main-content

## Swipe to navigate through the chapters of this book

Published in:

2015 | OriginalPaper | Chapter

# Mirror Symmetry in Physics: The Basics

Author: Callum Quigley

Publisher: Springer New York

## Abstract

These notes are aimed at mathematicians working on topics related to mirror symmetry, but are unfamiliar with the physical origins of this subject. We explain the physical concepts that enable this surprising duality to exist, using the torus as an illustrative example. Then, we develop the basic foundations of conformal field theory so that we can explain how mirror symmetry was first discovered in that context. Along the way we will uncover a deep connection between conformal field theories with (2,2) supersymmetry and Calabi-Yau manifolds.
Footnotes
1
The brief history of mirror symmetry that we are about to cover is far from complete, and the references we provide are far from exhaustive.

2
As the story goes, these results were presented at a conference at the MSRI in Berkeley where it was pointed out that one of their numbers was in disagreement with recent computations of mathematicians (using traditional and more rigorous techniques). However, an error in the computer code of the mathematicians was soon discovered, and the predictions on mirror symmetry were verified [47].

3
Note that rescaling z is equivalent to scaling the total area, A, and therefore is a Kähler (not a complex structure) deformation.

4
Physically, this follows from the fact that H = dB ≠ 0 would generate a finite energy density on X which would preclude the possibility of a Ricci-flat solution to the Einstein equations. Mathematically, it can be shown that H ≠ 0 requires that X is not Kähler, and so in particular will not be Calabi-Yau [36].

5
Reinstating factors of c, this becomes E 2 = c 2 p 2 + m 2 c 4, or at zero momentum simply E = mc 2.

6
For example a proton is composed of three quarks, each carrying their own intrinsic mass, but together they are responsible a mere 1 % of a the proton’s mass. The other 99 % arises from the internal binding energy (carried by gluons) that keeps the quarks from flying apart.

7
More precisely, in the natural units $$\hslash = c = 1$$ the tension of the string is T = (2π α′)−1, where α′ has dimensions of area. This sets the fundamental length scale of a string, $$\ell_{s} = \sqrt{\alpha '}$$. If we keep α′ explicit, as many physicists often do, then T-duality interchanges $$R \leftrightarrow \alpha '/R$$.

8
If the wrapping number were not integral, then the string must start and end at different points, and would no longer be closed.

9
More precisely, the integration cycle is (anything homologous to) the closed path that, when lifted to the covering space $$\mathbb{C}$$, connects the points z = 0 and $$z =\tau$$.

10
Sometimes these generalized rotations are split into spatial SO(d − 1) rotations and “boosts” along each of the d − 1 spatial directions.

11
Null separated points can only be reached by traveling at exactly c.

12
The second term in brackets is proportional to δ 3(0), and therefore infinite, but all is not lost. These sorts of infinities arise in many problems in quantum field theories, and they are a signal of our ignorance about physics at extremely short distance scales. Nevertheless, it is well-understood how to regulate and remove these infinite quantities from physically observable quantities. In this case, since we can only measure energy differences (with respect to the vacuum) the resolution is to simply drop this infinite vacuum energy.

13
Although d = 1 is also excluded, the notion of a conformal transformation is meaningless since every vector is necessarily parallel.

14
Unless we specify a global condition, we will now take the conformal group on $$\mathbb{C}$$ to mean the local one.

15
While the Virasoro algebra strongly constrains the structure of every CFT, the set of allowed fields and operators in a given theory must obey additional constraints such as locality and modular invariance. See [35] for further details.

16
For example, we can map the complex plane to a cylinder of radius R by the conformal mapping $$z \rightarrow w = R\log z.$$ However, in doing we break the scale invariance of the system by introducing the preferred length R, and this is reflected by a change in the vacuum energy density by an amount − c∕(24R 2).

17
To be precise, changing between cylinder and plane frames induces a shift in H by $$(c +\tilde{ c})/24$$ because of the anomalous (i.e. non-primary) transformation properties of T(z) and $$\tilde{T}(\bar{z})$$. For simplicity, throughout these notes we will ignore these subtle corrections, and refer the interested reader to the references for a more thorough treatment.

18
This is why we shifted the powers of z in the Laurent expansion of $$\mathcal{O}(z)$$ by h.

19
We will see by the end of this section that demanding invariance of the vacuum under the full Virasoro algebra is too strong a requirement.

20
In most cases of interest, the Virasoro Verma module is not irreducible and the corresponding CFT is neither rational nor unitary. However, one can also construct a CFT associated to the irreducible quotient. This is then a CFT with an interesting representation theory, especially there are central charges for which the simple Virasoro algebra is the symmetry algebra of a unitary rational CFT. A complete book is devoted to this subtle question [26]. We thank T. Creutzig for explaining this point to us.

21
Ironically, these states are usually referred to as highest weight states, in analogy with standard Lie theory.

22
If $$\mathcal{O}$$ is a descendent solely by the action of $$\tilde{L}_{-n}$$ then we consider it primary for this argument.

23
The normalization is set by the tension, T = (2π α′)−1. In this section we will work in units where α′ = 2, which will considerably simply the formulas. This convention differs from Sect. 2.4, where we used α′ = 1.

24
As a composite local operator, we should be careful in how we define T(z), because of potential singularities coming from the OPE. Typically, this ambiguity is handled by so-called normal ordering, by defining composite operators with the singular terms subtracted off. So, more precisely, the energy momentum operator is given by
$$\displaystyle{T(z) = -\frac{1} {2}\lim _{w\rightarrow z}\left (\partial X(z)\partial X(w) + \frac{1} {(z - w)^{2}}\right ).}$$
In what follows, we will always assume that composite operators are normal ordered.

25
Again, we are suppressing a constant additive shift in the spectrum.

26
Recall that we are now working in units with α′ = 2, and in general T-duality acts by $$R \leftrightarrow \alpha '/R$$.

27
More generally, for a multi-component fermion field $$\{\psi _{0}^{i},\psi _{0}^{j}\} =\delta ^{ij}$$.

28
Once again we are being sloppy about additive constants, which shift the zero point of the spectrum.

29
Additionally, supersymmetry groups contain R-symmetries as “internal” sub-groups, which act non-trivially on the supercharges only.

30
This is an example of an R-symmetry, alluded to in an earlier footnote.

31
This is the reason we have been referring to the N = 2 algebra in the singular, unlike the N = 1 cases.

32
In the Ramond sector, when ν = 0, we must also deal with G 0 ±, which also commute with L 0. We define Ramond groundstates to be those annihilated by both G 0 ±.

33
The terminology BPS multiplets is also used in this case.

34
The second condition is certainly required to be primary, but is not sufficient.

35
For simplicity, we suppress the normalization constant 1∕4π in front of the action.

36
This equivalence, together with an integral charge constraint, guarantees the existence of supersymmetry in the target space (as opposed to worldsheet) theory [3], since bosons originate in the NS sector and fermions originate in the R sector.

37
A nice discussion of this subtle point can be found in [20].

38
Similar statements can be made for the $$\mathcal{R}_{ac}$$ and $$\mathcal{R}_{ca}$$ rings, by using the spectral flow operators $$\mathcal{U}_{\pm 1,\mp 1}$$.

39
Alternatively, we could demand that the spectral flow operators $$\mathcal{U}_{\pm 1,0},\mathcal{U}_{0,\pm 1}$$ be mutually local with respect to the other operators in the theory, and the integrality condition will emerge automatically on the entire spectrum (not just the chiral sectors) [3].

40
To be precise, X is Kähler only when the B-field is closed. More generally, X can be bi-Hermitian [16] or equivalently Generalized Kähler [23].

41
The shift by c∕6 comes about by spectral flow from the Ramond sector.

Literature
1.
Alvarez-Gaume, L., Freedman, D.Z.: Geometrical structure and ultraviolet finiteness in the supersymmetric sigma model. Commun. Math. Phys. 80, 443 (1981). doi:10.1007/BF01208280
2.
Aspinwall, P.S., Greene, B.R., Morrison, D.R.: Calabi-Yau moduli space, mirror manifolds and space-time topology change in string theory. Nucl. Phys. B416, 414–480 (1994). doi:10.1016/0550-3213(94)90321-2
3.
Banks, T., Dixon, L.J., Friedan, D., Martinec, E.J.: Phenomenology and conformal field theory or can string theory predict the weak mixing angle? Nucl. Phys. B299, 613–626 (1988)
4.
Batyrev, V.V.: Dual polyhedra and mirror symmetry for Calabi-Yau hypersurfaces in toric varieties. J. Algebr. Geom. 3, 493–545 (1994)
5.
Batyrev, V.V., Borisov, L.A.: On Calabi-Yau complete intersections in toric varieties. In: eprint arXiv:alg-geom/9412017, p. 12017 (1994)
6.
Becker, K., Becker, M., Schwarz, J.: String Theory and M-Theory: A Modern Introduction. Cambridge University Press, Cambridge (2007)
7.
Belavin, A., Polyakov, A.M., Zamolodchikov, A.: Infinite conformal symmetry in two-dimensional quantum field theory. Nucl. Phys. B241, 333–380 (1984). doi:10.1016/0550-3213(84)90052-X
8.
Candelas, P., De La Ossa, X.C., Green, P.S., Parkes, L.: A pair of Calabi-Yau manifolds as an exactly soluble superconformal theory. Nucl. Phys. B359, 21–74 (1991). doi:10.1016/0550-3213(91)90292-6 CrossRef
9.
Candelas, P., Horowitz, G.T., Strominger, A., Witten, E.: Vacuum configurations for superstrings. Nucl. Phys. B258, 46–74 (1985). doi:10.1016/0550-3213(85)90602-9
10.
Candelas, P., Lynker, M., Schimmrigk, R.: Calabi-Yau manifolds in weighted P(4). Nucl. Phys. B341, 383–402 (1990). doi:10.1016/0550-3213(90)90185-G
11.
Deligne, P., Etingof, P., Freed, D.S., Jeffrey, L.C., Kazhdan, D., Morgan, J.W., Morrison, D.R., Witten, E.: Quantum Fields and Strings: A Course for Mathematicians, vols. 1, 2, pp. 1–1501. AMS, Providence (1999)
12.
Di Francesco, P., Mathieu, P., Senechal, D.: Conformal Field Theory, 890 p. Springer, New York (1997) CrossRef
13.
Dixon, L.J.: Some World Sheet Properties of Superstring Compactifications, on Orbifolds and Otherwise. In: Furlan, G., Pati, J.C., Sciama, D.W., Sezgin, E., Shafi, Q. (eds.) Unified Theories and Cosmology 1987: proceedings. ICTP Series in Theoretical Physics, vol. 4, 653p. Singapore, World Scientific (1988)
14.
Dixon, L.J., Harvey, J.A., Vafa, C., Witten, E.: Strings on orbifolds. Nucl. Phys. B261, 678–686 (1985). doi:10.1016/0550-3213(85)90593-0
15.
Friedan, D.: Nonlinear models in 2+ ε dimensions. Phys. Rev. Lett. 45, 1057–1060 (1980). doi:10.1103/PhysRevLett.45.1057
16.
Gates, S.J., Hull, C.M., Rocek, M.: Twisted multiplets and new supersymmetric nonlinear sigma models. Nucl. Phys. B248, 157 (1984)
17.
Ginsparg, P.H.: Applied conformal field theory. In: eprint arXiv:hep-th/9108028 (1988)
18.
Givental, A.: A mirror theorem for toric complete intersections. In: eprint arXiv:alg-geom/9701016 (1997)
19.
Givental, A.B., Givental, A.B.: Equivariant gromov-witten invariants. Int. Math. Res. Not. 13, 613–663 (1996)
20.
Greene, B.R.: String theory on Calabi-Yau manifolds. In: eprint arXiv:hep-th/9702155 (1996)
21.
Greene, B.R., Plesser, M.: Duality in Calabi-Yau moduli space. Nucl. Phys. B338, 15–37 (1990). doi:10.1016/0550-3213(90)90622-K
22.
Gross, M.: Mirror symmetry and the Strominger-Yau-Zaslow conjecture. In: eprint arXiv:1212.4220[math.AG] (2012)
23.
Gualtieri, M.: Generalized complex geometry. In: eprints arXiv:math/0401221 [math-dg] (2004)
24.
Hori, K., Katz, S., Klemm, A., Pandharipande, R., Thomas, R., et al.: Mirror Symmetry. Clay Mathematics Monographs, vol. 1. American Mathematical Society, Providence/Clay Mathematics Institute, Cambridge (2003)
25.
Hori, K., Vafa, C.: Mirror Symmetry. In: eprint arXiv:hep-th/0002222 (2000)
26.
Iohara, K., Koga, Y.: Representation Theory of the Virasoro Algebra. Springer Monographs in Mathematics. Springer, London (2011). doi: 10.​1007/​978-0-85729-160-8
27.
Kontsevich, M.: Enumeration of rational curves via Torus actions. In: eprint arXiv:hep-th/9405035 (1994)
28.
Kontsevich, M.: Homological algebra of mirror symmetry. In: eprint arXiv:alg-geom/9411018 (1994)
29.
Lerche, W., Vafa, C., Warner, N.P.: Chiral rings in N = 2 superconformal theories. Nucl. Phys. B324, 427 (1989). doi:10.1016/0550-3213(89)90474-4
30.
Lian, B., Liu, K., Yau, S.T.: Mirror Principle I. Asian J. Math. 1, 729 (1997)
31.
Lian, B.H., Liu, K., Yau, S.T.: Mirror Principle II. Asian J. Math. 3, 109 (1999)
32.
Lian, B., Liu, K., Yau, S.T.: Mirror Principle III. In: eprint arXiv:math/9912038 [math.AG] (1999)
33.
Lian, B.H., Liu, K., Yau, S.T.: Mirror Principle IV. In: eprint arxiv:math/9912038 [math.AG] (2000)
34.
Melnikov, I.V., Sethi, S., Sharpe, E.: Recent developments in (0,2) mirror symmetry. SIGMA 8, 068 (2012). doi:10.3842/SIGMA.2012.068 MathSciNet
35.
Polchinski, J.: String Theory. Vol. 1: An Introduction to the Bosonic String, 402 p. Cambridge University Press, Cambridge (1998)
36.
Strominger, A.: Superstrings with torsion. Nucl. Phys. B 274, 253 (1986). doi:10.1016/0550-3213(86)90286-5
37.
Strominger, A., Yau, S.T., Zaslow, E.: Mirror symmetry is T-duality. Nucl. Phys. B 479, 243–259 (1996)
38.
Vafa, C.: String vacua and orbifoldized L-G models. Mod. Phys. Lett. A4, 1169 (1989). doi:10.1142/S0217732389001350
39.
Vafa, C.: Topological mirrors and quantum rings. In: Yau, S.T. (ed.) Mirror Symmetry I, pp. 97–120 (1991). [hep-th/9111017]
40.
West, P.C.: Introduction to Supersymmetry and Supergravity, 425 p. World Scientific, Singapore (1990) CrossRef
41.
Witten, E.: Constraints on supersymmetry breaking. Nucl. Phys. B202, 253 (1982). doi:10.1016/0550-3213(82)90071-2 CrossRef
42.
Witten, E.: Topological sigma models. Commun. Math. Phys. 118, 411 (1988). doi:10.1007/BF01466725
43.
Witten, E.: On the structure of the topological phase of two-dimensional gravity. Nucl. Phys. B340, 281–332 (1990). doi:10.1016/0550-3213(90)90449-N CrossRef
44.
Witten, E.: Mirror manifolds and topological field theory. In: Yau, S.T. (ed.) Mirror Symmetry I, pp. 121–160 (1991). [hep-th/9112056]
45.
Witten, E.: Phases of N = 2 theories in two dimensions. Nucl. Phys. B 403, 159–222 (1993)
46.
Witten, E.: Chern-Simons gauge theory as a string theory. Prog. Math. 133, 637–678 (1995)
47.
Yau, S.T., Nadis, S.: The Shape of Inner Space: String Theory and the Geometry of the Universe’s Hidden Dimensions. Basic Books, New York (2010)
48.
Zumino, B.: Supersymmetry and Kahler manifolds. Phys. Lett. B87, 203 (1979). doi:10.1016/ 0370-2693(79)90964-X CrossRef