arXiv subjects

Eric D'Hoker

Publications and source records attributed to Eric D'Hoker.

At least 109 records · Page 6Linked to original sources

arXiv2002

Lectures on Two-Loop Superstrings

In these lectures, recent progress on multiloop superstring perturbation theory is reviewed. A construction from first principles is given for an unambiguous and slice-independent two-loop superstring measure on moduli space for even spin structure. A consistent choice of moduli, invariant under local worldsheet supersymmetry is made in terms of the super-period matrix. A variety of subtle new contributions arising from a careful gauge fixing procedure are taken into account. The superstring measure is computed explicitly in terms of genus two theta-functions and reveals the importance of a new modular object of weight 6. For given even spin structure, the measure exhibits a behavior under degenerations of the worldsheet that is consistent with physical principles. The measure allows for a unique modular covariant GSO projection. Under this GSO projection, the cosmological constant, the 1-, 2- and 3- point functions of massless supergravitons vanish pointwise on moduli space. A certain disconnected part of the 4-point function is shown to be given by a convergent integral on moduli space. A general consistent formula is given for the two-loop cosmological constant in compactifications with central charge c=15 and with N=1 worldsheet supersymmetry. Finally, some comments are made on possible extensions of this work to higher loop order.

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arXiv2002

Supersymmetric Gauge Theories and the AdS/CFT Correspondence

In these lecture notes we first assemble the basic ingredients of supersymmetric gauge theories (particularly N=4 super-Yang-Mills theory), supergravity, and superstring theory. Brane solutions are surveyed. The geometry and symmetries of anti-de Sitter space are discussed. The AdS/CFT correspondence of Maldacena and its application to correlation functions in the the conformal phase of N=4 SYM are explained in considerable detail. A pedagogical treatment of holographic RG flows is given including a review of the conformal anomaly in four-dimensional quantum field theory and its calculation from five-dimensional gravity. Problem sets and exercises await the reader.

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arXiv2001

Two-Loop Superstrings I, Main Formulas

An unambiguous and slice-independent formula for the two-loop superstring measure on moduli space for even spin structure is constructed from first principles. The construction uses the super-period matrix as moduli invariant under worldsheet supersymmetry. This produces new subtle contributions to the gauge-fixing process, which eliminate all the ambiguities plaguing earlier gauge-fixed formulas. The superstring measure can be computed explicitly and a simple expression in terms of modular forms is obtained. For fixed spin structure, the measure exhibits the expected behavior under degenerations of the surface. The measure allows for a unique modular covariant GSO projection. Under this GSO projection, the cosmological constant, the 1-, 2- and 3- point functions of massless supergravitons all vanish pointwise on moduli space without the appearance of boundary terms. A certain disconnected part of the 4-point function is shown to be given by a convergent, finite integral on moduli space. A general slice-independent formula is given for the two-loop cosmological constant in compactifications with central charge c=15 and N=1 worldsheet supersymmetry in terms of the data of the compactification conformal field theory. In this paper, a summary of the above results is presented with detailed constructions, derivations and proofs to be provided in a series of subsequent publications.

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arXiv2001

Two-Loop Superstrings IV, The Cosmological Constant and Modular Forms

The slice-independent gauge-fixed superstring chiral measure in genus 2 derived in the earlier papers of this series for each spin structure is evaluated explicitly in terms of theta-constants. The slice-independence allows an arbitrary choice of superghost insertion points q_1, q_2 in the explicit evaluation, and the most effective one turns out to be the split gauge defined by S_δ(q_1,q_2)=0. This results in expressions involving bilinear theta-constants M. The final formula in terms of only theta-constants follows from new identities between M and theta-constants which may be interesting in their own right. The action of the modular group Sp(4,Z) is worked out explicitly for the contribution of each spin structure to the superstring chiral measure. It is found that there is a unique choice of relative phases which insures the modular invariance of the full chiral superstring measure, and hence a unique way of implementing the GSO projection for even spin structure. The resulting cosmological constant vanishes, not by a Riemann identity, but rather by the genus 2 identity expressing any modular form of weight 8 as the square of a modular form of weight 4. The degeneration limits for the contribution of each spin structure are determined, and the divergences, before the GSO projection, are found to be the ones expected on physical grounds.

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arXiv2001

Two-Loop Superstrings III, Slice Independence and Absence of Ambiguities

The chiral superstring measure constructed in the earlier papers of this series for general gravitino slices is examined in detail for slices supported at two points x_α. In this case, the invariance of the measure under infinitesimal changes of gravitino slices established previously is strengthened to its most powerful form: the measure is shown, point by point on moduli space, to be locally and globally independent from the points x_α, as well as from the superghost insertion points p_a, q_αintroduced earlier as computational devices. In particular, the measure is completely unambiguous. The limit x_α= q_αis then well defined. It is of special interest, since it elucidates some subtle issues in the construction of the picture-changing operator Y(z) central to the BRST formalism. The formula for the chiral superstring measure in this limit is derived explicitly.

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arXiv2001

Two-Loop Superstrings II, The Chiral Measure on Moduli Space

A detailed derivation from first principles is given for the unambiguous and slice-independent formula for the two-loop superstring chiral measure which was announced in the first paper of this series. Supergeometries are projected onto their super period matrices, and the integration over odd supermoduli is performed by integrating over the fibers of this projection. The subtleties associated with this procedure are identified. They require the inclusion of some new finite-dimensional Jacobian superdeterminants, a deformation of the worldsheet correlation functions using the stress tensor, and perhaps paradoxically, another additional gauge choice, ``slice \hatμchoice'', whose independence also has to be established. This is done using an important correspondence between superholomorphic notions with respect to a supergeometry and holomorphic notions with respect to its super period matrix. Altogether, the subtleties produce precisely the corrective terms which restore the independence of the resulting gauge-fixed formula under infinitesimal changes of gauge-slice. This independence is a key criterion for any gauge-fixed formula and hence is verified in detail.

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arXiv2001

Three-Point Functions of Quarter BPS Operators in N=4 SYM

In a recent paper hep-th/0109064, quarter-BPS chiral primaries were constructed in the fully interacting four dimensional N=4 Super-Yang-Mills theory with gauge group SU(N). These operators are annihilated by four supercharges, and at order g^2 have protected scaling dimension and normalization. Here, we compute three-point functions involving these quarter-BPS operators along with half-BPS operators. The combinatorics of the problem is rather involved, and we consider the following special cases: (1) correlators < half half BPS > of two half-BPS primaries with an arbitrary chiral primary; (2) certain classes of < half quarter quarter > and < quarter quarter quarter > three-point functions; (3) three-point functions involving the Delta < 8 operators found in hep-th/0109064; (4) < half quarter quarter> correlators with the special quarter-BPS operator made of single and double trace operators only. The analysis in cases (1)-(3) is valid for general N, while (4) is a large N approximation. Order g^2 corrections to all three-point functions considered in this paper are found to vanish. In the AdS/CFT correspondence, quarter-BPS chiral primaries are dual to threshold bound states of elementary supergravity excitations. We present a supergravity discussion of two- and three-point correlators involving these bound states.

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arXiv1999

Lectures on Supersymmetric Yang-Mills Theory and Integrable Systems

We present a series of four self-contained lectures on the following topics: (I) An introduction to 4-dimensional 1\leq N \leq 4 supersymmetric Yang-Mills theory, including particle and field contents, N=1 and N=2 superfield methods and the construction of general invariant Lagrangians; (II) A review of holomorphicity and duality in N=2 super-Yang-Mills, of Seiberg-Witten theory and its formulation in terms of Riemann surfaces; (III) An introduction to mechanical Hamiltonian integrable systems, such as the Toda and Calogero-Moser systems associated with general Lie algebras; a review of the recently constructed Lax pairs with spectral parameter for twisted and untwisted elliptic Calogero-Moser systems; (IV) A review of recent solutions of the Seiberg-Witten theory for general gauge algebra and adjoint hypermultiplet content in terms of the elliptic Calogero-Moser integrable systems.

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arXiv1999

The Operator Product Expansion of N=4 SYM and the 4-point Functions of Supergravity

We give a detailed Operator Product Expansion interpretation of the results for conformal 4-point functions computed from supergravity through the AdS/CFT duality. We show that for an arbitrary scalar exchange in AdS(d+1) all the power-singular terms in the direct channel limit (and only these terms) exactly match the corresponding contributions to the OPE of the operator dual to the exchanged bulk field and of its conformal descendents. The leading logarithmic singularities in the 4-point functions of protected N=4 super-Yang Mills operators (computed from IIB supergravity on AdS(5) X S(5) are interpreted as O(1/N^2) renormalization effects of the double-trace products appearing in the OPE. Applied to the 4-point functions of the operators Ophi ~ tr F^2 + ... and Oc ~ tr FF~ + ..., this analysis leads to the prediction that the double-trace composites [Ophi Oc] and [Ophi Ophi - Oc Oc] have anomalous dimension -16/N^2 in the large N, large g_{YM}^2 N limit. We describe a geometric picture of the OPE in the dual gravitational theory, for both the power-singular terms and the leading logarithms. We comment on several possible extensions of our results.

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arXiv1999

Seiberg-Witten Theory and Calogero-Moser Systems

We present a brief account of a series of recent results on twisted and untwisted elliptic Calogero-Moser systems, and on their fundamental role in the Seiberg-Witten solution of gauge theories with one massive hypermultiplet in the adjoint representation of an arbitrary gauge algebra G.

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arXiv1999

AdS/CFT 4-point functions: How to succeed at z-integrals without really trying

A new method is discussed which vastly simplifies one of the two integrals over AdS(d+1) required to compute exchange graphs for 4-point functions of scalars in the AdS/CFT correspondence. The explicit form of the bulk-to-bulk propagator is not required. Previous results for scalar, gauge boson and graviton exchange are reproduced, and new results are given for massive vectors. It is found that precisely for the cases that occur in the AdS(5) X S(5) compactification of Type IIB supergravity, the exchange diagrams reduce to a finite sum of graphs with quartic scalar vertices. The analogous integrals in n-point scalar diagrams for n>4 are also evaluated.

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arXiv1999

Graviton exchange and complete 4-point functions in the AdS/CFT correspondence

The graviton exchange diagram for the correlation function of arbitrary scalar operators is evaluated in anti-de Sitter space, AdS(d+1). This enables us to complete the computation of the 4-point amplitudes of dilaton and axion fields in IIB supergravity on AdS5 X S5. By the AdS/CFT correspondence, we obtain the 4-point functions of the marginal operators Tr (F^2+...) and Tr(F F~ +...) in N=4, d=4 SU(N) SYM at large N, large g^2 N. The short distance asymptotics of the amplitudes are studied. We find that in the direct channel the leading power singularity agrees with the expected contribution of the stress-energy tensor in a double OPE expansion. Logarithmic singularities occur in the complete 4-point functions at subleading orders.

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arXiv1999

Graviton and gauge boson propagators in AdS(d+1)

We construct the gauge field and graviton propagators in Euclidean AdS(d+1) space-time by two different methods. In the first method the gauge invariant Maxwell or linearized Ricci operator is applied directly to bitensor ansatze for the propagators which reflect their gauge structure. This leads to a rapid determination of the physical part of the propagators in terms of elementary functions. The second method is a more traditional approach using covariant gauge fixing which leads to a solution for both physical and gauge parts of the propagators. The gauge invariant parts agree in both methods.

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arXiv1998

Gauge Boson Exchange in $AdS_{d+1}$

We study the amplitude for exchange of massless gauge bosons between pairs of massive scalar fields in Anti-de Sitter space. In the AdS/CFT correspondence this amplitude describes the contribution of conserved flavor symmetry currents to 4-point functions of scalar operators in the boundary conformal theory. A concise, covariant (Y2K-compatible) derivation of the gauge boson propagator in $\AdS_{d+1}$ is given. Techniques are developed to calculate the two bulk integrals over AdS space leading to explicit expressions or convenient, simple integral representations for the amplitude. The amplitude contains leading power and sub-leading logarithmic singularities in the gauge boson channel and leading logarithms in the crossed channel. The new methods of this paper are expected to have other applications in the study of the Maldacena conjecture.

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arXiv1998

General Scalar Exchange in AdS_d+1

The scalar field exchange diagram for the correlation function of four scalar operators is evaluated in anti-de Sitter space, $AdS_{d+1}$. The conformal dimensions $Δ_i$, $i=1,...,4$ of the scalar operators and the dimension $Δ$ of the exchanged field are arbitrary, constrained only to obey the unitarity bound. Techniques similar to those developed earlier for gauge boson exchange are used, but results are generally more complicated. However, for integer $Δ_i, Δ$, the amplitude can be presented as a multiple derivative of a simple universal function. Results simplify if further conditions hold, such as the inequalities, $Δ< Δ_1+Δ_3$ or $Δ<Δ_2+Δ_4$. These conditions are satisfied, with $<$ replaced by $\le$, in Type IIB supergravity on $AdS_5\times S_5$ because of selection rules from SO(6) symmetry. A new form of interaction is suggested for the marginal case of the inequalities. The short distance asymptotics of the amplitudes are studied. In the direct channel the leading singular term agrees with the double operator product expansion. Logarithmic singularities occur at sub-leading order in the direct channel but at leading order in the crossed channel. When the inequalities above are violated, there are also $(\log)^2$ singularities in the direct channel.

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arXiv1998

Field Theory Tests for Correlators in the AdS/CFT Correspondence

The order g^2 radiative corrections to all 2- and 3-point correlators of the composite primary operators Tr X^k are computed in {\cal N} = 4 supersymmetric Yang-Mills theory with gauge group SU(N). Corrections are found to vanish for all N. For k=2 this is a consequence of known superconformal nonrenormalization theorems, and for general k the result confirms an N-to-infinity, fixed large g^2N supergravity calculation and further conjectures in hep-th/9806074. A 3-point correlator involving {\cal N} = 4 descendents of Tr X^2 is calculated, and its order g^2 contribution also vanishes, giving evidence for the absence of radiative corrections in correlators of descendent operators.

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arXiv1997

Calogero-Moser Systems in SU(N) Seiberg-Witten Theory

The Seiberg-Witten curve and differential for ${\cal N}=2$ supersymmetric SU(N) gauge theory, with a massive hypermultiplet in the adjoint representation of the gauge group, are analyzed in terms of the elliptic Calogero-Moser integrable system. A new parametrization for the Calogero-Moser spectral curves is found, which exhibits the classical vacuum expectation values of the scalar field of the gauge multiplet. The one-loop perturbative correction to the effective prepotential is evaluated explicitly, and found to agree with quantum field theory predictions. A renormalization group equation for the variation with respect to the coupling is derived for the effective prepotential, and may be evaluated in a weak coupling series using residue methods only. This gives a simple and efficient algorithm for the instanton corrections to the effective prepotential to any order. The 1- and 2- instanton corrections are derived explicitly. Finally, it is shown that certain decoupling limits yield ${\cal N}=2$ supersymmetric theories for simple gauge groups $SU(N_1)$ with hypermultiplets in the fundamental representation, while others yield theories for product gauge groups $SU(N_1) \times ...\times SU(N_p)$, with hypermultiplets in fundamental and bi-fundamental representations. The spectral curves obtained this way for these models agree with the ones proposed by Witten using D-branes and M-theory.

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arXiv1997

Strong Coupling Expansions of SU(N) Seiberg-Witten Theory

We set up a systematic expansion of the prepotential for ${\cal N}=2$ supersymmetric Yang-Mills theories with SU(N) gauge group in the region of strong coupling where a maximal number of mutually local fields become massless. In particular, we derive the first non-trivial non-perturbative correction, which is the first term in the strong coupling expansion providing a coupling between the different U(1) factors, and which governs for example the strength of gaugino Yukawa couplings.

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