7D supersymmetric Yang-Mills on curved manifolds

Polydorou, Konstantina; Rocén, Andreas; Zabzine, Maxim

doi:10.1007/jhep12(2017)152

Cited by 17 publications

(73 citation statements)

References 48 publications

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“…In this case, reduces to

\begin{matrix} true \\ right \begin{matrix} d s_{10}^{2} & = e^{\frac{2}{3} ϕ} d s^{2} false(M_{7} false) + e^{- \frac{2}{3} ϕ} d s^{2} false(R^{1, 2} false) \\ e^{2 ϕ} & = {(k + \frac{q}{r})}^{- \frac{3}{2}} \\ H & = 14 e^{- \frac{4}{3} ϕ} {vol}_{3} \\ F_{2} & = - i q {vol}_{2} \\ F_{4} & = 4 i ψ . \end{matrix} \end{matrix}

Manifolds equipped with a G 2 ‐structure with purely scalar torsion are known as nearly‐parallel, or equivalently, weak G 2 ‐holonomy spaces. In particular, as determined in [] (see also ), generic Sasaki‐Einstein manifolds admit 2 such Killing spinors, generic tri‐Sasakians admit 3 and the round S 7 admits 8. Due to the four‐form flux

F_{4} \sim ψ

, the solution is invariant under

G_{2} \times S O false(1, 2 false)

or a subgroup thereof.…”

Section: Examples Of Backgroundsmentioning

confidence: 89%

“…This includes the canonical examples on S 7 and on Sasaki‐Einstein manifolds. We will mostly follow the notation of []. We decompose

\begin{matrix} true \\ right \begin{matrix} ε = χ \otimes ζ \otimes (\begin{matrix} 1 \\ 0 \end{matrix}) . \end{matrix} \end{matrix}

Using

\nabla_{μ} χ = \frac{1}{2} i τ_{0} γ_{μ} χ

, we obtain

\begin{matrix} true \\ right \begin{matrix} \nabla_{μ} ε = \frac{1}{2} τ_{0} {normalΓ}_{μ} Λ ε, Λ \equiv {normalΓ}_{089} = {scriptP}_{2} . \end{matrix} \end{matrix}

The off‐shell action on a weak G 2 ‐holonomy manifold is then given by

\begin{matrix} true \\ right S & = & left \int d^{7} x \sqrt{g_{false(7 false)}} (\frac{1}{2} F^{M N} F_{M N} - \bar{Ψ} (() {normalΓ}^{M} D_{M} - \frac{3}{2} τ_{0} Λ) normalΨ \\ left () + 8 τ_{0}^{2} ϕ^{a} ϕ_{a} + 2 τ_{0} false[ϕ^{a}, ϕ^{b} false] ϕ^{c} ε_{a b c} - K^{j} K_{j}) . \end{matrix}

The action is invariant under the following supersymmetry transformations:

\begin{matrix} true \\ right \begin{matrix} δ_{ε} A_{M} & = \bar{normalΨ} {normalΓ}_{M} ε \\ δ_{ε} Ψ & <...> \end{matrix} \end{matrix}

…”

Section: Gauge Theory On Curved 7‐branesmentioning

confidence: 99%

“…[19,42] This includes the canonical examples on S 7 and on Sasaki-Einstein manifolds. We will mostly follow the notation of [20]. We decompose 13…”

Section: The Action Is Invariant Under Gauge Transformations δmentioning

confidence: 99%

“…[2,8,19] In addition to S 7 , known admissible manifolds are proper G 2 -manifolds, Sasaki-Einstein spaces and tri-Sasakians. [20] A common feature is that all of these admit spinors satisfying a (non-generalized) Killing spinor equation, similar to the sphere. However, there is no reason to expect that supersymmetric field theories do not exist on more general spaces.…”

Section: Introductionmentioning

confidence: 99%

“…For comparison, in our notation the 32 × 32 gamma‐matrices of [] are given by

\begin{matrix} true \\ right {\overset{γ}{true}}_{μ}^{there} & left = γ_{μ} \otimes - σ_{2} \otimes σ_{1}, {\hat{γ}}_{8}^{there} = double-struck1 \otimes σ_{1} \otimes σ_{1} \\ right {\overset{γ}{true}}_{9}^{there} & left = double-struck1 \otimes σ_{3} \otimes σ_{1}, {\hat{γ}}_{0}^{there} = double-struck1 \otimes double-struck1 \otimes i σ_{2} . \end{matrix}

As a consequence,

{normalΛ}^{there} = 1 \otimes - i σ_{2} \otimes 1

, whereas

{normalΛ}^{here} = 1 \otimes 1 \otimes σ_{1}

.…”

mentioning

confidence: 99%

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Supersymmeric Gauge Theory on Curved 7‐Branes

Prins

2019

Fortschritte der Physik

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We construct novel 7d supersymmetric gauge theories which include a Chern‐Simons‐like term on curved spaces. In order to so, we examine the supersymmetry constraints for E7‐branes in type IIA* theory, rather than making use of an off‐shell supergravity. We find two classes of solutions to the constraints, expressed in terms of a G2‐structure with non‐vanishing intrinsic torsion. The supersymmetric gauge theories are then obtained by coupling flat gauge theory to such supersymmetric backgrounds. Various examples are given, including round and squashed S7 as well as S3×M4 with M4 hyperkähler.

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“…In this case, reduces to

\begin{matrix} true \\ right \begin{matrix} d s_{10}^{2} & = e^{\frac{2}{3} ϕ} d s^{2} false(M_{7} false) + e^{- \frac{2}{3} ϕ} d s^{2} false(R^{1, 2} false) \\ e^{2 ϕ} & = {(k + \frac{q}{r})}^{- \frac{3}{2}} \\ H & = 14 e^{- \frac{4}{3} ϕ} {vol}_{3} \\ F_{2} & = - i q {vol}_{2} \\ F_{4} & = 4 i ψ . \end{matrix} \end{matrix}

F_{4} \sim ψ

, the solution is invariant under

G_{2} \times S O false(1, 2 false)

or a subgroup thereof.…”

Section: Examples Of Backgroundsmentioning

confidence: 89%

“…This includes the canonical examples on S 7 and on Sasaki‐Einstein manifolds. We will mostly follow the notation of []. We decompose

\begin{matrix} true \\ right \begin{matrix} ε = χ \otimes ζ \otimes (\begin{matrix} 1 \\ 0 \end{matrix}) . \end{matrix} \end{matrix}

Using

\nabla_{μ} χ = \frac{1}{2} i τ_{0} γ_{μ} χ

, we obtain

\begin{matrix} true \\ right \begin{matrix} \nabla_{μ} ε = \frac{1}{2} τ_{0} {normalΓ}_{μ} Λ ε, Λ \equiv {normalΓ}_{089} = {scriptP}_{2} . \end{matrix} \end{matrix}

The off‐shell action on a weak G 2 ‐holonomy manifold is then given by

\begin{matrix} true \\ right S & = & left \int d^{7} x \sqrt{g_{false(7 false)}} (\frac{1}{2} F^{M N} F_{M N} - \bar{Ψ} (() {normalΓ}^{M} D_{M} - \frac{3}{2} τ_{0} Λ) normalΨ \\ left () + 8 τ_{0}^{2} ϕ^{a} ϕ_{a} + 2 τ_{0} false[ϕ^{a}, ϕ^{b} false] ϕ^{c} ε_{a b c} - K^{j} K_{j}) . \end{matrix}

The action is invariant under the following supersymmetry transformations:

\begin{matrix} true \\ right \begin{matrix} δ_{ε} A_{M} & = \bar{normalΨ} {normalΓ}_{M} ε \\ δ_{ε} Ψ & <...> \end{matrix} \end{matrix}

…”

Section: Gauge Theory On Curved 7‐branesmentioning

confidence: 99%

“…[19,42] This includes the canonical examples on S 7 and on Sasaki-Einstein manifolds. We will mostly follow the notation of [20]. We decompose 13…”

Section: The Action Is Invariant Under Gauge Transformations δmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…For comparison, in our notation the 32 × 32 gamma‐matrices of [] are given by

\begin{matrix} true \\ right {\overset{γ}{true}}_{μ}^{there} & left = γ_{μ} \otimes - σ_{2} \otimes σ_{1}, {\hat{γ}}_{8}^{there} = double-struck1 \otimes σ_{1} \otimes σ_{1} \\ right {\overset{γ}{true}}_{9}^{there} & left = double-struck1 \otimes σ_{3} \otimes σ_{1}, {\hat{γ}}_{0}^{there} = double-struck1 \otimes double-struck1 \otimes i σ_{2} . \end{matrix}

As a consequence,

{normalΛ}^{there} = 1 \otimes - i σ_{2} \otimes 1

, whereas

{normalΛ}^{here} = 1 \otimes 1 \otimes σ_{1}

.…”

mentioning

confidence: 99%

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Supersymmeric Gauge Theory on Curved 7‐Branes

Prins

2019

Fortschritte der Physik

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SYM on quotients of spheres and complex projective spaces

Lundin

Ruggeri

2022

J. High Energ. Phys.

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We introduce a generic procedure to reduce a supersymmetric Yang-Mills (SYM) theory along the Hopf fiber of squashed S2r−1 with U(1)r isometry, down to the ℂℙr−1 base. This amounts to fixing a Killing vector v generating a U(1) ⊂ U(1)r rotation and dimensionally reducing either along v or along another direction contained in U(1)r. To perform such reduction we introduce a ℤp quotient freely acting along one of the two fibers. For fixed p the resulting manifolds S2r−1/ℤp ≡ L2r−1(p, ±1) are a higher dimensional generalization of lens spaces. In the large p limit the fiber shrinks and effectively we find theories living on the base manifold. Starting from $$ \mathcal{N} $$ N = 2 SYM on S3 and $$ \mathcal{N} $$ N = 1 SYM on S5 we compute the perturbative partition functions on L2r−1(p, ±1) and, in the large p limit, on ℂℙr−1, respectively for r = 2 and r = 3. We show how the reductions along the two inequivalent fibers give rise to two distinct theories on the base. Reducing along v gives an equivariant version of Donaldson-Witten theory while the other choice leads to a supersymmetric theory closely related to Pestun’s theory on S4. We use our technique to reproduce known results for r = 2 and we provide new results for r = 3. In particular we show how, at large p, the sum over fluxes on ℂℙ2 arises from a sum over flat connections on L5(p, ±1). Finally, for r = 3, we also comment on the factorization of perturbative partition functions on non simply connected manifolds.

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7D supersymmetric Yang-Mills on hypertoric 3-Sasakian manifolds

Iakovidis

Qiu

Rocén

et al. 2020

J. High Energ. Phys.

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We study 7D maximally supersymmetric Yang-Mills theory on 3-Sasakian manifolds. For manifolds whose hyper-Kähler cones are hypertoric we derive the perturbative part of the partition function. The answer involves a special function that counts integer lattice points in a rational convex polyhedral cone determined by hypertoric data. This also gives a more geometric structure to previous enumeration results of holomorphic functions in the literature. Based on physics intuition, we provide a factorisation result for such functions. The full proof of this factorisation using index calculations will be detailed in a forthcoming paper.

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7D supersymmetric Yang-Mills on curved manifolds

Cited by 17 publications

References 48 publications

Supersymmeric Gauge Theory on Curved 7‐Branes

Supersymmeric Gauge Theory on Curved 7‐Branes

SYM on quotients of spheres and complex projective spaces

7D supersymmetric Yang-Mills on hypertoric 3-Sasakian manifolds

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