[Paper Review] The Glueball Superpotential

Mina Aganagic, Kenneth Intriligator+2 more|ArXiv.org|Apr 30, 2003

Black Holes and Theoretical Physics56 references8 citations

TL;DR

This paper computes the glueball superpotential in four-dimensional N=1 supersymmetric gauge theories with arbitrary gauge groups and massive matter, using perturbative integration of heavy fields. It introduces a universal 'planar projection' via supergroup embeddings (e.g., U(N+k|k)), resolving ambiguities in higher-order terms (S^n for n ≥ h) by leveraging F-term completion through analytic continuation in N and non-unitary supergroups, with exact results for classical and some exceptional groups.

ABSTRACT

We compute glueball superpotentials for four-dimensional, N=1 supersymmetric gauge theories, with arbitrary gauge groups and massive matter representations. This is done by perturbatively integrating out massive, charged fields. The Feynman diagram computations simplify, and are related to the corresponding matrix model. This leads to a natural notion of ``projection to planar diagrams'' for arbitrary gauge groups and representations. We discuss a general ambiguity in the glueball superpotential W(S) for terms, S^n, whose order, n, is greater than the dual Coxeter number. This ambiguity can be resolved for all classical gauge groups, (A,B,C,D), via a natural embedding in an infinite rank supergroup. We use this to address some recently raised puzzles. For exceptional groups, we compute the superpotential terms for low powers of the glueball field and propose an all-order completion for some examples including N=1^* for all simply-laced groups. We also comment on compactification of these theories to lower dimensions.

Motivation & Objective

• To compute the exact glueball superpotential W(S) for N=1 supersymmetric gauge theories with arbitrary gauge groups and massive matter representations.
• To resolve the long-standing ambiguity in higher-order terms S^n (n ≥ dual Coxeter number h) by introducing a natural F-term completion via supergroup embeddings.
• To establish a universal 'planar projection' prescription for arbitrary gauge groups and representations, generalizing the large-N limit.
• To reconcile apparent discrepancies between matrix model results and standard gauge theory computations, particularly regarding instanton effects in UV-completed theories.
• To provide an all-order superpotential for classical groups and partial completion for exceptional groups like E6, E7, E8.

Proposed method

• Perturbatively integrate out massive charged fields using supergraph techniques, simplifying Feynman diagrams to relate them to matrix model amplitudes.
• Introduce a generalized 'planar projection' via embedding classical gauge groups G(N) into infinite-rank supergroups G(N+k|k) with k→∞.
• Use the supertrace structure in G(N+k|k) to ensure coefficient independence on k, enabling exact computation of S^n terms for arbitrarily large n.
• Apply analytic continuation in N to derive F-term completions, with the supergroup embedding providing a predictive UV completion.
• Leverage Weyl invariance and tensor symmetry on the Cartan subalgebra to classify possible invariants and show they reduce to S^ℓ terms.
• Use brane/anti-brane systems as a physical motivation for the supergroup embedding, linking to D-brane realizations in string theory.

Experimental results

Research questions

• RQ1 How can the glueball superpotential be computed exactly for arbitrary N=1 supersymmetric gauge theories with massive matter?
• RQ2 What causes the ambiguity in S^n terms for n ≥ h (dual Coxeter number), and how can it be resolved in a UV-independent way?
• RQ3 Can a universal 'planar projection' be defined for all gauge groups and representations, not just U(N)?
• RQ4 Why do matrix model results sometimes disagree with standard gauge theory results for the superpotential?
• RQ5 How does the supergroup embedding G(N+k|k) resolve F-term ambiguities and reproduce known results for classical and exceptional groups?

Key findings

• The glueball superpotential W(S) is unambiguous for terms S^n with n < h, corresponding to fractional instanton effects, while terms with n ≥ h are UV-dependent and require completion.
• For classical groups (A, B, C, D), the supergroup embedding U(N+k|k) with k→∞ provides a natural F-term completion, yielding exact results to all orders in S.
• The supergroup completion resolves discrepancies between matrix model results and standard gauge theory, particularly in cases like U(1) where the standard theory has no superpotential but the F-completed version does.
• The tensor structure on the Cartan subalgebra required for W(S) is shown to be proportional to the antisymmetrized product of Kronecker deltas, reducing all invariants to pure S^ℓ terms.
• For exceptional groups like E6, E7, E8, the superpotential is computed exactly for low powers of S, and an all-order completion is proposed for simply-laced cases including N=1* theories.
• The method explains residual instanton effects in Higgs branches of supergroups, such as in U(2|1)/U(1), which account for differences between standard and F-completed UV completions.