observables can be expressed in a closed form in terms of an infinite matrix depending on the ’t Hooft coupling λ. These exact expressions can be used to generate the perturbative expansions at high orders in a very efficient way, and also to study analytically the leading behavior at strong coupling. We successfully compare these predictions to a direct Monte Carlo numerical evaluation of the matrix integral and to the Padé resummations derived from very long perturbative series, that turn out to be extremely stable beyond the convergence disk |λ| < π 2 of the latter.

Abstract We consider U(N) N $$ \mathcal{N} $$ = 4 super Yang-Mills theory and discuss how to extract the strong coupling limit of non-planar corrections to observables involving the 1 2 $$ \frac{1}{2} $$ -BPS Wilson loop. Our approach is based on a suitable saddle point treatment of the Eynard-Orantin topological recursion in the Gaussian matrix model. Working directly at strong coupling we avoid the usual procedure of first computing observables at finite planar coupling λ, order by order in 1/N, and then taking the λ ≫ 1 limit. In the proposed approach, matrix model multi-point resolvents take a simplified form and some structures of the genus expansion, hardly visible at low order, may be identified and rigorously proved. As a sample application, we consider the expectation value of multiple coincident circular supersymmetric Wilson loops as well as their correlator with single trace chiral operators. For these quantities we provide novel results about the structure of their genus expansion at large tension, generalising recent results in arXiv:2011.02885 .

Abstract Localization approach to N $$ \mathcal{N} $$ = 2 superconformal SU(N) × SU(N) quiver theory leads to a non-Gaussian two-matrix model representation for the expectation value of BPS circular SU(N) Wilson loop W $$ \left\langle \mathcal{W}\right\rangle $$ . We study the subleading 1/N 2 term in the large N expansion of W $$ \left\langle \mathcal{W}\right\rangle $$ at weak and strong coupling. We concentrate on the case of the symmetric quiver with equal gauge couplings which is equivalent to the ℤ2 orbifold of the SU(2N) N $$ \mathcal{N} $$ = 4 SYM theory. This orbifold gauge theory should be dual to type IIB superstring in AdS5 × (S 5 /ℤ2). We present a string theory argument suggesting that the 1/N 2 term in W $$ \left\langle \mathcal{W}\right\rangle $$ in the orbifold theory should have the same strong-coupling asymptotics λ 3/2 as in the N $$ \mathcal{N} $$ = 4 SYM case. We support this prediction on the gauge theory side by a numerical study of the localization matrix model. We also find a relation between the 1/N 2 term in the Wilson loop expectation value and the derivative of the free energy of the orbifold gauge theory on 4-sphere.

Abstract Starting with some known localization (matrix model) representations for correlators involving 1/2 BPS circular Wilson loop W $$ \mathcal{W} $$ in N $$ \mathcal{N} $$ = 4 SYM theory we work out their 1/N expansions in the limit of large ’t Hooft coupling λ. Motivated by a possibility of eventual matching to higher genus corrections in dual string theory we follow arXiv:2007.08512 and express the result in terms of the string coupling g s ∼ g YM 2 ∼ λ / N $$ {g}_{\mathrm{s}}\sim {g}_{\mathrm{YM}}^2\sim \lambda /N $$ and string tension T ∼ λ $$ T\sim \sqrt{\lambda } $$ . Keeping only the leading in 1/T term at each order in g s we observe that while the expansion of W $$ \left\langle \mathcal{W}\right\rangle $$ is a series in g s 2 / T $$ {g}_{\mathrm{s}}^2/T $$ , the correlator of the Wilson loop with chiral primary operators O J $$ {\mathcal{O}}_J $$ has expansion in powers of g s 2 / T 2 $$ {g}_{\mathrm{s}}^2/{T}^2 $$ . Like in the case of W $$ \left\langle \mathcal{W}\right\rangle $$ where these leading terms are known to resum into an exponential of a “one-handle” contribution ∼ g s 2 / T $$ \sim {g}_{\mathrm{s}}^2/T $$ , the leading strong coupling terms in WO J $$ \left\langle {\mathcal{WO}}_J\right\rangle $$ sum up to a simple square root function of g s 2 / T 2 $$ {g}_{\mathrm{s}}^2/{T}^2 $$ . Analogous expansions in powers of g s 2 / T $$ {g}_{\mathrm{s}}^2/T $$ are found for correlators of several coincident Wilson loops and they again have a simple resummed form. We also find similar expansions for correlators of coincident 1/2 BPS Wilson loops in the ABJM theory.

Abstract We consider a class of N $$ \mathcal{N} $$ = 2 conformal SU(N) SYM theories in four dimensions with matter in the fundamental, two-index symmetric and anti-symmetric representations, and study the corresponding matrix model provided by localization on a sphere S 4, which also encodes information on flat-space observables involving chiral operators and circular BPS Wilson loops. We review and improve known techniques for studying the matrix model in the large-N limit, deriving explicit expressions in perturbation theory for these observables. We exploit both recursive methods in the so-called full Lie algebra approach and the more standard Cartan sub-algebra approach based on the eigenvalue distribution. The sub-class of conformal theories for which the number of fundamental hypermultiplets does not scale with N differs in the planar limit from the N $$ \mathcal{N} $$ = 4 SYM theory only in observables involving chiral operators of odd dimension. In this case we are able to derive compact expressions which allow to push the small ’t Hooft coupling expansion to very high orders. We argue that the perturbative series have a finite radius of convergence and extrapolate them numerically to intermediate couplings. This is preliminary to an analytic investigation of the strong coupling behavior, which would be very interesting given that for such theories holographic duals have been proposed.