Higher-order gaugino condensates on a twisted $\mathbb T^4$: In the beginning was semi-classics
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2024
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| author | Anber, Mohamed M. Poppitz, Erich |
| author_facet | Anber, Mohamed M. Poppitz, Erich |
| contents | We compute the gaugino condensates, $\left\langle \prod_{i=1}^k \text{tr}(λλ)(x_i) \right\rangle $ for $1$ $\leq$ $k$ $\le$ $N-1$, in $SU(N)$ super Yang-Mills theory on a small four-dimensional torus $\mathbb{T}^4$, subject to 't Hooft twisted boundary conditions. Two recent advances are crucial to performing the calculations and interpreting the result: the understanding of generalized anomalies involving $1$-form center symmetry and the construction of multi-fractional instantons on the twisted $\mathbb T^4$. These self-dual classical configurations have topological charge $k/N$ and can be described as a sum over $k$ closely packed lumps in an instanton liquid. Using the path integral formalism, we perform the condensate calculations in the semi-classical limit and find, assuming gcd$(k,N)=1$, $\left\langle \prod_{i=1}^k \text{tr}(λλ)(x_i) \right\rangle = {\bf n}^{-1} \; N^2\left(16π^2 Λ^3\right)^k$, where $Λ$ is the strong-coupling scale and ${\bf n}$ is a normalization constant. We determine the normalization constant, using path integral, as ${\bf n} = N^2$, which is $N$ times larger than the normalization used in our earlier publication arXiv:2210.13568. This finding resolves the extra-factor-of-$N$ discrepancy encountered there, aligning our results with those obtained through direct supersymmetric methods on $\mathbb R^4$. The normalization constant ${\bf n}$ can be understood within the Euclidean path-integral framework as the Witten index $I_W$. From the Hamiltonian approach, it is well-established that $I_W = N$. While the value ${\bf n} = N^2$ correctly reproduces the condensate result, this discrepancy between the Hamiltonian and path-integral formulations calls for reconciliation. We attempt to provide a potential solution we outline in our discussion. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2408_16058 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Higher-order gaugino condensates on a twisted $\mathbb T^4$: In the beginning was semi-classics Anber, Mohamed M. Poppitz, Erich High Energy Physics - Theory High Energy Physics - Lattice High Energy Physics - Phenomenology We compute the gaugino condensates, $\left\langle \prod_{i=1}^k \text{tr}(λλ)(x_i) \right\rangle $ for $1$ $\leq$ $k$ $\le$ $N-1$, in $SU(N)$ super Yang-Mills theory on a small four-dimensional torus $\mathbb{T}^4$, subject to 't Hooft twisted boundary conditions. Two recent advances are crucial to performing the calculations and interpreting the result: the understanding of generalized anomalies involving $1$-form center symmetry and the construction of multi-fractional instantons on the twisted $\mathbb T^4$. These self-dual classical configurations have topological charge $k/N$ and can be described as a sum over $k$ closely packed lumps in an instanton liquid. Using the path integral formalism, we perform the condensate calculations in the semi-classical limit and find, assuming gcd$(k,N)=1$, $\left\langle \prod_{i=1}^k \text{tr}(λλ)(x_i) \right\rangle = {\bf n}^{-1} \; N^2\left(16π^2 Λ^3\right)^k$, where $Λ$ is the strong-coupling scale and ${\bf n}$ is a normalization constant. We determine the normalization constant, using path integral, as ${\bf n} = N^2$, which is $N$ times larger than the normalization used in our earlier publication arXiv:2210.13568. This finding resolves the extra-factor-of-$N$ discrepancy encountered there, aligning our results with those obtained through direct supersymmetric methods on $\mathbb R^4$. The normalization constant ${\bf n}$ can be understood within the Euclidean path-integral framework as the Witten index $I_W$. From the Hamiltonian approach, it is well-established that $I_W = N$. While the value ${\bf n} = N^2$ correctly reproduces the condensate result, this discrepancy between the Hamiltonian and path-integral formulations calls for reconciliation. We attempt to provide a potential solution we outline in our discussion. |
| title | Higher-order gaugino condensates on a twisted $\mathbb T^4$: In the beginning was semi-classics |
| topic | High Energy Physics - Theory High Energy Physics - Lattice High Energy Physics - Phenomenology |
| url | https://arxiv.org/abs/2408.16058 |