Exotic Codimension Two Branes

• Codimension two exotic branes are non-geometric, half-BPS objects defined by duality monodromies rather than conventional p-form charges, crucial for T-fold and U-fold constructions.
• Their classification relies on U-duality group representations and logarithmic harmonic functions that capture the monodromies in supergravity and string theory solutions.
• They play a central role in black hole microstate geometries and non-geometric flux compactifications, providing insights into duality symmetry breaking and moduli stabilization.

Codimension two exotic branes are non-geometric, typically half-BPS, extended objects in string and M-theory whose transverse space has effective dimension two. Their defining property is that their charge is encoded not by integrals of p-form field strengths (as for higher-codimension branes), but by discrete monodromies in the global duality symmetry group (e.g., SL(2,ℤ), O(d,d;ℤ), or full U-duality), corresponding to the holonomy of moduli around the defect. The backgrounds associated to codimension two exotic branes are characterized by multivalued patching via duality group transformations, rather than pure diffeomorphisms or gauge transformations, and thus these branes are integral to the structure of T-fold and U-fold non-geometric backgrounds. They play a central role in capturing nontrivial aspects of string dualities, in the completion of U-duality multiplets, and in the microscopic structure of black hole microstates.

1. Classification and Monodromies

The classification of codimension two exotic branes proceeds via their U-duality group representations and associated winding or isometry structure. In maximally supersymmetric settings (e.g., toroidal compactifications to D≥3), the relevant duality group is $G=E_{11-d(d)}(\mathbb{Z})$. In type II string theory and M-theory, the most prototypical examples include:

• Type IIB 7-branes: D7 (g_s^–1 tension), NS7 (g_s^–2), and further S-, T-, and U-duals with tension scaling as $T \sim g_s^{-n}$, $n\geq 1$ (Bergshoeff et al., 2011, Boer et al., 2012, Kleinschmidt, 2011).
• 5^2_2-branes: These are obtained by T-dualizing conventional NS5-branes along two isometric transverse directions; their worldvolume is five-dimensional, with two smeared isometries, and tension scaling as $T \sim g_s^{-2}(R_6R_7/\ell_s^2)^2$ (Boer et al., 2012, Kimura, 2016).
• Composite (p,q) branes: Including defect (p,q) five-branes, which are simultaneous bound states of defect NS5 and 5^2_2 branes and are classified by SL(2,ℤ)×SL(2,ℤ) monodromies on the torus (Kimura, 2014, Kimura, 2016).

The monodromy associated with a codimension two exotic brane is a conjugacy class in the relevant duality group: $$M_{\text{brane}} = U^{-1} \begin{pmatrix} 1 & 1 \ 0 & 1 \end{pmatrix} U, \quad U \in SL(2,\mathbb{Z})$$ for 7-branes (axio-dilaton monodromy), or

$$M_{5^2_2} = \begin{pmatrix} 1 & 0 \ -2\pi \sigma & 1 \end{pmatrix}$$

acting on the complexified Kähler modulus of a T^2 for the 5^2_2-brane (Sen, 22 Dec 2025, Boer et al., 2012, Kimura et al., 2023).

A key organizational principle is that in each dimension D, the number of half-BPS codimension two branes equals $\mathrm{dim}\,G - \mathrm{rank}\,G$, coinciding with the number of group-theoretically allowed monodromies (Bergshoeff et al., 2011).

2. Supergravity and String Backgrounds

Codimension two exotic branes manifest as (generically non-geometric) solutions to the supergravity equations with metric, dilaton, and antisymmetric tensor fields patched by duality transformations. In the canonical example, the four-dimensional solution for an exotic string is: $$ds^2_E = -dt^2 + dz^2 + \left(1-\frac{g_4^2}{2\pi}\ln\frac{r}{L_0}\right) \left(dr^2 + r^2 d\theta^2\right), \quad \tau(r,\theta) = i g_4^{-2} \left(1-\frac{g_4^2}{2\pi}\ln\frac{r}{L_0} e^{i\theta}\right)$$ where $r e^{i\theta} = x^1 + i x^2$ are the transverse coordinates and $\tau$ is the complex coupling (axio-dilaton) (Sen, 22 Dec 2025, Kleinschmidt, 2011).

For the 5^2_2-brane, the prototype string frame solution (smeared on the torus) is: $$ds^2 = H\,dx_{012345}^2 + H\,(dr^2 + r^2 d\theta^2) + H\,K^{-1} dx_{89}^2, \quad B_{89} = -\frac{\sigma\,\theta}{K},\quad K = H^2 + \sigma^2 \theta^2,\quad H = h_0 + \sigma \ln \frac{\mu}{r}$$ The monodromy as $\theta \to \theta + 2\pi$ is $B_{89} \mapsto B_{89} - 2\pi \sigma$, or an equivalent O(2,2;ℤ) transformation (Boer et al., 2012, Kimura et al., 2023, Kimura, 2014). All geometric quantities (metric, B, curvature, complex structure) are patched by the same monodromy in the generalized geometry or double field theory formalism (Kimura et al., 2023).

The harmonic functions in these backgrounds are always logarithmic in the radial coordinate r, leading to a nontrivial multi-valuedness (branch point at the brane). The complete non-geometric nature appears at the global, not local, level.

3. Duality Symmetry Breaking and Gauge Interpretation

In flat space, duality transformations (S-duality, T-duality, U-duality) are spontaneously broken discrete gauge symmetries in the presence of codimension two exotic branes. The physical observables are invariant under patching by monodromy matrices $M$, with the field configuration after encircling the brane related to the original by

$\Phi(\gamma+2\pi) = M \cdot \Phi(\gamma)$

for any field $\Phi$ transforming under the duality group (Sen, 22 Dec 2025, Kimura et al., 2023). This leads to observable Aharonov–Bohm phases for probes carrying duality charges: $$(Q, P) \to M_{(s,r)} (Q, P), \qquad \Delta \Phi = 2\pi[(Q',P')-(Q,P)] \in 2\pi \mathbb{Z}$$ This discrete AB phase directly demonstrates the gauge nature of duality in the presence of unshielded codimension two exotic branes (i.e., when the loop is outside the event horizon) (Sen, 22 Dec 2025).

For the seven-branes of type IIB, the SL(2,ℤ) duality acts on the axio-dilaton, and the corresponding monodromy encodes the spectrum and fusing rules of (p,q)-strings. For five-branes, both O(2,2;ℤ) (T-fold monodromy) and composite monodromies in higher duality groups arise (Boer et al., 2012, Kimura et al., 2023).

4. Phenomenology, Observability, and Non-Geometry

The physical detectability of codimension two exotic branes depends crucially on the macroscopic accessibility of the brane loop. For a loop of size $L$ and tension $T$, the Schwarzschild radius scales as $r_s \propto L$, so to avoid the brane being hidden behind a horizon, the tension must satisfy $T \ll L^{(D-3)/(1-D)}$ (Sen, 22 Dec 2025). This restricts observable branes to those with parametrically small tension in Planck units.

In practical terms, codimension two exotic branes generate non-geometric fluxes (Q-flux, R-flux), and serve as the sources for non-geometric compactifications. Their monodromies stabilize moduli and enable automorphic superpotential couplings in flux compactifications (Kleinschmidt, 2011, Bergshoeff et al., 2011, Fernandez-Melgarejo et al., 2018). They also provide the crucial physical setting in which T-fold and U-fold backgrounds have concrete brane sources: the patching of all background fields, including curvatures and complex structures, is given by the duality monodromy matrix (Kimura et al., 2023).

The semi-doubled gauged linear sigma model (GLSM) construction realizes these backgrounds in two-dimensional supersymmetric theories, reproducing the worldsheet instanton corrections predicted by double field theory (Kimura et al., 2018).

5. Bound States, Black Hole Microstates, and Duality Web

Codimension two exotic branes are essential in the context of black hole microstate geometries and the supertube effect. Standard bound state constructions of D- and NS-branes polarizing into higher-dimensional objects (supertubes, superstrata) inevitably generate exotic brane dipoles, whose worldvolumes lie along arbitrary closed curves in non-compact space. For instance, D4(6789)+D4(4589) can polarize into a 5^2_2(ψ4567;89) with monodromy around ψ, and their backreaction creates non-geometric T-folds (Boer et al., 2012, Park et al., 2015).

In the quiver quantum mechanics description of black hole bound states, codimension two exotic branes are associated with the Higgs branch, in contrast to codimension three branes corresponding to the Coulomb branch (Park et al., 2015). Thus, the large entropy attributed to stringy black holes is expected to have decisive contributions from microstates involving exotic brane dipoles.

In F-theory and related S- or U-fold frameworks, webs and junctions of exotic branes allow for generalizations of Hanany–Witten transitions, brane web constructions of gauge theories with exceptional symmetry, and higher-dimensional duality frames (Kimura, 2016, Kimura, 2014).

6. Origin, Mixed-Symmetry Potentials, and Effective Descriptions

In higher dimensions, codimension two exotic branes descend via toroidal reduction from parent objects in ten- or eleven-dimensions, including known branes and generalized KK monopoles (with multiple isometries). They electrically couple, at linearized level, to mixed-symmetry tensor potentials $A_{[m_1\ldots m_k],[n_1\ldots n_\ell],\ldots}$, which are dual, via higher-order duality relations, to the standard supergravity form fields (Bergshoeff et al., 2011, Fernandez-Melgarejo et al., 2018). For instance, the ten-dimensional field $D_{8,2}$ is dual to the B-field, and mixed-symmetry fields such as $D_{9,3}$ (for R-flux) or $D_{8,2}$ (for Q-flux) characterize the magnetic couplings and non-geometric fluxes (Fernandez-Melgarejo et al., 2018).

The proper background solutions require U-duality–covariant frameworks such as Double Field Theory (DFT) or Exceptional Field Theory (EFT), where brane charges, metric, and fluxes reside in representations of the duality group. All fields are consistently patched by duality monodromies; EFT solutions in the E_{7(7)}×ℝ^+ formalism unify many codimension two exotic branes as distinct section choices or U-duality rotations of a single extended-space solution (Otsuki et al., 2019, Berman et al., 2018).

7. Summary Table: Codimension Two Exotic Branes

Brane	Worldvolume	Tension Scaling	Monodromy Group
D7, NS7, 7₃	(1234567)	$g_s^{-n}$, n=1,2,3	SL(2,ℤ)_τ
5^2_2	(12345;67)	$g_s^{-2}(R_6R_7/\ell_s^2)^2$	SL(2,ℤ)_ρ/O(2,2;ℤ)
5^3_2, 3^4_3…	(12345;678…), etc.	$g_s^{-3}$, other scaling	U-duality (E_{d(d)}(ℤ))

Monodromy matrices act on the relevant axionic or geometric moduli and define the defect charge via their conjugacy class. All known examples can be generated by duality chains from standard branes and are essential for realizing the non-geometric sector of string/M-theory.

• (Sen, 22 Dec 2025) Exotic Branes and Symmetries of String Theory
• (Bergshoeff et al., 2011) Defect Branes
• (Kleinschmidt, 2011) Counting supersymmetric branes
• (Boer et al., 2012) Exotic Branes in String Theory
• (Kimura et al., 2023) On Geometries and Monodromies for Branes of Codimension Two
• (Kimura et al., 2018) Semi-doubled Gauged Linear Sigma Model for Five-branes of Codimension Two
• (Kimura, 2014) Defect (p,q) Five-branes
• (Kimura, 2016) Exotic Brane Junctions from F-theory
• (Park et al., 2015) Codimension-2 Solutions in Five-Dimensional Supergravity
• (Kimura, 2016) Supersymmetry Projection Rules on Exotic Branes
• (Fernandez-Melgarejo et al., 2018) Weaving the Exotic Web
• (Otsuki et al., 2019) Exotic Branes in M-Theory
• (Berman et al., 2018) Exotic Branes in Exceptional Field Theory: E_{7(7)} and Beyond