Pith. sign in

REVIEW 4 major objections 6 minor 75 references

Rank $Q$ E-string on a torus with flux

T0 review · 4 major / 6 minor · reviewed 2026-08-14 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read A 4d quiver with emergent symmetry reproduces E-string tori

desk verdict A checkable, genuinely new construction of rank-Q E-string torus models with one explicit loose end; the E-string identification is conditional on an observational gluing recipe and a missing index operator. read the letter →

arxiv 1908.03278 v2 pith:2LRPAX67 submitted 2019-08-08 hep-th

Sara Pasquetti , Shlomo S. Razamat , Matteo Sacchi , Gabi Zafrir This is my paper · ORCID
classification hep-th
keywords rank-QE-stringtoruscompactificationE8flux4dN=1quiveremergentsymmetryUSp(2Q)superconformalindexinterpolationkernel
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper argues that a single four-dimensional $\mathcal{N}=1$ quiver, called $E[USp(2Q)]$, is the building block for the rank-$Q$ E-string, a six-dimensional superconformal theory with $E_8\times SU(2)_L$ global symmetry, compactified on a torus with flux. The quiver's most notable feature is that one of its two $USp(2Q)$ symmetries is not visible in the ultraviolet: it emerges in the infrared as an enhancement of a product of $SU(2)$ flavor symmetries. Gluing copies of the block by gauging that emergent symmetry, following rules abstracted from the rank-one case, yields 4d theories whose global symmetries, anomalies, and superconformal indices match the predictions obtained by integrating the 6d anomaly polynomial. If correct, the construction turns strongly coupled 6d compactifications into calculable 4d quivers and connects them to known 3d theories and to elliptic hypergeometric identities.

What carries the argument

The central object is the $E[USp(2Q)]$ quiver: a chain of $USp(2n)$ gauge nodes for $n=1,\ldots,Q-1$, with a $USp(2Q)$ flavor symmetry at one end, diagonal and vertical $SU(2)$ flavors feeding each node, antisymmetric tensors, and flip singlets. The argument is carried by its supersymmetric index, which coincides with the interpolation kernel $K_c$; proven properties of that kernel, namely self-duality under exchanging the two $USp(2Q)$ fugacities, a flip-flip duality, a braid relation, and reduction identities, are reinterpreted as dualities and RG-flow statements for the quiver, and they provide the integral identities that make the gluing prescription consistent.

What would settle it

A decisive check is to evaluate both sides of the braid relation for $Q=2$ using only the quiver definition of $E[USp(2Q)]$ at generic fugacities; if the elliptic integral identity fails numerically for any assignment satisfying the balancing condition, the gluing rule cannot assemble the claimed tori.

Watch

Extended reading notes

Core claim

The central claim is that the $E[USp(2Q)]$ quiver flows to an infrared SCFT with global symmetry $USp(2Q)_x\times USp(2Q)_y\times U(1)_t\times U(1)_c$, with $USp(2Q)_y$ emerging from the product of $SU(2)$ flavor nodes along the tail. Coupling two octets of fundamental chirals to this block produces the basic tube for flux $(1/2,\ldots,1/2)$; gluing tubes by gauging the diagonal $USp(2Q)$ plus extra chiral and antisymmetric fields builds torus models. The paper verifies that, for flux vectors preserving $E_7$, $SO(14)$, $E_6\times SU(2)$, and related subgroups, the resulting anomalies agree with the six-dimensional predictions and the index expansions show the expected $E_8$ branchings. It further shows that dimensional reduction to 3d followed by real-mass flows maps the block to the $FM[SU(Q)]$, $FT[SU(Q)]$, and (up to flip fields) $T[SU(Q)]$ theories.

Load-bearing premise

The load-bearing premise is that the tube-gluing recipe learned from the rank-one case, namely how to couple the extra chiral fields and identify the two puncture symmetries, continues to work for every $Q$; the authors state this rule is ultimately motivated mostly by observation.

Editorial extensions

If this is right

  • The torus theories assembled from the block realize the predicted enhanced symmetries, such as $SU(2)_L\times E_7\times U(1)$ and $SU(2)_L\times SO(14)\times U(1)$, on loci of their conformal manifolds.
  • The braid relation turns different gluings into dual theories: for instance, the torus with flux $(2,2,0,0,0,0,0,0)$ is dual to the $E_7$-flux torus $(1,1,1,1,1,1,1,1)$.
  • Reducing $E[USp(2Q)]$ to 3d and deforming by real masses reaches $FM[SU(Q)]$, $FT[SU(Q)]$, and $T[SU(Q)]$, so dualities of those 3d models are inherited from the 4d block.
  • Gauging a symmetry that only exists in the infrared is a workable construction principle: the resulting strongly coupled models still pass anomaly and index checks.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Because the gluing rules are inferred from rank-one examples, the sharpest independent test would be a $Q=2$ or $Q=3$ index computation that uses only the quiver definition; if the braid and self-duality identities hold there, the pattern likely continues to all $Q$.
  • The absence of the $1_{-2}$ operator in the $E_7$-flux index suggests an extra cancellation mechanism, possibly from defects wrapping the torus; pinning it down would sharpen the map between 6d currents and 4d operators.
  • The same block reduces further to the kernel function appearing in 2d free-field correlation functions, so the construction may serve as a bridge from 6d SCFT compactifications to correlation-function identities in 2d CFT.
  • The enhancement of a product of $SU(2)$ symmetries to one $USp(2Q)$ hints at a general mechanism: chains of small flavor nodes can organize into a larger symplectic group in the IR, which could be probed by a-maximization in other quiver tails.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 6 minor

Summary. This paper proposes that the 4d N=1 quiver theory E[USp(2Q)] flows to an SCFT with global symmetry USp(2Q)_x × USp(2Q)_y × U(1)_t × U(1)_c, where the second USp(2Q) factor emerges in the IR as an enhancement of an SU(2)^Q symmetry of the UV Lagrangian. The authors then argue that gluing copies of this block, by gauging the two USp(2Q) symmetries together with chiral superfields, produces 4d models corresponding to rank-Q E-string compactified on a torus with various E8 fluxes. Evidence is provided by matching 't Hooft anomalies and the trial conformal anomalies a and c with 6d predictions, and by comparing low-order superconformal index expansions with expected E8 branching rules. The paper also identifies the index of E[USp(2Q)] with Rains' interpolation kernel, thereby giving a rigorous derivation of several index identities, and shows that the 3d reduction of E[USp(2Q)] is related to the T[SU(Q)] and FM[SU(Q)] theories.

Significance. If the central claims hold, the paper provides a systematic quiver construction of a large class of 4d SCFTs obtained from rank-Q E-string compactifications, including theories with emergent IR symmetries and with gauging of those emergent symmetries. The index identities inherited from Rains' work are externally proven and constitute a solid, checkable backbone for the proposal, and the anomaly computations are parameter-free in the sense that they do not fit free parameters to force agreement. The paper also makes concrete predictions for conformal anomalies and index spectra that can be tested independently. However, the central physical identification depends on an IR enhancement and a gluing dictionary that are asserted rather than derived, and on an unpublished reference for the expected 6d BPS spectrum, so the current evidence is suggestive and internally consistent but not conclusive.

major comments (4)
  1. [§4.1, §4.3–4.7] The tube-gluing recipe is introduced in Section 4.1 with the statement that it is 'an abstraction of rules that were observed to work in various examples... ultimately motivated mostly by observation,' and it is then used to construct every torus model in Sections 4.3–4.7. This is the load-bearing step connecting the quiver constructions to rank-Q E-string compactifications. The checks offered, anomaly matching and low-order index expansions, are necessary but not sufficient: they do not distinguish the proposed E-string interpretation from other possible 4d SCFTs with the same low-order data. Please provide an additional independent check, such as matching higher-order index terms, reproducing known 3d reductions, or deriving the gluing rule from the 6d boundary-condition picture; alternatively, the status of the E-string identification for Q>1 should be stated explicitly as a conjecture.
  2. [§4.3, Eq. (4.27)] For the Q=2 torus with flux z, the index expansion (4.27) contains no 1_{-2} operator at order pq, in apparent contradiction with the expected spectrum from the branching (4.28). The authors can only speculate that the operator is cancelled by defect contributions. A similar missing operator, (2,1)_{-3}, appears in the index (4.42) of the E6×SU(2) flux model. Since these missing operators are part of the claimed match between the 4d index and the 6d prediction, this discrepancy is directly relevant to the central claim and should be resolved or explicitly accounted for before the construction is presented as a derivation.
  3. [§3.1] The IR enhancement SU(2)^Q → USp(2Q)_y is the central dynamical premise of the paper, and it is precisely the enhanced symmetry that is later gauged in the tube gluing. The index-level evidence, based on Rains' Theorem 3.1, shows that the refined partition function is symmetric under exchange of the two sets of fugacities, but this does not by itself establish that the full SCFT possesses the enhanced symmetry as a genuine symmetry, nor does it rule out accidental symmetries or extra marginal deformations. Please either prove the enhancement in a controlled limit, or exhibit additional evidence such as matching chiral ring relations with USp(2Q)_y representations, or state clearly that the enhancement is an assumption on which the rest of the construction rests.
  4. [§4.3, paragraph before Eq. (4.28)] The expected 6d BPS operator content used to interpret the index expansions relies on the unpublished reference [57]. Since this expectation is used to judge whether the 4d index matches the 6d prediction, the manuscript should either include a self-contained derivation of this spectrum or explicitly flag the comparison as conditional on a forthcoming result. Until then, the discrepancy involving the missing 1_{-2} operator cannot be fully assessed.
minor comments (6)
  1. [§4.3, Eq. (4.20)] In the displayed equation after (4.20), the second line labels the c-anomaly as aE7,2n(c,t) rather than cE7,2n(c,t); please correct this typo.
  2. [§3.3] The sentence 'It is easy how this work in the Q = 2 case' should be rephrased, for example as 'It is easy to see how this works in the Q = 2 case.'
  3. [§3.2] The sentence 'Rains has proven in [32] that the fugacities of the SU(2)^Q_y symmetry are actually forming always characters of USp(2Q)_y' is grammatically unclear; please state precisely what invariance property of the index is proven.
  4. [§4.1] The U(1)_A symmetry is introduced as an accidental symmetry and then omitted from the subsequent discussion; please specify its charges and explain why it decouples from the anomaly and index checks.
  5. [§1 and Figure 1] The caption of Figure 1 says 'Each gauge node has USp(2n) symmetry,' but the text clarifies that the gauge groups run over n=1,...,Q-1 and the final node is a USp(2Q) flavor node; please make this consistent in the caption.
  6. [§5] In the 3d reduction, the notation T[USp(2Q)] is used for the reduced theory, but this may be confused with the standard T[SU(Q)] of Gaiotto-Witten; please add a note distinguishing the two.

Circularity Check

3 steps flagged · score 4.0 of 10

The Q>1 tube-gluing recipe is explicitly observational and imported from the authors' own prior work, and the index match relies on an unpublished self-citation with an admitted missing operator; otherwise the derivation is largely independent.

  1. ansatz smuggled in via citation [Section 4.1, first paragraph under 'The basic tube and gluing']
    "We note that the following discussion is an abstraction of rules that were observed to work in various examples, notably the rank 1 E-string [1], and some of its generalizations [16]. While there are arguments in favor of this picture, it is ultimately motivated mostly by observation."

    The tube and gluing rules are the only bridge between the E[USp(2Q)] block and the claimed rank-Q E-string compactifications on a torus. These rules are not derived here for Q>1; they are adopted from [1] and [16], both by the present authors, where they were themselves inferred from examples. Every torus model in Section 4 is built by applying this recipe (flux composition (4.2), superpotential (4.1), gauging both USp(2Q) puncture symmetries). Thus the central identification 'these quivers are rank-Q E-string compactifications' rests on a load-bearing ansatz imported via self-citation, although the paper is commendably explicit about the observational status.

  2. self citation load bearing [Section 4.3, paragraph before eq. (4.28)]
    "It is expected [57] (see also [58] and appendix E of [1]) that the lowest BPS operators contributing to the index of the 4d theory come from 6d conserved currents and the energy momentum tensor."

    The interpretation that turns the index expansion (4.27) into a check of the 6d prediction (4.28) is justified by reference [57], which is an unpublished work by two of the present authors (Beem, Razamat, Zafrir), together with [58] and appendix E of [1], also by the same group. Without this expectation, the claimed match between the 4d quiver index and the 6d E-string spectrum is not established. The paper then immediately admits that 'we are missing the 1_-2 operator', so the match is conditional exactly where the self-citation is load-bearing. The branching rule (4.28) itself is independent group theory, which limits the circularity.

1 more flagged steps
  1. other [Section 4.3, immediately after eq. (4.27)]
    "However, we are missing the 1−2 operator. It is not clear what eliminates it from the 4d theory, and it will be interesting to figure this out. One possibility is that it get canceled against defect operators wrapping the torus."

    This is an explicit admission of a missing operator required by the predicted 6d current multiplet decomposition (4.28). The proposed resolution is speculative ('one possibility'). This is not itself a circular step, but it is a stated limitation in the central verification: the 4d index does not fully reproduce the 6d expectation, so the claimed identification of the torus quivers with rank-Q E-string compactifications remains incomplete at the level of the presented evidence.

full rationale

The paper's core mathematical identities are not circular: the E[USp(2Q)] index is identified with Rains' interpolation kernel, and the key index identities (self-duality, flip-flip duality, braid relation) are theorems proven by Rains in reference [32]. The 6d anomaly predictions come from the anomaly polynomial of Ohmori-Shimizu-Tachikawa, independently of the present quiver constructions. The 4d anomaly computations are performed from the quiver field content and matched to these 6d predictions, not fitted to them, so there is no fitted-input-called-prediction circularity. The 3d reduction to T[USp(2Q)] and the subsequent flow to FM[SU(Q)] are derived by taking well-defined limits of the index. However, there are two load-bearing places where the argument leans on the authors' own prior or unpublished work. First, the tube-gluing recipe for general Q is stated to be 'an abstraction of rules that were observed to work in various examples' in [1] and [16], which are papers by the same authors; this recipe is the central mechanism connecting E[USp(2Q)] blocks to rank-Q E-string compactifications, and it is not independently proven for Q>1. Second, the interpretation of the torus index as matching 6d conserved-current contributions cites an unpublished paper by two of the authors, and the check is explicitly incomplete because the 1_{-2} operator is missing. These issues make the E-string identification conditional, but they do not make the derivation definitionally circular: the anomaly matches and the Rains-based dualities provide genuine independent content. A score of 4 reflects partial self-citation load-bearing with independent substance remaining.

Assumptions & free parameters 1 free parameters · 7 assumptions · 1 invented entities

The central construction rests on external inputs: the 6d E-string anomaly polynomial, the 5d reduction picture, localization formulae, and Rains' proven index identities. The paper adds a new quiver model but relies on an observational gluing prescription and an unproven IR enhancement. It introduces no free parameters fitted to data; the R-charge mixing coefficients are fixed by a-maximization.

free parameters (1)
  • Trial R-charge assignment R0 = 0 for q(n,n+1) and d(n); 2 for b_n, A(n), v(n)
    Chosen by hand as a convenient anomaly-free starting point; the physical R-charge is later fixed by a-maximization, so this is a convention rather than a fit to data.
assumptions (7)
  • domain assumption The 6d rank Q E-string SCFT exists with E8 x SU(2)_L global symmetry and anomaly polynomial as computed in [51].
    Used in Section 2.1 to predict 4d anomalies; the E-string existence and anomaly polynomial are standard string theory results but not proven inside the paper.
  • domain assumption The 5d reduction of the E-string with tuned holonomy flows to an IR-free USp(2Q) gauge theory with antisymmetric and eight fundamental hypermultiplets.
    Puncture contributions in Section 2.2 are computed from this 5d boundary-condition picture, citing [47,48]; if this reduction is wrong, the puncture anomaly inflow changes.
  • standard math Localization formulae give the exact superconformal index as a contour integral of elliptic gamma functions.
    Used in Appendix B and Section 3.1 to identify the E[USp(2Q)] index with Rains' interpolation kernel and to derive dualities.
  • standard math Rains' theorems on the interpolation kernel are correct and applicable to this physical context.
    The core self-duality, flip-flip duality, braid relation, and F4 invariance rest on Theorem 3.1, Proposition 3.5, Proposition 2.12, and Theorem 3.22 of [32].
  • ad hoc to paper The abstract tube-gluing rules obtained from observation in rank-one examples generalize to rank Q.
    Stated in Section 4.1 as 'ultimately motivated mostly by observation'; if not valid, the built tori are not actually E-string compactifications.
  • domain assumption a-maximization determines the superconformal R-symmetry, assuming no accidental abelian symmetries.
    Used in Sections 2.1 and 4.3 to compare central charges a and c; the authors explicitly note the caveat about accidental abelian symmetries.
  • ad hoc to paper The absence of certain expected index contributions is due to cancellations or defect contributions, e.g., the missing 1_{-2} operator for the E7-flux torus.
    The index expansion in Section 4.3 does not show the full expected 6d current multiplet, and the paper states it is not clear what eliminates the missing operator.
invented entities (1)
  • Emergent USp(2Q)_y IR symmetry of E[USp(2Q)]
    purpose: Provides the enhanced USp(2Q)xUSp(2Q)xU(1)^2 global symmetry and the self-duality; gluing two blocks gauges this emergent symmetry.
    Supported only by recombination of SU(2)^Q characters in the superconformal index and by assembly of gauge-invariant operators into an antisymmetric matrix A_y; there is no external falsifiable measurement outside this paper's framework.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Rank $Q$ E-string on a torus with flux." pith.science (2026). https://pith.science/paper/2LRPAX67

@misc{pith2026190803278,
  author       = {Pith},
  title        = {Pith review of: Rank $Q$ E-string on a torus with flux},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2LRPAX67}},
  note         = {Machine review of arXiv:1908.03278}
}
abstract

We discuss compactifications of rank $Q$ E-string theory on a torus with fluxes for abelian subgroups of the $E_8$ global symmetry of the $6d$ SCFT. We argue that the theories corresponding to such tori are built from a simple model we denote as $E[USp(2Q)]$. This model has a variety of non trivial properties. In particular the global symmetry is $USp(2Q)\times USp(2Q)\times U(1)^2$ with one of the two $USp(2Q)$ symmetries emerging in the IR as an enhancement of an $SU(2)^Q$ symmetry of the UV Lagrangian. The $E[USp(2Q)]$ model after dimensional reduction to $3d$ and a subsequent Coulomb branch flow is closely related to the familiar $3d$ $T[SU(Q)]$ theory, the model residing on an S-duality domain wall of $4d$ $\mathcal{N}=4$ $SU(Q)$ SYM. Gluing the $E[USp(2Q)]$ models by gauging the $USp(2Q)$ symmetries with proper admixtures of chiral superfields gives rise to systematic constructions of many examples of $4d$ theories with emergent IR symmetries. We support our claims by various checks involving computations of anomalies and supersymmetric partition functions. Many of the needed identities satisfied by the supersymmetric indices follow directly from recent mathematical results obtained by E. Rains.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

75 extracted references · 18 canonical work pages

  1. [57]

    C. Beem, S. S. Razamat, and G. Zafrir to appear

    work page

  2. [1]

    E – String Theory on Riemann Surfaces,

    H.-C. Kim, S. S. Razamat, C. Vafa, and G. Zafrir, “E – String Theory on Riemann Surfaces,” Fortsch. Phys. 66 no. 1, (2018) 1700074, arXiv:1709.02496 [hep-th]

    arXiv 2018

  3. [2]

    N=2 dualities,

    D. Gaiotto, “N=2 dualities,” JHEP 08 (2012) 034, arXiv:0904.2715 [hep-th] . – 50 –

    arXiv 2012

  4. [3]

    Wall-crossing, Hitchin Systems, and the WKB Approximation,

    D. Gaiotto, G. W. Moore, and A. Neitzke, “Wall-crossing, Hitchin Systems, and the WKB Approximation,” arXiv:0907.3987 [hep-th]

    arXiv

  5. [4]

    Sicilian gauge theories and N=1 dualities,

    F. Benini, Y. Tachikawa, and B. Wecht, “Sicilian gauge theories and N=1 dualities,” JHEP 1001 (2010) 088, arXiv:0909.1327 [hep-th]

    arXiv 2010

  6. [5]

    Four-Dimensional SCFTs from M5-Branes,

    I. Bah, C. Beem, N. Bobev, and B. Wecht, “Four-Dimensional SCFTs from M5-Branes,” JHEP 06 (2012) 005, arXiv:1203.0303 [hep-th]

    arXiv 2012

  7. [6]

    N = 1 theories of class Sk,

    D. Gaiotto and S. S. Razamat, “ N = 1 theories of class Sk,” JHEP 07 (2015) 073, arXiv:1503.05159 [hep-th]

    arXiv 2015

  8. [7]

    4d N = 1 from 6d (1, 0),

    S. S. Razamat, C. Vafa, and G. Zafrir, “4d N = 1 from 6d (1, 0),” JHEP 04 (2017) 064, arXiv:1610.09178 [hep-th]

    arXiv 2017

Show all 75 references
  1. [8]

    6d N = (1, 0) theories on T 2 and class S theories: Part I,

    K. Ohmori, H. Shimizu, Y. Tachikawa, and K. Yonekura, “6d N = (1, 0) theories on T 2 and class S theories: Part I,” JHEP 07 (2015) 014, arXiv:1503.06217 [hep-th]

    2015 arXiv

  2. [9]

    6d N = (1, 0) theories on S 1 /T2 and class S theories: part II,

    K. Ohmori, H. Shimizu, Y. Tachikawa, and K. Yonekura, “6d N = (1, 0) theories on S 1 /T2 and class S theories: part II,” JHEP 12 (2015) 131, arXiv:1508.00915 [hep-th]

    2015 arXiv

  3. [10]

    Brane webs, 5 d gauge theories and 6dN = (1, 0) SCFT’s,

    G. Zafrir, “Brane webs, 5 d gauge theories and 6dN = (1, 0) SCFT’s,” JHEP 12 (2015) 157, arXiv:1509.02016 [hep-th]

    2015 arXiv

  4. [11]

    S1/T 2 compactifications of 6dN = (1, 0) theories and brane webs,

    K. Ohmori and H. Shimizu, “ S1/T 2 compactifications of 6dN = (1, 0) theories and brane webs,” JHEP 03 (2016) 024, arXiv:1509.03195 [hep-th]

    2016 arXiv

  5. [12]

    4d N = 1 from 6dN = (1, 0) on a torus with fluxes,

    I. Bah, A. Hanany, K. Maruyoshi, S. S. Razamat, Y. Tachikawa, and G. Zafrir, “4d N = 1 from 6dN = (1, 0) on a torus with fluxes,” JHEP 06 (2017) 022, arXiv:1702.04740 [hep-th]

    2017 arXiv

  6. [13]

    Geometric engineering, mirror symmetry and 6d(1,0)→ 4d(N =2),

    M. Del Zotto, C. Vafa, and D. Xie, “Geometric engineering, mirror symmetry and 6d(1,0)→ 4d(N =2),” JHEP 11 (2015) 123, arXiv:1504.08348 [hep-th]

    2015 arXiv

  7. [14]

    F-theory and N = 1 SCFTs in four dimensions,

    D. R. Morrison and C. Vafa, “F-theory and N = 1 SCFTs in four dimensions,” JHEP 08 (2016) 070, arXiv:1604.03560 [hep-th]

    2016 arXiv

  8. [15]

    E 8 instantons on type-A ALE spaces and supersymmetric field theories,

    N. Mekareeya, K. Ohmori, Y. Tachikawa, and G. Zafrir, “E 8 instantons on type-A ALE spaces and supersymmetric field theories,” JHEP 09 (2017) 144, arXiv:1707.04370 [hep-th]

    2017 arXiv

  9. [16]

    D-type Conformal Matter and SU/USp Quivers,

    H.-C. Kim, S. S. Razamat, C. Vafa, and G. Zafrir, “D-type Conformal Matter and SU/USp Quivers,” JHEP 06 (2018) 058, arXiv:1802.00620 [hep-th]

    2018 arXiv

  10. [17]

    Compactifications of ADE conformal matter on a torus,

    H.-C. Kim, S. S. Razamat, C. Vafa, and G. Zafrir, “Compactifications of ADE conformal matter on a torus,” JHEP 09 (2018) 110, arXiv:1806.07620 [hep-th]

    2018 arXiv

  11. [18]

    Compactification of 6d minimal SCFTs on Riemann surfaces,

    S. S. Razamat and G. Zafrir, “Compactification of 6d minimal SCFTs on Riemann surfaces,” Phys. Rev. D98 no. 6, (2018) 066006, arXiv:1806.09196 [hep-th]

    2018 arXiv

  12. [19]

    4D Gauge Theories with Conformal Matter,

    F. Apruzzi, J. J. Heckman, D. R. Morrison, and L. Tizzano, “4D Gauge Theories with Conformal Matter,” JHEP 09 (2018) 088, arXiv:1803.00582 [hep-th]

    2018 arXiv

  13. [20]

    Compactifications of 6d N = (1, 0) SCFTs with non-trivial Stiefel-Whitney classes,

    K. Ohmori, Y. Tachikawa, and G. Zafrir, “Compactifications of 6d N = (1, 0) SCFTs with non-trivial Stiefel-Whitney classes,” JHEP 04 (2019) 006, arXiv:1812.04637 [hep-th]

    2019 arXiv

  14. [21]

    4d N=1 from 6d D-type N=(1,0),

    J. Chen, B. Haghighat, S. Liu, and M. Sperling, “4d N=1 from 6d D-type N=(1,0),” arXiv:1907.00536 [hep-th]

    1907 arXiv

  15. [22]

    ”Lagrangian

    A. Gadde, S. S. Razamat, and B. Willett, “”Lagrangian” for a Non-Lagrangian Field Theory withN = 2 Supersymmetry,” Phys. Rev. Lett. 115 no. 17, (2015) 171604, arXiv:1505.05834 [hep-th]. – 51 –

    2015 arXiv

  16. [23]

    A Lagrangian for the E 7 superconformal theory,

    P. Agarwal, K. Maruyoshi, and J. Song, “A Lagrangian for the E 7 superconformal theory,” JHEP 05 (2018) 193, arXiv:1802.05268 [hep-th]

    2018 arXiv

  17. [24]

    3d dualities from 2d free field correlators: recombination and rank stabilization,

    S. Pasquetti and M. Sacchi, “3d dualities from 2d free field correlators: recombination and rank stabilization,” arXiv:1905.05807 [hep-th]

    1905 arXiv

  18. [25]

    T[SU(N)] duality webs: mirror symmetry, spectral duality and gauge/CFT correspondences,

    A. Nedelin, S. Pasquetti, and Y. Zenkevich, “T[SU(N)] duality webs: mirror symmetry, spectral duality and gauge/CFT correspondences,” JHEP 02 (2019) 176, arXiv:1712.08140 [hep-th]

    2019 arXiv

  19. [26]

    Flipping the head of T [SU (N)]: mirror symmetry, spectral duality and monopoles,

    F. Aprile, S. Pasquetti, and Y. Zenkevich, “Flipping the head of T [SU (N)]: mirror symmetry, spectral duality and monopoles,” JHEP 04 (2019) 138, arXiv:1812.08142 [hep-th]

    2019 arXiv

  20. [27]

    S-Duality of Boundary Conditions In N=4 Super Yang-Mills Theory,

    D. Gaiotto and E. Witten, “S-Duality of Boundary Conditions In N=4 Super Yang-Mills Theory,” Adv. Theor. Math. Phys. 13 no. 3, (2009) 721–896, arXiv:0807.3720 [hep-th]

    2009 arXiv

  21. [28]

    Counting chiral primaries in N = 1, d=4 superconformal field theories,

    C. Romelsberger, “Counting chiral primaries in N = 1, d=4 superconformal field theories,” Nucl. Phys. B747 (2006) 329–353, arXiv:hep-th/0510060 [hep-th]

    2006 arXiv

  22. [29]

    An Index for 4 dimensional super conformal theories,

    J. Kinney, J. M. Maldacena, S. Minwalla, and S. Raju, “An Index for 4 dimensional super conformal theories,” Commun. Math. Phys. 275 (2007) 209–254, arXiv:hep-th/0510251 [hep-th]

    2007 arXiv

  23. [30]

    Applications of the Superconformal Index for Protected Operators and q-Hypergeometric Identities to N=1 Dual Theories,

    F. A. Dolan and H. Osborn, “Applications of the Superconformal Index for Protected Operators and q-Hypergeometric Identities to N=1 Dual Theories,” Nucl. Phys. B818 (2009) 137–178, arXiv:0801.4947 [hep-th]

    2009 arXiv

  24. [31]

    The supersymmetric index in four dimensions,

    L. Rastelli and S. S. Razamat, “The supersymmetric index in four dimensions,” J. Phys. A50 no. 44, (2017) 443013, arXiv:1608.02965 [hep-th]

    2017 arXiv

  25. [32]

    Multivariate Quadratic Transformations and the Interpolation Kernel,

    E. M. Rains, “Multivariate Quadratic Transformations and the Interpolation Kernel,” SIGMA 14 (2018) 019, arXiv:1408.0305 [math.CA]

    2018 arXiv

  26. [33]

    Localization techniques in quantum field theories,

    V. Pestun et al., “Localization techniques in quantum field theories,” J. Phys. A50 no. 44, (2017) 440301, arXiv:1608.02952 [hep-th]

    2017 arXiv

  27. [34]

    Electric - magnetic duality in supersymmetric nonAbelian gauge theories,

    N. Seiberg, “Electric - magnetic duality in supersymmetric nonAbelian gauge theories,” Nucl. Phys. B435 (1995) 129–146, arXiv:hep-th/9411149 [hep-th]

    1995 arXiv

  28. [35]

    On the elliptic beta function,

    V. P. Spiridonov, “On the elliptic beta function,” Uspekhi Mat. Nauk 56 (2001) 181–282

    2001

  29. [36]

    Transformations of elliptic hypergeometric integrals,

    E. M. Rains, “Transformations of elliptic hypergeometric integrals,” Ann. of Math. (2) 171 no. 1, (2010) 169–243, arXiv:math/0309252 [math]

    2010 arXiv

  30. [37]

    Hyperbolic hypergeometric functions,

    F. V. de Bult, “Hyperbolic hypergeometric functions,” Ph.D. thesis

  31. [38]

    Comments on 3d Seiberg-like dualities,

    F. Benini, C. Closset, and S. Cremonesi, “Comments on 3d Seiberg-like dualities,” JHEP 10 (2011) 075, arXiv:1108.5373 [hep-th]

    2011 arXiv

  32. [39]

    S-duality and 2d Topological QFT,

    A. Gadde, E. Pomoni, L. Rastelli, and S. S. Razamat, “S-duality and 2d Topological QFT,” JHEP 03 (2010) 032, arXiv:0910.2225 [hep-th]

    2010 arXiv

  33. [40]

    An elliptic hypergeometric integral with W (F4) symmetry,

    F. V. de Bult, “An elliptic hypergeometric integral with W (F4) symmetry,” Ramanujan 25 (2011) 1

    2011

  34. [41]

    Superconformal indices for N = 1 theories with multiple duals,

    V. P. Spiridonov and G. S. Vartanov, “Superconformal indices for N = 1 theories with multiple duals,” Nucl. Phys. B824 (2010) 192–216, arXiv:0811.1909 [hep-th]

    2010 arXiv

  35. [42]

    An E7 Surprise,

    T. Dimofte and D. Gaiotto, “An E7 Surprise,” JHEP 10 (2012) 129, arXiv:1209.1404 [hep-th]. – 52 –

    2012 arXiv

  36. [43]

    USp(2N c) SQCD3 with antisymmetric: dualities and symmetry enhancements,

    A. Amariti and L. Cassia, “USp(2N c) SQCD3 with antisymmetric: dualities and symmetry enhancements,” JHEP 02 (2019) 013, arXiv:1809.03796 [hep-th]

    2019 arXiv

  37. [44]

    A tale of exceptional 3 d dualities,

    S. Benvenuti, “A tale of exceptional 3 d dualities,” JHEP 03 (2019) 125, arXiv:1809.03925 [hep-th]

    2019 arXiv

  38. [45]

    From 3 d dualities to 2d free field correlators and back,

    S. Pasquetti and M. Sacchi, “From 3 d dualities to 2d free field correlators and back,” arXiv:1903.10817 [hep-th]

    1903 arXiv

  39. [46]

    Multipoint correlation functions in Liouville field theory and minimal Liouville gravity,

    V. A. Fateev and A. V. Litvinov, “Multipoint correlation functions in Liouville field theory and minimal Liouville gravity,” Theor. Math. Phys. 154 (2008) 454–472, arXiv:0707.1664 [hep-th]

    2008 arXiv

  40. [47]

    Branes, Calabi-Yau spaces, and toroidal compactification of the N=1 six-dimensional E(8) theory,

    O. J. Ganor, D. R. Morrison, and N. Seiberg, “Branes, Calabi-Yau spaces, and toroidal compactification of the N=1 six-dimensional E(8) theory,” Nucl.Phys. B487 (1997) 93–127, arXiv:hep-th/9610251 [hep-th]

    1997 arXiv

  41. [48]

    Five-dimensional SUSY field theories, nontrivial fixed points and string dynamics,

    N. Seiberg, “Five-dimensional SUSY field theories, nontrivial fixed points and string dynamics,” Phys. Lett. B388 (1996) 753–760, arXiv:hep-th/9608111 [hep-th]

    1996 arXiv

  42. [49]

    Superconformal fixed points with E(n) global symmetry,

    J. A. Minahan and D. Nemeschansky, “Superconformal fixed points with E(n) global symmetry,” Nucl. Phys. B489 (1997) 24–46, arXiv:hep-th/9610076 [hep-th]

    1997 arXiv

  43. [50]

    Chiral compactifications of 6-D conformal theories,

    C. S. Chan, O. J. Ganor, and M. Krogh, “Chiral compactifications of 6-D conformal theories,” Nucl. Phys. B597 (2001) 228–244, arXiv:hep-th/0002097 [hep-th]

    2001 arXiv

  44. [51]

    Anomaly polynomial of E-string theories,

    K. Ohmori, H. Shimizu, and Y. Tachikawa, “Anomaly polynomial of E-string theories,” JHEP 08 (2014) 002, arXiv:1404.3887 [hep-th]

    2014 arXiv

  45. [52]

    Supersymmetric Boundary Conditions in N=4 Super Yang-Mills Theory,

    D. Gaiotto and E. Witten, “Supersymmetric Boundary Conditions in N=4 Super Yang-Mills Theory,” J. Statist. Phys. 135 (2009) 789–855, arXiv:0804.2902 [hep-th]

    2009 arXiv

  46. [53]

    The Exact superconformal R symmetry maximizes a,

    K. A. Intriligator and B. Wecht, “The Exact superconformal R symmetry maximizes a,” Nucl. Phys. B667 (2003) 183–200, arXiv:hep-th/0304128 [hep-th]

    2003 arXiv

  47. [54]

    Exact superpotentials, quantum vacua and duality in supersymmetric SP(N(c)) gauge theories,

    K. A. Intriligator and P. Pouliot, “Exact superpotentials, quantum vacua and duality in supersymmetric SP(N(c)) gauge theories,” Phys. Lett. B353 (1995) 471–476, arXiv:hep-th/9505006 [hep-th]

    1995 arXiv

  48. [55]

    Bootstrapping the superconformal index with surface defects,

    D. Gaiotto, L. Rastelli, and S. S. Razamat, “Bootstrapping the superconformal index with surface defects,” JHEP 01 (2013) 022, arXiv:1207.3577 [hep-th]

    2013 arXiv

  49. [56]

    An SU(2) Anomaly,

    E. Witten, “An SU(2) Anomaly,” Phys. Lett. B117 (1982) 324–328. [,230(1982)]

    1982

  50. [58]

    Geometrization of relevance,

    S. S. Razamat, “Geometrization of relevance,” talk at ‘Avant-garde methods for quantum field theory and gravity, Nazareth 2/2019 (https://phsites.technion.ac.il/the-fifth-israeli-indian-conference-on-string-theory/program/)

    2019

  51. [59]

    The N = 1 superconformal index for class S fixed points,

    C. Beem and A. Gadde, “The N = 1 superconformal index for class S fixed points,” JHEP 04 (2014) 036, arXiv:1212.1467 [hep-th]

    2014 arXiv

  52. [60]

    3d dualities from 4d dualities,

    O. Aharony, S. S. Razamat, N. Seiberg, and B. Willett, “3d dualities from 4d dualities,” JHEP 07 (2013) 149, arXiv:1305.3924 [hep-th]

    2013 arXiv

  53. [61]

    From 4d superconformal indices to 3d partition functions,

    F. A. H. Dolan, V. P. Spiridonov, and G. S. Vartanov, “From 4d superconformal indices to 3d partition functions,” Phys. Lett. B704 (2011) 234–241, arXiv:1104.1787 [hep-th] . – 53 –

    2011 arXiv

  54. [62]

    Reducing the 4d Index to the S3 Partition Function,

    A. Gadde and W. Yan, “Reducing the 4d Index to the S3 Partition Function,” JHEP 12 (2012) 003, arXiv:1104.2592 [hep-th]

    2012 arXiv

  55. [63]

    Exact Results for Wilson Loops in Superconformal Chern-Simons Theories with Matter,

    A. Kapustin, B. Willett, and I. Yaakov, “Exact Results for Wilson Loops in Superconformal Chern-Simons Theories with Matter,” JHEP 03 (2010) 089, arXiv:0909.4559 [hep-th]

    2010 arXiv

  56. [64]

    The Exact Superconformal R-Symmetry Extremizes Z,

    D. L. Jafferis, “The Exact Superconformal R-Symmetry Extremizes Z,” JHEP 05 (2012) 159, arXiv:1012.3210 [hep-th]

    2012 arXiv

  57. [65]

    Notes on SUSY Gauge Theories on Three-Sphere,

    N. Hama, K. Hosomichi, and S. Lee, “Notes on SUSY Gauge Theories on Three-Sphere,” JHEP 03 (2011) 127, arXiv:1012.3512 [hep-th]

    2011 arXiv

  58. [66]

    SUSY Gauge Theories on Squashed Three-Spheres,

    N. Hama, K. Hosomichi, and S. Lee, “SUSY Gauge Theories on Squashed Three-Spheres,” JHEP 05 (2011) 014, arXiv:1102.4716 [hep-th]

    2011 arXiv

  59. [67]

    SUSY monopole potentials in 2+1 dimensions,

    F. Benini, S. Benvenuti, and S. Pasquetti, “SUSY monopole potentials in 2+1 dimensions,” JHEP 08 (2017) 086, arXiv:1703.08460 [hep-th]

    2017 arXiv

  60. [68]

    Seiberg dualities and the 3d/4d connection,

    V. Niarchos, “Seiberg dualities and the 3d/4d connection,” JHEP 07 (2012) 075, arXiv:1205.2086 [hep-th]

    2012 arXiv

  61. [69]

    Correlation functions in Liouville theory,

    M. Goulian and M. Li, “Correlation functions in Liouville theory,” Phys. Rev. Lett. 66 (1991) 2051–2055

    1991

  62. [70]

    Expectation values of local fields for a two-parameter family of integrable models and related perturbed conformal field theories,

    P. Baseilhac and V. A. Fateev, “Expectation values of local fields for a two-parameter family of integrable models and related perturbed conformal field theories,” Nucl. Phys. B532 (1998) 567–587, arXiv:hep-th/9906010 [hep-th]

    1998 arXiv

  63. [71]

    N = 1 conformal dualities,

    S. S. Razamat and G. Zafrir, “ N = 1 conformal dualities,” arXiv:1906.05088 [hep-th]

    1906 arXiv

  64. [72]

    Dual boundary conditions in 3d SCFTs,

    T. Dimofte, D. Gaiotto, and N. M. Paquette, “Dual boundary conditions in 3d SCFTs,” JHEP 05 (2018) 060, arXiv:1712.07654 [hep-th]

    2018 arXiv

  65. [73]

    4d Index to 3d Index and 2d TQFT,

    F. Benini, T. Nishioka, and M. Yamazaki, “4d Index to 3d Index and 2d TQFT,” Phys. Rev. D86 (2012) 065015, arXiv:1109.0283 [hep-th]

    2012 arXiv

  66. [74]

    Global Properties of Supersymmetric Theories and the Lens Space,

    S. S. Razamat and B. Willett, “Global Properties of Supersymmetric Theories and the Lens Space,” Commun. Math. Phys. 334 no. 2, (2015) 661–696, arXiv:1307.4381 [hep-th]

    2015 arXiv

  67. [75]

    Elliptic hypergeometric sum/integral transformations and supersymmetric lens index,

    A. P. Kels and M. Yamazaki, “Elliptic hypergeometric sum/integral transformations and supersymmetric lens index,” SIGMA 14 (2018) 013, arXiv:1704.03159 [math-ph] . – 54 –

    2018 arXiv

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.