Summing the Instantons: Quantum Cohomology and Mirror Symmetry in Toric Varieties

David R. Morrison; M. Ronen Plesser

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Summing the Instantons: Quantum Cohomology and Mirror Symmetry in Toric Varieties

David R. Morrison, M. Ronen Plesser

High Energy Physics - Theory · Submitted 30 Dec 1994 · 72 references

Pith's verdict: The paper that actually sums the toric instanton series directly, without assuming mirror symmetry, and gets the monomial-divisor map out as a theorem under stated hypotheses.

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CitationDavid R. Morrison, M. Ronen Plesser. "Summing the Instantons: Quantum Cohomology and Mirror Symmetry in Toric Varieties." arXiv:hep-th/9412236 (1994). https://pith.science/paper/3SQ6CINM

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This is the paper that does the direct computation rather than routing through mirror symmetry. Witten's GLSM gives compact toric instanton moduli spaces; Morrison and Plesser sum the series explicitly and get exact correlators for both V and the Calabi-Yau hypersurface M. The quantum restriction formula (4.31) and the global monomial-divisor map (5.32) are the two results I would actually cite. The proof of the mirror map via uniqueness of graded expectation functions on R0(V) ≅ P0(Λ) is a clean argument, and the fact that the instanton-derived relations (5.13) match the Jacobian-ring relations (5.29) exactly is the kind of coincidence that is not a coincidence. That is the real content of the paper, and it holds up on reading.

Referee report

Significance

This is a substantial and, in my judgment, correct advance. The paper does what its title advertises: it computes instanton corrections directly, without mirror symmetry, and sums them. The two examples are not decorative — the quintic Yukawa coupling (4.24) and the four two-parameter couplings (4.28) agree exactly with the mirror-symmetry results of Candelas, de la Ossa, Font, Katz, and Morrison, which is a strong independent check on the whole machinery. The quantum restriction formula (4.31)/(4.33) and the global monomial-divisor mirror map (5.32) are the paper's most durable products, and the four-phase agreement in §5.3 (Eqs. (5.19)–(5.21) all reproducing (4.28)) is a genuinely nontrivial internal consistency check. Appendix B is a model of honesty: it exhibits a singular V where the algebraic relations fail to generate the full ideal, and shows the instanton sum still works. The paper is also careful to distinguish what is proved from what is assumed, which is not universal in this literature.

Major objections (4)

[§3.8, Eqs. (3.68)–(3.70)] The identification of the GLSM operator σ_a with the divisor class (η_a)_n on each moduli space, with an n-independent normalization, is assumed rather than derived: 'We will assume this holds exactly (the normalization factor is independent of n), though we do not have a complete justification for this choice.' The same paragraph assumes the Euler-class normalization (3.69). Every instanton coefficient in (3.68) and (3.70), and hence every summed correlator in (3.67), inherits this map, so the assumption is load-bearing. I think the paper actually contains the ingredients to close most of the gap and should say so explicitly rather than leave it as an assumption. First, the n = 0 sector is fixed by classical intersection numbers, and §2's argument that the two-point function is not renormalized pins the pairing, hence the overall constant, at n = 0. Second, §2 establishes that a graded Frobenius algebra of finite length has a graded expectation function unique up to a single scalar; combined with the §3.9 derivation of the relations (3.83) at every n, this excludes an n-dependent normalization, since the relations determine the ring and the expectation function is then fixed up to one q-independent constant. Third, Example 1 yields the pure monomial Y_{5m+4} = q^m in (3.71), which an n-dependent normalization would generically spoil. Please make this chain explicit; as written, a reader is left with the impression that the central computation rests on an unargued convention.
[§4.3, Eq. (4.20) and the (−1)^{d_0} weighting] The treatment of pointlike instantons in the M model is the weakest link in the paper's otherwise tight argument. The sign (−1)^{d_0} is obtained from a fermionic-reordering argument, and the text concedes that 'the two extra χ zero modes are more subtle to treat, but will not affect this discussion.' The conclusion that pointlike instantons do no more than renormalize the couplings, t_a = τ_a + (1/2)Q_{a0} plus exponentially suppressed corrections, is asserted rather than computed; the exponential corrections to (4.20) are explicitly left for future work. This matters for the abstract's claim of 'explicit solutions' for M: what is actually obtained is a solution in the GLSM's algebraic coordinates and algebraic gauge, not in the canonical coordinates of the nonlinear model. The paper is candid about this in §4.3 and §6, and the mirror comparison in §5.4 is legitimately carried out in algebraic gauge, so the physics conclusion is not in doubt. But the abstract should be qualified to say that the M-model solution is algebraic-coordinate, and the sign discussion in §4.3 should either be expanded or pointed to a place where the χ zero modes are treated.
[§5.4, Eqs. (5.35)–(5.37)] The proof of the monomial-divisor mirror map establishes equality of correlators only up to a function s(q): 'It seems likely that s(q) must in fact be a constant, but we have no general proof of this.' The reframing — that correlators are sections of a line bundle, so equality up to s(q) is still a proof of mirror symmetry at the level of topological field theory — is reasonable and, I think, correct, but it is a reframing, and the abstract's unqualified 'we prove some conjectures about mirror symmetry' does not carry it. The proof also rests on hypotheses (i) smooth V and (ii) simplicial mirror fan, both stated in §5.4 but neither reflected in the abstract. I would ask that the abstract and the opening of §5.4 state the qualification explicitly: equality holds up to a meromorphic normalization function whose zeros and poles lie on the singular locus, under the two stated hypotheses. The argument that s(q) is fixed by its singularities, given that both singularity loci are known and agree, is suggestive but not a proof, and should be labeled as such.
[§3.5–§3.6 and §4.2, Eqs. (4.11)–(4.15), Figure 3] The exactness of the one-loop twisted superpotential (3.36) — the claim that it receives no nonperturbative corrections — is deferred to reference [7], listed as work in progress. This claim underlies the determination of the singular locus in §3.5 and §4.2 and hence the corrected phase diagram of Figure 3. The situation is better than it first appears: the operator relations (3.52) are re-derived independently in §3.9 from the combinatorics of the instanton moduli spaces, without invoking exactness, so the correlator computations do not depend on the unpublished argument. The singular locus does depend on it, but §5.4 checks the A-model singular locus against the principal A-determinant of the B model and finds agreement, which is a meaningful independent confirmation. I would ask the authors to state this division of labor explicitly — which results rest on [7] and which are independently established — so that a reader can see that the central claims survive even if the exactness argument of [7] were to require modification.

Minor objections (6)

[§3.6 and §3.9] The symbols δ_i, ξ_i, and η_a are used with overlapping meanings: δ_i is introduced in (3.51) as the operator corresponding to ξ_i, while η_a generates the cohomology of the moduli space via (3.64). A short dictionary at the start of §3.9, stating which symbol lives in the ring of operators and which on the moduli space, would save the reader real effort.
[§3.7] The statement that 'the ratio of determinants can be ±1; in the present case it is in fact just 1' is asserted without justification. Since the sign of the determinant ratio feeds directly into the sign conventions used later in §4.3, one sentence of explanation, or a pointer to where it is shown, would be worthwhile.
[§5.3] The rule that contributions from sectors with unbroken discrete subgroup H must be divided by |H| is applied in the examples (the Z_5 in Example 1, the Z_2 and Z_4 in Example 2) but is not derived. A sentence explaining why the overcounting factor is exactly |H|, rather than something else, would strengthen the treatment of the nongeometric phases.
[Figure 3] The caption should identify the axes (r_1, r_2) and label the enclosed central region, which the text describes as inherently strongly coupled in the sense that no instanton expansion converges there. As drawn, the figure is hard to read without the surrounding text.
[§6] The paper notes that consistency of the construction requires agreement between the two expressions for the quantum cohomology of M, from §4 (Eq. (4.31)) and §5 (Eq. (5.14)), and that this has not been shown explicitly. The two examples do verify it, and it would help to say so in §6 rather than leave the impression that the agreement is untested.
[§4.4, Eq. (4.28)] The overall normalization of the X_j is fixed by matching the n = 0 term to the classical intersection calculation. It would be useful to state this explicitly at the point where the sum is performed, since the reader otherwise has to reconstruct the normalization convention from (4.2) and (4.26).

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The paper

The paper claims

In the gauged linear sigma model the instanton moduli spaces for a toric target are themselves compact toric varieties, so the whole instanton series can be summed in closed form, giving exact quantum cohomology for toric varieties and for.

The reading

The paper sets out to compute the instanton corrections to correlation functions in the topological A-twisted $\sigma$ model when the target is a toric variety $V$, or a Calabi–Yau hypersurface $M$ inside one, without appealing to mirror symmetry. The authors' route is Witten's gauged linear $\sigma$ model: in the linear theory the moduli space of degree-$n$ instantons is a compact toric quotient built from spaces of holomorphic sections of line bundles on the worldsheet, so each instanton coefficient becomes an ordinary intersection number in a toric variety that can be evaluated combinatorially. Because those spaces are compact, the sum over all instanton numbers is not merely formal — it produces explicit rational functions of the exponentiated couplings $q_a = e^{2i}$, whose only singularities are poles, and which can therefore be continued across the whole parameter space. For smooth $V$ the result reproduces and explains Batyrev's algebraic presentation of the quantum cohomology ring, with the relations $x_i^{Q_{ai}} = q_a$ derived rather than postulated. For the hypersurface $M$ the authors obtain a quantum restriction formula expressing the correlators of $M$ in terms of those of $V$, and a second, more symmetric formulation (the $V^+$ model, the total space of the canonical bundle with a superpotential switched off) that allows instanton expansions about any limiting point, including Landau–Ginzburg and hybrid phases. The payoff the authors claim is a proof, at the level of topological correlation functions, of mirror-symmetry conjectures for hypersurfaces in smooth toric varieties, including the monomial-divisor mirror map.

What carries the argument

The load-bearing mechanism is the replacement of the nonlinear $\sigma$ model's noncompact space of holomorphic maps by the linear model's compact toric instanton moduli space $M_n = (Y_n - F_n)/T$, where the excluded set $F_n$ is read off from the same fan data and moment-map constants $r_a$ that define $V$ itself. Intersections in $M_n$ are computed by the Stanley–Reisner relations of that toric variety, so each instanton coefficient is a monomial counting problem. Two further devices carry the argument: the operator relations $x_i^{Q_{ai}} = q_a$ (equivalently $x_i^{d_i^*} = _a q_a^{n_a^*}$ for every $n^*$ with all $d_i^* 0$), proved by a point-splitting argument that relates the $n$-th and $(n+n^*)$-th moduli spaces; and the introduction of an extra chiral field with charges $Q_{a0} = -_i Q_{ai}$, which makes the model Calabi–Yau, turns the parameter space into a symmetric object where all anomaly coefficients vanish, and yields the trace functional $(O) = O _0 O$ that converts the $V^+$ computation into the $M$ computation via $O _M = (-_0 O)$.

Extended reading notes

Core claim

The central claim is that the A-model correlation functions of a toric variety, and of a Calabi–Yau hypersurface in one, are exactly computable by summing instantons in the linear model, and that the resulting algebraic relations between the generators are precisely the relations that appear in the mirror's Jacobian ring. Concretely, the authors show that the degree-$n$ instanton moduli space is $M_n = (Y_n - F_n)/T$ with $Y_n = i H^0(O(d_i))$ and $d_i = _a Q_{ai} n_a$, a compact toric variety whose intersection numbers give each instanton coefficient, with an Euler-class factor $_{d_i<0} _i^{-d_i-1}$ when some $d_i$ is negative. Summing these coefficients yields explicit rational functions: for the quintic in $^4$ the single Yukawa coupling is $5/(1+5^5 q)$; for the two-parameter example the four Yukawa couplings are given in closed form with denominator $(1-2^8 q_1)^2 - 2^{18} q_1^2 q_2$, and the authors verify these agree with the mirror computations of Candelas and collaborators under the substitution the monomial-divisor map prescribes. For the hypersurface they prove a quantum restriction formula, expressing an $M$-model correlator as $O (-K)/(1-(-K)^p)$ evaluated in the quantum cohomology ring of $V$, and they show that the resulting ring agrees with the restricted chiral ring of the conjectured mirror hypersurface, matching the principal discriminant's components face by face with the components of the A-model singular locus.

Load-bearing premise

The identification of the linear model's operator $_a$ with the divisor class $_a$ on every instanton moduli space, with a normalization that does not vary with instanton number, is assumed rather than proved; the authors state in section 3.8 that they 'do not have a complete justification for this choice,' and every instanton coefficient depends on it.

Editorial extensions

If this is right

For any smooth toric variety, the quantum cohomology ring is generated by divisor classes subject to the relations $x_i^{Q_{ai}} = q_a$, with no further relations needed, giving a complete and checkable algorithm for the ring and its correlators.
The A-model correlators of a Calabi–Yau hypersurface $M V$ are determined by those of $V$ through the quantum restriction formula, so the notoriously hard Gromov–Witten problem for hypersurfaces reduces to a toric intersection computation.
Instanton expansions can be computed about any limiting point of the moduli space, so the Landau–Ginzburg, orbifold and hybrid phases are all reachable, and the expansions in different phases are shown to agree after analytic continuation.
Under the hypotheses that $V$ is smooth and the mirror fan is simplicial, the monomial-divisor mirror map of Aspinwall, Greene and Morrison is proved at the level of topological correlation functions, with the map given globally as $x_i c_i _i$ and $q_a _i (c_i/c_0)^{Q_{ai}}$.
The components of the A-model singular locus, indexed by faces of the polytope, are matched one-to-one with the irreducible components of the principal discriminant of the mirror's defining polynomial, as described by Kapranov's toric generalization of Horn's results.

Reading between the lines

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

Because the summed correlators are rational functions with poles only along the discriminant, the paper effectively gives a global coordinate system on the whole moduli space rather than a large-radius asymptotic expansion; a reader could use this to test whether the analytic continuation through the strongly coupled region enclosed by the singular curve is really single-valued, which the authors
The authors note that their coefficients are not the standard Gromov–Witten invariants because they work in 'algebraic' coordinates and gauge; extracting the canonical coordinates would amount to computing the instanton corrections to $t_a = _a + Q_{a0}/2$ from pointlike instantons, which they leave open. That inversion is the natural next computation, and it is what would let the results be compa
The observation that the linear model's instanton spaces are compactifications of the map spaces by pointlike instantons suggests a general principle: any sigma model whose target sits inside a Fano or toric quotient admits a compactified moduli problem, so the same summation may work for complete intersections and for hypersurfaces in arbitrary Fano varieties, as the authors themselves suggest in
The proof strategy — match the ring of relations on one side with a Jacobian ring on the other, then invoke uniqueness of the graded expectation function up to scale — is a template that does not depend on the details of toric geometry, and could plausibly be applied to other mirror constructions where both chiral rings are presented by generators and relations.

Original abstract

We use the gauged linear sigma model introduced by Witten to calculate instanton expansions for correlation functions in topological sigma models with target space a toric variety $V$ or a Calabi--Yau hypersurface $M \subset V$. In the linear model the instanton moduli spaces are relatively simple objects and the correlators are explicitly computable; moreover, the instantons can be summed, leading to explicit solutions for both kinds of models. In the case of smooth $V$, our results reproduce and clarify an algebraic solution of the $V$ model due to Batyrev. In addition, we find an algebraic relation determining the solution for $M$ in terms of that for $V$. Finally, we propose a modification of the linear model which computes instanton expansions about any limiting point in the moduli space. In the smooth case this leads to a (second) algebraic solution of the $M$ model. We use this description to prove some conjectures about mirror symmetry, including the previously conjectured ``monomial-divisor mirror map'' of Aspinwall, Greene, and Morrison.

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arXiv listing abstract v2. Results and the main claim are underlined; show every role.

We use the gauged linear sigma model introduced by Witten to calculate instanton expansions for correlation functions in topological sigma models with target space a toric variety $V$ or a Calabi--Yau hypersurface $M \subset V$. In the linear model the instanton moduli spaces are relatively simple objects and the correlators are explicitly computable; moreover, the instantons can be summed, leading to explicit solutions for both kinds of models. In the case of smooth $V$, our results reproduce and clarify an algebraic solution of the $V$ model due to Batyrev. In addition, we find an algebraic relation determining the solution for $M$ in terms of that for $V$. Finally, we propose a modification of the linear model which computes instanton expansions about any limiting point in the moduli space. In the smooth case this leads to a (second) algebraic solution of the $M$ model. We use this description to prove some conjectures about mirror symmetry, including the previously conjectured ``monomial-divisor mirror map'' of Aspinwall, Greene, and Morrison.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We use the gauged linear sigma model introduced by Witten to calculate instanton expansions for correlation functions in topological sigma models with target space a toric variety $V$ or a Calabi--Yau hypersurface $M \subset V$.	method 0.71	0.92	0.11	asserted 1.00
s1	In the linear model the instanton moduli spaces are relatively simple objects and the correlators are explicitly computable; moreover, the instantons can be summed, leading to explicit solutions for both kinds of models.	method 0.66	0.77	0.21	asserted 1.00
s2	In the case of smooth $V$, our results reproduce and clarify an algebraic solution of the $V$ model due to Batyrev.	result 0.88	0.76	0.11	asserted 1.00
s3	In addition, we find an algebraic relation determining the solution for $M$ in terms of that for $V$.	result 0.96	0.96	0.17	asserted 1.00
s4	Finally, we propose a modification of the linear model which computes instanton expansions about any limiting point in the moduli space.	contribution 0.82	0.93	0.13	asserted 0.72
s5	In the smooth case this leads to a (second) algebraic solution of the $M$ model.	result 0.97	0.91	0.12	asserted 0.99
s6	We use this description to prove some conjectures about mirror symmetry, including the previously conjectured ``monomial-divisor mirror map'' of Aspinwall, Greene, and Morrison.	result 0.85	0.92	0.40	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 0.93 · evidence from proof or derivation 1.00.

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Figures

**Figure 1.** Classical phase diagram. As we decrease the value of r we will find quantum corrections to these results. There is a natural deformation of the low-energy model (given by deforming the K¨ahler class) which one expects to correspond to the deformation of the ultraviolet theory by changing r. Thus we can expect that at least for sufficiently large r (we use this language somewhat loosely; what we mean is r deep in KV , so that all of the gauge couplings are small) the lowenergy theory is given by the nonlinear sigma model with target space V and some K¨ahler form. The quantum…

**Figure 2.** Phase diagram for M. r1 ≥ 0, 2r1 + r2 = 0

**Figure 3.** Fully corrected phase diagram for M. r the singular locus approaches the cones predicted in subsection 3.2 above. For finite r there are instanton corrections as predicted in section two and computed above; these introduce a θ dependence in the singular locus (which is a holomorphic divisor in q space). For the remainder of this section we once more restrict attention to the “geometric” phases (values of r in the interior of Kq). (This includes the phases we have labeled I and II in example 2.) 4.3. The Topological Model As in the previous section, our main interest will be in the…

Figures reproduced from arXiv:hep-th/9412236 by the authors.

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Forward citations

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How later papers used it

92 later papers cite it

Evidence: substantial. 83 of the citing papers share no author with it, and 24 of those carry a confident label from Pith's citation classifier.

Pith read 20 of the 264 citing sentences, choosing only those whose citation markers point at this paper alone, and wrote the paraphrases from them. The summary, the counts, the evidence level, the links and the quotes come from code; the summary counts the papers listed below. A later paper is listed under a label when Pith's citation classifier gives one of its sentences that label at probability 0.8 or more (0.95 for support), Pith's reading finds that the citing paper itself did it, and a second check confirms the paraphrase against the sentence. This section lists uses only. It does not list criticism or disagreement, so their absence here says nothing about how the paper has been received. Papers that share an author with this one are listed apart, under "By the same authors". A quote is the citing authors' own sentence, shown when their paper's licence is CC BY, CC BY-SA or CC0. JSON

4 later papers by other authors are listed below: 3 used its method or tool and 1 compared its own results with it. 1 more paper by the same authors is listed: 1 used its method or tool.

Uses its method or tool 4 of 17 papers with this label · 1 by the same authors

Applied this paper's methods to the Coulomb branch, using the one-loop-exact twisted effective superpotential.

Quantum cohomology from mixed Higgs-Coulomb branches (2023) · 0.91 · paraphrased: the licence does not allow quoting
Employed the strategy developed in this paper for (2,2) theories to solve the A/2-twisted M model.

Testing the (0,2) mirror map (2018) · 0.98 · paraphrased: the licence does not allow quoting
Changed to the algebraic coordinate of this paper's GLSM.

Hemisphere Partition Function and Analytic Continuation to the Conifold Point (2016) · 0.80 · paraphrased: the licence does not allow quoting

By the same authors

Used techniques developed in this paper for abelian GLSMs to analyze the phase structure of the quantum-corrected Kähler moduli spaces of linear determinantal varieties in projective space.

Nonabelian 2D Gauge Theories for Determinantal Calabi-Yau Varieties (2012) (same authors) · 0.89 · paraphrased: the licence does not allow quoting

Compares with it 1 of 6 papers with this label

Said that on the (2,2) locus its own result reduces to that obtained in this paper and given in eqn.

Summing the Instantons in Half-Twisted Linear Sigma Models (2008) · 0.98 · paraphrased: the licence does not allow quoting

How the literature uses it

91 arXiv papers cite this in 275 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 173 sentences without a confident label

Builds on it 7 papers

This (0,2) extension of the quantum restriction formula of [ref] is derived in section [ref].

Summing the Instantons in Half-Twisted Linear Sigma Models · A Summary of the Results · 1.00
Expanding on earlier work [ref] , I focus on a particular infinite sequence of constructions corresponding to Calabi–Yau hypersurfaces in Hirzebruch scrolls, the rich mathematical history of the latter [ref] affording their study from multiple different points of view, including their realization within GLSM models [ref] — which is where we start.

Beyond Algebraic Superstring Compactification · A Showcasing Deformation Family · 0.95
In this paper, we have generalized the Coulomb-branch based computations of quantum cohomology described in [ref] to cases with mixed Higgs-Coulomb branches, discussing in detail the case of GLSMs for hypersurfaces in projective spaces, $ℙ^n[d]$.

Quantum cohomology from mixed Higgs-Coulomb branches · Conclusions · 0.99

Uses its method or tool 17 papers

By using the GLSM fields and localization, the correlation functions $⟨O[η]^k⟩$ can also be calculated in the IR by the methods of [ref] for the Higgs contribution and those of [ref] for the Coulomb contribution.

Quantum cohomology from mixed Higgs-Coulomb branches · IR correlation functions · 1.00
In order to solve the A/2-twisted $M$ model we employ the strategy developed in [ref] for (2,2) theories and later extended to a class of (0,2) linear models in [ref] .

Testing the (0,2) mirror map · A/2 correlators · 0.98
To construct these, we first note that the action on the $α_m$ is the same as in the (2,2) case, and there is a well-known way to pick invariant coordinates [ref] .

A (0,2) Mirror Map · GLSM moduli space of reflexively plain models · 0.88

Compares with it 6 papers

On the (2,2) locus $M_(0) = K = -∑_i,a Q_i^a σ_a$, and our result reduces to that obtained in [ref] and given in eqn.

Summing the Instantons in Half-Twisted Linear Sigma Models · Localization and Quantum Restriction · 0.98
This matches precisely the results of [ref] .

The equivariant A-twist and gauged linear sigma models on the two-sphere · Geometric and orbifold phases in the $=0$ and $m^F=0$ limit · 1.00
This result was previously obtained in [ref] using the topologically A-twisted version of the GLSM which doesn't require the supersymmetric localization and the counterterm analysis that we did.

2D BPS Rings from Sphere Partition Functions · Relations and the ring · 1.00

Cites it as support 2 papers

This was confirmed for Calabi-Yau hypersurface examples in [ref] by summing the gauge instanton sums.

Quantum cohomology from mixed Higgs-Coulomb branches · Introduction · 0.98
This sum-rule in turn implies that the initial toric manifold (and therefore its squashed deformation) is non-compact [ref] .

Squashed toric sigma models and mock modular forms · Introduction and summary · 0.96

Cites it as background 35 papers

We have drawn heavily upon [ref] , [ref] , [ref] , [ref] in preparing this review.

Gromov-Witten invariants and localization · Introduction · 0.91
While this is the natural choice of coordinates from the GLSM perspective, in general these are related in a non-trivial fashion to the natural coordinates of the nonlinear model: see the discussion around equation (4.20) in [ref] , for example. ] $r$ has nontrivial RG flow, and will flow between different phases of the GLSM.

Quantum cohomology from mixed Higgs-Coulomb branches · Introduction · 0.80
For over three decades now, the standard vehicle to this end is Witten's gauged linear sigma model (GLSM) [ref] , where the space of ground-states is defined as the subset of the field space where a well-specified potential is minimized, at the intersection [equation] Here, $W(X)$ is the superpotnential function of chiral $(2,2)$-superfields[This brief discussion is limited to the simplest, $(2,2)$-supersymmetric case; $(0,2)$-generalizations are described in the literature [ref] , while the ultimate, minimally $(0,1)$-supersymmetric extensions remain largely unexplored.], $X_i$, with lowest component fields $x_i$.

Ricci-Flat Mirror Hypersurfaces in Spaces of General Type · Extending Gauged Linear Sigma Models · 0.81

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

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D. R. Morrison and M. R. Plesser, work in progress

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