Supergravity backgrounds for four-dimensional maximally supersymmetric Yang-Mills

Abstract

In this note, we describe supersymmetric backgrounds for the four-dimensional maximally supersymmetric Yang-Mills theory. As an extension of the method of Festuccia and Seiberg to sixteen supercharges in four dimensions, we utilize the coupling of the gauge theory to maximally extended conformal supergravity. Included among the fields of the conformal supergravity multiplet is the complexified coupling parameter of the gauge theory; therefore, backgrounds with spacetime varying coupling -such as appear in F-theory and Janus configurations -are naturally included in this formalism. We demonstrate this with a few examples from past literature.

Six-dimensional origin of N $$ \mathcal{N} $$ = 4 SYM with duality defects

“…To see this flavor symmetry, note that the 2d action for the chiral bosons β i along C (4.10) contains the coupling 20 27) 19 This uses the fact that ∂b + Z ∧ J + is anti imaginary self-dual, so we must set ∂b + Z = 0, whereas∂b + Z ∧ J + is imaginary self-dual. 20 The rest of the chiral bosons action is trivially invariant under the symmetries we describe.…”

“…Unlike the case studied here, which could be understood in terms of D3-instantons with duality surface defects, in those cases, the theories are 2d chiral supersymmetric theories, with point-like defects on the compactification curve, where the D3-branes intersect 7-branes in F-theory. More recently a discussion of supergravity backgrounds for 4d N = 4 SYM appeared in [20]. Also, some duality defects for N = 2 theories were proposed in [21].…”

“…We assume the existence of a 6d Killing spinor E independent on the fiber coordinates x, y, with negative chirality. Such a spinor can be decomposed, as explained in appendix A.2, into 20) where E + , E − are 4d spinors of positive and negative 4d chirality respectively and η ± are constant 2d chiral spinors which serve as a basis for the decomposition. The 6d Killing spinor equations (2.17) then reduce to the 4d Killing spinor equations,…”

“…Under the twisted symmetry group it decomposes to the sum of irreducible representations 20) corresponding to an imaginary self-dual (1, 2)-form h (1,2) , a holomorphic one-form h (1,0) and a holomorphic three-form h (3,0) . The twist charge q L /2 indicates the difference in the holomorphic and anti-holomorphic degree: q L /2 = q − p for a form Λ (p,q) .…”

Deformation of $$ \mathcal{N} $$ = 4 SYM with varying couplings via fluxes and intersecting branes

“…(See also section 3.4 in [2]) It is a solution of the SUSY conditions (2.14)-(2.18) with the parameters depending only on x 3 . ISO(1, 2) is the Poincaré group acting on x 0,1,2 and SO(3) × SO(3) acts on x 4,5,6 and x 7,8,9 . The metric is assumed to be flat and the non-trivial components of the couplings in the action are given as follows: 5 τ = 4πa(c + i) = τ 0 + 4πD e i2ψ , (2.23)…”