arXiv subjects

P. J. Heslop

Publications and source records attributed to P. J. Heslop.

15 recordsLinked to original sources

arXiv2026

Hidden Conformal Symmetry in AdS$_2\times$S$^2$ Beyond Tree Level

Correlators of a certain one-dimensional superconformal field theory dual to hypermultiplets in AdS$_2\times$S$^2$ exhibit a hidden four-dimensional conformal symmetry which allows one to repackage all tree-level 4-point correlators into a single four-dimensional object corresponding to a contact diagram arising from a massless $ϕ^4$ theory in AdS$_2\times$S$^2$. This theory serves as a toy model for IIB string theory in AdS$_5\times$S$^5$ and is interesting in its own right because AdS$_2\times$S$^2$ describes the near-horizon limit of extremal black holes in four dimensions. We argue that after acting with an $SU(1,1)\times SU(2)$ Casimir, all one-loop correlators can similarly be encoded by a four-dimensional function which arises from a one-loop scalar bubble diagram in AdS$_2\times$S$^2$, explaining how the hidden conformal symmetry extends beyond tree level. Finally, we conjecture a scalar effective field theory with a derivative interaction in AdS$_2\times$S$^2$ whose Witten diagrams should directly reproduce 4-point correlators to all loops without acting with Casimirs.

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arXiv2025

A formula for the block expansion in free CFTs and applications to ${\cal N}=4$ SYM at strong coupling

An explicit analytic formula is presented that computes the conformal (super-)block decomposition of any free scalar or half-BPS diagram in 1d, 2d or 4d CFTs, with various supersymmetries, including none. We prove our formula by exploiting a connection between conformal blocks and symmetric polynomials. Then we give a direct application of our result to the study of four-point correlators in ${\cal N}=4$ SYM at strong coupling. In particular, we give a CFT proof of the tree-level Witten diagram representation of $\langle {\cal O}_2^2{\cal O}_2^2 {\cal O}_q{\cal O}_q\rangle$ on AdS$_5\times$S$^5$, providing new and highly non-trivial checks of the AdS/CFT correspondence. Our method works for a more general class of multi-particle correlators and can be used to bootstrap new results at strong coupling.

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arXiv2010

A note on N=8 counterterms

The most difficult counterterms to construct in any supersymmetric theory are those that cannot be written as full superspace integrals of gauge-invariant integrands. In $D=4$ maximal supergravity it has been known for some time that there are just three of these at the linearised level. In this article we discuss these counterterms again from the point of view of representations of the superconformal group. In particular, we show that the only independent invariants constructed from shortened superconformal multiplets in $D=4$ are BPS.

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arXiv2008

Diagonal free field matrix correlators, global symmetries and giant gravitons

We obtain a basis of diagonal free field multi-matrix 2-point correlators in a theory with global symmetry group G. The operators fall into irreducible representations of G. This applies for gauge group U(N) at finite N. For composites made of n fundamental fields, this is expressed in terms of Clebsch-Gordan coefficients for the decomposition of the n-fold tensor products of the fundamental field representation in terms of G \times S_n representations. We use this general construction in the case of the SL(2) sector of \cN=4 SYM. In this case, by using oscillator constructions, we reduce the computation of the relevant Clebsch-Gordans coupling infinite dimensional discrete series irreps of SL(2) to a problem in symmetric groups. Applying these constructions we write down gauge invariant operators with a Fock space structure similar to that arising in a large angular momentum limit of worldvolume excitations of giant gravitons. The Fock space structure emerges from Clebsch multiplicities of tensor products of symmetric group representations. We also give the action of the 1-loop dilatation operator of \cN=4 SYM on this basis of multi-matrix operators.

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arXiv2008

Diagonal multi-matrix correlators and BPS operators in N=4 SYM

We present a complete basis of multi-trace multi-matrix operators that has a diagonal two point function for the free matrix field theory at finite N. This generalises to multiple matrices the single matrix diagonalisation by Schur polynomials. Crucially, it involves intertwining the gauge group U(N) and the global symmetry group U(M) with Clebsch-Gordan coefficients of symmetric groups S_n. When applied to N=4 super Yang-Mills we consider the U(3) subgroup of the full symmetry group. The diagonalisation allows the description of a dual basis to multi-traces, which permits the characterisation of the metric on operators transforming in short representations at weak coupling. This gives a framework for the comparison of quarter and eighth-BPS giant gravitons of AdS_5 x S^5 spacetime to gauge invariant operators of the dual N=4 SYM.

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arXiv2006

N=4 Superconformal Characters and Partition Functions

Character formulae for positive energy unitary representations of the N=4 superconformal group are obtained through use of reduced Verma modules and Weyl group symmetry. Expansions of these are given which determine the particular representations present and results such as dimensions of superconformal multiplets. By restriction of variables various `blind' characters are also obtained. Limits, corresponding to reduction to particular subgroups, in the characters isolate contributions from particular subsets of multiplets and in many cases simplify the results considerably. As a special case, the index counting short and semi-short multiplets which do not form long multiplets found recently is shown to be related to particular cases of reduced characters. Partition functions of N=4 super Yang Mills are investigated. Through analysis of these, exact formulae are obtained for counting half and some quarter BPS operators in the free case. Similarly, partial results for the counting of semi-short operators are given. It is also shown in particular examples how certain short operators which one might combine to form long multiplets due to group theoretic considerations may be protected dynamically.

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arXiv2005

More on La Grande Bouffe: towards higher spin symmetry breaking in AdS

We discuss higher spin gauge symmetry breaking in AdS space from a holographic prespective. Indeed, the AdS/CFT correspondence implies that N=4 SYM theory at vanishing coupling constant is dual to a theory in AdS which exhibits higher spin gauge symmetry enhancement. When the SYM coupling is non-zero, the current conservation condition becomes anomalous, and correspondingly the local higher spin symmetry in the bulk gets spontaneously broken. In agreement with previous results and holographic expectations, we find that the Goldstone mode responsible for the symmetry breaking in AdS has a non-vanishing mass even in the limit in which the gauge symmetry is restored. Moreover, we show that the mass of the Goldstone mode is exactly the one predicted by the correspondence. Finally, we obtain the precise form of the higher spin supercurrents in the SYM side.

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arXiv2004

Aspects of superconformal field theories in six dimensions

We introduce the analytic superspace formalism for six-dimensional $(N,0)$ superconformal field theories. Concentrating on the $(2,0)$ theory we write down the Ward identities for correlation functions in the theory and show how to solve them. We then consider the four-point function of four energy momentum multiplets in detail, explicitly solving the Ward identities in this case. We expand the four-point function using both Schur polynomials, which lead to a simple formula in terms of a single function of two variables, and (a supersymmetric generalisation of) Jack polynomials, which allow a conformal partial wave expansion. We then perform a complete conformal partial wave analysis of both the free theory four-point function and the AdS dual four-point function. We also discuss certain operators at the threshold of the series a) unitary bound, and prove that some such operators may not develop anomalous dimensions, by finding selection rules for certain three-point functions. For those operators which are not protected, we find representations with which they may combine to become long.

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arXiv2004

Superconformal field theories in analytic superspace

We summarise recent work on superconformal field theories using analytic superspace. All operators of N=4 SYM can be given as unconstrained superfields on analytic superspace. We show how to write down operators as superfields on analytic superspace and how to completely solve the Ward indentities for their correlation functions. We discuss the non-renormalisation of certain operators, and of some of their correlation functions. We discuss the relationship between harmonic and analytic superspace. Finally we discuss applications of these techniques to superconformal field theory in 6 dimensions.

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arXiv2003

Aspects of N=4 SYM

The properties of gauge-invariant composite operators and their correlation functions in N=4 SYM are discussed in the analytic superspace formalism. A complete classification of the different types of operators in the theory is given. Operators can be either protected or unprotected according to whether they do not or do have anomalous dimensions, and the analytic superspace formalism allows one to identify which type a given operator is in a straightforward manner. A simple discussion is given of the behaviour of reducible multiplets at threshold. It is pointed out that there is a class of ``semi-protected'' operators which do not have anomalous dimensions but which do not necessarily have non-renormalised three-point functions when the other two operators in the correlator are protected, although two-point functions of such operators are non-renormalised. A complete discussion of superconformal invariants in analytic superspace is given. The paper includes a modified discussion of the transformation rules of analytic superfields which clarifies the $U(1)_Y$ properties of operators and correlation functions and, in particular, explicit examples are given of three-point correlation functions which violate this symmetry. A tensor, $\cE$, invariant under $SL(n|m)$ but not under $GL(n|m)$, is introduced and used in the discussion of $U(1)_Y$ and in the construction of invariants.

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arXiv2003

Integral invariants in N=4 SYM and the effective action for coincident D-branes

The construction of supersymmetric invariant integrals is discussed in a superspace setting. The formalism is applied to D=4, N=4 SYM and used to construct the F^2, F^4 and (F^5 + \del^2 F^4) terms in the effective action of coincident D-branes. The results are in agreement with those obtained by other methods. A simple derivation of the abelian \del^4 F^4 invariant is given and generalised to the non-abelian case. We also find some double-trace invariants. The invariants are interpreted in terms of superconformal multiplets: the F^2 and F^4 terms are given by one-half BPS multiplets, the (F^5+\del^2F^4) arises as a full superspace integral of the Konishi multiplet K and the abelian \del^4 F^4 term comes from integrating the fourth power of the field strength superfield. Counterparts of the abelian invariants are exhibited for the D=6,(2,0) tensor multiplet and the D=3, N=8 scalar multiplet. The method is also applied to D=4, N=8 supergravity. All invariants in the linearised theory (with SU(8) symmetry) which arise from partial superspace integrals are constructed.

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arXiv2002

Four-point functions in N=4 SYM

A new derivation is given of four-point functions of charge $Q$ chiral primary multiplets in N=4 supersymmetric Yang-Mills theory. A compact formula, valid for arbitrary $Q$, is given which is manifestly superconformal and analytic in the internal bosonic coordinates of analytic superspace. This formula allows one to determine the spacetime four-point function of any four component fields in the multiplets in terms of the four-point function of the leading chiral primary fields. The leading term is expressed in terms of $1/2 Q(Q-1)$ functions of two conformal invariants and a number of single variable functions. Crossing symmetry reduces the number of independent functions, while the OPE implies that the single-variable functions arise from protected operators and should therefore take their free form. This is the partial non-renormalisation property of such four-point functions which can be viewed as a consequence of the OPE and the non-renormalisation of three-point functions of protected operators.

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arXiv2002

Superfield representations of superconformal groups

Representations of four dimensional superconformal groups are constructed as fields on many different superspaces, including super Minkowski space, chiral superspace, harmonic superspace and analytic superspace. Any unitary irreducible representation can be given as a field on any one of these spaces if we include fields which transform under supergroups. In particular, on analytic superspaces, the fields are unconstrained. One can obtain all representations of the N=4 complex superconformal group $PSL(4|4)$ with integer dilation weight from copies of the Maxwell multiplet on $(4,2,2)$ analytic superspace. This construction is compared with the oscillator construction and it is shown that there is a natural correspondence between the oscillator construction of superconformal representations and those carried by superfields on analytic superspace.

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arXiv2001

OPEs and 3-point correlators of protected operators in N=4 SYM

Two- and three-point correlation functions of arbitrary protected operators are constructed in N=4 SYM using analytic superspace methods. The OPEs of two chiral primary multiplets are given. It is shown that the $n$-point functions of protected operators for $n\leq4$ are invariant under $U(1)_Y$ and it is argued that this implies that the two- and three-point functions are not renormalised. It is shown explicitly how unprotected operators can be accommodated in the analytic superspace formalism in a way which is fully compatible with analyticity. Some new extremal correlators are exhibited.

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arXiv2001

A note on composite operators in N=4 SYM

We discuss composite operators in N=4 super Yang-Mills theory and their realisations as superfields on different superspaces. The superfields that realise various operators on analytic superspace may be different in the free, interacting and quantum theories. In particular, in the quantum theory, there is a restricted class of operators that can be written as analytic tensor superfields. This class includes all series B and C operators in the theory as well as some series A operators which saturate the unitarity bounds. Operators of this type are expected to be protected from renormalisation.

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