arXiv subjects

M. P. Mattis

Publications and source records attributed to M. P. Mattis.

15 recordsLinked to original sources

arXiv1999

Multi-Instanton Calculus and the AdS/CFT Correspondence in N=4 Superconformal Field Theory

We present a self-contained study of ADHM multi-instantons in SU(N) gauge theory, especially the novel interplay with supersymmetry and the large-N limit. We give both field- and string-theoretic derivations of the N=4 supersymmetric multi-instanton action and collective coordinate integration measure. As a central application, we focus on certain n-point functions G_n, n=16, 8 or 4, in N=4 SU(N) gauge theory at the conformal point (as well as on related higher-partial-wave correlators); these are correlators in which the 16 exact supersymmetric and superconformal fermion zero modes are saturated. In the large-N limit, for the first time in any 4-dimensional theory, we are able to evaluate all leading-order multi-instanton contributions exactly. We find compelling evidence for Maldacena's conjecture: (1) The large-N k-instanton collective coordinate space has the geometry of a single copy of AdS_5 x S^5. (2) The integration measure on this space includes the partition function of 10-dimensional N=1 SU(k) gauge theory dimensionally reduced to 0 dimensions, matching the description of D-instantons in Type IIB string theory. (3) In exact agreement with Type IIB string calculations, at the k-instanton level, G_n = \sqrt{N} g^8 k^{n-7/2} e^{2πikτ} \sum_{d|k} d^{-2} F_n(x_1,...,x_n), where F_n is identical to a convolution of n bulk-to-boundary supergravity propagators.

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arXiv1999

Breakdown of Cluster Decomposition in Instanton Calculations of the Gluino Condensate

A longstanding puzzle concerns the calculation of the gluino condensate <{trλ^2\over 16π^2}> = cΛ^3 in N=1 supersymmetric SU(N) gauge theory: so-called weak-coupling instanton (WCI) calculations give c=1, whereas strong-coupling instanton (SCI) calculations give, instead, c=2[(N-1)!(3N-1)]^{-1/N}. By examining correlators of this condensate in arbitrary multi-instanton sectors, we cast serious doubt on the SCI calculation of <{trλ^2\over 16π^2}> by showing that an essential step --- namely cluster decomposition --- is invalid. We also show that the addition of a so-called Kovner-Shifman vacuum (in which <{trλ^2\over 16π^2}> = 0) cannot straightforwardly resolve this mismatch.

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arXiv1998

Multi-Instantons and Maldacena's Conjecture

We examine certain n-point functions G_n in {\cal N}=4 supersymmetric SU(N) gauge theory at the conformal point. In the large-N limit, we are able to sum all leading-order multi-instanton contributions exactly. We find compelling evidence for Maldacena's conjecture: (1) The large-N k-instanton collective coordinate space has the geometry of AdS_5 x S^5. (2) In exact agreement with type IIB superstring calculations, at the k-instanton level, $G_n = \sqrt{N} g^8 k^{n-7/2} e^{-8π^2 k/g^2}\sum_{d|k} d^{-2} \times F_n(x_1,...,x_n)$, where F_n is identical to a convolution of n bulk-to-boundary SUGRA propagators.

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arXiv1998

Yang-Mills Instantons in the Large-N Limit and the AdS/CFT Correspondence

We examine a certain 16-fermion correlator in {\cal N}=4 supersymmetric SU(N) gauge theory in 4 dimensions. Generalizing recent SU(2) results of Bianchi, Green, Kovacs and Rossi, we calculate the exact N-dependence of the effective 16-fermion vertex at the 1-instanton level, and find precise agreement in the large-N limit with the prediction of the type IIB superstring on AdS_5 x S^5. This suggests that the string theory prediction for the 1-instanton amplitude considered here is not corrected by higher-order terms in the alpha' expansion.

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arXiv1998

The Instanton Hunter's Guide to Supersymmetric SU(N) Gauge Theories

We present a compendium of results for ADHM multi-instantons in SU(N) SUSY gauge theories, followed by applications to N=2 supersymmetric models. Extending recent SU(2) work, and treating the N=1 and N=2 cases in parallel, we construct: (i) the ADHM supermultiplet, (ii) the multi-instanton action, and (iii) the collective coordinate integration measure. Specializing to N=2, we then give a closed formula for F_k, the k-instanton contribution to the prepotential, as a finite-dimensional collective coordinate integral. This amounts to a weak-coupling solution, in quadratures, of the low-energy dynamics of N=2 SQCD, without appeal to duality. As an application, we calculate F_1 for all SU(N) and any number of flavors N_F; for N_F<2N-2 and N_F=2N-1 we confirm previous instanton calculations and agree with the proposed hyper-elliptic curve solutions. For N_F=2N-2 and N_F=2N with N>3 we obtain new results, which in the latter case we do not understand how to reconcile with the curves.

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arXiv1997

Supersymmetry and the Multi-Instanton Measure II: From N=4 to N=0

Extending recent N=1 and N=2 results, we propose an explicit formula for the integration measure on the moduli space of charge-n ADHM multi-instantons in N=4 supersymmetric SU(2) gauge theory. As a consistency check, we derive a renormalization group relation between the N=4, N=2, and N=1 measures. We then use this relation to construct the purely bosonic (``N=0'') measure as well, in the classical approximation in which the one-loop small-fluctuations determinants is not included.

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arXiv1997

Supersymmetry and the Multi-Instanton Measure

We propose explicit formulae for the integration measure on the moduli space of charge-n ADHM multi-instantons in N=1 and N=2 supersymmetric gauge theories. The form of this measure is fixed by its (super)symmetries as well as the physical requirement of clustering in the limit of large spacetime separation between instantons. We test our proposals against known expressions for n < 3. Knowledge of the measure for all n allows us to revisit, and strengthen, earlier N=2 results, chiefly: (1) For any number of flavors N_F, we provide a closed formula for F_n, the n-instanton contribution to the Seiberg-Witten prepotential, as a finite-dimensional collective coordinate integral. This amounts to a solution, in quadratures, of the Seiberg-Witten models, without appeal to electric-magnetic duality. (2) In the conformal case N_F=4, this means reducing to quadratures the previously unknown finite renormalization that relates the microscopic and effective coupling constants, τ_{micro} and τ_{eff}. (3) Similar expressions are given for the 4-derivative/8-fermion term in the gradient expansion of N=2 supersymmetric QCD.

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arXiv1997

Instantons, Higher-Derivative Terms, and Nonrenormalization Theorems in Supersymmetric Gauge Theories

We discuss the contribution of ADHM multi-instantons to the higher-derivative terms in the gradient expansion along the Coulomb branch of N=2 and N=4 supersymmetric SU(2) gauge theories. In particular, using simple scaling arguments, we confirm the Dine-Seiberg nonperturbative nonrenormalization theorems for the 4-derivative/8-fermion term in the two finite theories (N=4, and N=2 with N_F=4).

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arXiv1997

Multi-Instantons, Three-Dimensional Gauge Theory, and the Gauss-Bonnet-Chern Theorem

We calculate multi-instanton effects in a three-dimensional gauge theory with N=8 supersymmetry and gauge group SU(2). The k-instanton contribution to an eight-fermion correlator is found to be proportional to the Gauss-Bonnet-Chern integral of the Gaussian curvature over the centered moduli-space of charge-k BPS monopoles, \tilde{M}_{k}. For k=2 the integral can be evaluated using the explicit metric on \tilde{M}_{2} found by Atiyah and Hitchin. In this case the integral is equal to the Euler character of the manifold. More generally the integral is the volume contribution to the index of the Euler operator on \tilde{M}_{k}, which may differ from the Euler character by a boundary term. We conjecture that the boundary terms vanish and evaluate the multi-instanton contributions using recent results for the cohomology of \tilde{M}_{k}. We comment briefly on the implications of our result for a recently proposed test of M(atrix) theory.

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arXiv1997

Instantons, Three-Dimensional Gauge Theory, and the Atiyah-Hitchin Manifold

We investigate quantum effects on the Coulomb branch of three-dimensional N=4 supersymmetric gauge theory with gauge group SU(2). We calculate perturbative and one-instanton contributions to the Wilsonian effective action using standard weak-coupling methods. Unlike the four-dimensional case, and despite supersymmetry, the contribution of non-zero modes to the instanton measure does not cancel. Our results allow us to fix the weak-coupling boundary conditions for the differential equations which determine the hyper-Kahler metric on the quantum moduli space. We confirm the proposal of Seiberg and Witten that the Coulomb branch is equivalent, as a hyper-Kahler manifold, to the centered moduli space of two BPS monopoles constructed by Atiyah and Hitchin.

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arXiv1996

On Mass-Deformed N=4 Supersymmetric Yang-Mills Theory

We construct the n-instanton action for the above model with gauge group SU(2), as a function of the collective coordinates of the general self-dual configurations of Atiyah, Drinfeld, Hitchin and Manin (ADHM). We calculate the quantum modulus u = at the 1-instanton level, and find a discrepancy with Seiberg and Witten's proposed exact solution. As in related models (N=2, N_F=3 or 4), this discrepancy may be resolved by modifying their proposed relation between $\tilde u$ (the parameter in the elliptic curve) and u.

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arXiv1996

On N=2 Supersymmetric QCD with 4 Flavors

Seiberg and Witten's proposed solution of N=2 SQCD with N_c=2 and N_F=4 is known to conflict with instanton calculations in three distinct ways. Here we show how to resolve all three discrepancies, simply by reparametrizing the elliptic curve in terms of quantities $τ^0_{eff}$ and $\tilde{u}$ rather than $τ$ and $u = < Tr A^2 > $. SL(2,Z) invariance of the curve is preserved. However, there is now an infinite ambiguity in the relation between $τ^0_{eff}$ and $τ$ and between $\tilde{u}$ and $u$, corresponding to an infinite number of unknown coefficients in the instanton expansion. Thus the reinterpreted curve (unlike the cases N_F<4) no longer determines the quantum modulus u as a function of the classical VEV a.

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arXiv1996

A Two-Instanton Test of the Exact Solution of N=2 Supersymmetric QCD

The exact solution of N=2 supersymmetric QCD found by Seiberg and Witten includes a prediction for all multi-instanton contributions to the low-energy dynamics. In the case where the matter hypermultiplets are massless, the leading non-perturbative contribution at weak coupling comes from the two-instanton sector. We calculate this contribution from first principles using the exact two-instanton solution of the self-dual Yang-Mills equation found by Atiyah, Drinfeld, Hitchin and Manin. We find exact agreement with the predictions of Seiberg and Witten for all numbers of flavours. We also confirm their predictions for the case of massive hypermultiplets.

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arXiv1996

Multi-Instanton Check of the Relation Between the Prepotential F and the Modulus u in N=2 SUSY Yang-Mills Theory

By examining multi-instantons in N=2 supersymmetric SU(2) gauge theory, we derive, on very general grounds, and to all orders in the instanton number, a relationship between the prepotential F(Phi), and the coordinate on the quantum moduli space u= . This relation was previously obtained by Matone in the context of the explicit Seiberg-Witten low-energy solution of the model. Our findings can be viewed as a multi-instanton check of the proposed exact results in supersymmetric gauge theory.

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arXiv1996

Multi-Instanton Calculus in $N=2$ Supersymmetric Gauge Theory

The Seiberg-Witten solution of N=2 supersymmetric SU(2) gauge theory may be viewed as a prediction for the infinite family of constants F_n measuring the n-instanton contribution to the prepotential F. Here we examine the instanton physics directly, in particular the contribution of the general self-dual solution of topological charge n constructed by Atiyah, Drinfeld, Hitchin and Manin (ADHM). In both the bosonic and supersymmetric cases, we determine both the large- and short-distance behavior of all the fields in this background. This allows us to construct the exact classical interaction between n ADHM (super-)instantons mediated by the adjoint Higgs. We calculate the one- and two-instanton contributions to the low-energy Seiberg-Witten effective action, and find precise agreement with their predicted values of F_1 and F_2.

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