[Paper Review] Localization of Vortex Partition Functions in $\mathcal{N}=(2,2) $ Super Yang-Mills theory

Yutaka Yoshida|arXiv (Cornell University)|Jan 5, 2011

Black Holes and Theoretical Physics15 references16 citations

TL;DR

This paper computes the vortex partition function in $χ=(2,2)$ $U(N)$ super Yang-Mills theory with $N$ fundamental chiral multiplets using equivariant localization. By deforming the $χ=(4,4)$ theory via mass terms and applying localization to the $Q$-exact action, the partition function localizes to $N$-tuple one-dimensional partitions, yielding a K-theoretic expression in terms of twisted masses and equivariant parameters, with the abelian case reducing to an exponential generating function related to $\mathbb{C}P^{N-1}$ Gromov-Witten invariants.

ABSTRACT

In this article, we study the localizaiton of the partition function of BPS vortices in $\mathcal{N}=(2,2)$ $U(N)$ super Yang-Mills theory with $N$-flavor on $\R^2$. The vortex partition function for $\mathcal{N}=(2,2)$ super Yang-Mills theory is obtained from the one in $\mathcal{N}=(4,4)$ super Yang-Mills theory by mass deformation. We show that the partition function can be written as $Q$-exact form and integration in the partition functions is localized to the fixed points which are related to $N$-tuple one dimensional partitions of positive integers.

Motivation & Objective

• To compute the vortex partition function in $\mathcal{N}=(2,2)$ $U(N)$ super Yang-Mills theory with $N$ fundamental chiral multiplets.
• To extend the method of equivariant localization to non-abelian vortex moduli spaces in two-dimensional $\mathcal{N}=(2,2)$ theories.
• To derive the K-theoretic vortex partition function from the equivariant character of the K"{a}hler quotient space describing the vortex moduli.
• To establish a connection between the vortex partition function and the $J$-function of $\mathrm{Gr}(N_c, N_f)$ in the large $N_f$ limit.

Proposed method

• Use of $Q$-exact deformation to localize the path integral to fixed points in the vortex moduli space.
• Reduction from $\mathcal{N}=(4,4)$ to $\mathcal{N}=(2,2)$ theory via mass deformation and large mass limit of the superpotential.
• Application of equivariant localization to the $U(N)$ gauge theory with $N$ fundamental chiral multiplets and twisted masses.
• Computation of the tangent space character of the vortex moduli space $\mathcal{M}_{k,N}$ as a quotient of representation spaces.
• Derivation of the K-theoretic partition function via characters of $V$, $W$, and $Q$, with $V$ encoding $N$-tuple partitions.
• Taking the $\beta \to 0$ limit to recover the cohomological (field theory) partition function from the K-theoretic form.

Experimental results

Research questions

• RQ1 How does the vortex partition function in $\mathcal{N}=(2,2)$ $U(N)$ super Yang-Mills theory with $N$ fundamental flavors behave under equivariant localization?
• RQ2 What is the structure of the vortex moduli space $\mathcal{M}_{k,N}$ in the presence of twisted masses, and how does it decompose under the $U(1)^N$ symmetry?
• RQ3 Can the K-theoretic vortex partition function be derived from the equivariant character of the K"{a}hler quotient construction of the moduli space?
• RQ4 How does the abelian limit of the partition function relate to the $J$-function of $\mathbb{C}P^{N-1}$?
• RQ5 What is the relation between the non-abelian vortex partition function and the $J$-function of the Grassmannian $\mathrm{Gr}(N_c, N_f)$ in the large $N_f$ limit?

Key findings

• The vortex partition function in $\mathcal{N}=(2,2)$ $U(N)$ theory localizes to fixed points labeled by $N$-tuples of one-dimensional partitions of positive integers.
• The K-theoretic vortex partition function is given by $ Z^\text{K-theoretic} = \sum_{k=0}^\infty e^{2\pi(r+i\theta)k} \sum_{k_1+\cdots+k_N=k} \prod_{i,\tilde{i}=1}^N \prod_{l_i=1}^{a_i} \prod_{j_i=1}^{b_i} \frac{1}{(1 - Q_{\tilde{i}i} q^{k_{\tilde{i}}+1-l_i})(1 - Q_{\tilde{i}i} q^{1-j_i})} $, with $ a_i = \min(k_i, k_{\tilde{i}}) $, $ b_i = \max(k_i - k_{\tilde{i}}, 0) $.
• In the abelian case ($N=1$), the partition function reduces to $ Z = \exp\left( \frac{e^{2\pi(r+i\theta)}}{\epsilon} \right) $, matching the generating function of $\mathbb{C}P^0$ Gromov-Witten invariants.
• The cohomological limit ($\beta \to 0$) yields the field theory partition function: $ Z = \sum_{k=0}^\infty e^{2\pi(r+i\theta)k} \sum_{k_1+\cdots+k_N=k} \prod_{i,\tilde{i}=1}^N \prod_{l_i=1}^{a_i} \prod_{j_i=1}^{b_i} \frac{1}{((k_{\tilde{i}}+1-l_i)\epsilon - m_{i\tilde{i}})((1-j_i)\epsilon - m_{i\tilde{i}})} $.
• The results suggest a generalization to non-abelian vortices via the $J$-function of $\mathrm{Gr}(N_c, N_f)$, particularly in the strong coupling limit where the moduli space becomes the space of holomorphic maps to $\mathrm{Gr}(N_c, N_f)$.
• The partition function for $U(1)$ with $N_f$ flavors is shown to be related to the $J$-function of $\mathbb{C}P^{N_f-1}$, providing a geometric interpretation of the generating function.