Quantum Higgs vacua and the conformal field theory of the Higgs branch in two-dimensional (4,4) supersymmetric NS5-brane gauge theories

This problem concerns the open questions posed in Witten's 1997 study of the conformal field theory (CFT) on the Higgs branch of two-dimensional $\mathcal{N}=(4,4)$ supersymmetric gauge theories, which describe NS5-brane systems via discrete light-cone quantization (DLCQ). The central question is whether the $k=1$ theory, $U(1)$ gauge theory with a single charged hypermultiplet whose charged matter has no classical Higgs branch, flows in the infrared to an isolated, massive 'quantum Higgs vacuum'; Witten asked for decisive arguments settling existence or nonexistence, noting that the existence of such a vacuum is what would give a single Type IIA fivebrane an intrinsic interacting limit. He conjectured that, if it exists, the Higgs-branch CFT of the $k=1$ fivebrane is the symmetric-product orbifold CFT deformed from worldsheet theta-angle $\theta=\pi$ to $\theta=0$, a deformation expected to be strongly coupled. The questions Witten left open on these points, together with the related questions they raise, are stated in the Problem Statement below; no decisive argument settling them is known in the literature to date.

Background

The NS5-brane is a solitonic extended object of type II string theory, carrying magnetic charge under the Neveu-Schwarz two-form field, whose six-dimensional worldvolume theory at low energy is a non-gravitational 'little string theory'. Its discrete light-cone quantization (DLCQ, a definition of a theory in which a light-like direction is compactified so that states carry integer units of light-cone momentum) is a two-dimensional gauge theory with $\mathcal{N}=(4,4)$ supersymmetry (eight supercharges): $U(n)$ gauge theory with an adjoint hypermultiplet and $k$ fundamental hypermultiplets, where $n$ counts light-cone momentum units and $k$ the number of fivebranes. Hypermultiplets are the supersymmetric matter multiplets whose scalar fields parametrize the Higgs branch, the moduli space of vacua obtained by giving expectation values to matter scalars; here that space is the moduli space $M_{n,k}$ of $n$ instantons (self-dual solutions of the four-dimensional Yang-Mills equations, parametrized algebraically by the ADHM (Atiyah-Drinfeld-Hitchin-Manin) construction) of an $SU(k)$ gauge theory. For $k=1$ this space is exactly the symmetric product $\mathrm{Sym}^n(\mathbb{R}^4)$, the configuration space of $n$ unordered points in $\mathbb{R}^4$.

Witten (1997) analyzed the conformal field theory (CFT) on this Higgs branch, motivated by a puzzle about the Coulomb branch, the competing moduli space parametrized by vector-multiplet scalars, which for the fivebrane has the geometry of a semi-infinite tube and is therefore hard to interpret as the target of a CFT. He studied an 'induced' gauge theory, obtained by dropping the vector-multiplet kinetic term so that integrating out the hypermultiplets generates the effective action; it has a controlled expansion in $1/k$ (in the induced-theory literature the number $k$ of fundamental hypermultiplets is often called the flavor number $N_f$, and the gauge rank $n$ the color number $N_c$) and a central charge $\hat c = k-1$, giving a useful description of the Higgs branch near its origin for large $k$. He then identified a set of precise open questions centered on a single fivebrane, $k=1$. The crux is that for $n=k=1$ the $U(1)$ gauge theory has one charged hypermultiplet, whose scalar is forced to vanish by the D-term equation so that the charged matter contributes no classical Higgs branch; the branch present classically is carried by the adjoint hypermultiplet, which for $U(1)$ is free and gives the copy $M_{1,1}=\mathbb{R}^4$ of the moduli space (and $\mathrm{Sym}^n(\mathbb{R}^4)$ for general $n$, as above). Nevertheless the fivebrane interpretation suggests that the charged sector develops a 'quantum Higgs vacuum', an isolated massive vacuum existing only quantum mechanically, with central charge $\hat c=0$ (a gapped, infrared-trivial theory rather than a nontrivial CFT). Assuming such a vacuum, Witten conjectured that the Higgs-branch CFT of the $k=1$ fivebrane with $n$ units of momentum is not the free symmetric-product orbifold CFT but its marginal deformation that changes the worldsheet theta-angle of the effective $U(1)$ gauge description of the codimension-four coincidence singularities from the free value $\theta=\pi$ to $\theta=0$, a deformation that preserves the symmetries but is probably strongly coupled. He posed this as a route to computing fivebrane observables, alongside questions about the birational equivalence $M_{n,k} \sim \mathrm{Sym}^n M_{1,k}$, the $1/k$ expansion and its relation to four-dimensional $SU(k)$ gauge theory, critical-$k$ thresholds for nontrivial infrared fixed points in the three-dimensional-QED-type analogues, and generalizations to nonabelian groups and to fivebranes transverse to a four-torus $T^4$ rather than $\mathbb{R}^4$.

The question is closely tied to the D1-D5 system (a bound state of D1-branes, one-dimensional extended objects on which open strings can end, with D5-branes, their five-dimensional counterparts; its near-horizon geometry underlies the AdS$_3$/CFT$_2$ holographic duality between string theory on three-dimensional anti-de Sitter space and a two-dimensional CFT): the IR fixed point of the same two-dimensional gauge theory is the D1-D5 CFT, whose symmetric-product orbifold point, $\mathrm{Sym}^N(T^4)$, is the candidate dual of string theory on AdS$_3$ with one unit of NS-NS flux (flux of the two-form field strength through the AdS3 factor). Results in that setting therefore bear directly on the fivebrane questions. Prior work bearing on the computability of the deformed theory and its D1-D5 relative has studied the protected spectrum, the $1/N_f$ (equivalently large-$k$) expansion of the induced gauge theory, the integrability of the planar one-loop dilatation operator, and dual descriptions of the D1-D5/AdS$_3$ system (References 4, 8-12).

Problem Statement

This problem concerns the questions Witten (1997) left open about the quantum Higgs branch of NS5-brane systems as described by two-dimensional $\mathcal{N}=(4,4)$ gauge theories, centered on the case of a single fivebrane, $k=1$:

(a) Existence of the $k=1$ quantum Higgs vacuum. Does the charged sector of the $n=k=1$ theory, $U(1)$ gauge theory in two dimensions with $\mathcal{N}=(4,4)$ supersymmetry, a single charged hypermultiplet and a free adjoint hypermultiplet, whose charged hypermultiplet is forced to vanish by the D-term equation so that the charged matter has no classical Higgs branch, flow in the infrared to an isolated, massive quantum Higgs vacuum with $\hat c=0$, distinct from the classical $\mathbb{R}^4$ branch carried by the free adjoint hypermultiplet? Witten asked for decisive arguments showing that such a vacuum does or does not exist, noting that either answer matters for the theory of fivebranes.

(b) The $\theta=0$ Higgs-branch CFT and its computability. Assuming the $n=k=1$ vacuum exists, is the Higgs-branch CFT of the $k=1$ fivebrane with $n$ units of light-cone momentum indeed the symmetric-product orbifold CFT on $\mathrm{Sym}^n(\mathbb{R}^4)$ deformed from worldsheet theta-angle $\theta=\pi$ (the free orbifold) to $\theta=0$, where the angle is that of the effective $U(1)$ gauge theory with two equally charged hypermultiplets describing the codimension-four coincidence locus? Can this strongly coupled theory be controlled well enough to compute fivebrane observables?

(c) The birational map. The $n$-instanton moduli space $M_{n,k}$ is birational to the symmetric product $\mathrm{Sym}^n M_{1,k}$, where $M_{1,k}$ is the one-instanton moduli space and 'birational' means that the two spaces are isomorphic on dense open subsets; however, this equivalence was not accompanied by a useful description of the birational transformation involved. Construct such an explicit map in a form usable in conformal-field-theory computations.

(d) The $1/k$ expansion. Develop the large-$k$ (equivalently $1/N_f$) expansion of the fivebrane Higgs-branch theory beyond leading order, and determine whether it yields a controlled $1/k$ expansion of four-dimensional $SU(k)$ gauge theory.

(e) Critical thresholds. In three-dimensional quantum electrodynamics (QED$_3$), the flow to a nontrivial infrared fixed point is argued to occur only above a critical number of matter flavors. By analogy, characterize the precise thresholds and conditions — for example critical values of $k$ — under which nontrivial infrared fixed points appear in the nonabelian and higher-dimensional analogue models discussed in the paper (QED$_3$-type theories and their supersymmetric versions), which motivate the induced-gauge-theory analysis of the two-dimensional theory.

(f) Generalizations. Extend the analysis beyond the $U(1)$ case: understand the $U(n)$ theory with $k>1$ fundamental hypermultiplets as a marginal deformation of an orbifold of $n$ copies of the $U(1)$ theory with $k$ charged hypermultiplets (a more far-reaching version of the birational relation in (c)); give an exact embedding of the marginal twist (theta-angle) parameter in the underlying ultraviolet $U(n)$ theory, where it is so far characterized only in an effective $U(1)$ description near the codimension-four coincidence locus; and treat fivebranes transverse to a four-torus $T^4$ rather than $\mathbb{R}^4$, where the relevant moduli spaces are instanton moduli spaces of $U(k)$ Yang-Mills theory on $T^4$.

Questions (a)-(f) are posed separately, each corresponding to one of the questions Witten left open; each is resolved or refuted independently of the others, and a complete answer to one of them does not resolve the remainder (part (f) collects three related extensions, which may likewise be answered separately).

References 8-12 bear most directly on these questions, and none of them settles the part it touches: Reference 8 computes the one-loop dilatation operator of the induced theory, which controls the perturbative expansion around the free orbifold point rather than the strongly coupled $\theta=0$ theory; Reference 9 offers brane-dynamics and supersymmetric-partition-sum arguments that its authors present as reasons to believe rather than as a decisive proof, so it does not by itself settle (a); and References 10-12 derive the dual of the neighbouring D1-D5 / AdS$_3$ system with pure NS-NS flux, whose symmetric-product orbifold is free (or, in the pure NS-NS case, marginally deformed) rather than the type IIA NS5-brane Higgs-branch CFT on $\mathrm{Sym}^n(\mathbb{R}^4)$, so they do not settle (b) or (f).

References

1. Edward Witten, "On The Conformal Field Theory Of The Higgs Branch", JHEP 07 (1997) 003. arXiv:hep-th/9707093, doi:10.1088/1126-6708/1997/07/003, https://arxiv.org/abs/hep-th/9707093
2. John Brodie, "Two Dimensional Mirror Symmetry From M-theory", Nucl. Phys. B 517 (1998) 36-52. arXiv:hep-th/9709228, doi:10.1016/S0550-3213(97)00755-4, https://arxiv.org/abs/hep-th/9709228
3. O. Aharony, M. Berkooz, S. Kachru, N. Seiberg, E. Silverstein, "Matrix Description of Interacting Theories in Six Dimensions", Adv. Theor. Math. Phys. 1 (1998) 148-157. arXiv:hep-th/9707079, doi:10.4310/atmp.1997.v1.n1.a5, https://arxiv.org/abs/hep-th/9707079
4. Ofer Aharony, Micha Berkooz, "IR Dynamics of d=2, N=(4,4) Gauge Theories and DLCQ of "Little String Theories"", JHEP 10 (1999) 030. arXiv:hep-th/9909101, doi:10.1088/1126-6708/1999/10/030, https://arxiv.org/abs/hep-th/9909101
5. Ofer Aharony, "A brief review of "little string theories"", Class. Quantum Grav. 17 (2000) 929-938. arXiv:hep-th/9911147, doi:10.1088/0264-9381/17/5/302, https://arxiv.org/abs/hep-th/9911147
6. R. Dijkgraaf, "Instanton Strings and HyperKaehler Geometry", Nucl. Phys. B 543 (1999) 545-571. arXiv:hep-th/9810210, doi:10.1016/S0550-3213(98)00869-4, https://arxiv.org/abs/hep-th/9810210
7. R. Dijkgraaf, E. Verlinde, M. Vonk, "On the partition sum of the NS five-brane", 2002. arXiv:hep-th/0205281, https://arxiv.org/abs/hep-th/0205281
8. Olof Ohlsson Sax, Alessandro Sfondrini, Bogdan Stefanski, "Integrability and the Conformal Field Theory of the Higgs branch", JHEP 06 (2015) 103. arXiv:1411.3676, doi:10.1007/JHEP06(2015)103, https://arxiv.org/abs/1411.3676
9. Jeffrey A. Harvey, David Kutasov, Sungjay Lee, "Comments on Quantum Higgs Vacua", 2014. arXiv:1406.6000, https://arxiv.org/abs/1406.6000
10. Lorenz Eberhardt, Matthias R. Gaberdiel, Rajesh Gopakumar, "The Worldsheet Dual of the Symmetric Product CFT", JHEP 04 (2019) 103. arXiv:1812.01007, doi:10.1007/JHEP04(2019)103, https://arxiv.org/abs/1812.01007
11. Lorenz Eberhardt, Matthias R. Gaberdiel, Rajesh Gopakumar, "Deriving the AdS$_3$/CFT$_2$ Correspondence", JHEP 02 (2020) 136. arXiv:1911.00378, doi:10.1007/JHEP02(2020)136, https://arxiv.org/abs/1911.00378
12. Lorenz Eberhardt, "A perturbative CFT dual for pure NS-NS AdS$_3$ strings", J. Phys. A 55 (2022) 064001. arXiv:2110.07535, doi:10.1088/1751-8121/ac47b2, https://arxiv.org/abs/2110.07535